DE RHAM COHOMOLOGY OF DIFFERENTIAL MODULES ON ALGEBRAIC VARIETIES

Yves ANDRE and Francesco BALDASSARRI

TABLE OF CONTENTS

I. Regularity in several variables. §1 Geometric models of divisorially valued function fields. §2 Logarithmic differential operators. §3 Regular connections along a divisor. §4 Extensions with logarithmic poles. §5 Regular connections: the global case. §6 Exponents. Appendix A: A letter of Ph. Robba (Nov. 2, 1984). Appendix B: Models and log schemes.

II. Irregularity in several variables. §1 Spectral norms. §2 The generalized Poincaré-Katz rank of irregularity. §3 Some consequences of the Turrittin-Levelt theorem. §4 Newton polygons. §5 Stratification of the singular locus by Newton polygons. §6 Formal decomposition of an integrable connection at a singular divisor. §7 Cyclic vectors, indicial polynomials and tubular neighborhoods.

III. Direct images (the Gauss-Manin connection). §1 Elementary fibrations. §2 Review of connections and de Rham cohomology. §3 Dévissage. §4 Generic finiteness of direct images. §5 Generic base change for direct images. §6 Coherence of the cokernel of a regular connection. §7 Regularity and exponents of the cokernel of a regular connection. §8 Proof of the main theorems: finiteness, regularity, monodromy, base

change (in the regular case). Appendix C: Berthelot’s comparison theorem on O_{X}- vs. D_{X}-linear duals. Appendix D: Introduction to Dwork’s algebraic dual theory.

IV. Complex and p-adic comparison theorems. §1 Review of analytic connections and De Rham cohomology. §2 Abstract comparison criteria. §3 Comparison theorem for algebraic vs. complex-analytic cohomology. §4 Comparison theorem for algebraic vs. rigid-analytic cohomology (regular

coefficients).

§5 Rigid-analytic comparison theorem in relative dimension one. §6 Comparison theorem for algebraic vs. rigid-analytic cohomology (irreg-

ular coefficients).

§7 The relative non-archimedean Turrittin theorem. Appendix E: Riemann’s existence problem: an elementary approach.

General Introduction.

This is a study of algebraic differential modules in several variables, and of some of their relations with analytic differential modules. Let us explain its source.

The idea of computing the cohomology of a manifold, in particular its Betti numbers, by means of differential forms goes back to E. Cartan and G. De Rham. In the case of a smooth complex algebraic variety X, there are three variants:

i) using the De Rham complex of algebraic differential forms on X, ii) using the De Rham complex of holomorphic differential forms on the

analytic manifold X^{an} underlying X,

iii) using the De Rham complex of C^{∞} complex differential forms on the differentiable manifold X^{dif} underlying X^{an}, which turn out to be equivalent. Namely, one has canonical isomorphisms of hypercohomology:

H_{DR}(X) ≅ H_{DR}(X^{an}) ≅ H_{DR}(X^{dif}) .

While the second isomorphism is a simple sheaf-theoretic consequence of the Poincaré lemma, which identifies both vector spaces with the complex cohomology H(X^{top}, C) of the topological space underlying X, the first isomorphism is a deeper result of A. Grothendieck, which shows in particular that the Betti numbers can be computed algebraically. This result has been generalized by P. Deligne to the case of nonconstant coefficients: for any algebraic vector bundle M on X endowed with an integrable regular connection, one has canonical isomorphisms

H_{DR}(X, M) ≅ H_{DR}(X^{an}, M^{an}) .

The notion of regular connection is a higher dimensional generalization of the classical notion of fuchsian differential equations (only regular singularities).

These results were to have a significant influence on the later development of the theory of D-modules. The crucial point in their proof is a comparison between a meromorphic De Rham complex and a De Rham complex with essential singularities, carried out by using Hironaka’s resolution of singularities. This also turns out to be the main point in the proof of the non-archimedean counterparts of these comparison theorems, established respectively by R. Kiehl and by the second author (under the extra assumption that the exponents of M are algebraic numbers).

This point became much better understood when Z. Mebkhout showed, in the complex case, that the gap between meromorphic and essentially singular De Rham complexes is measured by a certain perverse sheaf (“positivity of the irregularity”). Thanks to this deep theorem, it is no longer necessary to invoke resolution of singularities, as a deus ex machina, to investigate any serious question of ramification arising in the theory of D-modules, for example in the proof of the Grothendieck-Deligne theorem.

However, the situation is not so nice in the non-archimedean setting, where no analogue of the positivity theorem is known. This motivated us to pose the following problem, which was the starting point of this work:

(∗) to give an “elementary” proof of the comparison theorems, which is “formal” enough to apply both in the complex-analytic and rigid-analytic situations.

Of course, such a proof should not only avoid Hironaka’s theorem, but also any monodromy argument.

Our solution (IV.3,4) is inspired by M. Artin’s proof of the comparison theorem between algebraic and complex-analytic étale cohomology. It relies on three new tools:

i) an algebraic construction of Deligne’s canonical extensions (I.4), ii) a dévissage, inspired by Artin’s technique of elementary fibrations, which

permits the reduction of many problems on direct images to the relative onedimensional case (III.3),

iii) an abstract model of “comparison theorem” in the setting of differential algebra (IV.2).

Settling this problem also led us to revisit some fundamental results on direct images of regular differential modules by a smooth morphism. For instance, we found elementary and algebraic proofs of the so-called finiteness, base change, regularity and monodromy theorems (III.8). As a by-product of these techniques, we present an elementary proof of the generalized Riemann existence theorem for coverings (IV.App.E) which uses neither resolution, nor any extension theorem for analytic coherent sheaves. The first chapter (I) provides a self-contained exposition of the algebraic theory of regularity in several variables, and (III.App.D) proposes a new definition of relative algebraic De Rham cohomology with compact supports. This is based on a strong comparison result (III.App.C), due to Berthelot, between two different notions of the dual of a differential module. Then (III.App.D) shows that our relative algebraic De Rham cohomology with compact supports coincides with Dwork’s algebraic dual theory, which is a powerful tool in the study of direct images in the case of elementary fibrations.

What happens if one drops the assumption that the connection on M is regular? In the complex setting, Deligne pointed out that the natural mapping

H_{DR}(X, M) → H_{DR}(X^{an}, M^{an})

may be neither injective nor surjective. In strong contrast, the second author conjectured ten years ago that, in the non-archimedean setting, the isomorphism

H_{DR}(X, M) ≅ H_{DR}(X^{an}, M^{an})

holds even in the case of an irregular connection (at least if X and M together with its connection are defined over some number field). Different approaches (Baldassarri, B. Chiarellotto) have established the conjecture for curves, but failed in the higher dimensional case.

The final result of this book is a complete proof of this general non-archimedean comparison theorem for De Rham cohomology (IV.6.1). As a corollary, we obtain that the functor of p-adic analytification of connections (defined over some number field) is fully faithful (IV.6.8).

The strategy of the proof is the same as in the case of regular connections, but our dévissages à la Artin are much more delicate in the general case. Indeed, we found that a detailed study of irregularity in several variables was a necessary preliminary step. This study is carried out in chapter (II), where we introduce the stratification of Z by Newton polygons, prove a semicontinuity theorem for Newton polygons (II.5) and obtain a formal decomposition of an integrable connection at a singular divisor Z (II.6).

Note on the style. Chapters I, II, III are purely algebraic; I, II are noncohomological. The partition into four chapters is based on logical grounds; however, we advise the reader to read I, III and the first half of IV before II and the second half of IV, which are much more technical (although most of the new results of this work are contained in the latter parts).

We have consistently tried to be as elementary and down-to-earth as possible, even at the cost of elegance or conciseness. For instance, we work mainly with vector bundles with integrable connection and with smooth morphisms, so that neither the general theory of holonomic modules nor the language of derived categories is needed (except in appendices C and D, independent of the rest of the text).

Acknowledgements. This work was partially sponsored by the European Network Arithmetic Algebraic Geometry (contract FMRX-CT96-0006 (DG 12 BDCN)). The first (resp. second) author thanks the University of Padova (resp. Paris 6) for hospitality and support during the preparation of this book.

Yves André  Francesco Baldassarri
Institut de Mathématiques  Dipartimento di Matematica
4 place Jussieu  Università di Padova
Tour 46-00, 5ème étage, case 247  Via Belzoni 7
F-75252 Paris Cedex 05 - France  I-35131 Padova - Italy
e-mail: andre@math.jussieu.fr  e-mail: baldassa@math.unipd.it

Chapter I.

Regularity in several variables.

Table of contents:

§1 Geometric models of divisorially valued function fields.

§2 Logarithmic differential operators.

§3 Regular connections along a divisor.

§4 Extensions with logarithmic poles.

§5 Regular connections: the global case.

§6 Exponents.

Appendix A: A letter of Ph. Robba (Nov. 2, 1984).

Appendix B: Models and log schemes.

Introduction.

The central topic of this chapter is the notion of regularity in several variables. For an algebraic integrable connection ∇ on the complement of a divisor Z in an algebraic variety X, the notion of regularity along Z may be defined, or characterized, in at least four different algebraic ways:

a) in terms of the iterated action of any single vector field D generically transversal to Z: the order of the poles occurring in the action of D^{n} is at most n + constant,

b) by the fact that the logarithmic differential operators of increasing order act with poles of bounded order at the generic point of Z,

c) via the classical notion of regularity in one variable, applied to the restriction of ∇ to sufficiently many smooth curves in X intersecting Z transversally,

d) by the existence of an extension with logarithmic poles along Z.

Although the equivalence of these points of view is “well-known”, the proofs given in the literature are always transcendental, and usually rely on Hironaka’s resolution of singularities (and are sometimes incomplete, cf. (5.9) below). J. Bernstein in his notes [Bn] requested an algebraic proof for these equivalences.

The aim of this chapter is to provide a purely algebraic, elementary, systematic treatment of these questions.

Let us describe in more detail the content of each section. Throughout this chapter the letter K will denote an algebraically closed field of characteristic 0.

In section 1, we discuss divisorially valued function fields over K, and their geometric models, namely smooth K-models carrying a marked smooth divisor. The local structure of morphisms of models is analyzed with some care. We study the existence of étale tubular neighborhoods rectifying a given transversal vector field. In appendix B, where we profited from suggestions by A.Ogus, these results are recast in the now standard framework of log geometry.

In section 2, we examine the functoriality of logarithmic differential operators.

In the next section, we concentrate on differential modules over a function field F over K, that are regular along a divisorial valuation v of F. For any model (X, Z) of (F, v), we prove the equivalence of a) - d).

In section 4, we fix a section τ : K/Z ⟶ K of the canonical projection, with τ(0) = 0. We consider a divisor Z with strict normal crossings in a smooth K-variety X, and an integrable connection ∇ on X \ Z, regular along Z. We show that the connection admits a unique extension with logarithmic poles along Z and exponents in Im τ. Our proof is purely algebraic, in contrast to the Manin-Deligne construction [De, II, 5.4 and 5.7] which relies on monodromy and on the consideration of multivalued sections of moderate growth.

In section 5, we introduce the global notion of regularity. We start from a birational definition, and give the following equivalent characterizations of a regular connection ∇ on X:

e) the restriction of ∇ to every smooth curve in X is regular in the classical sense,

f) ∇ is regular along the one-codimensional part of the boundary of X in some normal compactification \overline{X}. Our proofs are purely algebraic and do not use resolution of singularities.

In the last section, we define the exponents of a regular connection along a divisor and show that they coincide with the classical exponents of the differential module induced on any curve meeting the divisor transversally. We also introduce a global, birational notion of exponents of a regular connection, give several characterizations, and study its behaviour under inverse images and finite direct images.

Robba’s result, reproduced in appendix A, will be useful at several places in this book. It is a pleasure to publish without any change his letter from many years ago.

§1 Geometric models of divisorially valued function fields.

1.1 Models.

We set here: F = a function field over K, i.e. a finitely generated extension field of K; d = tr deg_{K}F; v = a divisorial valuation of F/K, i.e. a discrete valuation of F, trivial over K, normalized by the condition that its value group is exactly Z, with valuation ring R_{v} ⊃ K, maximal ideal 𝔪_{v} and residue field k(v) ⊃ K, such that

tr deg_{K}k(v) = tr deg_{K}F − 1 .

A uniformizing parameter of R_{v} will usually be denoted by t_{v}, or simply by t. We will denote by (F̂, v̂) the completion of (F, v), and by R̂_{v} = Rv̂, 𝔪̂_{v} = 𝔪v̂ the valuation ring and maximal ideal of v̂. We will refer to (F, v) (resp. to (F̂, v̂)) as a (divisorially) valued function field over K (resp. a field of formal functions in d variables over K).

Definition 1.1.1. A smooth (affine) K-model of (F, v) is a pair (X, Z) consisting of a smooth (affine) connected K-variety X, with generic point η_{X}, and of a smooth closed irreducible subvariety Z, of generic point η_{Z}, together with isomorphisms of K-algebras κ(η_{X}) ≅ F, O_{X,η_{Z}} ≅ R_{v}, fitting in the natural commutative diagram

O_{X,η_{Z}} ↪ κ(η_{X})
≅↓  ↓≅
R_{v} ↪ F

We shall also simply write model instead of smooth K-model. In this situation, κ(η_{Z}) = k(v). We will regard Z as a closed reduced subscheme of X, and denote its ideal sheaf by J_{Z}.

Lemma 1.1.2. Smooth K-models (X, Z) of (F, v) exist.

Indeed, one may write R_{v} as a localization of a finitely generated integral Kalgebra R; then v is the valuation associated to a prime divisor Z′ of the normalization X′ of Spec R. A sufficiently small open neighborhood X of η_{Z′} in X′, together with Z = Z′ ∩ X, fulfill the requirements.

Lemma 1.1.3. Let (X, Z) and (X′, Z′) be two models of (F, v). Then there exists open neighborhoods U of η_{Z} in X and U′ of η_{Z′} in X′, and an isomorphism ϵ : U⟶^{∼}U′ that identifies Z ∩ U with the inverse image of Z′ ∩ U′ by ϵ.

Proof. We may first replace X (resp. X′) by an open affine neighborhood of η_{Z} (resp. η_{Z′}). Then we may assume that O(X) and O(X′) are contained in R_{v} ⊂ F. Let 𝔭_{Z} (resp. 𝔭_{Z′}) be the prime ideal of O(X) (resp. O(X′)) corresponding to Z (resp. Z′), so that R_{v} = O(X)_{𝔭_{Z}} = O(X′)_{𝔭_{Z′}} and O(X) ∩ 𝔪_{v} = 𝔭_{Z}, O(X′) ∩ 𝔪_{v} = 𝔭_{Z′}. Upon replacing X by a smaller open neighborhood of η_{Z}, we may then assume that there is a morphism ϕ : X ⟶ X′, with ϕ(η_{Z}) = η_{Z′}. Since ϕ_{η_{Z}}^{♯} : O_{X′,η_{Z′}} ⟶^{∼}O_{X,η_{Z}}, we get the result by [EGA I, Prop. 5.6.4]. Q.E.D.

Definition 1.1.4. A coordinatized K-model (X, Z, \underline{x}) of (F, v) is a smooth affine K-model (X, Z) endowed with global étale coordinates \underline{x} = (x_{1}, …, x_{d}) on X such that J_{Z} = x_{1}O_{X}. The étale coordinates (t_{v} = x_{1}, x_{2}, …, x_{d}) on X are said to be adapted to Z.

Lemma 1.1.5. Any model (X, Z) of (F, v) admits a Zariski covering by coordinatized models.

Proof. Follows from [EGA IV, 4, Cor. 17.12.2, d]. Q.E.D.

Remark 1.1.6. In the situation of definition 1.1.1 (resp. 1.1.4), we set X̂ = the smooth formal K-scheme X̂_{Z}, completion of X along Z. We say that (X̂, Z) (resp. (X̂, Z, \underline{x})) is a smooth (affine) formal model (resp. a coordinatized formal model) of (F̂, v̂).

1.2 Morphisms of models.

We consider another divisorially valued function field (G, w) over K. Let (Y, W) be a model of (G, w).

Definition 1.2.1. A morphism f : (X, Z) ⟶ (Y, W) is a morphism of K-

schemes f^{∘} : X ⟶ Y (sometimes abusively denoted also by f) such that f^{∘}(X \ Z) ⊂ Y \ W.

Let (X, Z)⟶^{f}(Y, W) be a morphism of models.

Definition 1.2.2. We say that f is a closed immersion (resp. locally closed

immersion) if f^{∘} : X ⟶ Y is a closed immersion (resp. locally closed immersion) of K-schemes, and X, W meet transversally along Z.

We say that f is log dominant if f^{∘} : X ⟶ Y is dominant and f^{∘}(η_{Z}) = η_{W}.

Each of these notions is stable under composition.

Remark 1.2.3. A morphism of K-models f : (X, Z) ⟶ (Y, W) such that

f^{∘}(Z) ⊂ W is log dominant if and only if f^{∘} is flat at η_{Z}.

Proof. Assume f^{∘} is flat at η_{Z}. Then f^{∘} is certainly dominant and dim O_{X,η_{Z}} −

dim O_{Y,f^{∘}(η_{Z})} = dim O_{f^{∘}^{−1}(f^{∘}(η_{Z})),η_{Z}} ≥ 0. So, dim O_{Y,f^{∘}(η_{Z})} ≤ 1, and since f^{∘}(η_{Z}) ∈

W, f^{∘}(η_{Z}) = η_{W}. Assume conversely that f is log dominant. Then O_{X,η_{Z}} is flat over the DVR O_{Y,η_{W}}, since O_{Y,η_{W}} ↪ O_{X,η_{Z}} is injective. Q.E.D.

Definition 1.2.4. Let s = f^{∘}(η_{Z}) and consider

f^{∘}^{♯} : O_{Y,s} ⟶ O_{X,η_{Z}} ;

let y ∈ O_{Y,s} be a local equation of W in Y at s. The multiplicity of f is the

positive integer e_{f} = v(f^{∘}^{♯}(y)). If e_{f} = 1, we say that f is simple.

Remark 1.2.5. The multiplicity of f is then the multiplicity of the point η_{Z} in

the fiber f^{∘}^{−1}(f^{∘}(η_{Z})). So, f is simple if and only if f^{∘} is transversal to W at η_{Z}

[EGA IV, 17.13.3]. Notice that if f is simple, then f^{∘}(Z) ⊂ W.

Lemma 1.2.6.

i) f is simple if f^{∘} : X ⟶ Y is smooth at η_{Z} and f^{∘}(η_{Z}) ⊂ W;

ii) if f is simple and log dominant, then f^{∘} : X ⟶ Y is smooth at η_{Z}; iii) a locally closed immersion of models is simple;

iv) if f^{∘} : X ⟶ Y is a closed immersion and f is simple, there are open neighborhoods U of η_{Z} in X and V of η_{W} in Y such that (U, U ∩ Z) ⟶ (V, V ∩ W) is a closed immersion.

Proof. i) and iii) follow from remark 1.2.5. ii) follows from [EGA IV, 17.5.1]. iv) follows from remark 1.2.5 and [EGA IV, 9.9.2 (ii)], applied to F = O_{X}. Q.E.D.

The following obvious lemma is only stated for ease of reference.

Lemma 1.2.7. Let (X, Z)⟶^{f}(Y, W) be a log dominant morphism. We have a commutative diagram of inclusions

(1.2.7.1)  \begin{matrix}R_{w} & ⟶ & R_{v} \\ ↓ & & ↓ \\ G & ⟶ & F \end{matrix}

such that 𝔪_{v} ∩ R_{w} = 𝔪_{w} and 𝔪_{w}R_{v} = 𝔪_{v}^{e_{f}}. In particular, the K-embedding of fields G ↪ F is compatible with the valuations in the sense that v_{|G} = e_{f}w, and induces an extension of residue fields k(w) ↪ k(v); e_{f} = e(v/w) is then the relative ramification index of v over w. Conversely, any diagram (1.2.7.1) arises from a log dominant morphism of models.

Proposition 1.2.8. Let f : (X, Z) ⟶ (Y, W) be a morphism of models with

X a curve, such that f^{∘}(Z) ⊂ W. Then there exists, étale-locally on X, a factorization

(X, Z)⟶^{i}(S, D)⟶^{g}(Y, W) ,

where (S,D) is a model, i is a closed immersion, g is log dominant and g^{∘}_{|S\D} : S \ D ⟶ Y \ W is smooth. One has e_{f} = e_{g}.

Proof.

a) In the canonical graph factorization of f

(X, Z)⟶^{(1,f)}(X × Y, X × W)⟶^{p}(Y, W) ,

the second projection p is log dominant. We may replace f by (1, f), hence assume that the underlying morphism X ⟶ Y is a closed immersion, and that X ×_{Y} W is the point Z with multiplicity e. The problem is local at Z (for the étale topology), both in X and in Y.

b) There exists a system of coordinates (x_{1}, …, x_{n}) on some étale neighborhood Y′ of Z in Y, such that the inverse image W′ of W is given by x_{1} = 0 and the inverse image X′ of X is given by x_{3} = … = x_{n} = 0, x_{1} = x_{2}^{e} (this is easily seen by choosing first x_{3}, …, x_{n} such that the surface x_{3} = … = x_{n} = 0 contains a Zariski neighborhood of Z in X and meets W transversally at Z, which reduces us to the familiar situation of the intersection of two smooth curves on a surface; see also part b) of the proof of (6.5.5) below). Since (Y′, W′) ⟶ (Y, W) is log dominant and smooth outside W′, we may replace (Y, W) by (Y′, W′).

c) Let us now consider the e-fold covering π : Y′ ⟶ Y ramified at x_{1} = 0 defined by x_{1} ↦ x_{1}^{e} (the other x_{i}’s being unchanged). The inverse image of X is the union of the curves X_{ζ} : x_{3} = … = x_{n} = 0, x_{1} = ζx_{2}, where ζ runs among the e-th roots of unity. We may then replace (Y, W) by (Y′, W′ : x_{1} = 0) and (X, Z) by (X_{ζ}, Z). This yields the desired factorization, and the equality e_{f} = e_{g} is immediate. Q.E.D.

Remark 1.2.9. The theory of log schemes allows to prove proposition 1.2.8 without the restriction that X is a curve (App. B), but we shall not need this stronger result.

Remark 1.2.10. Similar definitions are understood for formal schemes of the type considered in (1.1.6). Analogous results hold.

1.3 Tubular neighborhoods.

Definition 1.3.1. Let X be a normal K-variety and Z be a closed subvariety of X of pure codimension 1. We say that X is a tubular neighborhood of Z, if there is a commutative diagram of K-varieties

(1.3.1.1)  \begin{matrix}X & ↩^{i} & Z \\ f ↓ & ↙ g \\ S \end{matrix}

where

i) i is the canonical closed immersion; ii) f is smooth of (pure) relative dimension 1; iii) g is étale. We will say that (X⟶^{f}S, t) is a coordinatized tubular neighborhood of Z in X (and t is a relative coordinate) if (1.3.1.1) can be completed into the commutative diagram

(1.3.1.2)

X  ↩^{i} Z
ϕ ↙ f ↓ ↙ g
A_{S}^{1} ⟶  S
pr_{1} ↓  ↓
A_{K}^{1} ⟶ SpecK

where ϕ is étale, the square is fibered, pr_{1} ∘ ϕ = t, and J_{Z} = tO_{X}. The (resp. coordinatized) tubular neighborhood (1.3.1.1) (resp. (1.3.1.2)) will be said to be strict if S = Z, g is the identity map, and the fibers of f are geometrically connected.

Lemma 1.3.2. For any model (X, Z) of (F, v), there exists a Zariski (resp. étale) open cover of Z in X consisting of (resp. strict) coordinatized tubular neighborhoods (U⟶^{f}S, t) of the inverse image of Z in U.

Proof. For any x ∈ Z, we may shrink (X, Z) to an affine coordinatized neighborhood of x as in (1.1.4). Then f : X ⟶ S := SpecK[x_{2}, …, x_{d}] is smooth of pure relative dimension 1, and g : Z ⟶ S is étale.

We now show that the statement for the étale topology follows from the one for the Zariski topology. Given the diagram (1.3.1.2) and a closed point x ∈ Z, there is an étale neighborhood S′ ⟶ S of s = g(x) ∈ S, and a closed point s′ ∈ S′ above s such that in the diagram obtained by base change

(1.3.2.1)

X_{S′}  ↩^{i_{S′}} Z_{S′}
ϕ_{S′} ↙ f_{S′} ↓ ↙ g_{S′}
A_{S′}^{1} ⟶  S′
pr_{1} ↓  ↓
A_{K}^{1} ⟶ SpecK  ,

the morphism g_{S′} admits a section ψ : S′ ⟶ Z_{S′}, with ψ(s′) a closed point of Z_{S′} lying above x [BLR, 2.2, Prop.14]. We then replace Z_{S′} by S′, g_{S′} by 1_{S′}, and i_{S′} by i_{S′} ∘ ψ. So, we obtain a diagram

X_{S′}  ↩^{i′} S′
f_{S′} ↓ ↙ 1_{S′}
S′

which is a strict tubular neighborhood of S′, but for the condition on (geometric) connectedness of the fibers of f_{S′}. To satisfy this, we replace X_{S′} by the union of the connected components of i′(y) in f_{S′}^{−1}(y), for all points y ∈ S′, which is open in X_{S′} [EGA IV, 15.6.5]. Since K is algebraically closed, we conclude, using [EGA IV, 9.7.7, i], that all fibers of f_{S′} are in fact geometrically connected. Q.E.D.

Proposition 1.3.3. Let (C, P) ⟶ (X, Z) be a closed immersion of models, with C a curve. There exist an open neighborhood U of P in X and a coordinatized tubular neighborhood (U⟶^{f}S, t) of Z ∩ U, such that, in the notation of (1.3.1.2) with (X, Z) replaced by (U, Z ∩ U), C ∩ U = f^{−1}(g(P)).

Proof. Since C and Z meet transversally at the closed point P, [EGA IV, 17.13.8.1] gives an isomorphism

J_{P} /J_{P}^{2} ≅ (J_{Z}/J_{Z}^{2} ⊗_{O_{Z}} κ(P)) ⊕ (J_{C}/J_{C}^{2} ⊗_{O_{C}} κ(P))

where by J_{Y} we denote the O_{X}-ideal corresponding to the closed subscheme Y of X. Since J_{P} /J_{P}^{2} ≅ Ω_{X/K}^{1} ⊗_{O_{X}} κ(P), for any system of sections x_{1} (resp. x_{2}, …, x_{d}) of O_{X} in a neighborhood U of P, representing a system of generators of J_{Z}/J_{Z}^{2} ⊗_{O_{Z}} κ(P) (resp. of J_{C}/J_{C}^{2} ⊗_{O_{C}} κ(P)), (x_{1}, …, x_{d}) are étale coordinates at P. We shrink U around U ∩ Z, so that the morphism f : U ⟶ S := SpecK[x_{2}, …, x_{d}] is a tubular neighborhood of Z, locally at f(P). Certainly C ∩ U ⊂ f^{−1}(f(P)). But f has connected fibers, so we have equality. Q.E.D.

These considerations extend without difficulty to the case of a non-connected (but still smooth) divisor Z.

1.4 Integral curves.

1.4.1. In the situation of a tubular neighborhood 1.3.1, let ∂ ∈ Γ(X, \underline{Der}_{X/K}) be a non-zero derivation such that

(1.4.1.1)  < ∂, f^{∗}ω >= 0 ,

for any section ω of Ω_{S/K}^{1}; we will say that (1.3.1.1), or simply X⟶^{f}S, is a family of integral curves of ∂ in X (based on S).

The commutative diagram (1.3.1.2) defines a splitting

(1.4.1.2)  Ω_{X/K}^{1} = O_{X}dt ⊕ f^{∗}Ω_{S/K}^{1} .

If (X⟶^{f}S, t) is a coordinatized tubular neighborhood of Z and ∂, as in (1.4.1.1), also satisfies < ∂, dt >= 1, we write ∂ = ∂_{t,f}, or simply ∂ = \frac{∂}{∂t}, when f is clearly understood.

Definition 1.4.2. Let (X, Z) be a model and D ∈ Γ(X, \underline{Der}_{X/K}). We say that D is transversal to Z at ξ ∈ Z if for one (hence for any) local equation l ∈ O_{X,x} of Z at ξ, Dl is a unit in O_{X,ξ}. In that case, we also say that lD has a simple zero along Z at ξ. We say that D is transversal to (resp. has a simple zero along) Z, if D is transversal to (resp. has a simple zero along) Z at every point ξ ∈ Z.

For instance, if (X⟶^{f}S, t) is a coordinatized tubular neighborhood of Z in X, ∂_{t,f} is transversal to Z.

Similar definitions are understood in case (X̂_{Z}, Z) is a smooth K-model of (F̂, v̂) and D ∈ Γ(X̂_{Z}, \underline{Der}_{X̂_{Z}/K}).

Proposition 1.4.3. Let (X, Z) be a model of (F, v) and D ∈ Γ(X, \underline{Der}_{X/K}) be transversal to (resp. have a simple zero along) Z at the point z ∈ Z. A necessary and sufficient condition for the existence of a family of integral curves (U⟶^{f}S, t) of D on some open neighborhood U of z in X, is that the field of constants F^{D} of D in F satisfy

(1.4.3.1)  trdeg_{K}F^{D} = trdeg_{K}F − 1 .

Proof. We first notice that F^{D} \ {0} ⊂ R_{v}^{×}. In fact, if x_{1} is a parameter of R_{v}, and if g = x_{1}^{n}u, with n > 0 and u ∈ R_{v}^{×}, were a non-zero element of F^{D}, the equation 0 = Dg = nx_{1}^{n−1}(Dx_{1})u + x_{1}^{n}Du, would give a contradiction both in case D is transversal and in case D has a simple zero along v. So, by reduction modulo 𝔪_{v}, we get on injection π : F^{D} ↪ k(v) ≅ κ(Z), which is an algebraic extension of fields, by (1.4.3.1). We then pick a local equation x_{1} ∈ O_{X,z} for Z, and choose x_{2}, …, x_{d} in F^{D} ∩ π^{−1}(O_{Z,z}), so that (π(x_{2}), …, π(x_{d})), is a system of parameters of O_{Z,z}. By the argument used in the proof of a) ⇒ d) in [EGA IV, Cor. 17.12.2], upon replacing X by a Zariski neighborhood of η_{Z}, we may assume that x_{1}, …, x_{d} are étale coordinates on X, adapted to Z. Upon further Zariski localization at z, we obtain a coordinatized tubular neighborhood (X⟶^{f}S, x_{1}) of Z in X as in (1.3.1.2), where D = u\frac{∂}{∂x_{1}} (resp. D = ux_{1} \frac{∂}{∂x_{1}} ), with u ∈ Γ(X, O_{X}) never zero on Z if D is transversal to (resp. has a simple zero along) Z. Q.E.D.

Corollary 1.4.4. Let (F, v) be a function field over K equipped with a divisorial valuation v. Let ∂ ∈ Der(F/K) be such that ∂R_{v} ⊂ 𝔪_{v} but ∂𝔪_{v} ⊄ 𝔪_{v}^{2}, and assume trdeg_{K}F^{∂} = trdeg_{K}F − 1. There exists a model (X, Z) of (F, v) such that ∂ ∈ Γ(X, \underline{Der}_{X/K}) has a simple zero along Z and a coordinatized tubular

neighborhood (X⟶^{f}S, t) of Z in X, which is a family of integral curves of ∂ in X.

In the case of a smooth formal K-model (X̂_{Z}, Z) of (F̂, v̂) one can be more precise.

Lemma 1.4.5. Let (X̂_{Z}, Z) be a smooth K-model of (F̂, v̂).

i) If there exists D ∈ Γ(X̂_{Z}, \underline{Der}_{X̂_{Z}/K}) transversal to Z, then there exists an isomorphism

(1.4.5.1)  X̂_{Z} ⟶^{∼}Â_{K}^{1} × Z ,

such that the composite map

(1.4.5.2)  Z ↪ X̂_{Z} ⟶^{∼}Â_{K}^{1} × Z⟶^{pr_{2}}Z

is the identity on the K-scheme Z. ii) Assume an isomorphism as in i) exists. Let D ∈ Γ(X̂_{Z}, \underline{Der}_{X̂_{Z}/K}) be transversal to (resp. have a simple zero along) Z. Then there exists a unique isomorphism as in i) in which D corresponds to the derivation (\frac{d}{dt}, 0) (resp. (t\frac{d}{dt}, 0)), where t denotes the canonical coordinate on Â_{K}^{1} ≅ SpfK[[t]].

Proof. i) The existence of D transversal to Z implies the triviality of the conormal sheaf : N_{Z/X} = J_{Z}/J_{Z}^{2} ≅ O_{Z}. The sheaf OX̂_{Z} on Z coincides with the completion of the O_{Z}-algebra S_{O_{Z}}^{·} (N_{Z/X}) in the topology defined by the ideal of augmentation. So, we get an isomorphism (1.4.5.1) satisfying the requirement in i).

ii) We treat the case of D transversal to Z first. We modify the given isomorphism (1.4.5.1) so as to satisfy the further condition on the derivation corresponding to D. Uniqueness of this modification allows one to assume that X̂_{Z} is affine. We set Z = SpecA, X̂_{Z} = SpfA[[x]] and Dx = ∑_{j≥0} a_{j}x^{j}, a_{j} ∈ A, a_{0} ∈ A^{×}. We first show that there is a unique map (of sets)

\begin{matrix}A & ⟶ A[[x]] \\ a & ⟼ φ_{a}(x) \end{matrix}

such that φ_{a}(0) = a and Dφ_{a}(x) = 0. We write φ_{a}(x) = a + ∑_{j>0} u_{j}x^{j}, Da = ∑_{h≥0} b_{h}x^{h}, b_{h} ∈ A, and impose

0 = Dφ_{a}(x) = Da + ∑_{j>0} ju_{j}x^{j−1} ∑_{i≥0} a_{i}x^{i} + ∑_{j>0}(Du_{j})x^{j} .

This gives

u_{1} = −b_{0}/a_{0} ∈ A^{×}
(h + 1)a_{0}u_{h+1} = −b_{h} − ∑_{j≤h^{i+j=h+1}} ju_{j}a_{i} − coefficient of x^{h} in ∑_{j=1}^{h}(Du_{j})x^{j} ,

from which we see that u_{1}, … u_{h} ∈ A uniquely determine u_{h+1} ∈ A. The map a ⟼ φ_{a} is necessarily a K-algebra homomorphism, and it is injective. If A′ denotes the image of A in A[[x]] via a ⟼ φ_{a}, we have A[[x]] = A′[[x]] and we see that we might have assumed from the beginning that DA = 0. We now look for t = ∑_{j>0} u_{j}x^{j}, u_{j} ∈ A, such that Dt = 1. This gives

\begin{aligned}1 & = ∑_{j>0} ju_{j}x^{j−1} ∑_{i≥0} a_{i}x^{i} \\ u_{1} & = \frac{1}{a_{0}} ∈ A′ \\ 0 & = ∑_{i+j=h+1} ju_{j}a_{i} , h > o , \end{aligned}

which again shows that u_{1}, …, u_{h} ∈ A uniquely determine u_{h+1} ∈ A.

We now discuss the case of D having a simple zero along Z. We are given the isomorphism (1.4.5.1) (again we assume Z = SpecA affine) and the standard coordinate t on Â_{K}^{1}. We apply the first part of this ii) paragraph to the derivation t^{−1}D. We may then assume that D = (u\frac{d}{dt}, 0), for a unit u ∈ A[[t]]^{×}. We look for a parameter x of A[[t]] such that Dx = x. We have Dt = ut. If u = 1, we take x = t. Otherwise, let 1 − u = βt^{r}, with β ∈ A[[t]], r ∈ Z_{>0}. We set x = αt, for a unit α ∈ A[[t]]^{×}. The condition D(αt) = αt is equivalent to Dα = α(1 − u) = αβt^{r}. So, α = exp βt^{r+1}/(r + 1) and x = t exp βt^{r+1}/(r + 1) is the only solution. Q.E.D.

Corollary 1.4.6. Let (F̂, v̂) be a formal function field over K. Let ∂ ∈ Der(F̂/K) be such that ∂Rv̂ ⊂ Rv̂ but ∂𝔪v̂ ⊄ 𝔪v̂ (resp. ∂Rv̂ ⊂ 𝔪v̂ but ∂𝔪v̂ ⊄ 𝔪2 ; v̂). Let Z = SpecA be any smooth affine K-model of k(v̂). There exists an open dense subset U of Z and a smooth affine formal K-model (X̂_{U}, U) of (F̂, v̂), with X̂_{U} = Â_{K}^{1} × U, such that ∂ = (\frac{d}{dt}, 0) (resp. ∂ = (t\frac{d}{dt}, 0)), t denoting the canonical coordinate on Â_{K}^{1} ≅ SpfK[[t]].

Definition 1.4.7. A derivation ∂ ∈ Der(F̂/K) such that ∂Rv̂ ⊂ Rv̂ but ∂𝔪v̂ ⊄ 𝔪v̂ (resp. ∂Rv̂ ⊂ 𝔪v̂ but ∂𝔪v̂ ⊄ 𝔪2 ; v̂) is called transversal to v̂ (resp. is said to have a simple zero at v̂).

So, ∂ is transversal to v̂ if and only if tv̂∂ has a simple zero at v̂.

§2 Logarithmic differential operators.

2.1. Let (X, Z) be a model of the divisorially valued function field (F, v). We denote by j the immersion of X \ Z in X.

We define \underline{Der}_{X/K,Z} (resp. \underline{Diff}_{X/K,Z}^{n}, for any n = 0, 1, …) as the subsheaf of \underline{Der}_{X/K} (resp. \underline{Diff}_{X/K}^{n}) of the sections that preserve, as operators on O_{X}, the J_{Z}-adic filtration. Finally, \underline{Diff}_{X/K,Z} = ∪_{n}\underline{Diff}_{X/K,Z}^{n}. The stalk of \underline{Der}_{X/K,Z} (resp. \underline{Diff}_{X/K,Z}^{n}) at η_{Z} is the R_{v}-module (2.1.1)

\begin{matrix}Der_{v}(F/K) = & \{∂ ∈ Der(F/K) | ∂𝔪_{v} ⊂ 𝔪_{v} \} \\ (resp. Diff_{v}^{n}(F/K) = & \{L ∈ Diff^{n}(F/K) | L𝔪_{v}^{i} ⊂ 𝔪_{v}^{i} , ∀i = 0, 1, …, n \}). \end{matrix}

Lemma 2.1.2. The O_{X}-modules \underline{Der}_{X/K,Z} and \underline{Diff}_{X/K,Z}^{n} are locally free of finite type. The filtration of \underline{Diff}_{X/K,Z}^{n} by the order of differential operators is canonically split. If (t_{v} = x_{1}, x_{2}, …, x_{d}) are étale coordinates on X adapted to Z, a basis of sections of \underline{Der}_{X/K,Z} (resp. of \underline{Diff}_{X/K,Z}^{n}) is given by \{x_{1} \frac{∂}{∂x_{1}} , \frac{∂}{∂x_{2}} , …, \frac{∂}{∂x_{d}} \} (resp. by \{(x_{1} \frac{∂}{∂x_{1}} )^{α_{1}}(\frac{∂}{∂x_{2}} )^{α_{2}} … (\frac{∂}{∂x_{d}})^{α_{d}} | ∑_{i=1}^{d} α_{i} ≤ n \}). Such a basis of \underline{Der}_{X/K,Z} will be said to be adapted to Z.

Proof. Let X = SpecA and L ∈ Γ(X, \underline{Diff}_{X/K,Z}^{n}). We can write L =

∑_{i=0}^{n} P_{i}(\frac{∂}{∂x_{1}} )^{i}, with P_{i} = ∑_{\underline{β}∈N^{d−1}} a(i) ; \underline{β}(\frac{∂}{∂x_{2}} )^{β_{2}} … (\frac{∂}{∂x_{d}})^{β_{d}}, and a(i) ; \underline{β} ∈ A. We

first prove that a(i) ; \underline{β} ∈ x_{1}^{i}A, ∀i = 0, …, n and ∀\underline{β} = (β_{2}, …, β_{d}) ∈ N^{d−1}. Assume

a(i) ; \underline{β} ∈ x_{1}^{i}A, ∀i < j ≤ n and ∀\underline{β} ∈ N^{d−1}. Let \underline{β} ∈ N^{d−1}; we compute

L(x_{1}^{j}\underline{x}\underline{β}) = ∑_{i=0}^{j−1} j(j − 1) … (j − i + 1)x_{1}^{j−i}P_{i}(\underline{x}\underline{β}) + j!P_{j}(\underline{x}\underline{β}) ∈ x_{1}^{j}A .

It follows from the inductive assumption that P_{j}(\underline{x}\underline{β}) ∈ x_{1}^{j}A. Therefore, a(j) ; \underline{β} ∈ x_{1}^{j}A, ∀\underline{β} ∈ N^{d−1}. Hence, an A-basis of Γ(X, \underline{Diff}_{X/K,Z}^{n}) is given by the operators \{x_{1}^{α_{1}}(\frac{∂}{∂x_{1}} )^{α_{1}}(\frac{∂}{∂x_{2}} )^{α_{2}} … (\frac{∂}{∂x_{d}})^{α_{d}} | ∑_{i=1}^{d} α_{i} ≤ n \}. Formulas (2.1.2.1)

x^{m}(\frac{∂}{∂x})^{m} = x\frac{∂}{∂x}(x\frac{∂}{∂x} − 1) … (x\frac{∂}{∂x} − m + 1) = (x\frac{∂}{∂x})^{m} + ∑_{j<m} b_{m,j}(x\frac{∂}{∂x})^{j}

and their reciprocals

(2.1.2.2)  (x\frac{∂}{∂x})^{m} = x^{m}(\frac{∂}{∂x})^{m} + ∑_{j<m} a_{m,j}x^{j}(\frac{∂}{∂x})^{j} ,

with a_{m,j}, b_{m,j} ∈ Z, permit us to conclude. Q.E.D. Remark 2.1.3. If X is a curve, the fiber of \underline{Diff}_{X/K,Z}^{n} at the point Z has a canonical basis over K: (1, x\frac{∂}{∂x}, …, (x\frac{∂}{∂x})^{n}) (which does not depend on the choice of the local coordinate x at Z).

2.1.4. Let (X, Z) be a smooth K-model and let D ∈ Γ(X, \underline{Der}_{X/K}). The following facts are obviously equivalent:

i) D satisfies (1.4.3.1) and there exists a local equation x of Z such that Dx = x; ii) there exists an adapted basis (D_{1}, …, D_{d}) of (\underline{Der}_{X/K,Z})_{η_{Z}} with D = D_{1}. If the previous conditions hold, then D_{j}F^{D} ⊂ F^{D} and D_{j}x = 0, for j = 2, …, d.

2.1.5. One defines similarly the sheaves (resp. modules) \underline{Der}_{X̂_{Z}/K,Z} and \underline{Diff}_{X̂_{Z}/K,Z}^{n}

(resp. Derv̂(F̂/K) and Diff n ; v̂(F̂/K)) on a smooth formal model (X̂_{Z}, Z) of (F̂, v̂). A system of étale coordinates (x_{1}, …, x_{d}) on X̂_{Z} is adapted to Z if there is an isomorphism (1.4.5.1) such that x_{1} corresponds to the standard coordinate on Â_{K}^{1} and (x_{2}, …, x_{d}) to étale coordinates on Z. The basis (x_{1} \frac{∂}{∂x_{1}} , \frac{∂}{∂x_{2}} , …, \frac{∂}{∂x_{d}}) of \underline{Der}_{X̂_{Z}/K,Z} (and of Derv̂(F̂/K)) is then adapted to Z. According to (1.4.5), any D ∈ Derv̂(F̂/K) having a simple zero at v̂ is such that Dx = x for some parameter x of R̂_{v} and is part of an adapted basis (D = D_{1}, …, D_{d}) of Derv̂(F̂/K). Once again, D_{j}F̂^{D} ⊂ F̂^{D} and D_{j}x = 0, for j = 2, …, d. A result analogous to lemma 2.1.2 holds. Notice that the ring Diff_{cont}(F̂/K) of v̂-adically continuous F̂/K-differential operators of F̂ is generated as a (“left”) F̂-vector space by Diffv̂(F̂/K) =

∪_{n}Diff n ; v̂(F̂/K).

2.2. The sheaf Ω_{X/K}^{1}(log Z) of differentials with logarithmic singularities at Z is the locally free O_{X}-module locally generated by \frac{dx_{1}}{x_{1}} , dx_{2}, …, dx_{d} for any system (t_{v} = x_{1}, x_{2}, …, x_{d}) of étale coordinates on X adapted to Z. We view it as a subsheaf of j_{∗}Ω_{X\Z}^{1}, and as a natural dual for \underline{Der}_{X/K,Z}.

We define Ω_{X/K}^{h}(log Z) = ⋀^{h} Ω_{X/K}^{1}(log Z), viewed as a subsheaf of j_{∗}Ω_{X\Z}^{h}. It is clear that Ω_{X/K}^{•}(log Z) is a subcomplex of j_{∗}Ω_{X\Z}^{•}, called the De Rham complex of X with logarithmic singularities at Z.

Lemma 2.2.1. Ω_{X/K}^{•}(log Z) is the smallest subcomplex of j_{∗}Ω_{X\Z}^{•} containing Ω_{X/K}^{•}, stable under wedge product, and such that \frac{df}{f} is a local section of Ω_{X/K}^{1}(log Z) whenever f is a local section of j_{∗}O_{X\Z} . A local section ω of j_{∗}Ω_{X\Z}^{h} is a section of Ω_{X/K}^{h}(log Z) if and only if ω and dω have a simple pole at Z.

See [De, II, 3.2 and 3.3.1] (the proof in given in the analytic context but is easily translated into algebraic terms, working locally for the étale topology on X; it works more generally for divisors with strict normal crossings).

2.3. We consider a model (Y, W) of another divisorially valued function field (G, w) over K, and a morphism of models f : (X, Z) ⟶ (Y, W). We denote by k the immersion of Y \ W in Y.

It follows from the description (2.2.1) that the functoriality map f^{∗}(k_{∗}Ω_{Y \W}^{•}) ⟶ j_{∗}Ω_{X\Z}^{•} induces a map

(2.3.1)  f^{∗}Ω_{Y/K}^{•}(log W) ⟶ Ω_{X/K}^{•}(log Z).

Using the fact that Ω_{X/K}^{1}(log Z) is locally free, we see by duality that the functoriality map \underline{Der}_{X/K} ⟶ f^{∗}\underline{Der}_{Y/K} induces a map of O_{X}-modules

(2.3.2)  T_{f} : \underline{Der}_{X/K,Z} ⟶ f^{∗}\underline{Der}_{Y/K,W} .

Furthermore, using (2.1.2), we deduce that the functoriality map \underline{Diff}_{X/K}^{n} ⟶ f^{∗}\underline{Diff}_{Y/K}^{n} induces a map of O_{X}-modules

(2.3.3)  T_{f} : \underline{Diff}_{X/K,Z}^{n} ⟶ f^{∗}Diff_{Y/K,W}^{n} .

If g : (Y, W) ⟶ (T, D) is another morphism of models, we have

f^{∗}(T_{g}) ∘ T_{f} = T_{g∘f} ,

and T_{1} = 1. From these, we deduce maps of R_{v}-modules

(2.3.4)  \begin{matrix}T_{f} : Der_{v}(F/K) ⟶ Der_{w}(G/K) ⊗_{R_{w}} R_{v} , \\ T_{f} : Diff_{v}^{n}(F/K) ⟶ Diff_{w}^{n}(G/K) ⊗_{R_{w}} R_{v} , \end{matrix}

and similar maps for completions.

Let us examine the special cases of closed immersions and log dominant f : (X, Z) ⟶ (Y, W).

Proposition 2.4.

i) If f is a closed immersion, there is an exact sequence

(2.4.1)  0 ⟶ \underline{Der}_{X/K,Z} ⟶^{T_{f}}f^{∗}\underline{Der}_{Y/K,W} ⟶ N_{X/Y}^{∨} ⟶ 0,

where N_{X/Y}^{∨} denotes the normal sheaf. ii) If f is log dominant, there is an open neighborhood U of η_{Z} in X and an exact sequence

(2.4.2) 0 ⟶ (\underline{Der}_{X/Y})_{|U} ⟶ (Der_{X/K,Z})_{|U} ⟶^{T_{f}}(f^{∗}\underline{Der}_{Y/K,W})_{|U} ⟶ 0,

where \underline{Der}_{X/Y} denotes the dual sheaf of Ω_{X/Y}^{1} . In particular, if we set

Der_{v}(F/G) = Der_{v}(F/K) ∩ Der(F/G) ,

we may identify T_{f}(∂) with the restriction of ∂ ∈ Der_{v}(F/K) to an F-valued derivation of G, and we have an exact sequence

(2.4.3) 0 ⟶ Der_{v}(F/G) ⟶ Der_{v}(F/K)⟶^{T_{f}}Der_{w}(G/K) ⊗_{R_{w}} R_{v} ⟶ 0

We may choose an adapted basis (D_{1}, …, D_{d}) of Der_{v}(F/K), with (D_{1|G}, …, D_{r|G}) an adapted basis of Der_{w}(G/K), while D_{i|G} = 0, for i = r + 1, …, d.

Proof. i) The immersion of X into Y is regular, so that there is an exact sequence 0 ⟶ \underline{Der}_{X/K} ⟶ f^{∗}\underline{Der}_{Y/K} ⟶ N_{X/Y}^{∨} ⟶ 0. Hence the problem is localized at Z. Since W and X meet transversally at Z, there exists in a neighborhood of every point of Z in Y a system of étale coordinates (x_{1}, x_{2}, …, x_{n}) such that x_{1} = 0 is a local equation for W, (x_{r+1} = 0, …, x_{n} = 0) is a local equation for X, and (\frac{∂}{∂x_{r+1}} , …, \frac{∂}{∂x_{n}}) is a local basis of sections of N_{X/Y}^{∨}. Then according to (2.1.2), (x_{1} \frac{∂}{∂x_{1}} , \frac{∂}{∂x_{2}} , …, \frac{∂}{∂x_{n}}) is a local basis of sections of f^{∗}\underline{Der}_{Y/K,W}, (x_{1} \frac{∂}{∂x_{1}} , \frac{∂}{∂x_{2}} , …, \frac{∂}{∂x_{r}}) is a local basis of sections of \underline{Der}_{X/K,Z}, and the result follows. ii) There is an exact sequence 0 ⟶ \underline{Der}_{X/Y} ⟶ \underline{Der}_{X/K} ⟶ f^{∗}\underline{Der}_{Y/K}. We have to show that in some open neighborhood U of η_{Z} in X, Im(\underline{Der}_{X/Y} ⟶ \underline{Der}_{X/K}) ⊂ \underline{Der}_{X/K,Z}, and T_{f} is surjective. We may replace X and Y by étale neighborhoods of η_{Z} and η_{W} respectively. In particular, we may assume that W ⟶ Y is a section of a morphism h : Y ⟶ S = W (1.3.2) with smooth connected one-dimensional fibers. We may assume that, if f_{η_{S}} denotes the fiber of the S-morphism f above η_{S}, f_{η_{S}} is smooth except above the point η_{Z} of the curve Y_{η_{S}}. Locally for the étale topology, there is a system of coordinates (x_{1}, …, x_{n}) on X adapted to Z, a system of coordinates (z_{2}, …, z_{r}) on S and a coordinate y on Y_{η_{S}} satisfying: y ∘ f = x_{1}^{e} on X_{η_{S}}, and z_{i} ∘ h ∘ f = x_{i} for i = 2, …, r.

Then we have x_{1}^{e−1}dx_{1} = dx_{2} = … = dx_{r} = 0 in Ω_{X/Y}^{1} . It is now clear that (\frac{∂}{∂x_{r+1}} , …, \frac{∂}{∂x_{n}}) is a local basis of sections of \underline{Der}_{X/Y}, (x_{1} \frac{∂}{∂x_{1}} , \frac{∂}{∂x_{2}} , …, \frac{∂}{∂x_{n}}) is a local basis of sections of \underline{Der}_{X/K,Z}, (x_{1} \frac{∂}{∂x_{1}} , \frac{∂}{∂x_{2}} , …, \frac{∂}{∂x_{r}}) is a local basis of sections of f^{∗}\underline{Der}_{Y/K,W}, and the result follows.

An exact sequence similar to (2.4.3) holds for fields of formal functions.

Remark 2.5. In case ii), we see that Der_{X/Y} is flat at η_{Z} (in contrast to Ω_{X/Y}^{1} if e ≥ 2). On U, the dual of \underline{Der}_{X/Y} is Ω_{X/Y}^{1} (log Z), the image of Ω_{X}^{1}(log Z) in j_{∗}Ω_{(X\Z)/(Y \W)}^{1}. Notice that although dx_{1} ≠ 0 in Ω_{X/Y}^{1} if e ≥ 2, we have dx_{1}/x_{1} = e^{−1}f^{∗}dy/y = 0 in Ω_{X/Y}^{1} (log Z).

§3 Regularity along a divisor.

3.1 Differential modules.

3.1.1. Let F/K be a function field in d variables. An F/K-differential module is a module over Diff(F/K) which is finitely dimensional as a vector space over F; this dimension is called the rank of the differential module. Similarly, if F̂/K is a formal function field, an F̂/K-differential module is a module over Diff_{cont}(F̂/K) which is finitely dimensional as a vector space over F̂. (When using these definitions, it is convenient to drop our general assumption that K is algebraically closed).

If E_{1} and E_{2} are differential modules with respect to the same (formal) function field, there are canonically defined differential modules E_{1} ⊗_{F} E_{2} and \underline{Hom}_{F}(E_{1}, E_{2}).

3.1.2. Let V be a smooth connected variety over K, F = κ(V). A connection on a coherent O_{V} -module E is an additive mapping

(3.1.2.1)  ∇ : E ⟶ E ⊗_{O_{V}} Ω_{V}^{1}

satisfying Leibniz rule. It is called integrable if the induced O_{V} -linear mapping

(3.1.2.2)  ∇ : \underline{Der}_{V} ⟶ \underline{End}_{K}E

is compatible with the Lie bracket [ , ]; E is then necessarily locally free (cf.

[Ka1] and (III,2.1) for more detail).

The integrability ensures that the generic fiber

(3.1.2.3)  ∇ : \underline{Der}(F/K) ⟶ \underline{End}_{K}E_{η_{V}}

extends to a structure of F/K-differential module E on E_{η_{V}}; we say that (E, ∇) is a model of E on V.

If (E_{1}, ∇_{1}) and (E_{2}, ∇_{2}) are models of the F/K-differential modules E_{1} and E_{2} respectively, there are canonically defined models (E_{1}, ∇_{1}) ⊗ (E_{2}, ∇_{2}) (connection: ∇_{1}⊗1+1⊗∇_{2}) and \underline{Hom}((E_{1}, ∇_{1}), (E_{2}, ∇_{2})) (connection: −(∇_{1})^{t}⊗ 1 + 1 ⊗ ∇_{2}) of E_{1} ⊗_{F} E_{2} and \underline{Hom}_{F}(E_{1}, E_{2}) respectively .

An O_{V} -linear mapping E_{1} ⟶ E_{2} is called horizontal if it commutes with the connections.

3.1.3. Let us consider the situation of (1.2.7.1). Any G/K-differential module E uniquely extends to a F/K-differential module E_{F}. Viceversa, if [F : G] < ∞, any F/K-differential module E′ may be regarded as a G/K-differential module, which we denote by _{G}E′ in that capacity. A similar notation will be used for F̂/K or Ĝ/K-differential modules.

There is a canonical morphism (_{G}E′)ˆ ⟶Ĝ (E′ ^{ˆ (v)}), where E′ ^{ˆ (v)} denotes the v-completion of E′. In fact, the canonical morphism (_{G}E′)ˆ ⟶ ⊕_{v|w} Ĝ(E′ ^{ˆ (v)}) is an isomorphism.

3.1.4. Inverse and direct images of integrable connections will be discussed in detail in (III,2.1). It suffices here to say that if (X, Z)⟶^{f}(Y, W) is a log dominant morphism of models, corresponding to (1.2.7.1), then for any model (E, ∇) of E on Y \ W, f^{∗}(E, ∇) is a model of E_{F} on X \ Z. If moreover f is finite étale on X \ Z, then for any model (E′, ∇′) of E′ on X \ Z, f_{∗}(E′, ∇′) is a model of _{G}E′ on Y \ W.

3.2 Regularity: review of the one-variable case.

Let C be a field of characteristic 0. We consider the field of formal Laurent series C((x)) with the x-adic valuation v.

Definition 3.2.1. A C((x))/C-differential module Ê is called regular if it contains a C[[x]]-lattice stable under x\frac{d}{dx}.

Let \overline{C} be the algebraic closure of C and C ⊂ C′ ⊂ \overline{C} be an intermediate extension. For a C((x))/C-differential module Ê we set Ê_{C′} = C′((x))⊗_{C((x))} Ê, a C′((x))/C′-differential module.

The following results are classical (see [Man] or [DGS,III.8]).

Proposition 3.2.2. A C((x))/C-differential module Ê of rank μ is regular if and only if the following equivalent conditions are satisfied:

a) for every m ∈ Ê, the smallest C[[x]]-submodule of Ê containing m and stable under x\frac{d}{dx} is finitely generated;

b) there is an integer c and a C[[x]]-lattice in Ê stable under x^{c}Diff_{v}(C((x))/C);

c) for some (resp. any) cyclic vector m ∈ Ê, the coefficients b_{j} of the associated differential equation (x\frac{d}{dx})^{μ}m − ∑_{j<μ} b_{j}(x\frac{d}{dx})^{j}m = 0 belong to C[[x]] (in this situation we also say that the differential equation is regular);

d) there is a finite subset A of \overline{C} such that for any intermediate extension C(A) ⊂ C′ ⊂ \overline{C} there is a decomposition

(3.2.2.1)  Ê_{C′} ≅ ⊕_{α∈A}C′((x)) ⊗_{C′} KerÊ_{C′} (x\frac{d}{dx} − α)^{μ} .

Moreover, if such a decomposition exists over C′, it is is canonical, and the factor C′((x)) ⊗_{C′} KerÊ_{C′} (x\frac{d}{dx} − α)^{μ} = C′((x)) ⊗_{C(α)} KerÊ_{C(α)}(x\frac{d}{dx} − α)^{μ} does

not change if α is replaced by α + n, n ∈ Z . The subset A + Z of \overline{C} is uniquely determined by the condition

(3.2.2.2)  β ∈ A + Z if and only if KerÊ\overline{C} (x\frac{d}{dx} − β)^{μ} ≠ 0 .

Proposition 3.2.3. Let Ê_{1} and Ê_{2} be C((x))/C-differential modules.

i) If Ê sits in an exact sequence of C((x))/C-differential modules

0 ⟶ Ê_{1} ⟶ Ê ⟶ Ê_{2} ⟶ 0

then Ê is regular if and only if Ê_{1} and Ê_{2} are regular. ii) If Ê_{1} and Ê_{2} are regular, so are Ê_{1} ⊗_{C((x))} Ê_{2} and \underline{Hom}_{C((x))}(Ê_{1}, Ê_{2}).

3.2.4. Let F/K be a function field in one variable (K not necessarily algebraically closed, but closed in F) and v be a discrete valuation of F, trivial on K. Let ∂ be a K-linear derivation of F̂ transversal to v (1.4.7), and let C = F̂^{∂} be the constant subfield. Then C ≅ k(v) is a finite extension of K, and there is a canonical isomorphism (R̂_{v}, v̂, ∂) ≅ (C[[x]], x−adic valuation, \frac{d}{dx}) ( also (1.4.6)).

A F/K-differential module E is called regular at v if the F̂/C-differential module Ê = E ⊗_{F} F̂ is regular.

Notice that if ∂ comes from a derivation of F, then F^{∂} = K, which may be a proper subfield of F̂^{∂}.

3.2.5. In the situation of (3.1.3), it is clear on the definition that Ê (resp. Ê′ if [F̂ : Ĝ] < ∞) is regular if and only if ÊF̂ (resp. ĜÊ′) is regular. Similarly, E (resp. E′ if [F : G] < ∞) is regular at w (resp. at every v | w) if and only if E_{F} (resp. _{G}E′) is regular at v (resp. w). Note that if a divisorial valuation v of F induces the trivial valuation on G, then for any E, E_{F} is regular at v.

3.3 The case of a function field in several variables.

3.3.1. Let (F̂, v) be a field of formal functions over K (assumed again to be algebraically closed), and (D_{1}, …, D_{d}) an adapted basis of Der_{v}(F̂/K) ( (2.1.5)). We set C = F̂^{D_{1}} so that F̂ = C((x)), for a parameter x of R̂_{v} such that D_{1}x = x, D_{j}C ⊂ C and D_{j}x = 0, j = 2, …, d.

Let Ê be a F̂/K-differential module of rank μ. Let us assume that the induced C((x))/C-differential module Ê_{/C} is regular, and consider its decomposition (3.2.2.1), after possibly replacing C by a finite extension. For α ∈ A and ν = 0, 1, …, set K_{α}^{(ν)} = KerÊ(D_{1} − α)^{ν}, a C-vector subspace of Ê.

Lemma 3.3.2. For all j = 2, …, d, ν = 0, 1, …, and α ∈ A

D_{j}K_{α}^{(ν)} ⊆ K_{α}^{(ν+1)} .

Proof. Let ∂ ∈ {D_{2}, …, D_{d}}. The statement is trivial for ν = 0. We proceed by induction: assume the statement holds for ν < ν_{0} (ν_{0} ≥ 1) and let m ∈ K_{α}^{(ν_{0})}. So, (D_{1} − α)^{ν_{0}}m = 0 and (D_{1} − α)m ∈ K_{α}^{(ν_{0}−1)}. Then (D_{1} − α)^{ν_{0}+1}∂m = (D_{1} − α)^{ν_{0}}∂(D_{1} − α)m + (D_{1} − α)^{ν_{0}}(∂α)m = 0, since ∂α ∈ C and by the induction assumption. Q.E.D.

Proposition 3.3.3. i) There exists a F̂-basis \underline{e} of Ê such that

(3.3.3.1)  D_{j}\underline{e} = \underline{e}G_{j} ,

with G_{j} ∈ M_{μ×μ}(C) for all j = 1, …, d. ii) For any F̂-basis \underline{e} of Ê as in i), the eigenvalues of G_{1} are in K.

Proof. Let us first assume that C is sufficiently big as to contain the set A in part d) of (3.2.2). It follows from lemma 3.3.2 that D_{j}K_{α}^{(μ)} ⊆ K_{α}^{(μ)}, for all j = 2, …, d, and this implies i) under our assumption on C. Let us turn to ii) . For j = 2, …, d, one has

D_{j}(G_{1}) = G_{j}G_{1} − G_{1}G_{j} ,

whence D_{j}(G_{1}^{m}) = G_{j}G_{1}^{m} − G_{1}^{m}G_{j}, and D_{j}Tr(G_{1}^{m}) = Tr(D_{j}(G_{1}^{m})) = 0. Therefore, the coefficients of the characteristic polynomial of G_{1} are constant, i.e. in K. In particular A is a subset of K, so that there was in fact no need to enlarge C in i). Q.E.D.

Corollary 3.3.4. For any F̂/K-differential module Ê, the following conditions are equivalent:

i) there exists an R̂_{v}-lattice ER̂_{v} in Ê and an integer c such that Diff_{v}(F̂/K)ER̂_{v} ⊂ x_{1}^{−c}ER̂_{v}; ii) for every R̂_{v}-lattice ER̂_{v} in Ê, there is an integer c such that Diff_{v}(F̂/K)ER̂_{v} ⊂ x_{1}^{−c}ER̂_{v}; iii) for some D ∈ Der_{v}(F̂/K) with a simple zero at (or transversal to) v the induced F̂/F̂^{D}-differential module Ê_{/F̂^{D}} is regular. iv) for every D ∈ Der_{v}(F̂/K) with a simple zero at (or transversal to) v the induced F̂/F̂^{D}-differential module Ê_{/F̂^{D}} is regular;

v) there exists an R̂_{v}-lattice ER̂_{v} in Ê such that Diff_{v}(F̂/K)ER̂_{v} ⊂ ER̂_{v}.

Proof. i) ⇒ iv) follows from the characterization b) of proposition 3.2.2 of regularity in one variable. iv) ⇒ iii) is trivial. iii) ⇒ v) follows from i) of (3.3.3), using (2.1.5) to complete D into an adapted basis of Der_{v}(F̂/K) and taking as lattice ER̂_{v} the R̂_{v}-span of the basis \underline{e}. v) ⇒ i) is obvious. At last, if E′ ; R̂_{v}

is another lattice, there exists a positive integer c′ such that x_{1}^{c′}E′ ; R̂_{v} ⊂ ER̂_{v} ⊂

x_{1}^{−c′}E′ ; R̂_{v}. Then Diff_{v}(F̂/K)ER̂_{v} ⊂ x_{1}^{−c−2c′}ER̂_{v}. This shows that i) ⇔ ii).

Q.E.D.

Definition 3.3.5. A F̂/K-differential module Ê is called regular if it satisfies the equivalent conditions of (3.3.4). If (F, v) is a divisorially valued function field over K with completion F̂, an F/K-differential module E is called regular at v if Ê is regular.

3.4 Regular connections along a divisor.

3.4.1. Let (X, Z) be a model of a divisorially valued function field (F, v). Let E be a F/K-differential module, and let (E, ∇) be a model of E on X \ Z ( (3.1)). The integrable connection ∇ is said to be regular at (or along) Z if and only if E is regular at v.

Proposition 3.4.2. Assume that (x_{1}, …, x_{d}) is an adapted system of coordinates for (X, Z) (cf. (1.1.4)). Then ∇ is regular at Z if and only if there exists a coherent extension Ẽ ⊂ j_{∗}E of E on X and an integer c such that \underline{Diff}_{X/K,Z} Ẽ ⊂ x_{1}^{−c}Ẽ.

Proof. Note that \underline{Diff}_{X/K,Z} acts naturally on j_{∗}E as a sub-K_{X}-algebra of j_{∗}(\underline{Diff}_{X\Z/K}). The “if”-part comes from criterion (3.3.4,i). Conversely, let

Ẽ ⊂ j_{∗}E be any coherent extension of E on X, and let us choose the integer c as in (3.3.4.ii) applied to the completion of the lattice Ẽ_{η_{Z}}. Then there exists a closed subset Z′ ⊂ Z of codimension ≥ 2 in X such that j^{′∗}\underline{Diff}_{X/K,Z} Ẽ ⊂ x_{1}^{−c}j^{′∗}Ẽ,

where j′ denotes the embedding of X \ Z′ in X. It then suffices to replace Ẽ by j′_{∗}j^{′∗}Ẽ, which is coherent according to [EGA IV, 5.11.4]. Q.E.D.

Proposition 3.4.3. Let (X, Z) be a model and (E, ∇), (E_{1}, ∇_{1}), (E_{2}, ∇_{2}) be coherent sheaves with integrable connections on X \ Z.

i) If E sits in a horizontal exact sequence 0 ⟶ E_{1} ⟶ E ⟶ E_{2} ⟶ 0 , then ∇ is regular at Z if and only if ∇_{1} and ∇_{2} are regular at Z. ii) If ∇_{1} and ∇_{2} are regular at Z, so are (E_{1}, ∇_{1})⊗(E_{2}, ∇_{2}) and \underline{Hom}((E_{1}, ∇_{1}), (E_{2}, ∇_{2})).

Proof. This follows from (3.2.3).

Proposition 3.4.4. Let (X, Z)⟶^{f}(Y, W) be a morphism of models such that f(Z) ⊂ W. Let (E, ∇) be a coherent sheaf with integrable connection on Y \ W. If ∇ is regular along W, then f^{∗}(∇) is regular along Z. Conversely, if f^{∗}(∇) is regular along Z and if f is log dominant, then ∇ is regular along W.

Proof. We use the characterization (3.4.2) of regularity along a divisor and the functoriality (2.3.2) of logarithmic differential operators. By definition of the inverse image, we have the equality, for any L ∈ \underline{Diff}_{X/K,Z} and any

local section m of E, ∇(L)f^{∗}(m) = f^{∗}(∇(T_{f}L)m). Let Ẽ ⊂ k_{∗}E be a coherent extension of E on Y as in (3.4.2). We thus have

\underline{Diff}_{X/K,Z}f^{∗}Ẽ = f^{∗}(∇(T_{f} \underline{Diff}_{X/K,Z})Ẽ) ⊂ f^{∗}(∇(Diff_{Y/K,W})Ẽ) .

Let y be a local equation of W at f(η_{Z}). If ∇ is regular at W, there is an integer c such that \underline{Diff}_{Y/K,W} Ẽ ⊂ y^{−c}Ẽ. We obtain that \underline{Diff}_{X/K,Z}f^{∗}Ẽ ⊂ y^{−c}f^{∗}Ẽ ⊂

x_{1}^{−ce_{f}} f^{∗}Ẽ, hence f^{∗}(∇) is regular along Z. The second assertion follows from (3.1.4). and (3.2.5). Q.E.D.

Proposition 3.4.5. The property of being regular at Z is local at η_{Z} for the fppf topology on X.

Proof. If X′ ⟶ X is any fppf neighborhood of η_{Z}, we can extend it into a log dominant morphism f : (X′, Z′) ⟶ (X, Z) of smooth K-models. If f^{∗}(E, ∇) is regular along Z′, (E, ∇) is regular along Z, by the previous statement. Q.E.D.

3.4.6 Examples.

3.4.6.1. Take Y = the (x, y)-plane with the x-axis (y = 0) removed, W = y-axis (x = 0), X = the diagonal (x = y) in the (x, y)-plane, Z = the origin. Let ∇ be the connection on E = O_{Y \W}, whose formal solution is e^{1/y}. It is

regular at W (in fact, it has no singularity at all along W), but its restriction to X \ Z is not regular at Z. However the embedding of X \ Z into Y \ W does not extend to a morphism of models (X, Z) ⟶ (Y, W).

3.4.6.2. Consider the morphism f : (x, y) ⟼ (u, v) = (xy, y) of (X, Z) = (A^{2}, y = 0) into (Y, W) = (A^{2}, v = 0). The connection ∇ on E = O_{Y \W} with formal solution e^{u/v} pulls back to the connection f^{∗}∇ killing e^{x}. So, f^{∗}∇ is regular along Z, while ∇ is irregular along W. Notice that f is a birational map, but it is not flat at η_{Z}.

Theorem 3.4.7. Let (X, Z) be a model and (E, ∇) be a coherent sheaf with integrable connection on X \ Z. The following properties are equivalent:

i) ∇ is regular at Z;

ii) for any locally closed immersion of models (C, P)↪^{i}(X, Z), with C a curve, i_{C\P}^{∗} (∇) is regular at P; iii) there is an open neighborhood U of η_{Z} in X and a tubular neighborhood U⟶^{f}S of Z ∩ U, such that ∇_{|U_{s}} is regular at Z_{s} for a Zariski-dense set of closed points s ∈ S.

Proof. The implication i) ⇒ ii) is part of (3.4.4). The implication ii) ⇒ iii) is immediate. Let us prove iii) ⇒ i). We may replace U by any affine connected neighborhood of η_{Z}. This allows us to assume that

a) there is a transversal derivation ∂ = ∂_{t,f} ∈ Γ(U, \underline{Der}_{U/K}) such that f is a family of integral curves of ∂; in particular, by (1.4.1.2), ∂ lies in the image of Γ(U_{s}, \underline{Der}_{U_{s}/K}) for every s ∈ S;

b) there is a section m of E over U \ Z such that E is free with basis (m, ∇(t∂)m, …, ∇(t∂)^{μ−1}m) (by the lemma of the cyclic vector). Let us write the associated differential equation (t∂)^{μ}m − ∑_{j<μ} b_{j}(t∂)^{j}m = 0, with b_{j} ∈ O(U \Z). By assumption the restriction of b_{j} to U_{s} \Z_{s} is holomorphic at every point of the finite set Z_{s} (Fuchs’ criterion of regularity, c) of (3.2.2)). Since this holds for a dense set of points s ∈ S, this implies that b_{j} has no pole at η_{Z}, which means that (E, ∇) is regular at Z. Q.E.D.

In (II.4), we shall show that in order to check the regularity of ∇, it suffices to find sufficiently many locally closed curves C ⊂ X, so that the points C ∩ Z are dense in Z, for which ∇_{|C\C∩Z} is regular.

3.4.8 Example. Take X = the (x, y)-plane, Z = x-axis (y = 0), C = any smooth curve cutting Z transversally (only) at the origin. (So (C, P) ↪ (X, Z) is a closed immersion of models.) Let ∇_{1} be the connection on E = O_{X\Z}, whose formal solution is e^{x/y} (i.e. ∇_{1}(1) = −\frac{dx}{y} + x\frac{dy}{y^{2}} ). It is not regular at Z, but its restriction to C \ P is regular at P. This shows the necessity of considering a dense set of points P ∈ Z.

3.4.9. Let X be a smooth connected variety over K, F = κ(X). Let j : X ⟶ \overline{X} be a dominant open immersion of X in a normal K-variety \overline{X}, and let \overline{Z} be a (not necessarily irreducible) subvariety of \overline{X} \ X of pure codimension one in \overline{X}. If (E, ∇) is a coherent sheaf with integrable connection on X, we say that ∇ is regular at \overline{Z} if its generic fiber E is regular at the divisorial valuations of F defined by the irreducible components of \overline{Z}.

3.4.10. This definition applies especially in the case where \overline{X} and \overline{Z} are smooth, but \overline{Z} has several connected components. This geometric setting is a useful generalization of the notion of smooth K-model. The discussion of logarithmic differential operators remains the same in this context. All the results of this section 3.4 (except (3.4.5) which requires an obvious modification) extend in a straightforward manner to this slightly more general setting.

§4 Extensions with logarithmic poles.

We fix here a section τ : K/Z ⟶ K of the canonical projection K ⟶ K/Z, such that τ(0) = 0.

4.1. Let X be a smooth connected K-variety and Z = ⋃_{i=1}^{s} Z_{i} be a divisor with (strict) normal crossings in X, sum of smooth connected divisors Z_{i}. Let ϵ : X′ ↪ X be an open subvariety of X, containing X \ Z and the generic point of each Z_{i}; we set Z′_{i} = Z_{i} ∩ X′, Z′ = ⋃_{i=1}^{s} Z′_{i}. Then X′ = X \ T, where T is a finite union ⋃ T_{α} of closed irreducible subvarieties T_{α} of some Z_{i}, with codim_{Z_{i}}T_{α} ≥ 1, hence codim_{X}T_{α} ≥ 2, ∀α. Let us decompose j into ϵ ∘ j′ where

X \ Z = X′ \ Z′↪^{j′}X′↪^{ϵ}X.

Lemma 4.2. For any h = 0, 1, …, we have the equality of sheaves

ϵ^{∗}j_{∗}Ω_{(X\Z)/K}^{h} = j′_{∗}Ω_{(X′\Z′)/K}^{h} ,

and of their subsheaves

ϵ^{∗}Ω_{X/K}^{h}(log Z) = Ω_{X′/K}^{h}(log Z′) .

Moreover, the canonical morphism

Ω_{X/K}^{h}(log Z) ⟶ ϵ_{∗}Ω_{X′/K}^{h}(log Z′)

[EGA 0_{I}, 4.4.3.2] is an isomorphism.

Proof. The equality ϵ^{∗}j_{∗}Ω_{(X\Z)/K}^{h} = j′_{∗}Ω_{(X′\Z′)/K}^{h} follows from [EGA IV, Prop. 5.9.4] applied to F = Ω_{X/K}^{h}. The second equality is clear from the explicit local description of Ω_{X/K}^{h}(log Z) and Ω_{X′/K}^{h}(log Z′) respectively. Taking into account the fact that Ω_{X/K}^{h}(log Z) is locally free, the third isomorphism follows according to [EGA 0_{IV}, 16.4.5] (prof_{T} Ω_{X/K}^{h}(log Z) = prof_{T} O_{X} = inf_{x∈T} prof_{O_{X,x}}O_{X,x} = inf_{x∈T} dimO_{X,x} ≥ 2 ). Q.E.D.

4.3. Let M be a torsion-free coherent O_{X}-module, and let ∇ be an integrable X/K-connection with logarithmic singularities along Z

(4.3.1)  ∇ : M ⟶ Ω_{X/K}^{1}(logZ) ⊗ M .

Notice that the underlying coherent O_{X\Z}-module with connection (M, ∇)_{|X\Z} is regular along Z. In particular, M_{|X\Z} is locally free. Since M is torsion-free, it follows that there exists an open subvariety X′ as in (4.1) containing X \ Z and the generic point of each Z_{j}, such that M_{|X′} is locally free. We denote by i′_{j} the locally closed embedding Z′_{j} ↪ X. In this situation, there is a well-defined endomorphism

Res_{Z′_{j}}∇ : i′_{j}^{∗}M ⟶ i′_{j}^{∗}M ,

called the residue of ∇ at Z′_{j}. It is induced, locally, by the action of ∇(x_{j} \frac{∂}{∂x_{j}}), where (x_{1}, …, x_{d}) denotes a system of local coordinates such that Z_{i} = V (x_{i}) [De, II.3.8]. The characteristic polynomial P_{j}(t) ∈ O(Z′_{j})[t] of Res′_{Z′_{j}}∇ turns out to be constant along Z′_{j}, i.e. P_{j}(t) ∈ K[t] (cf. [De, II.3.10]).

Definition 4.4. We denote by MIC^{τ} (X(logZ)) the category of torsion-free coherent O_{X}-modules with integrable X/K-connection with logarithmic singularities along Z, such that the roots of P_{j}(t) (j = 1, …, s) are in the image of τ. If (E, ∇) is a coherent O_{X\Z}-module with integrable connection, and if (Ẽ, ˜∇) is an object of MIC^{τ} (X(logZ)) such that Ẽ is locally free and (Ẽ, ˜∇)_{|X\Z} ≅ (E, ∇), we say that (Ẽ, ˜∇) is a τ-extension of (E, ∇) on X (with logarithmic poles on Z).

The condition that Ẽ is locally free is not automatic ^{(1)}: indeed if X = the x, y plane, and Z : xy = 0, the ideal of the origin carries a natural connection with

(1) we are grateful to H. Esnault for pointing out this subtlety to us. logarithmic singularities at Z, with residue = 0; its restriction outside the origin is the trivial connection.

Before we discuss the existence of τ-extensions, let us recall a few general facts about extensions of locally free sheaves. Let Y be a normal variety, T ⊂ Y a closed subvariety of codimension ≥ 2, ϵ : Y \ T ↪ Y the open immersion, F a locally free O_{Y \T} -module of finite rank. Then

i) ϵ_{∗}F is a reflexive^{(1)} coherent O_{Y} -module, cf. [EGA IV, Cor. 5.11.4] and [Se, Prop. 7],

ii) ϵ_{∗}F is the unique reflexive extension of F, up to unique isomorphism [Se, Prop. 7, and §3, Rem.2]; the point is that if F̃ is a reflexive coherent O_{Y} -module, then

ϵ_{∗}ϵ^{∗}F̃ = ϵ_{∗}\underline{Hom}(ϵ^{∗}F̃^{∨}, O_{Y \T}) = \underline{Hom}(F̃^{∨}, ϵ_{∗}O_{Y \T}) = \underline{Hom}(F̃^{∨}, O_{Y}) = F̃ ;

(applying this to F̃ = (ϵ_{∗}F)^{∨∨}, one deduces that (ϵ_{∗}F)^{∨∨} = ϵ_{∗}(ϵ^{∗}F̃) = ϵ_{∗}F, i.e. the reflexivity of ϵ_{∗}F stated in i)).

iii) ϵ_{∗}F need not be locally free if rk F > 1; for example, if Y = A^{3}, T = the origin, π : Y \ T → P^{2} the natural projection, then F = π^{∗}Ω_{P^{2}}^{1} does not extend to a locally free O_{Y} -module [Se, 5.(a)].

We now revisit a result of Gérard and Levelt [GL].

Proposition 4.5. For any object (M, ∇) of MIC^{τ} (X(logZ)), M is locally free if and only if it is reflexive.

Due to the previous remarks (and lemma 4.2), this is equivalent to

Proposition 4.6. Let (E, ∇) be a coherent O_{X\Z}-module with integrable connection, and let (Ẽ′, ˜∇′) ∈ MIC^{τ} (X′(logZ′)) be a τ-extension of (E, ∇) on X′ (as in (4.3)). Then (Ẽ, ˜∇) := (ϵ_{∗}Ẽ′, ϵ_{∗} ˜∇′) ∈ MIC^{τ} (X(logZ)) is a τ-extension of (E, ∇) on X.

Proof. Using lemma 4.2, one sees that (Ẽ, ˜∇) is equipped with an integrable connection with logarithmic singularity as in (4.3.1), and it is clear that (Ẽ, ˜∇) is an object of MIC^{τ} (X(logZ)). The problem is to show that the (reflexive coherent) extension Ẽ of the locally free O_{X′}-module Ẽ′ is locally free. For any closed point x ∈ X, we denote by (Ô_{X,x}, 𝔪̂_{X,x}) the completion of the local ring (O_{X,x}, 𝔪_{X,x}). It is enough to prove that, for any closed point z ∈ Z,

ˆẼ_{z} := Ẽ_{z} ⊗ Ô_{X,z} is a free Ô_{X,z}-module. We recall that Ô_{X,z} ≅ K[[x_{1}, …, x_{d}]], and that we may assume that in this isomorphism, x_{1}, …, x_{d} are étale coordinates in X at z, and x_{1} … x_{s} = 0, s ≥ 2, is a local equation at z of the divisor Z

(1) i.e. isomorphic to its bidual. Any locally free module of finite rank is obviously reflexive. The converse is not true if the rank > 1. in X. We now apply the theory of [GL, §1]: the Ô_{X,z}-module ˆ Ẽ_{z} is torsion free,

and ∇(x_{j} \frac{∂}{∂x_{j}}) are commuting K-derivations of ˆ Ẽ_{z}. Let ^{s}∇_{j} be the semisimple

part of the action of ∇(x_{j} \frac{∂}{∂x_{j}}) on the filtered K-vector space (ˆẼ_{z}, (𝔪̂_{X,x}^{n})_{n}), in the sense of loc. cit.; these operators determine a new integrable connection s∇ on ˆ Ẽ_{z}, with logarithmic poles along x_{1} … x_{s} = 0, by ^{s}∇(x_{j} \frac{∂}{∂x_{j}}) = ^{s}∇_{j}; in particular, the operators ^{s}∇_{j} mutually commute. There is a finite-dimensional

K-vector space V ⊂ ˆ Ẽ_{z} stable under the ^{s}∇_{j}, and such that V ⊕ 𝔪̂_{X,z}ˆẼ_{z} = ˆ Ẽ_{z}. We have ν := dimV ≥ μ := generic rank of E, and the problem is to prove ν = μ.

Let \underline{e} = (e_{1}, …, e_{ν}) be a common basis of eigenvectors: ^{s}∇_{j}e_{k} = α_{jk}e_{k}. We notice that α_{jk} is a root of P_{j} whenever the image of e_{k} in Ẽ_{z} ⊗ Ô_{Z_{j},z} does not vanish.

Lemma 4.6.1. The following conditions are equivalent:

i) for every (j, k), α_{jk} is a root of P_{j}, ii) every α_{jk} lies in the image of τ,

iii) ˆ Ẽ_{z} is free.

It is clear that i) implies ii), and the last remark shows that iii) implies i), since

\underline{e} is a basis of ˆ Ẽ_{z} if ˆ Ẽ_{z} is free. The implication ii) ⇒ iii) is due to Gérard-Levelt [GL, Thm. 3.4.i]. For convenience, let us sketch the argument. If ν ≠ μ, we can write e_{r+1} = λ_{1}e_{1} + … + λ_{r}e_{r}, with λ_{i} ∈ Frac Ô_{X,z} (not all in K) and r minimal. Applying ^{s}∇_{j}, we get α_{j,r+1}e_{r+1} = ∑_{k≤r}(x_{j} \frac{∂}{∂x_{j}} λ_{k} + α_{jk}λ_{k})e_{k}, and on combining these equalities:

x_{j} \frac{∂}{∂x_{j}} λ_{1} + (α_{j1} − α_{j,r+1})λ_{1} = … = x_{j} \frac{∂}{∂x_{j}} λ_{r} + (α_{jr} − α_{j,r+1})λ_{r} = 0 .

Let us remark that for any λ ∈ Frac Ô_{X,z}, λ ≠ 0, β_{j} ∈ K, the simultaneous equations x_{j} \frac{∂}{∂x_{j}} λ = β_{j}λ imply that all β_{j} are integers. We draw that if some λ_{k} ≠ 0, then α_{jk} = α_{j,r+1} for all j, so that λ_{k} ∈ K. This shows λ_{k} ∈ K for all k, a contradiction. Q.E.D.

Let us show that in our situation, condition i) of the lemma is satisfied. We fix (j, k). If e_{k} does not vanish in Ẽ_{z} ⊗ Ô_{Z_{j},z}, we are done. Otherwise, there exists i ≠ j such that e_{k} does not vanish in Ẽ_{z} ⊗ Ô_{Z_{i},z} (by construction of V). In order to show that α_{jk} is a root of P_{j}, we may localize Ô_{X,z} at the prime (x_{i}, x_{j}) and thus reduce to the case of the completed local ring Ô_{X,η_{i,j}} of the generic point η_{i,j} of Z_{i} ∩ Z_{j}. This is a complete regular local ring of dimension 2. It is known that any reflexive module of finite type over such a ring is free. In particular, Ẽ_{z} ⊗ Ô_{X,η_{i,j}} is free. As in the implication iii) ⇒ i) of the lemma, we conclude that α_{jk} is a root of P_{j}. This achieves the proof of (4.5) and (4.6). Q.E.D.

Proposition 4.7. Let (M_{n}, ∇_{n}), for n = 1, 2, be objects of MIC^{τ} (X(logZ)), with M_{n} locally free, and let φ : M_{1|X\Z} ⟶ M_{2|X\Z} be a horizontal isomorphism on X \ Z. Then φ extends (uniquely) to an isomorphism \overline{φ} : M_{1} ⟶ M_{2} in MIC^{τ} (X(logZ)).

Proof. Let z be a closed point of Z. Upon replacing X by an affine open neighborhood of z, we may assume X = SpecA affine and étale over A_{K}^{d} with coordinates (x_{1}, …, x_{d}), Z = V (x_{1} · . . . · x_{s}), and for n = 1, 2, M_{n} = M_{n} ⊗ O_{X}, with M_{n} a free A-module of rank μ, M_{1} = ⊕_{j=1}^{μ}Ae_{j}, M_{2} = ⊕_{j=1}^{μ}Ae′_{j}. We have

(4.7.1)  \begin{aligned}∇_{1}(x_{i} \frac{∂}{∂x_{i}})\underline{e} & = \underline{e}G^{(i)} \\ ∇_{2}(x_{i} \frac{∂}{∂x_{i}})\underline{e}′ & = \underline{e}′H^{(i)} \end{aligned}

for i = 1, …, d, where G^{(i)}, H^{(i)} ∈ M_{μ}(A), and there exists B ∈ GL(μ, A[\frac{1}{x_{1}·...·x_{s}}]), namely the matrix of φ w.r.t. the bases \underline{e}, \underline{e}′, such that

(4.7.2)  x_{i} \frac{∂B}{∂x_{i}} = BG^{(i)} − H^{(i)}B

for i = 1, …, d. We prove that B ∈ GL(μ, A). It is sufficient to prove that for i = 1, …, s, B ∈ M_{μ}(A_{(x_{i})}). Let C_{i} = A_{(x_{i})}/x_{i}A_{(x_{i})} be the residue field of the DVR A_{(x_{i})} and assume N_{i} < 0 is the order of pole of B at Z_{i} = V (x_{i}). Let B_{N_{i}} ≠ 0, G_{0}^{(i)}, H_{0}^{(i)} ∈ M_{μ}(C_{i}) be the reductions of x_{i}^{−N_{i}}B, G^{(i)}, H^{(i)} modulo x_{i}A_{(x_{i})}. From

(4.7.3)  x_{i} \frac{∂(x_{i}^{−N_{i}}B)}{∂x_{i}} = x_{i}^{−N_{i}}B(G^{(i)} − N_{i}) − x_{i}^{−N_{i}}H^{(i)}B

we deduce in M_{μ}(C_{i})

(4.7.4)  N_{i}B_{N_{i}} = B_{N_{i}}G_{0}^{(i)} − H_{0}^{(i)}B_{N_{i}}

for all i = 1, …, s. But det(tI_{μ} − G_{0}^{(i)}), det(tI_{μ} − H_{0}^{(i)}) are the characteristic polynomials of Res_{Z_{i}}∇, Res_{Z_{i}}∇′, respectively, hence their roots are in the image of τ. The linear transformation

X ⟼ XG_{0}^{(i)} − H_{0}^{(i)}X

of M_{μ}(C_{i}) into itself cannot have integral non-zero characteristic roots. Hence N_{i} < 0 is impossible. Q.E.D.

Proposition 4.8. Let (X, Z) be a model of a divisorially valued function field (F, v), and let E be a coherent O_{X\Z}-module with integrable connection ∇, regular along Z. Then there exists an affine neighborhood U of η_{Z} such that the restriction of (E, ∇) to U \ (Z ∩ U) admits a τ-extension on U.

Proof. We may restrict X around the generic point of Z, hence assume that X = SpecA is étale over A_{K}^{d} with coordinates (x_{1}, …, x_{d}), Z = V (x_{1}) irreducible, and that E_{|X\Z} ≅ Γ(X \ Z, E) ⊗ O_{X\Z}, with Γ(X \ Z, E) = ⊕_{j=1}^{μ}A[\frac{1}{x_{1}} ]e_{j} a free A[\frac{1}{x_{1}} ]-module of rank μ. The valuation ring R_{v} of (F, v) is the localized ring A_{(x_{1})}. The completion (F̂, v̂) of (F, v) is isomorphic to κ(Z)((x_{1})) equipped with the x_{1}-adic valuation, and Rv̂ corresponds to κ(Z)[[x_{1}]].

Let us consider the F̂/K-differential module Ê = Γ(X\Z, E)⊗F̂. According to proposition 3.3.3, the assumption of regularity implies the existence of an F̂- basis ˆ \underline{e} of Ê such that

(4.8.1)  ˆ∇(x_{1} \frac{∂}{∂x_{1}} )ˆ\underline{e} = ˆ \underline{e}T , ˆ∇(\frac{∂}{∂x_{i}})ˆ\underline{e} = ˆ \underline{e}H_{i} , for i = 2, …, d ,

with T ∈ M_{μ}(K) the eigenvalues of T being in the image of τ, and H_{i} ∈ M_{μ}(κ(Z)). Let us write ˆ \underline{e} = \underline{e}P, with P ∈ GL(μ, F̂). Let Q ∈ GL(μ, F) be a matrix sufficiently close v̂-adically to P so that

(4.8.2)  \begin{matrix}v̂(P − Q) > −v̂(P^{−1}) \\ v̂(I_{μ} − P^{−1}Q) > −v̂(P) − v̂(P^{−1}) (≥ 0) \\ v̂(I_{μ} − Q^{−1}P) > max(0, v̂(P)) . \end{matrix}

We then set ˜ \underline{e} = \underline{e}Q = ˆ \underline{e}P^{−1}Q, and calculate

∇(x_{1} \frac{∂}{∂x_{1}} )˜\underline{e} = ˆ \underline{e}(T P^{−1}Q − P^{−1} \frac{x_{1}∂P}{∂x_{1}} P^{−1}Q + P^{−1} \frac{x_{1}∂Q}{∂x_{1}}) =

˜\underline{e}(Q^{−1}P T P^{−1}Q − Q^{−1} \frac{x_{1}∂P}{∂x_{1}} P^{−1}Q + Q^{−1} \frac{x_{1}∂Q}{∂x_{1}}) =

˜\underline{e}(Q^{−1}P T P^{−1}Q − Q^{−1} \frac{x_{1}∂P}{∂x_{1}} (P^{−1}Q − I_{μ}) − Q^{−1} \frac{x_{1}∂(P − Q)}{∂x_{1}}) =

˜\underline{e}(T + (Q^{−1}P − I_{μ})T + T (P^{−1}Q − I_{μ}) + (Q^{−1}P − I_{μ})T (P^{−1}Q − I_{μ})+

P^{−1} \frac{x_{1}∂P}{∂x_{1}} (P^{−1}Q − I_{μ}) + (P^{−1} − Q^{−1})\frac{x_{1}∂(P^{−1}Q − I_{μ})}{∂x_{1}}−

−P^{−1} \frac{x_{1}∂(P − Q)}{∂x_{1}} + (P^{−1} − Q^{−1})\frac{x_{1}∂(P − Q)}{∂x_{1}}) .

By (4.8.2), we see that

∇(x_{1} \frac{∂}{∂x_{1}} )˜\underline{e} ≡ ˜ \underline{e}T modulo x_{1} ∑_{i=1}^{μ} Rv̂˜\underline{e}_{i} .

Therefore,

(4.8.3)  ∇(x_{1} \frac{∂}{∂x_{1}} )˜\underline{e} = ˜ \underline{e}G̃^{(1)} ,

with G̃^{(1)} ∈ M_{μ}(R_{v}), and the eigenvalues of the image of the matrix G̃^{(1)} in M_{μ}(κ(Z)) modulo x_{1}R_{v} belong to the image of τ.^{(1)} A similar calculation for i = 2, …, d, shows that

(4.8.4)  ∇(\frac{∂}{∂x_{i}})˜\underline{e} = ˜ \underline{e}G̃^{(i)} ,

with G̃^{(i)} ∈ M_{μ}(R_{v}). Since ˜ \underline{e} = \underline{e}Q, with Q ∈ GL(μ, F), ˜ \underline{e} represents a basis of sections of Γ(V, E), on a dense open subset V of X\Z of the form V = X\(Z∪Y), where Y = ⋃_{α} Y_{α} is a finite union of irreducible divisors Y_{α} of X, such that Y_{α} ⊄ Z, ∀α. We may also assume that V is so small that the matrices G̃^{(i)} are holomorphic on V. We claim that the proposition holds with U = X \ Y. Indeed, the matrices G^{(i)} are in M_{μ}(A), and the eigenvalues of the image of G^{(1)} modulo x_{1}A in M_{μ}(κ(Z)) belong to the image of τ. We then extend trivially E to X, by putting Ẽ = ∑_{i=1}^{μ} O_{X}j_{∗}e_{i}. The connection ∇ then obviously extends to a connection ˜∇ on Ẽ with logarithmic singularities along Z. The residue of ˜∇ along Z is represented in the basis (i_{Z}^{∗}(j_{∗}e_{1}), …, i_{Z}^{∗}(j_{∗}e_{μ})) by the matrix G^{(1)} modulo x_{1}A, hence (Ẽ, ˜∇) is an object of MIC^{τ} (X(log Z)).

(1) As for (4.8.2), we may be a little more explicit by noticing that if Q ∈ M_{μ}(F) is such that

v̂(P − Q) = h − (μ − 1)v̂(P) + max(v̂(det P), −μv̂(P)) ,

with h > 0, then:

(1) v̂(Q) = v̂(P);

(2) v̂(det Q − det P) ≥ h + max(v̂(det P), −μv̂(P));

(3) v̂(det Q) = v̂(det P), hence Q ∈ GL(μ, F);

(4) v̂(Q^{−1}) = v̂(P^{−1});

(5) v̂(Q^{−1} − P^{−1}) ≥ h − v̂(P) ≥ h + v̂(P^{−1});

(6) v̂(I_{μ} − Q^{−1}P) , v̂(I_{μ} − P^{−1}Q) ≥ h.

Theorem 4.9. In the general situation (4.1), any coherent O_{X\Z}-module with integrable connection, regular along Z, admits a τ-extension on X, unique up to unique isomorphism.

Proof. Let (E, ∇) be a coherent O_{X\Z}-module with integrable connection, regular along Z. We already proved that a τ-extension (Ẽ, ˜∇) of a coherent O_{X\Z}-module (E, ∇) with integrable connection is unique if it exists. By (4.8), there exists an affine neighborhood U_{i} of η_{Z_{i}} such that the restriction of (E, ∇) to U_{i} \ (Z ∩ U_{i}) admits a τ-extension (E, ∇) on U_{i}. (E, ∇) itself and these τextensions glue together, and get a τ-extension of (E, ∇) on X′ = (X \Z)∪⋃ U_{i}. We apply at last proposition 4.6 and get a τ-extension of (E, ∇) on X. Q.E.D.

Remark 4.10. Since τ is not additive, the formation of the τ-extension is not compatible with ⊗, \underline{Hom}, and is not functorial in general. However:

Proposition 4.11. Let (X, Z)⟶^{f}(Y, W) be a simple morphism of models. Let (E, ∇) be a coherent sheaf with integrable connection on Y \ W, regular along W, and let (Ẽ, ˜∇) be its τ-extension on Y. Then f^{∗}(Ẽ, ˜∇) is the τ-extension of f^{∗}(E, ∇) on X.

Proof. f^{∗}(Ẽ, ˜∇) is an extension with logarithmic poles along Z and the problem is to show that the eigenvalues of Res_{Z}f^{∗}(∇) are in the image of τ. It is enough to show that the characteristic polynomials P_{W} and P_{Z} of Res_{W} ∇ and Res_{Z}f^{∗}(∇) respectively coincide in K[t]. This is clear if f is a closed immersion of models, since P_{Z} is a specialization of P_{W} in this case. In general, in the canonical graph factorization of f

(X, Z) ⟶^{i = (1,f)}(X × Y, X × W)⟶^{p}(Y, W) ,

i is simple, hence a closed immersion of models in some neighborhood of i(η_{Z}) (1.2.5). We are reduced to the trivial case where f = p is a projection. Q.E.D.

§5 Regular connections: the global case.

5.1. Let X be a smooth connected variety over K, F = κ(X). Let E be a F/K-differential module, and let (E, ∇) be a model of E on X. Then E or ∇ is said to be regular or Fuchsian if E is regular at every divisorial valuation v of F. For instance, the trivial connection (O_{X}, d_{X/K}) is regular. Since any such v admits a smooth K-model (X, Z), this notion of (global) regularity reduces to the notion of regularity along a divisor (previous section), by using various models. One should however keep in mind the birational character of this notion.

Proposition 5.2. Let (E, ∇), (E_{1}, ∇_{1}), (E_{2}, ∇_{2}) be coherent sheaves with integrable connections on X.

i) If E sits in a horizontal exact sequence 0 ⟶ E_{1} ⟶ E ⟶ E_{2} ⟶ 0, then ∇ is regular if and only if ∇_{1} and ∇_{2} are regular. ii) If ∇_{1} and ∇_{2} are regular, so are (E_{1}, ∇_{1})⊗(E_{2}, ∇_{2}) and \underline{Hom}((E_{1}, ∇_{1}), (E_{2}, ∇_{2})).

Proof. This follows from (3.2.3) (or (3.4.3)).

Proposition 5.3. The property of being regular is invariant under inverse image by dominant morphisms and under direct image by finite étale morphisms. In particular, it is local at η_{X} for the fppf (hence for the étale) topology on X.

Proof. This follows from (3.1.4) and (3.2.5) (for the case of a dominant morphism f, notice that f induces an embedding of function fields κ(Y) ↪ κ(X), and that any divisorial valuation of κ(Y) extends to a divisorial valuation of κ(X)).

Proposition 5.4. Let X′ be a normal K-variety, and let (E, ∇) be a coherent sheaf with integrable connection on a smooth open dense subset X ⊂ X′. Let P be a closed point of ∂X = X′ \X. Assume that ∇ is regular at every component of ∂X of codimension 1 in X′ passing through P. If \overline{C}⟶^{h}X′ is any morphism from a smooth curve \overline{C}, such that C = h^{−1}(X) is the complement of a point Q with h(Q) = P in X′, then ∇_{|C} is regular at Q.

Proof. We may replace X′ by any affine neighborhood of P, and assume that ∂X ∩ \overline{C} = \{P \}. Let \{x_{1}, x_{2} …\} be a finite set of generators of the ideal of ∂X in O(X′) such that x_{i} ≠ 0 on C. We may and shall replace ∂X by the reduced divisor defined by the ideal √(Πx_{i}). There is a closed subset T ⊂ ∂X such that Z̃ = ∂X \ T is a disjoint union of smooth divisors Z_{i} in X̃ = X′ \ T defined by the equation x_{i} = 0. We denote by j′ (resp. j̃, resp. k) the open immersion of X̃ into X′ (resp. X into X̃, resp. C into \overline{C}), and set j = j′ ∘ j̃. If P ∉ T, the result follows from (3.4.4) (which relies upon (3.4.2)). In order to settle the case P ∈ T, we use the existence of an extension (Ẽ ⊂ j̃_{∗}E, ˜∇) of (E, ∇) on X̃ with logarithmic poles along Z̃ (4.9). We need the

Lemma 5.5. The functoriality map h^{∗}(j_{∗}Ω_{X/K}^{1}) ⟶ k_{∗}Ω_{C/K}^{1} induces a map

(5.5.1)  h^{∗}j′_{∗}Ω_{X̃/K}^{1}(log Z̃) ⟶ Ω_{\overline{C}/K}^{1}(log Q).

Indeed, j′_{∗}Ω_{X̃/K}^{1}(log Z̃) is generated by Ω_{X′/K}^{1} and by the sections \frac{dx_{i}}{x_{i}} (taking into account the fact that j′_{∗}OX̃ = O_{X′}), and Ω_{\overline{C}/K}^{1}(log Q) contains

h^{∗}(\frac{dx_{i}}{x_{i}}) = \frac{d(x_{i}∘h)}{x_{i}∘h} ( (2.2.1)).

Let us turn back to the proof of (5.4). Due to the functoriality (5.5), the mapping ˜∇ gives rise to a composite mapping

h^{∗}j′_{∗} ˜∇ : h^{∗}j′_{∗}Ẽ ⟶ h^{∗}j′_{∗}Ẽ ⊗_{O\overline{C}} h^{∗}j′_{∗}Ω_{X̃/K}^{1}(log Z̃) ⟶ h^{∗}j′_{∗}Ẽ ⊗_{O\overline{C}} Ω_{\overline{C}/K}^{1}(log Q) which is an extension of ∇_{|C} with logarithmic poles at Q. Hence ∇_{|C} is regular at Q. Q.E.D.

Remark 5.6. The use of extensions with logarithmic poles is not essential in the proof of (5.4). We sketch a variant, which uses instead jets with logarithmic poles (cf. App. B.6). The sheaf of OX̃-algebras P_{X̃/K,Z̃}^{n} is generated by the \frac{j_{n}x_{i}}{x_{i}} (cf. (B.6.2)); as a left OX̃-module, it is the dual of \underline{Diff}_{X̃/K,Z̃}^{n}. The argument of (5.5) gives a map of functoriality

(5.6.1)  h^{∗}j′_{∗}P_{X̃/K,Z̃}^{n} ⟶ P_{\overline{C}/K,Q}^{n}.

We can then imitate (3.4.5): let Ẽ ⊂ k_{∗}E be a coherent extension of E on X̃ as in (3.4.2). Then for all n, the mapping ˜∇ gives rise to a composite mapping

h^{∗}j′_{∗} ˜∇ : h^{∗}j′_{∗}Ẽ ⟶ (Πx_{i})^{−c}h^{∗}j′_{∗}Ẽ ⊗_{O\overline{C}} h^{∗}j′_{∗}P_{X̃/K,Z̃}^{n} ⟶ x^{−c′}h^{∗}j′_{∗}Ẽ ⊗_{O\overline{C}} P_{\overline{C}/K,Q}^{n}

for some c′ independent of n; hence ∇_{|C} is regular at Q. (Note however that since j′_{∗}Ω_{X̃/K}^{1}(log Z̃) is not flat in general, it is not clear from (5.5.1) whether \underline{Der}_{\overline{C}/K} ⟶ h^{∗}\underline{Der}_{X′/K} induces maps of O_{X′}-modules T_{h} : \underline{Der}_{\overline{C}/K,Q} ⟶ h^{∗}j′_{∗}(\underline{Der}_{X̃/K,Z̃}) and T_{h} : \underline{Diff}_{\overline{C}/K,Q}^{n} ⟶ h^{∗}j′_{∗}(\underline{Diff}_{X̃/K,Z̃}^{n}).) We shall give still another proof of (5.4) in (6.5.6).

Theorem 5.7. Let (E, ∇) be a coherent sheaf with integrable connection on a smooth connected K-variety X. The following properties are equivalent:

i) ∇ is regular; ii) for any morphism C⟶^{h}X, with C a smooth K-curve, h^{∗}(∇) is regular;

iii) for any locally closed immersion of a smooth K-curve C↪^{i}X, i^{∗}(∇) is regular.

Proof. iii) ⇒ ii): If the image of h is a point, the result is trivial (h^{∗}(∇) is trivial), so we may assume that h is finite. The reduction to the case of a locally closed immersion uses (5.3). ii) ⇒ i): let v be a divisorial valuation of κ(X), and let (X′, Z′) be a model of (κ(X), v). We may shrink X′ and X, and assume that X = X′ \ Z′. Then ii) implies that ∇ is regular Z′, according to (3.4.7).

i) ⇒ iii): we may extend X to a proper normal variety \overline{X} and i to a morphism \overline{C}⟶^{h}\overline{X}, where \overline{C} is the normalization of closure of C in \overline{X}. We conclude by (5.4). Q.E.D.

Corollary 5.8. Let f : X ⟶ Y be a morphism of smooth varieties, and (E, ∇) be a coherent sheaf with regular connection on Y. Then f^{∗}(∇) is regular. The converse is true if f is dominant. In particular, the property of (E, ∇) of being regular is local at η_{X} for the fppf topology.

Proof. Immediate from (5.7). Q.E.D.

Remark 5.9. The proofs of (5.7) and (5.8) found in the literature are usually much more complicated and use Hironaka’s resolution of singularities. The proof in [De] is incomplete, since it uses an assertion about non-existence of confluences which is false as it stands [loc. cit. 4.1.2, 1.24]; this point is corrected in the erratum (April 1971). The treatment in [Bn] of the problem of inverse images relies on the test of curves which is stated in a wrong way: in [loc. cit. 4 (iii)], it is asserted that regularity at a divisor Z may be checked by considering one curve which cuts Z transversally ( counterexample (3.4.8)).

We now show that the birational notion of global regularity actually involves only finitely many divisorial valuations of the function field.

Theorem 5.10. Let X be a smooth connected K-variety, and let j : X ⟶ \overline{X} be an open dominant immersion of X in a proper normal K-variety \overline{X}; we denote by (\overline{X})^{sm} the smooth part of \overline{X}. Let (E, ∇) be a coherent sheaf with integrable connection on X. The following properties are equivalent:

i) ∇ is regular; ii) ∇ is regular at every irreducible component \overline{Z} of \overline{X} \ j(X) of codimension 1 (see (3.4.9)); iii) for every irreducible component \overline{Z} of \overline{X} \ j(X) of codimension 1, there is an open neighborhood U of η_{Z} in (\overline{X})^{sm} and a tubular neighborhood U⟶^{f}S of Z ∩ U, such that ∇_{|U_{s}} is regular at Z_{s} for a Zariski-dense set of closed points s ∈ S.

Proof. The implication i) ⇒ iii) follows from (5.8), while iii) ⇒ ii) follows from (3.4.7) (since \overline{X} is normal, the smooth part (\overline{Z})^{sm} of Z is open dense). Let us prove ii) ⇒ (∇ is regular at every divisorial valuation v of K(\overline{X})). Since \overline{X} is proper, v is centered at some point x ∈ \overline{X} of codimension ≥ 1. More precisely, there is a model (X′, Z′) of (K(\overline{X}), v) and a birational map X′⟶^{ϵ}\overline{X} such that ϵ(η_{Z′}) = x. We have to check the regularity of ϵ^{∗}∇ on the inverse image of X, along Z′. We may assume x ∈ (\overline{X} \ X) and ϵ^{−1}(\overline{X} \ X) ⊂ Z′. Let P be any closed point in ϵ(Z′) ⊂ (x) . We see via (3.4.7) that it is enough to show that for any morphism \overline{C}⟶^{h}\overline{X} such that C = h^{−1}(X) is the complement of a point Q with h(Q) = P in a smooth curve \overline{C}, ∇_{|C} is regular at Q if ∇ is regular at every component \overline{Z} of \overline{X} \ X containing P and of codimension 1 in \overline{X}. We conclude by (5.4). Q.E.D.

§6 Exponents.

6.1 Indicial polynomials.

Definition 6.1.1. Let C be a field of characteristic 0, and let Λ ∈ C((x))[\frac{d}{dx}] be a scalar differential operator. The indicial polynomial of Λ at x = 0 is the unique ϕ = ϕ_{Λ,0} ∈ C[s] such that for every s ∈ Z,

Λx^{s} = ϕ(s)x^{s+r} + higher order terms in x .

The roots (in \overline{C}) of ϕ_{Λ,0} are called the (Fuchs) exponents of Λ at x = 0.

For Λ ∈ C(x)[\frac{d}{dx}], and θ ∈ C (resp. θ = ∞), one defines in the same way the indicial polynomial and the exponents ϕ_{Λ,θ} Λ at x = θ.

Let us write Λ, as in the previous definition, in the form ∑_{i=0}^{μ} γ_{i}(x\frac{d}{dx})^{i}, γ_{i} ∈ C((x)), and set

(6.1.1.1)

r = min ord_{0}γ_{i} ,
ν = max{i such that ord_{0}γ_{i} = r} .

Clearly, ϕ(s) = ∑_{i=0}^{ν}(x^{−r}γ_{i})_{|x=0}s^{i}.

6.1.2 We now assume that C is a function field over an algebraically closed field K.

Proposition 6.1.3. Let Λ be as before, and assume C((x))[\frac{d}{dx}]/C((x))[\frac{d}{dx}]Λ underlies a C((x))/K-differential module. Then the roots of ϕ are in K.

This is a consequence of Robba’s result reproduced in appendix A; however, we present here a proof which avoids Turrittin’s theorem.

Proof. i) Let us write Λ_{f} = ∑_{i=0}^{ν}(\frac{γ_{i}}{γ_{ν}})(x\frac{d}{dx})^{i}. Then Λ_{f} is regular, and Λ = γ_{ν}(Λ_{f} + xΛ′), where Λ′ ∈ C[[x]][x\frac{d}{dx}]. Clearly, ϕ_{Λ,0}(s) = (x^{−r}γ_{ν})(0) · ϕ_{Λ_{f} ,0}(s), and ϕ_{Λ_{f} ,0}(s) is the (usual) indicial polynomial at 0 of the regular differential operator Λ_{f}. Hence, we are reduced to the regular case.

ii) For a regular Λ, we may change the cyclic basis of C((x))[\frac{d}{dx}]/C((x))[\frac{d}{dx}]Λ into a basis in which ∇(x\frac{d}{dx}) is represented by a constant matrix in Jordan normal form (after replacing C by a finite extension if necessary). The roots of ϕ are the diagonal entries of this matrix. Then Robba’s argument goes through, and shows that the exponents belong to K. Q.E.D.

Corollary 6.1.4. Let Λ ∈ C(x)[\frac{d}{dx}] be such that C(x)[\frac{d}{dx}]/C(x)[\frac{d}{dx}]Λ underlies a C(x)/K)-differential module. Then the roots of the indicial polynomial of Λ at any θ ∈ C ∪ ∞, are in K.

Proof. The indicial polynomial (and the exponents) of Λ at θ, resp. at ∞, may be defined using a change of variable x ⟼ x′ = x − θ (θ ∈ C), resp. x ⟶ x′ = \frac{1}{x}. One can apply (6.1.3) to Λ viewed as an element of C((x′))[\frac{d}{dx′}] Q.E.D.

6.2 Exponents: review of the one variable regular case.

Let again C be a field of characteristic 0 and consider a regular C((x))/Cdifferential module Ê as in (3.2.2). We apply the notation of section 1 to (F̂, v) = (C((x)), x − adic valuation), despite the fact that C is not necessarily algebraically closed. Definition 6.2.1. The elements of A, considered up to addition by an integer, are called the exponents of Ê. The subset A + Z of \overline{C} is denoted by Exp(Ê). Following the last assertion in (3.2.2), Exp(Ê) is canonically attached to Ê.

The following results are classical (see [Man] or [DGS, III.8]).

Proposition 6.2.2. i) Exp(Ê) coincides, modulo Z, with the roots of the indicial polynomial of the differential operator associated with any cyclic vector m as in c) of (3.2.2). ii) Let ER̂_{v} be an R̂_{v}-lattice of Ê stable by Diff_{v}(F̂/K), as in v) of (3.3.4). Let (D_{1}, …, D_{d}) be an adapted basis of Der_{v}(F̂/K); ∇(D_{1}) induces a K-linear endomorphism Res_{v}∇ of the finite dimensional C-vector space ER̂_{v}/𝔪̂_{v}ER̂_{v}. Then Exp(Ê) coincides, modulo Z, with the eigenvalues of Res_{v}∇.

Proposition 6.2.3. Let Ê_{1} and Ê_{2} be regular C((x))/C-differential modules.

i) If Ê sits in an exact sequence of C((x))/C-differential modules

0 ⟶ Ê_{1} ⟶ Ê ⟶ Ê_{2} ⟶ 0

then Exp(Ê) = Exp(Ê_{1}) ∪ Exp(Ê_{2}). ii) Exp(Ê_{1} ⊗_{C((x))} Ê_{2}) = Exp(Ê_{1}) + Exp(Ê_{2}) and Exp(\underline{Hom}_{C((x))}(Ê_{1}, Ê_{2})) = −Exp(Ê_{1}) + Exp(Ê_{2}).

6.2.4. Let F/K be a function field in one variable (K not necessarily algebraically closed), and let v be a non-trivial valuation of F trivial on K. Then C := k(v) is a finite extension of K, and there is an element x of 𝔪_{v} such that (F̂, v̂) = (C((x)), x−adic valuation), R̂_{v} = C[[x]].

If E is a F/C-differential module, regular at v, we set Exp_{v}(E) = Exp(Ê).

Lemma 6.2.5. In the situation of (3.1.3), where Ê is a Ĝ/K- differential module, we have Exp(ÊF̂) = e(v/w)Exp(Ê) + Z. If [F̂ : Ĝ] < ∞, we have Exp(Ê′) = e(v/w)Exp(ĜÊ′) for any F̂/K-differential module Ê′.

Proof. Let x be a generator of 𝔪_{v} such that y = x^{e(v/w)} is a generator of 𝔪_{w}, so that F̂ ≅ C((x)), Ĝ ≅ C′((y)), with C′ = C ∩ Ĝ. Using i) of (6.2.3), it suffices to consider the case when Ê ≅ C′((y)) ⊗_{C′} KerÊ(y \frac{d}{dy} − α)^{μ}. By reason of dimension, we then have ÊF̂ ≅ C((x)) ⊗_{C} KerÊF̂ (y \frac{d}{dy} − α)^{μ}. Since x\frac{d}{dx} = e(v/w)y \frac{d}{dy}, this implies the first assertion. To deduce from it (with

Ê = ĜÊ′) the second one, it suffices to show that (ĜÊ′)F̂ ≅ (Ê′)^{[F̂:Ĝ]}. As before, we may assume that Ê′ ≅ C((x))⊗_{C} KerÊ′(x\frac{d}{dx} −α) , and we are reduced to the case of the trivial C((x))/C-differential module C((x)), for which it is immediate that (_{C′((y))}C((x)))_{C((x))} ≅ C((x))^{[C((x)):C′((y))]}.

Notice that if v is a divisorial valuation of a function field F over K which induces the trivial valuation on the subfield G ⊂ K, then for any G/K-differential module E, Exp_{v}(E_{F}) = Z.

6.3 The case of a function field in several variables.

Let (F̂, v) be a field of formal functions over K, and (D_{1}, …, D_{d}) an adapted basis of Der_{v}(F̂/K) . We set C = F̂^{D_{1}} so that F̂ = C((x)), for a parameter x of R̂_{v} such that D_{1}x = x, D_{j}C ⊂ C and D_{j}x = 0, for j = 2, …, d.

Definition 6.3.1. The exponents (at v) of a regular F̂/K-differential module Ê are the exponents of the induced F̂/C-differential module (in one variable x) Ê_{/C}. We set Exp_{v}(Ê) = Exp(Ê_{/C}) = A + Z, where A is the set of eigenvalues of the matrix G_{1} appearing in (3.3.3).

Proposition 6.3.2. Exp_{v}(Ê) ⊂ K.

This follows from (3.3.3) as well as from (6.1.3), (6.2.2).

6.4 Exponents of a regular connection along a divisor.

Let (X, Z) be a model of a divisorially valued function field (F, v). Let E be a F/K-differential module regular at v, and let (E, ∇) be a model of E on X \ Z ( (3.1)).

Definition 6.4.1. The exponents of ∇ along Z are the exponents of Ê at v. We set Exp_{Z}(∇) = Exp_{v}(Ê).

Proposition 6.4.2. Let (X, Z) be a model and (E, ∇), (E_{1}, ∇_{1}), (E_{2}, ∇_{2}) be coherent sheaves with integrable regular connections on X \ Z.

i) If E sits in a horizontal exact sequence 0 ⟶ E_{1} ⟶ E ⟶ E_{2} ⟶ 0, then Exp_{Z}(∇) = Exp_{Z}(∇_{1}) ∪ Exp_{Z}(∇_{2}). ii) Exp_{Z}((E_{1}, ∇_{1}) ⊗ (E_{2}, ∇_{2})) = Exp_{Z}(∇_{1}) + Exp_{Z}(∇_{2}), Exp_{Z}(\underline{Hom}((E_{1}, ∇_{1}), (E_{2}, ∇_{2}))) = −Exp_{Z}(∇_{1}) + Exp_{Z}(∇_{2}).

Proof. This follows from (6.2.3).

Proposition 6.4.3. Let (X, Z)⟶^{f}(Y, W) be a morphism of models such that f(Z) ⊂ W. Let (E, ∇) be a coherent sheaf with integrable connection on Y \ W, regular along W. Then Exp_{Z}(f^{∗}(∇)) = e_{f}Exp_{W} (∇) + Z.

Proof. i) If f is log dominant, the result follows from (6.2.5). ii) Let us next assume that X is a curve. We may replace X by any connected étale neighborhood of Z. Using the factorization (1.2.7), and the previous case, we are reduced to the case where f is a closed immersion of models. In this case, the result follows from (4.11). iii) In the general case, we can find a closed immersion (C, P)⟶^{i}(X, Z) with C a curve. Then e_{f} = e_{i∘f} and the result follows from ii). Q.E.D.

6.4.5 Example. As in (3.4.6), take Y = the (x, y)-plane with the x-axis (y = 0) removed, W = y-axis (x = 0) X = the diagonal (x = y) in the (x, y)-plane Z = the origin. For a, b ∈ K, and let ∇_{a,b} be the connection on E = O_{Y \W}, whose formal

solution is x^{a}y^{b}. It is regular at W with Exp_{W} (∇_{a,b}) = a + Z; its restriction to X \ Z is also regular at Z, but has Exp_{Z} = a + b + Z. Again, the point is that the embedding of X \ Z into Y \ W does not extend to a morphism of models (X, Z) ⟶ (Y, W).

6.4.6. The definition of exponents along a divisor generalizes in an obvious way in the situation of (3.4.9): let X be a smooth connected variety over K, F = κ(X). Let j : X ⟶ \overline{X} be a dominant open immersion of X in a normal K-variety \overline{X}, and let \overline{Z} be a (not necessarily irreducible) subvariety of \overline{X} \ X. If (E, ∇) is a coherent sheaf with integrable connection on X, we denote by Exp\overline{Z}(∇) the set of exponents of the generic fiber E at the divisorial valuations of F defined by the irreducible components of \overline{Z} of pure codimension one in \overline{X}.

6.5 Global exponents.

6.5.1. Let X be a smooth connected variety over K, F = κ(X). Let E be a regular F/K-differential module, and let (E, ∇) be a model of E on X. The exponents of E or ∇ are the exponents of E at any divisorial valuation of F.

Notation 6.5.2. We denote by ZExp(∇) (resp. QExp(∇)) the additive subgroup (resp. sub-Q-vector space) of K generated by the exponents of ∇.

Notice that ZExp(∇) contains 1, and that QExp(∇) = Q ⊗_{Z} ZExp(∇).

Proposition 6.5.3. Let (E, ∇), (E_{1}, ∇_{1}), (E_{2}, ∇_{2}) be coherent sheaves with regular connections on X.

i) If E sits in a horizontal exact sequence 0 ⟶ E_{1} ⟶ E ⟶ E_{2} ⟶ 0, then ZExp(∇) = ZExp(∇_{1}) + ZExp(∇_{1}). ii) ZExp((E_{1}, ∇_{1}) ⊗ (E_{2}, ∇_{2})) ⊂ ZExp(∇_{1}) + ZExp(∇_{2}), ZExp(\underline{Hom}((E_{1}, ∇_{1}), (E_{2}, ∇_{2}))) ⊂ ZExp(∇_{1}) + ZExp(∇_{2}).

Proof. This follows from (6.4.2).

Proposition 6.5.4. QExp(∇) is invariant under inverse image by dominant morphisms and under direct image by finite étale morphisms. In particular, its formation is local at η_{X} for the fppf (hence for the étale) topology on X.

Proof. This follows from (6.2.5).

Proposition 6.5.5. Let X′ be a normal K-variety, and let (E, ∇) be a coherent sheaf with regular connection on an open dense subset X ⊂ X′. Let P be a closed point of ∂X = X′ \ X. Let \overline{C}⟶^{h}X′ be a morphism from a smooth curve \overline{C}, such that C = h^{−1}(X) is the complement of a point Q with h(Q) = P in \overline{C}. Then Exp_{Q}(∇_{|C}) ⊂ ZExp(∇). If moreover X′ is smooth and ∂X = ∪Z_{i} is a divisor with strict normal crossings, then Exp_{Q}(∇_{|C}) is contained in the subgroup of K generated by 1 and the sets Exp_{Z_{i}}(∇).

Proof. a) Let us first prove the second assertion. In this situation, we have the τ-extension (Ẽ, ˜∇) on X′ at our disposal (4.9). We may assume that Ẽ is free, that each component Z_{i} is globally described by an equation x_{i} = 0, and that Ω_{\overline{C}/K}^{1}(log Q) is free with basis \frac{dx}{x} for some global coordinate x on \overline{C}. The

functoriality h^{∗}Ω_{X′/K}^{1}(log ∂X) ⟶ Ω_{\overline{C}/K}^{1}(log Q) induces a linear map

T_{h}^{∗} : \frac{h^{∗}Ω_{X′/K}^{1}(log ∂X)}{h^{∗}Ω_{X′/K}^{1}} ⟶ \frac{Ω_{\overline{C}/K}^{1}(log Q)}{Ω_{\overline{C}/K}^{1}}

of skyscraper sheaves of K-vector spaces (concentrated at Q). A basis of \frac{h^{∗}Ω_{X′/K}^{1}(log ∂X)}{h^{∗}Ω_{X′/K}^{1}}

(resp. \frac{Ω_{\overline{C}/K}^{1}(log Q)}{Ω_{\overline{C}/K}^{1}}) at Q is given by the classes of the h^{∗}(\frac{dx_{i}}{x_{i}}) (resp. by the class

of \frac{dx}{x}), and one has T_{h}^{∗}(h^{∗}(\frac{dx_{i}}{x_{i}})) ≡ e_{i} \frac{dx}{x} mod. Ω_{\overline{C}/K}^{1} for some intersection multiplicities e_{i} ∈ Z_{≥0} (e_{i} is the valuation of the image in O_{\overline{C},Q} of a local equation of Z_{i} in O_{X′,P}).

Now ˜∇ induces a composite map

(h^{∗}Ẽ)_{Q} ⊗ κ(Q) ⟶^{u=(...,u_{i},...)} (h^{∗}Ẽ)_{Q} ⊗ \frac{h^{∗}Ω_{X′/K}^{1}(log ∂X)}{h^{∗}Ω_{X′/K}^{1}}

⟶^{v=1⊗T_{h}^{∗}} (h^{∗}Ẽ)_{Q} ⊗ \frac{Ω_{\overline{C}/K}^{1}(log Q)}{Ω_{\overline{C}/K}^{1}} ≅ (h^{∗}Ẽ)_{Q} ⊗ κ(Q) .

The map u_{i} is nothing but the specialization of Res_{Z_{i}}∇ at P, and v∘u is nothing but Res_{Q}h^{∗}∇. Since the Res_{Z_{i}}∇ mutually commute, the endomorphisms v∘u_{i} = e_{i}u_{i} of the finite dimensional K = κ(Q)-vector space (h^{∗}Ẽ)_{Q}⊗κ(Q) also commute and the eigenvalues of ∑_{i} v ∘ u_{i} are of the form ∑_{i} e_{i}λ_{i}, for eigenvalues λ_{i} of u_{i}. We conclude that Exp_{Q}(h^{∗}∇) ⊂ (∑ e_{i} Exp_{Z_{i}}(∇)) + Z.

b) Let us now turn to the first assertion. By factoring h via its graph and localizing around P, we reduce to the case where h is a closed immersion, P = Q. Let \overline{X} ⊂ P^{N} be a normal projective completion of X′, D = \overline{X} \ X, \overline{\overline{C}} := closure of C in P^{N}. Let d = dimX. Then, for δ >> 0, a system {Y_{1}, …, Y_{d−2}} of d − 2 hypersurfaces of degree δ passing through \overline{\overline{C}} and sufficiently general, verifies:

(i) \overline{S} = Y_{1} ∩ … ∩ Y_{d−2} ∩ \overline{X} is a proper normal surface with (isolated) singularities contained in Y_{1} ∩ … ∩ Y_{d−2} ∩ \overline{X}^{sing},

(ii) S′ = Y_{1} ∩ … ∩ Y_{d−2} ∩ X′ contains \overline{C},

(iii) S = Y_{1} ∩ … ∩ Y_{d−2} ∩ X is a smooth open subset of S′ containing C, (iv) S′ cuts D transversally except at a finite set of points.

For lack of reference, we give some detail. Let ϵ : Bl ⟶ \overline{X} be the blowing-up of \overline{X} centered at \overline{\overline{C}}, with exceptional divisor E. We set O_{Bl}(1) = ϵ^{∗}Ker(O\overline{X} ⟶ O\overline{\overline{C}}). Then for δ >> 0, ϵ^{∗}O\overline{X}(δ) ⊗ O_{Bl}(1) is very ample; a basis of sections defines an embedding of Bl into P^{M}. Since

ϵ_{∗}(ϵ^{∗}O\overline{X}(δ) ⊗ O_{Bl}(1)) ≅ O\overline{X}(δ) ⊗ Ker(O\overline{X} ⟶ O\overline{\overline{C}}) ≅ Ker(O\overline{X}(δ) ⟶ O\overline{\overline{C}}(δ)) ,

the linear system of hypersurfaces of degree δ in \overline{X} passing through \overline{\overline{C}} gives rise to an embedding

\overline{X} \ \overline{\overline{C}} ≅ Bl \ E ↪ P^{M} .

Besides, we note that ϵ^{−1}(\overline{X}^{sing}) and ϵ^{−1}(D^{sing}) are of codimension ≥ 2 in Bl. The above properties then follow from Bertini’s theorem applied to Bl [Kl].

The local theory of the embedded resolution of singularities of curves on a surface shows that there is a smooth surface S′′, a smooth curve \overline{C}′ and a birational morphism ϵ : S′′ ⟶ S′ inducing an isomorphism \overline{C}′ ≅ \overline{C}, such that (ϵ^{−1}(∂S))^{red} is a divisor with normal crossings and that \overline{C}′ cuts (ϵ^{−1}(∂S))^{red} transversally. In fact, S′′ is obtained by performing a sequence of monoidal transformations (centered at closed points), \overline{C}′ is the iterated strict transform of \overline{C} and cuts (ϵ^{−1}(∂S))^{red} only at some point Q′ of the exceptional divisor E′ of the last monoidal transformation. If we perform the corresponding sequence of monoidal transformations (centered at the same points) on X′ instead of S′, we get a commutative diagram

S′′ ⟶^{ϵ} S′
ι′ ↓  ι ↓
X′′ ⟶^{ϵ′} X′

where ι′ is a closed immersion; the exceptional divisor E′′ of the last monoidal transformation restricts to E′ on S′′, hence meets S′′ transversally in a neighborhood of Q′ in E′. Applying (6.4.3), we get

\begin{aligned}Exp_{Q}(∇_{|C}) & = Exp_{Q′}(ϵ^{∗}ι^{∗}∇_{|\overline{C}′\Q′}) = Exp_{E′}(ϵ^{∗}ι^{∗}∇) = \\ Exp_{E′}(ι^{′∗}ϵ^{′∗}∇) & = Exp_{E′′}(ϵ^{′∗}∇) ⊂ ZExp(ϵ^{′∗}∇) = ZExp(∇) . \end{aligned}

Q.E.D.

Remark 6.5.6. The argument in part b) also provides another proof of (5.4), replacing the reference to (6.4.3) by a reference to (3.4.4).

Open question 6.5.7. Is it possible, in the situation of (6.5.5) (without assumption of normal crossings), that Exp_{Q}(∇_{|C}) is always contained in the subgroup of K generated by 1 and the set Exp_{∂X}(∇)? In fact, the argument in part b) reduces the question to the case when X is a surface.

Theorem 6.5.8. We fix an additive subgroup Σ of K containing 1. Let (E, ∇) be a coherent sheaf with regular connection on a smooth connected K-variety X. The following properties are equivalent:

i) the exponents of ∇ are in Σ; ii) for any morphism C⟶^{h}X, with C a smooth K-curve, the exponents of h^{∗}(∇) are in Σ; iii) for any locally closed immersion of a smooth K-curve C↪^{i}X, the exponents of i^{∗}(∇) are in Σ.

Proof. iii) ⇒ ii): If the image of h is a point, the result is trivial (h^{∗}(∇) is trivial), so we may assume that h is finite. The reduction to the case of a locally closed immersion uses (6.2.5). ii) ⇒ i): let v be a divisorial valuation of κ(X), and let (X′, Z′) be a model of (κ(X), v). We may shrink X′ and X, and assume that X = X′ \ Z′. The result follows from (6.4.3).

i) ⇒ iii): is a corollary of the first assertion in (6.5.5). Q.E.D.

Corollary 6.5.9. Let f : X ⟶ Y be a morphism of smooth varieties, and (E, ∇) be a coherent sheaf with regular connection on Y. Then ZExpf^{∗}(∇) ⊂ ZExp(∇).

Proof. Immediate from (6.5.8). Q.E.D.

Open question 6.5.10. Let j : X ⟶ \overline{X} be an open dominant immersion of X in a proper normal K-variety \overline{X}. Is it true that Exp(∇) is the subgroup of K generated by 1 and the sets Exp\overline{Z}(∇), for every irreducible component \overline{Z} of \overline{X} \ j(X), of codimension 1 in \overline{X}? This would be true if the previous open question has a positive answer. In this direction, we have:

Theorem 6.5.11. Assume that \overline{X} is smooth (and proper) and that X is the complement of a divisor \overline{Z} with strict normal crossings in \overline{X}. Then Exp(∇) is the subgroup of K generated by 1 and the set Exp\overline{Z}(∇).

Proof. Let us prove that for any divisorial valuation v of K(\overline{X}), the exponents of the generic fiber of ∇ are in the subgroup Σ of K generated by 1 and the set Exp\overline{Z}(∇). Since \overline{X} is proper, v is centered at some point x ∈ \overline{X}. More precisely, there is a model (X′, Z′) of (K(\overline{X}), v) and a birational map X′⟶^{ϵ}\overline{X} such that ϵ(η_{Z′}) = x. We have to show that Exp_{Z′}(ϵ^{∗}∇) ⊂ Σ. Let P be any closed point in ϵ(Z′) ⊂ (x) . By (6.5.8), it suffices to show that for any morphism \overline{C}⟶^{h}\overline{X} such that C = h^{−1}(X) is the complement of a point Q with h(Q) = P in a smooth curve \overline{C}, Exp_{Q}(∇_{|C}) ⊂ Σ. This follows from the second assertion of (6.5.5). Q.E.D.

Appendix A: A letter of Philippe Robba (Nov. 2, 1984).

Mon cher Francesco,

Soit y = (y_{1}, …, y_{n}). Soit M un K(x, y)-module différentiel intégrable libre de rang s.

Considérons y comme des paramètres et M comme un K(y)(x)-module différentiel. Alors

Théorème. Soit a ∈ K. Les exposants de ce module différentiel en a sont indépendants de y.

Démonstration. Prenons a = 0. Soit F = ( F_{1} ; ⋮ ; F_{s} ) une base de M. Les

matrices de dérivation sont

\frac{∂}{∂x}F = HF , \frac{∂}{∂y_{i}} F = G_{i}F

où H, G_{i} ∈ Mat(s, K(x, y)). L’hypothèse que M est intégrable se traduit par

\frac{∂}{∂y_{i}} H + HG_{i} = \frac{∂}{∂x}G_{i} + G_{i}H

pour 1 ≤ i ≤ n. D’après le théorème de Turrittin, il existe m tel que si z = x^{m} et si l’on considère M comme K(y)^{alg}((z))-module différentiel, il existe une base F̂ telle que la matrice de dérivation Ĥ associée à \frac{∂}{∂z},

\frac{∂}{∂z} F̂ = ĤF̂

soit sous forme de Jordan. Les quantités Res_{0}(Ĥ_{jj}) ∈ K(y)^{alg} sont m fois les exposants de M. On doit donc montrer que les Res_{0}(\frac{∂}{∂y_{i}} Ĥ_{jj}) sont nuls.

On a

\frac{∂}{∂y_{i}} Ĥ = \frac{∂}{∂z} Ĝ_{i} + Ĝ_{i}Ĥ − ĤĜ_{i} .

On sait que Res_{0}(\frac{∂}{∂z} Ĝ_{i}) = 0. Il reste donc à considérer les termes diagonaux de Ĝ_{i}Ĥ − ĤĜ_{i}.

Considérons la décomposition de Ĥ en ses sous-blocs

Ĥ = ( \begin{matrix}Ĥ^{1} & 0 & 0 & … & 0 \\ 0 & Ĥ^{2} & 0 & … & 0 \\ … & … & … & … & … \\ 0 & 0 & 0 & … & 0 \\ 0 & 0 & 0 & … & Ĥ^{r} \end{matrix} )

où Ĥ^{j} est de la forme

Ĥ^{j} = ( \begin{matrix}η_{j} & 1 & 0 & … & 0 & 0 \\ 0 & η_{j} & 1 & 0 & … & 0 \\ … & … & … & … & … & … \\ … & … & … & … & … & … \\ & & & … & η_{j} & 1 \\ 0 & 0 & 0 & … & 0 & η_{j} \end{matrix} ) .

Si l’on considère la décomposition par blocs correspondante de Ĝ_{i}

Ĝ_{i} = ( \begin{matrix}Ĝ_{i}^{11} & Ĝ_{i}^{12} & … & … \\ Ĝ_{i}^{21} & Ĝ_{i}^{22} & … & … \\ … & … & … & … \\ … & … & … & Ĝ_{i}^{rr} \end{matrix} )

les blocs diagonaux de Ĝ_{i}Ĥ − ĤĜ_{i} seront Ĝ_{i}^{jj}Ĥ^{j} − Ĥ^{j}Ĝ_{i}^{jj}. Si Ĥ^{j} est de dimension s_{j} et si α_{j} = Res_{0}(η_{j}), on aura

Tr ∘ Res_{0}(\frac{∂}{∂y_{i}} Ĥ^{j}) = s_{j} \frac{∂}{∂y_{i}} α_{j} = Res_{0} ∘ Tr(Ĝ_{i}^{jj}Ĥ^{j} − Ĥ^{j}Ĝ_{i}^{jj}) = 0 ,

et donc \frac{∂}{∂y_{i}} α_{j} = 0. C.Q.F.D.

Philippe

Copie à Bernie.

Appendix B: Models and log schemes.

B.1. The notion of smooth model of a divisorially valued function field is a somewhat ad hoc elementary notion, not too far from the birational point of view. A broader and more conceptual setting is provided by the theory of log schemes of Kazuya Kato ([KK], [IO], [KF]). In this appendix, we recast some of our results of §§1,2 in this framework.

B.2. We identify a smooth K-model (X, Z) (and, more generally, any K-variety X carrying a marked divisor Z) with the log scheme (X, M_{Z}), whose log structure is given in terms of the inclusion j : X \ Z ⟶ X by M_{Z} = O_{X} ∩ j_{∗}O_{X\Z}^{×} ↪^{α_{Z}}O_{X}. A morphism of two such structures f : (X, Z) ⟶ (Y, W) in the sense of log

schemes, is a morphism of K-schemes f^{∘} : X ⟶ Y such that the set-theoretic

inverse image of W by f^{∘} is contained in Z. This is compatible with our definition 1.2.1 of a morphism of models.

If X is smooth and Z is a divisor with reduced normal crossings^{(1)} in X, (X, M_{Z}) is a fine saturated log scheme, smooth over K, i.e. over the trivial log scheme (SpecK, K^{×}) associated to SpecK [KF, Cor. 4.9]. More precisely, we consider the category LSch_{(SpecK,K^{×})}^{fs} of fine saturated log schemes over K defined in [KF, §2]. We denote it LV_{K} for short. The category of K-schemes may be regarded as a full subcategory of LV_{K}, by identifying a K-scheme X with the log scheme (X, O_{X}^{×}). In that sense, for any object (X, M) of LV_{K}, we have a canonical morphism (X, M) ⟶ X of log schemes over K. It will be convenient to regard SpecK as a smooth K-model.

B.2.1 Example. Let X = SpecK and M = K^{×} ⊕N ⟶ K be the log structure (x, n) ⟼ x if n = 0 and (x, n) ⟼ 0 otherwise. Then the canonical morphism (X, M) ⟶ X is not log smooth. In fact, its relative dimension is 0, while the rank of ω_{(X,M)/X}^{1} is 1 [IO, Exp. II, Cor. 3.15].

It may be easily checked that any morphism f : (X, Z) ⟶ (Y, W) of smooth K-models is exact [KK, §4]. It is in fact integral [KK, Cor. 4.4 (ii)]. In particular, a morphism of smooth K-models is a closed immersion in the sense of log schemes if and only if it is an exact closed immersion. If f is smooth, the underlying morphism of schemes is flat [KK, Cor. 4.5].

We recall from [IO] that the characteristic monoid (resp. abelian sheaf) C_{X} (resp. C_{X}^{gp}) of a log scheme (X, M) is the étale sheaf of monoids (M/O_{X}^{×}) (resp. the abelian sheaf (M/O_{X}^{×})^{gp}).

Proposition B.3. Let (X, M) be an objet of LV_{K} log smooth over K, such that the rank of C_{X}^{gp} is everywhere ≤ 1. Then X is étale locally a smooth K-model.

(1) In this appendix we think in terms of the étale topology.

Proof. We first prove that étale locally at any closed point x of X, there exist strict (log) smooth morphisms (X, M)⟶^{φ}A_{K}^{1 log} = Spec(K[N]) = (A_{K}^{1}, M_{t=0}). We use [KK, Example 3.7]. We may assume that there exists a finitely generated integral monoid P and an étale morphism X ⟶ SpecK[P] such that M = P O_{X}^{×}. The map P ⟶ C_{X,x} is then surjective. We have to consider two cases:

1) The rank of C_{X,x}^{gp} is 0. Then also C_{X,x}^{gp} = C_{X,x} = 0.

2) The rank of C_{X,x}^{gp} is 1, so that C_{X,x} ≅ N.

In case 1), the structural morphism (X, M) ⟶ K is strict. Therefore x is a smooth point of X and any smooth morphism of a neighborhood of x to A_{K}^{1} \ {t = 0} gives what we need. In case 2), we pick an element p ∈ P such that p ⟼ 1, under the canonical map P ⟶ C_{X,x} ≅ N. The corresponding morphism SpecK[P] ⟶ SpecK[N] is then log smooth while the composition X ⟶ SpecK[P] ⟶ SpecK[N] is strict.

Now, étale locally, the log structure of (X, M) is the inverse image of the log structure of A_{K}^{1 log}, hence it coincides with the log structure M_{Z}, where Z is the smooth divisor φ^{−1}(t = 0). Q.E.D.

Lemma B.4.1. Let f : (X, Z) ⟶ (Y, N) be a morphism of smooth objects in LV_{K}, with (X, Z) a smooth K-model.

(i) f is an exact closed immersion if and only if étale locally on Y it is a closed immersion of smooth K-models as defined in (1.2.2). (ii) Assume (Y, N) = (Y, W) is also a smooth K-model. Then f is a closed immersion of log schemes if and only if it is a closed immersion of K-models as defined in (1.2.2).

Proof. We first prove the second part.

(ii) By definition f is a closed immersion of usual schemes. Let z be any closed point of Z and let t be a local equation of W in Y at f(z). Then M_{W,f(z)} = O_{Y,f(z)}^{×} ⊕ t^{N} and M_{Z,z} = f^{∗}(M_{W})_{z} = O_{X,z}^{×} ⊕ (f^{∗}t)^{N}. So, f^{∗}t is a local equation for Z at z in X.

(i) Since f is strict, and the rank of C_{X}^{gp} is at most 1, f(X) is contained in the open subset of Y where the rank of C_{Y}^{gp} is at most 1. By proposition B.3, this open set is covered by étale neighborhoods that are smooth K-models when equipped with the inverse image log structure. By [IO, Exp. II, 2.1] and [KF, 3.4], a closed (resp. an exact closed) immersion of log schemes remains such after strict base change in LV_{K}. After strict étale localization to the previous étale neighborhoods, we get exact closed immersions of log schemes, that are smooth K-models, hence the result according to (ii). Q.E.D.

Lemma B.4.2. Let f : (X, M) ⟶ (Y, N) be a morphism locally of finite type (as a morphism of schemes) in LV_{K}. Then there exists étale locally on X a factorization in LV_{K}

(X, M)⟶^{i}(S, L)⟶^{g}(Y, N) ,

with i an exact closed immersion and g smooth. If f is a closed immersion, g can be taken to be étale.

Proof. This is proven in [HK, (2.9.2), (2.9.3)], except for the condition that (S, L) should be saturated. So now, assume (S, L) is only fine and (X, M)⟶^{i}(S, L) is an exact closed immersion. We follow the argument of [KK, prop. 4.10 (1)]. Etale locally on X, there is a chart (P_{X} ⟶ M, Q_{S} ⟶ L, h : Q ⟶ P) of i with h surjective. Let Q′ = (h^{gp})^{−1}(P) where h^{gp} : Q^{gp} ⟶ P^{gp}; since P is saturated, so is Q′. Let then

S′ = S ×_{Spec(Z[Q])} Spec(Z[Q′]) ,

and endow S′ with the inverse image L′ of the canonical log structure of Spec(Z[Q′]). Then, as in loc.cit., we see that i factors into

(X, M)⟶^{i′′}(S′, L′)⟶^{i′}(S, L) ,

with i′′ an exact closed immersion and i′ étale. Q.E.D.

B.4.3 Example. Let (X, Z) = (SpecK[t], t = 0), and let f : (X, Z) ⟶ X be the canonical morphism. Then f is not smooth, but it decomposes into the diagonal embedding i : (X, Z) ⟶ (X ×_{K} X = SpecK[x, y], y = 0), i(t) = (t, t), which is a closed immersion, followed by the projection p : (SpecK[x, y], y = 0) ⟶ X, (x, y) ⟼ x, which is smooth.

Corollary B.4.4. Let f : (X, Z) ⟶ (Y, W) be a morphism of smooth Kmodels. Then there exists étale locally on X a factorization in LV_{K}

(X, Z)⟶^{i}(S, D)⟶^{g}(Y, W) ,

where (S,D) is a smooth K-model, i is a closed immersion and g smooth.

Proof. We first decompose f, étale locally on X, into

(X, Z)⟶^{i}(S, L)⟶^{g}(Y, W) ,

with i an exact closed immersion and g smooth. We then observe that (S, L) is smooth, and apply lemma B.4.1 to further reduce i, étale locally on S, to an exact closed immersion of smooth K-models. Q.E.D.

B.5. We now characterize geometrically smoothness for a morphism of Kmodels.

Proposition B.5.1. Let (X, Z)⟶^{f}(Y, W) be a morphism of smooth K-models

such that f^{∘}(Z) ⊂ W. Then the following are equivalent:

i) f is smooth at η_{Z},

ii) f^{∘} is flat at η_{Z}, iii) f is log dominant in the sense of (1.2.2).

Proof. i) ⇒ ii) by [KK, Cor. 4.5]. ii) ⇔ iii) by remark 1.2.3. We prove iii) ⇒ i) We must prove that the O_{X,η_{Z}} -module ω_{(X,Z)/(Y,W),η_{Z}}^{1} is free. It is clear, by definition, that

(B.5.1.1)  ω_{(X,Z)/(Y,W),η_{Z}}^{1} /torsion = Ω_{X/Y,η_{Z}}^{1} /torsion .

By [H3, Chap. II, Lemma 8.9] is suffices to prove that

dim_{κ(X)}ω_{(X,Z)/(Y,W),η_{Z}}^{1} ⊗_{O_{X,η_{Z}}} κ(X) = dim_{κ(Z)}ω_{(X,Z)/(Y,W),η_{Z}}^{1} ⊗_{O_{X,η_{Z}}} κ(Z) .

Now the l.h.s. equals dim_{κ(X)}Ω_{κ(X)/κ(Y)}^{1} = dimX−dimY, since the log structure is trivial at the generic point of X. The r.h.s. equals Ω_{X/Y,η_{Z}}^{1} ⊗_{O_{X,η_{Z}}} κ(Z) = Ω_{κ(Z)/κ(W)}^{1} by (B.5.1.1). But dim_{κ(Z)}Ω_{κ(Z)/κ(W)}^{1} = dimZ − dimW. Hence they are equal. Q.E.D.

B.5.2 Examples.

B.5.2.1. Let X be the affine (x, y)-plane deprived of the y-axis (x = 0), and Z be the divisor y = 0 in X. Let Y be the affine (x, y)-plane and W be its y-axis. The morphism (X, Z)⟶^{f}(Y, W), (x, y) ⟼ (xy, y), is not smooth since it is not flat. B.5.2.2. Let X be the affine (x, y)-plane and let Z be the divisor y = 0 in X. Let Y be the projective x-line and W be its point at infinity. The projection (X, Z)⟶^{f}(Y, W), (x, y) ⟼ x, is smooth, but f(Z) ⊂ Y \ W.

B.6 Logarithmic differential operators. Let f : (X, M) ⟶ (Y, N) be a morphism in LV_{K}. We refer to [KK, Rmk. 5.8] and to [Og, 1.1] for the description of how the Grothendieck theory of differential operators may be developed for the morphism f. Namely, let

Δ : (X, M) ⟶ (X, M) ×_{(Y,N)} (X, M)

be the diagonal embedding in LV_{K}, which is a closed immersion. One applies lemma B.4.2 to decompose Δ étale locally on X into an exact closed immersion

Δ′ : (X, M) ⟶ (X′, M′) ,

of sheaf of ideals J_{X′}, followed by an étale morphism. Let P_{(X,M)/(Y,N)}^{n} = (P_{Δ′}^{n}, M_{n}) be the n-the infinitesimal neighborhood of X in X′ in the scheme theoretic sense, equipped with the log structure M_{n} inverse image of M′. One checks that this local definition makes global sense and provides a canonical nilpotent exact closed immersion Δ^{(n)} : (X, M) ⟶ P_{(X,M)/(Y,N)}^{n} and two canon-

ical projections p_{1}^{(n)} and p_{2}^{(n)} : P_{(X,M)/(Y,N)}^{n} ⟶ (X, M) in LV_{K}, such that

p_{i}^{(n)} ∘ Δ^{(n)} = 1_{(X,M)}, for i = 1, 2. The sheaf of jets of order n of f is the augmented O_{X}-algebra

(B.6.1)  P_{(X,M)/(Y,N)}^{n} = p_{1∗}^{(n)}(O_{P_{(X,M)/(Y,N)}^{n}}) ≅ Δ^{′−1}(O_{X′}/J_{X′}^{n+1}) ,

equipped with the canonical ring homorphisms

(B.6.2)  O_{X} ⟶^{j_{n}=p_{2}^{(n)∗}}P_{(X,M)/(Y,N)}^{n} ⟶^{Δ^{(n)∗}}O_{X} .

It is filtered by the powers of the augmentation ideal KerΔ^{(n)∗}, and these data fit in an inverse system via the natural projections

(B.6.3)  P_{(X,M)/(Y,N)}^{n} ⟶ P_{(X,M)/(Y,N)}^{n−1} ,

with kernel (KerΔ^{(n)∗})^{n}. One finds

(B.6.4)  ω_{(X,M)/(Y,N)}^{1} = KerΔ^{(n)∗}/(KerΔ^{(n)∗})^{2} ≅ Δ^{′−1}(J_{X′}/J_{X′}^{2}) ,

for any n ≥ 1

Definition B.6.5. For a morphism f as before, we define the sheaf of relative differential operators of order n as

(B.6.5.1)  \underline{Diff}_{(X,M)/(Y,N)}^{n} = Hom_{O_{X}} (P_{(X,M)/(Y,N)}^{n}, O_{X}) ,

and the sheaf of relative derivations as

(B.6.5.2)  \underline{Der}_{(X,M)/(Y,N)} = Hom_{O_{X}} (ω_{(X,M)/(Y,N)}^{1}, O_{X}) .

There are natural morphisms

(B.6.5.3)  P_{X/Y}^{n} ⟶ P_{(X,M)/(Y,N)}^{n}

and

(B.6.5.4)  Ω_{X/Y}^{1} ⟶ ω_{(X,M)/(Y,N)}^{1} .

One defines as in the classical case a ring structure in

(B.6.5.5)  \underline{Diff}_{(X,M)/(Y,N)} = ∪_{n∈N}\underline{Diff}_{(X,M)/(Y,N)}^{n} ,

a natural injective morphism

(B.6.5.6)  \underline{Der}_{(X,M)/(Y,N)} ↪ \underline{Diff}_{(X,M)/(Y,N)}^{1} ,

and a left \underline{Diff}_{(X,M)/(Y,N)}-structure on O_{X}, compatible with the classical action of \underline{Diff}_{X/Y}, via the natural morphism, dual to (B.6.5.3),

(B.6.5.7)  \underline{Diff}_{(X,M)/(Y,N)} ⟶ \underline{Diff}_{X/Y} .

and inducing, dually to (B.6.5.4),

(B.6.5.8)  \underline{Der}_{(X,M)/(Y,N)} ⟶ \underline{Der}_{X/Y} .

Proposition B.6.6. Let (X, Z) be a smooth K-model. Then ω_{(X,Z)/K}^{1} (resp. \underline{Der}_{(X,Z)/K}) identifies via (B.6.5.4) (resp. (B.6.5.8)) to Ω_{X/K}^{1}(logZ) (resp. \underline{Der}_{X/K,Z}). Furthermore, (B.6.5.7) is injective and identifies \underline{Diff}_{(X,Z)/K} with \underline{Diff}_{X/K,Z}.

Proof. We may assume that (X, Z) is coordinatized with adapted coordinates (x_{1}, …, x_{d}). Then both ω_{(X,Z)/K}^{1} and Ω_{X/K}^{1}(logZ) are freely generated by \{\frac{dx_{1}}{x_{1}} , dx_{2}, …, dx_{d}\}, so the first assertion is clear.

The polynomial structure of \underline{Diff}_{(X,Z)/K} is proved as in [Og,1.1], with the simplification due to the assumption that K is a field of characteristic zero, so that the divided power calculus is not needed. Injectivity of \underline{Diff}_{(X,Z)/K}^{n} ⟶ \underline{Diff}_{X/K}^{n}, and therefore of (B.6.5.5), follows by lack of torsion. The identification of \underline{Diff}_{(X,Z)/K} with \underline{Diff}_{X/K,Z} then follows from lemma 2.1.2.

With this definition, the functoriality of \underline{Diff}_{(X,M)/(Y,N)}^{n} is formal.

Proposition B.7. Under the equivalent conditions of proposition (B.5.1), there is an exact sequence of free O_{X,η_{Z}} -modules of finite type: (B.7.1) 0 ⟶ ω_{(Y,W)/K,η_{W}}^{1} ⊗_{O_{X,η_{W}}} O_{X,η_{Z}} ⟶ ω_{(X,Z)/K,η_{Z}}^{1} ⟶ ω_{(X,Z)/(Y,W),η_{Z}}^{1} ⟶ 0 .

In the situation of ii) of proposition 2.4,

ω_{(X,Z)/(Y,W),η_{Z}}^{1} = Ω_{X/Y,η_{Z}} /torsion ,

hence

\begin{matrix}\underline{Der}_{(X,Z)/(Y,W),η_{Z}} = Hom_{O_{X,η_{Z}}} (ω_{(X,Z)/(Y,W),η_{Z}}^{1} , O_{X,η_{Z}}) \\ = Hom_{O_{X,η_{Z}}} (Ω_{X/Y,η_{Z}} , O_{X,η_{Z}}) = \underline{Der}_{X/Y,η_{Z}} . \end{matrix}

Then

Der_{v}(F/G) = \underline{Der}_{X/Y,η_{Z}} = \underline{Der}_{(X,Z)/(Y,W),η_{Z}}

and the exact sequence (2.4.3) is dual to (B.7.1).

Chapter II.

Irregularity in several variables.

Table of contents:

§1 Spectral norms.

§2 The generalized Poincaré-Katz rank of irregularity.

§3 Some consequences of the Turrittin-Levelt theorem.

§4 Newton polygons.

§5 Stratification of the singular locus by Newton polygons.

§6 Formal decomposition of an integrable connection at a singular divisor.

§7 Cyclic vectors, indicial polynomials and tubular neighborhoods.

Introduction.

In this chapter, we tackle the study of irregularity in several variables. This domain is far less explored than the island of regularity.

A systematic study has been undertaken by Y. Laurent and Z. Mebkhout [L], [Me4], [LMe]. Their main theme is the geometric construction, based upon the idea of the V -filtration, of some algebraic cycles which turn out to be positive and to correspond, through the “comparison theorem”, to the perverse sheaf of irregularity and its Gevrey filtration introduced in [Me2]; in the process, they attach a Newton polygon to any holonomic D-module along a hypersurface, and even a collection of Newton polygons indexed by the components of hte characteristic variety of the sheaf of irregularity.

Our approach is rather more algebraic, with a flavor of ultrametric analysis. It is close in spirit to chapter I.

In that chapter, we saw that one of the characterizations of regularity along a divisor is the condition that logarithmic differential operators of increasing order act with poles of bounded order at the generic point of Z. The consideration of this order of growth leads, in the general case, to the notion of the Poincaré-Katz rank ρ_{Z}(∇), which may be viewed as the coarsest measure of irregularity of an integrable connection ∇ at a singular divisor Z.

We introduce and study a finer invariant, the Newton polygon NP_{Z}(∇). It has the following properties: a) there is a dense open subset U of Z such that the Newton polygon (in the usual sense) of the restriction of ∇ to any curve meeting Z transversally inside U coincides with NP_{Z}(∇),

b) the maximal slope of NP_{Z}(∇) is equal to ρ_{Z}(∇).

More precisely, we construct a partition of the singular locus Z by constructible subsets, the Newton polygon partition, together with a bunch of secondary Newton polygons, with the property that a polygon N is the Newton polygon of the restriction of the connection to “almost every curve” passing through the stratum attached to N. Moreover, we prove that the associated function on Z (assumed to be smooth), with polygonal values, is semicontinuous. One of our main technical tools is a version of the Turrittin-Levelt decomposition in the relative setting. Unfortunately, the relation between the Laurent-Mebkhout polygons and ours remains unclear for us (except for the “principal” one). It also remains to understand the link with C. Sabbah’s analytic study of irregularity in two variables [S].

Let us now describe in more detail the content of each section.

In section 1 we recall some properties of the spectral norm of a non-archime-dean Banach algebra A over the field K endowed with the trivial valuation (so, the topology of K is discrete), in some special situations. Namely, we are given a complete discretely valued extension field (F̂, v̂) of K and a finite dimensional vector space M over F̂: on arbitrarily choosing the absolute value a^{−1} ∈ (0, 1) of a uniformizer of F̂ (resp. an F̂-basis of M to whose elements we give norm one) both F̂ and M may be naturally regarded as K-Banach spaces. Typically A = L_{K}(M), the K-algebra of continuous K-linear endomorphisms of M, endowed with the operator norm || ||_{L_{K}(M)}.

In section 2, we define a measure of the defect of regularity along an irreducible divisor, the Poincaré-Katz rank of a differential module (M, ∇) on X \ Z, for a smooth model (X, Z) of (F, v). To do this, we first take the formal v-completions (F̂, v̂) of (F, v) and (M̂, ˆ∇) of the generic fiber (M, ∇) of (M, ∇). We define the Poincaré-Katz rank of (M̂, ˆ∇) along v̂ (and of (M, ∇) along Z) in terms of the spectral norms of all covariant derivations preserving the ideal of the valuation v̂ of F̂ (in one variable, the use of spectral norms in this context already appeared in the work of G. Christol and B. Dwork [CD]). We then give an asymptotic formula for the rank, involving the action of logarithmic differential operators of increasing order at the generic point of the singular divisor. We examine the behaviour of the rank under inverse (resp. finite direct) images, and some exactness properties related to the notion of rank. We also examine the case of a confluence (2.5.8). From this down-to-earth, birational and ultrametric point of view, the Poincaré-Katz rank appears as a more elementary invariant than the irregularity (Fuchs number). In the sequel of the chapter, we shall in fact reprove in a different (indirect) way most of these properties of the rank, as consequences of results on Newton polygons.

In section 3, using Turrittin’s theorem, we show that the rank may be read algebraically in terms of the iterated action of any vector field transversal to the singular divisor, and we define the slope filtration of M̂.

In section 4, we review Newton polygons in one variable, study their variation in a family of integral curves (constructibility, semicontinuity), and clarify some aspects of the phenomenon of confluence. Using the slope filtration, we then introduce the Newton polygon of M̂; its maximal slope is the Poincaré-Katz rank. We examine some functorialities.

In the last three sections, which are of a less elementary nature, we give up the birational point of view. We then study the variation of Newton polygons when one restricts the connection to various curves transversal to Z, not necessarily belonging to a family of integral curves. Using Gauss maps, we establish property a) stated above (5.3.1). This allows us to give complements to some of our results of (I) about regular connections (5.3.3), (5.3.5). We introduce the (principal) Newton polygon of an integrable connection at a singular divisor and study its behaviour under inverse image. We then introduce secondary Newton polygons (5.4) and prove the semicontinuity theorem (5.4.4).

In section 6, we establish, under a technical assumption, a formal decomposition of integrable connections in several variables at a singular divisor.

The last section contains technical results about cyclic vectors and indicial polynomials which will be crucial in the last chapter. In particular, we give sufficient (and essentially necessary) conditions for the existence, in the neighborhood of a singular divisor, of a cyclic vector with respect to a transversal derivation.

§1 Spectral norms.

1.1. Let F be a function field over a field K of characteristic 0, and let v be a divisorial valuation of F/K. Let (F̂, v = v̂) denote the completion. We fix a real constant a > 1, and consider the non-archimedean absolute value | |_{F̂,a} = | |_{a} on F̂ defined by |y|_{a} = a^{−v(y)}. For any matrix A = (a_{i,j}) ∈ M_{ν×ν′}(F̂), we set ||A||_{F̂,a} = ||A||_{a} := sup_{i,j} |a_{i,j}|_{a}, a norm of F̂-Banach space on M_{ν×ν′}(F̂) such that

(1.1.1)  ||AB||_{a} ≤ ||A||_{a}||B||_{a}

whenever multiplication is possible.

Let (F, v)/(G, w) be an extension of divisorially valued function fields (or fields of formal functions); this means that we have an extension of fields G ⊂ F, that 𝔪_{v} ∩ R_{w} = 𝔪_{w} and 𝔪_{w}R_{v} = 𝔪_{v}^{e}, where e = e(v/w) is the relative ramification index. Then for y ∈ Ĝ, we have

(1.1.2)  |y|_{Ĝ,a}^{e(v/w)} = |y|_{F̂,a} ,

and similarly for matrix-norms.

Let M be a finite dimensional F̂-vector space. Any choice of a basis \underline{e} of M over F̂ determines on M a norm || ∑ a_{i}e_{i}||_{M,a,\underline{e}} = sup_{i} |a_{i}|_{a} of Banach space, both over F̂ and over K endowed with the trivial absolute value (inducing the discrete topology on K). Two norms | |_{1}, | |_{2} on a vector space V are strongly equivalent if there exist positive constants C_{1}, C_{2} such that C_{1}|y|_{1} ≤ |y|_{2} ≤ C_{2}|y|_{1}, for all y ∈ V; we write | |_{1} ∼ | |_{2} if that is the case. If \underline{f} = \underline{e}A is another basis of M̂, the norms || ||_{M,a,\underline{e}} and || ||_{M,a,\underline{f}} are strongly equivalent, since for any y ∈ M

(1.1.3)  ||A||_{a}^{−1}||y||_{M,a,\underline{e}} ≤ ||y||_{M,a,\underline{f}} ≤ ||A^{−1}||_{a}||y||_{M,a,\underline{e}} .

In particular, (F̂, | |_{a}) is a K-Banach space. Since F̂ = C((x)), for a uniformizer x of R_{v} and a function field C ⊂ R_{v} over K, {x^{i}}_{i∈Z} is an orthogonal (topological) basis of (F̂, | |_{a}) over K.

In the situation of formula 1.1.2, for a Ĝ-vector space N freely generated by \underline{g}, and NF̂ := N ⊗Ĝ F̂,

(1.1.4)  || ||_{NF̂,a,\underline{g}⊗1} = || ||_{N,a,\underline{g}}^{e(v/w)} .

Under the same assumptions, the Ĝ-vector space obtained by restriction of scalars from a finite-dimensional F̂-vector space M, will be denoted by ĜM. If \underline{e} (resp.

\underline{f}) is a basis of the F̂-vector space M (resp. of the Ĝ-vector space F̂), \underline{f}\underline{e} := (f_{i}e_{j}) is a basis of ĜM. Under the extra assumption that Ĝ is of finite index in F̂,

(1.1.5)  || ||_{M,a,\underline{e}} ∼ || ||_{ĜM,a,\underline{f}\underline{e}}^{e(v/w)} .

1.2. Let (M, || ||_{M}), (N, || ||_{N}) be two K-Banach spaces. The K-vector space L_{K}(M, N) of continuous K-linear maps from M to N is a K-Banach space in the usual norm

||L||_{L_{K}(M,N)} := sup{||L(y)||_{N}/||y||_{M} | y ∈ M , y ≠ 0 } ,

for L ∈ L_{K}(M, N). The ring L_{K}(M) := L_{K}(M, M) of continuous K-linear endomorphisms of M becomes then a K-Banach algebra with respect to the norm || ||_{L_{K}(M)}. Strongly equivalent norms on M and N induce strongly equivalent norms on L_{K}(M, N).

We denote by (M ˆ⊗N, || ||_{M ˆ⊗N}) the topological tensor product of (M, || ||_{M}) and (N, || ||_{N}): it is the completion of (M ⊗ N, || ||_{M⊗N}), where

||x||_{M⊗N} = inf max_{i} ||m_{i}||_{M}||n_{i}||_{N} ,

over all representations of x ∈ M ⊗ N as x = ∑_{i} m_{i} ⊗ n_{i}. If M, N are finite dimensional F̂-vector spaces, || ||_{M} = || ||_{M,a,\underline{e}}, || ||_{N} = || ||_{N,a,\underline{f}}, for an F̂-basis \underline{e} of M (resp. \underline{f} of N), we have a canonical projection

M ˆ⊗_{K}N ⟶ M ⊗F̂ N ,

and a canonical injection

LF̂(M, N) ↪ L_{K}(M, N) .

One easily checks, using orthogonal bases, that

(1.2.1)  ||x||_{M⊗F̂N,a,\underline{e}⊗\underline{f}} = inf_{x′⟼x} ||x′||_{M ˆ⊗_{K}N} = min_{x′⟼x} ||x′||_{M ˆ⊗_{K}N} ,

and

(1.2.2)  ||x||_{LF̂(M,N),a,\underline{e}^{∨}⊗\underline{f}} = ||x||_{L_{K}(M,N)} .

For L ∈ L_{K}(M), and M as in the previous formulas, the norm ||L||_{L_{K}(M)} will be also denoted by ||L||_{M,a,\underline{e}}.

1.3. Let (A, || ||_{A}) be a Banach algebra with identity over K. As in the classical case and with the same proof [T.S., 1.2.3], one shows that for any f ∈ A, the limit lim_{n→∞}(||f^{n}||_{A})^{1/n} exists in R_{≥0}; it is called the spectral norm ||f||_{A,sp} of f. We have

(1.3.1)  ||f||_{A,sp} := lim_{n→∞}(||f^{n}||_{A})^{1/n} = inf_{n≥1}(||f^{n}||_{A})^{1/n} .

The theory of analytic spaces of Berkovich [Bk] offers the possibility of dealing with the spectrum Σ_{f} ⊂ A^{1} of f as in the classical case [loc. cit., Chap. 7]. In particular, [loc. cit., 7.1.2] shows that ||f||_{A,sp} is determined by Σ_{f}.

Proposition 1.3.2. Let J is a square-zero bilateral ideal of A and u be an element of J. The spectral norm of f + u coincides with the one of f.

The proposition is a consequence of the following

Lemma 1.3.3. Let notation be as in (1.3.2), (\overline{A} = A/J , || ||\overline{A}) be the quotient K-Banach algebra, and f ⟼ \overline{f} be the canonical projection. For any f ∈ A

||f||_{A,sp} = ||\overline{f}||_{\overline{A},sp} .

Proof. We first observe that if z is in J, then (1+z)(1−z) = (1−z)(1+z) = 1 in A. Clearly, Σ_{f′} ⊂ Σ_{f}. We have to show that A^{1} \ Σ_{f′} ⊂ A^{1} \ Σ_{f}. For a (Berkovich) point x ∈ A^{1}, H(x) denotes the corresponding complete valuation field (cf. loc.cit.): we have to prove that if x is such that 1 ⊗ T (x) − f′ ⊗ 1 is invertible in B ˆ⊗H(x), then 1 ⊗ T (x) − f ⊗ 1 is invertible in A ˆ⊗H(x). There is a y ∈ A ˆ⊗H(x) such that u = 1 − (1 ⊗ T (x) − f ⊗ 1)y ∈ J and v = 1 − y(1 ⊗ T (x) − f ⊗ 1) ∈ J. But then 1 − u and 1 − v are invertible in A and (1 ⊗ T (x) − f ⊗ 1)y(1 + u) = (1 + v)y(1 ⊗ T (x) − f ⊗ 1) = 1. Therefore, y(1 + u) = (1 + v)y and 1 ⊗ T (x) − f ⊗ 1 is invertible in A ˆ⊗H(x). Therefore, x ∉ Σ_{f}. Q.E.D.

In the particular case of (A, || ||_{A}) = (L_{K}(M), || ||_{L_{K}(M)}), where (M, || ||_{M}) = (M, || ||_{M,a,\underline{e}}), with M a finite dimensional F̂-vector space and \underline{e} a basis of it, ||L||_{A,sp}, will be denoted by ||L||_{M,sp}. It is in fact clearly independent of the choice of the basis \underline{e}.

Lemma 1.4. Let (M, || ||_{M}) and (N, || ||_{N}) be Banach spaces over K.

i) If D (resp. D′) is an element of L_{K}(M) (resp. of L_{K}(N)),

||D ˆ⊗_{K}D′||_{M ˆ⊗_{K}N,sp} = ||D||_{M,sp}||D′||_{N,sp} ,
||L_{K}(D, D′)||_{L_{K}(M,N),sp} = ||D||_{M,sp}||D′||_{N,sp} ,
||D ˆ⊗_{K}1_{N} + 1_{M} ˆ⊗_{K}D′||_{M ˆ⊗_{K}N,sp} ≤ max(||D||_{M,sp}, ||D′||_{N,sp}) ,

||L_{K}(1_{M}, D′) − L_{K}(D, 1_{N})||_{L_{K}(M,N),sp} ≤ max(||D||_{M,sp}, ||D′||_{N,sp}) .

If ||D||_{M,sp} ≠ ||D′||_{N,sp}, equality holds in the previous formulas. ii) Assume that in i) M (resp. N) is a F̂-vector space of finite dimension and that || ||_{M} (resp. || ||_{N}) is the norm associated to a F̂-basis of M (resp. N), and assume that

(1.4.1)

D ˆ⊗_{K}1_{N} + 1_{M} ˆ⊗_{K}D′ kills the closed K-vector subspace
Ker(M ˆ⊗_{K}N ⟶ M⊗F̂N) of M ˆ⊗_{K}N.

Then D ˆ⊗_{K}1_{N}+1_{M} ˆ⊗_{K}D′ induces a continuous K-linear operator F ∈ L_{K}(M⊗F̂N). We have

||F ||_{M⊗F̂N,sp} = ||D ˆ⊗_{K}1_{N} + 1_{M} ˆ⊗_{K}D′||_{M ˆ⊗_{K}N,sp} ≤ max(||D||_{M,sp}, ||D′||_{N,sp}) , with equality holding if ||D||_{M,sp} ≠ ||D′||_{N,sp}. iii) Assumptions are as in ii), with (1.4.1) replaced by

(1.4.2)

L_{K}(1_{M}, D′) − L_{K}(D, 1_{N}) is stable on the closed K-vector subspace
HomF̂(M, N) of L_{K}(M, N).

Then L_{K}(1_{M}, D′) − L_{K}(D, 1_{N}) induces a continuous K-linear operator G ∈ L_{K}(HomF̂(M, N)). We have

||G||_{HomF̂(M,N),sp} = ||L_{K}(1_{M}, D′)−L_{K}(D, 1_{N})||_{L_{K}(L_{K}(M,N)),sp} ≤ max(||D||_{M,sp}, ||D′||_{N,sp}) ,

with equality holding if ||D||_{M,sp} ≠ ||D′||_{N,sp}.

Proof. To prove i), we first note that the operators D ˆ⊗_{K}1_{N} and 1_{M} ˆ⊗_{K}D′ on M ˆ⊗_{K}N such that D ˆ⊗_{K}1_{N} ∘ 1_{M} ˆ⊗_{K}D′ = D ˆ⊗_{K}D′ (resp. L_{K}(1_{M}, D′) and L_{K}(D, 1_{N}) on L_{K}(M, N), such that L_{K}(1_{M}, D′) ∘ L_{K}(D, 1_{N}) = L_{K}(D, D′)) commute. On the other hand, ||D ˆ⊗_{K}1_{N}||_{M ˆ⊗_{K}N,sp} = ||D||_{M,sp}, ||1_{M} ˆ⊗_{K}D′||_{M ˆ⊗_{K}N,sp} = ||D′||_{N,sp} (resp. ||L_{K}(D, 1_{N})||_{L_{K}(M,N),sp} = ||D||_{M,sp}, ||L_{K}(1_{M}, D′)||_{L_{K}(M,N),sp} = ||D′||_{N,sp}).

The first two formulas of i) certainly hold with a ≤ replacing equality. The opposite inequality is checked directly. In fact ||m ⊗ n||_{M⊗N} = ||m||_{M}||n||_{N} and ||D ⊗ D′(m ⊗ n)||_{M⊗N} = ||D(m) ⊗ D′(n)||_{M⊗N} = ||D(m)||_{M}||D(n)||_{N}, and similarly for L_{K}(M, N). The remaining statements of i) are proven using the previous rmark and the arguments of [vR, p. 222-223].

ii) (resp. iii)) follows from formula 1.2.1 (resp. 1.2.2).

Lemma 1.5. Let

(1.5.1)  (P, || ||_{P}) = (M, || ||_{M}) ⊕ (N, || ||_{N})

be an orthogonal direct sum of K-Banach spaces and let F ∈ L_{K}(P) be an operator such that F (M) ⊂ M. Let D = F_{|M} ∈ L_{K}(M) and D′ ∈ L_{K}(N) be the operators induced by F. Then

||F ||_{P,sp} = max(||D||_{M,sp}, ||D′||_{N,sp}) .

Proof. Let us consider the operator L′ = D ⊕ D′ ∈ L_{K}(P). Obviously, ||L′||_{P,sp} = max(||D′||_{N,sp}, ||D||_{M,sp}). On the other hand, L = L′ + H, where H is an operator that kills M and sends P to M. Such operators form a bilateral ideal J, with J^{2} = 0, in the sub-K-Banach algebra A of L_{K}(P, P) of operators preserving M. Since J ⊂ R_{A}, we may apply lemma 1.3.3 to deduce ||L||_{P,sp} = ||L′||_{P,sp}.

Lemma 1.6. i) Let (M, || ||_{M}) = (M, || ||_{M,a,\underline{e}}) and Ĝ ⊂ F̂ be as in (1.1.5). Then, for any L ∈ L_{K}(M) = L_{K}(ĜM)

||L||_{M,sp} = ||L||_{ĜM,sp}^{e(v/w)} .

ii) Under the assumptions of i), let (M, || ||_{M}) = (F̂, | |_{a}) and let (D_{1}, …, D_{d}) be an adapted R_{v}-basis of Der_{v}(F̂/K) (see (I.2.1.2)). Then

||D_{i}||_{F̂,sp} = ||D_{i}||_{ĜF̂,sp} = 1 for every i = 1, …, d .

Proof. Left to the reader.

1.7. We prefer a valuative version of the previous notions. Let M be a finite F̂-vector space and M_{0} be a lattice in M, i.e. an R_{v}-submodule spanned by a F̂-basis \underline{e} of M. It is clear from formula 1.1.3, that the Banach norm || ||_{M} = || ||_{M,a,\underline{e}} on M is determined by M_{0} and a. So, we define T_{v,M_{0}} = T_{v,\underline{e}} : M ⟶ Z ∪ \{∞\} (resp. T_{v,M_{0}} = T_{v,\underline{e}} : L_{K}(M) ⟶ Z ∪ \{∞\}, resp. v : M_{ν×ν′}(F̂) ⟶ Z ∪ \{∞\}) by T_{v,\underline{e}}(m) = − log_{a} ||m||_{M} (resp. T_{v,\underline{e}}(L) = − log_{a} ||L||_{L_{K}(M)}, resp. v(A) = − log_{a} ||A||_{a} = inf v(a_{i,j}), for A = (a_{i,j})). We call T_{v,M_{0}} a valuative (Banach) norm on M (resp. L_{K}(M)). Whenever it makes sense,

(1.7.1)  \begin{aligned}v(A + B) & ≥ min(v(A), v(B)) , \\ v(AB) & ≥ v(A) + v(B) . \end{aligned}

and (1.7.2)

v_{sp}(L) = v_{M,sp}(L) := − log_{a} ||L||_{a,sp} = lim_{n→∞} \frac{T_{v,M_{0}}(L^{n})}{n} = sup_{n≥1} \frac{T_{v,M_{0}}(L^{n})}{n} ,

is independent of the chosen lattice M_{0} of M. We call v_{M,sp}(L), the spectral valuation of L ∈ L_{K}(M).

Lemma 1.8. Let a_{i,n}, for i = 1, …, d and n ≥ 0, be real numbers such that, for any i, lim_{n→∞} a_{i,n}/n := ρ_{i} exists in R. Then

(∗)_{d}  lim_{n→∞} \frac{1}{n} sup ; ∑_{i} n_{i}=n ∑_{i=1}^{d} a_{i,n_{i}} = sup(ρ_{1}, …, ρ_{d}) ,

and

(∗∗)_{d}  lim_{n→∞} \frac{1}{n} sup ; ∑_{i} n_{i}≤n ∑_{i=1}^{d} a_{i,n_{i}} = sup(0, ρ_{1}, …, ρ_{d}) .

Proof. (C. Mariconda) Clearly the conjunction of (∗∗)_{1} and (∗)_{d} implies (∗∗)_{d}. On the other hand, the inductive step (∗)_{d} ⇒ (∗)_{d+1} follows from (∗)_{2}. Since (∗)_{1} is trivial, we are reduced to proving (∗)_{2} and (∗∗)_{1} .

We first prove (∗)_{2}. We set a_{1,n} = a_{n}, a_{2,n} = b_{n}. Certainly, for all n > 0,

\frac{1}{n} sup(a_{0} + b_{n}, …, a_{n} + b_{0}) ≥ sup(a_{0}/n + b_{n}/n, a_{n}/n + b_{0}/n) ,

so that the inequality “≥” holds.

Let us fix ϵ > 0. There exists n_{ϵ} such that

(ρ_{1} − ϵ)k ≤ a_{k} ≤ (ρ_{1} + ϵ)k , (ρ_{2} − ϵ)k ≤ b_{k} ≤ (ρ_{2} + ϵ)k ∀k ≥ n_{ϵ} .

For n ≥ 2n_{ϵ} we then have

sup(a_{n_{ϵ}} + b_{n−n_{ϵ}}, …, a_{n−n_{ϵ}} + b_{n_{ϵ}}) ≤ (max(ρ_{1}, ρ_{2}) + ϵ)n .

From this we deduce, for n ≥ 2n_{ϵ},

\frac{sup(a_{0} + b_{n}, …, a_{n} + b_{0})}{n} ≤ sup(\frac{sup(a_{0} + b_{n}, …, a_{n_{ϵ}−1} + b_{n−n_{ϵ}+1})}{n},

\frac{sup(a_{n} + b_{0}, …, a_{n−n_{ϵ}+1} + b_{n_{ϵ}−1})}{n}, \frac{sup(a_{n_{ϵ}} + b_{n−n_{ϵ}}, …, a_{n−n_{ϵ}} + b_{n_{ϵ}})}{n}) ≤

sup(\frac{sup(a_{0}, …, a_{n_{ϵ}−1})}{n} + \frac{sup(b_{n}, …, b_{n−n_{ϵ}+1})}{n}, \frac{sup(a_{n}, …, a_{n−n_{ϵ}+1})}{n}+

\frac{sup(b_{0}, …, b_{n_{ϵ}−1})}{n}, \frac{sup(a_{n_{ϵ}} + b_{n−n_{ϵ}}, …, a_{n−n_{ϵ}} + b_{n_{ϵ}})}{n}) ≤

sup(\frac{sup(a_{0}, …, a_{n_{ϵ}−1})}{n} + ρ_{2} + ϵ, \frac{sup(b_{0}, …, b_{n_{ϵ}−1})}{n} + ρ_{1} + ϵ, sup(ρ_{1}, ρ_{2}) + ϵ) .

Since, for the fixed ϵ,

lim_{n→∞} \frac{sup(a_{0}, …, a_{n_{ϵ}−1})}{n} = lim_{n→∞} \frac{sup(b_{0}, …, b_{n_{ϵ}−1})}{n} = 0 ,

formula (∗)_{2} follows.

We now prove (∗∗)_{1}. We set a_{1,n} = a_{n}, and assume lim_{n→∞} a_{n}/n = ρ ∈ R. We prove that

(∗∗)_{1}  lim_{n→∞} \frac{1}{n} sup(a_{0}, …, a_{n}) = sup(0, ρ) .

Certainly, for all n > 0, \frac{1}{n} sup(a_{0}, …, a_{n}) ≥ sup(a_{0}/n, a_{n}/n), so that the inequality “≥” holds. Let us fix ϵ > 0. There exists n_{ϵ} such that

(ρ − ϵ)k ≤ a_{k} ≤ (ρ + ϵ)k ∀k ≥ n_{ϵ} .

For n ≥ n_{ϵ} we then have sup(a_{n_{ϵ}}, …, a_{n}) ≤ (ρ + ϵ)n. From this we deduce, for n ≥ n_{ϵ},

\frac{sup(a_{0}, …, a_{n})}{n} = \frac{sup(sup(a_{0}, …, a_{n_{ϵ}−1}), sup(a_{n_{ϵ}}, …, a_{n}))}{n} =

sup(\frac{sup(a_{0}, …, a_{n_{ϵ}−1})}{n}, \frac{sup(a_{n_{ϵ}}, …, a_{n})}{n}) .

Since, for the fixed ϵ, lim_{n→∞} \frac{sup(a_{0},...,a_{nϵ−1})}{n} = 0 , (∗∗)_{1} follows. Q.E.D.

Remarks 1.9. 1.9.1. The lemma also holds (with essentially the same proof) if one replaces “ lim_{n→∞}” by “lim sup_{n→∞}” everywhere in the statement.

1.9.2. If in the previous lemma we assume that the sequences n ⟼ a_{i,n} − ρ_{i} n, for i = 1, …, d, are bounded from above, the same is obviously true for the sequence n ⟼ (sup_{∑_{i} n_{i}=n} ∑_{i=1}^{d} a_{i,n_{i}}) − sup(ρ_{1}, …, ρ_{d}) n (resp. n ⟼

(sup_{∑_{i} n_{i}≤n} ∑_{i=1}^{d} a_{i,n_{i}}) − sup(0, ρ_{1}, …, ρ_{d}) n ).

§2 The generalized Poincaré-Katz rank of irregularity.

2.1. Let (F, v) be a divisorially valued function field over K, with completion (F̂, v) (I.1.1). So, v is a discrete valuation of F, trivial over K, normalized by the condition that its value group is exactly Z. Let (M, ∇) be an F̂/K-differential module, where ˆ∇ is viewed as an F̂-linear map Diff_{cont}(F̂/K) ⟶ End_{K}(M) (cf. (I.3.1.1)).

Definition 2.1.1. We define the Poincaré-Katz rank of (M, ∇) (at v) as

(2.1.1.1)  ρ_{v}(M) = ρ_{v}((M, ∇)) = − inf v_{sp}(∇(D)) ( ≥ 0 ) ,

where the infimum is taken over all D ∈ Der_{v}(F̂/K). For a F/K-differential module (M, ∇), we set ρ_{v}(M) = ρ_{v}(M, ∇) := ρ_{v}(M̂, ˆ∇) and call it again the Poincaré-Katz rank of (M, ∇) at v.

Let \underline{e} = (e_{1}, …, e_{μ}) be an F̂-basis of M. Let (D_{1}, D_{2}, …, D_{d}) be an adapted R_{v}-basis of Der_{v}(F̂/K) and for any \underline{α} ∈ N^{d}

∇(\underline{D}\underline{α})\underline{e} = \underline{e}G\underline{α} .

Lemma 2.1.2.

T_{v,\underline{e}}(∇(\underline{D}\underline{α})) = inf\{v(G\underline{β}) : β_{i} ≤ α_{i} ∀i\} ( ≤ 0 ) .

In particular

v(G\underline{α}) ≥ ∑_{i=1}^{d} α_{i} min(0, v(G_{\underline{1}_{i}})) ,

and v_{sp}(∇(D)) ≤ 0 for all D ∈ Der_{v}(F̂/K).

Proof. The first assertion may be checked directly. As for the second, it follows from the fact that || ||_{L_{K}(M)} is a K-algebra norm, so that

T_{v,\underline{e}}(∇(\underline{D}\underline{α})) ≥ ∑_{i=1}^{d} α_{i}T_{v,\underline{e}}(∇(D_{i})) .

Q.E.D. Proposition 2.1.3.

ρ_{v}(M̂) = lim_{n→∞} \frac{1}{n} sup_{L∈Diff_{v}^{n}(F̂/K)} −T_{v,\underline{e}}(∇(L)) = sup_{i} −v_{sp}(∇(D_{i})) .

Proof. Since || ||_{L_{K}(M)} is a K-algebra norm

inf_{L∈Diff_{v}^{n}(F̂/K)} T_{v,\underline{e}}(∇(L)) = inf ; ∑_{i} n_{i}≤n ∑_{i} T_{v,\underline{e}}(∇(D_{i}^{n_{i}})) .

It follows from lemma 1.8 that lim_{n→∞} \frac{1}{n} inf ; ∑_{i} n_{i}≤n ∑_{i} T_{v,\underline{e}}(∇(D_{i}^{n_{i}})) exists and equals

inf_{i} lim_{n→∞} \frac{1}{n}T_{v,\underline{e}}(∇(D_{i}^{n})) = inf_{i} v_{sp}(∇(D_{i})). Q.E.D. Corollary 2.1.3.1. We have

ρ_{v}(M̂) = lim_{n→∞} h_{v,\underline{e}}(n)/n ,

with

h_{v,\underline{e}}(n) = sup ; ∑_{i=1}^{d} α_{i}≤n −v(G\underline{α}) ,

Proof. From lemma 2.1.2. Q.E.D. Remark 2.1.3.2. By remark 1.9.1 we may also say that

ρ_{v}(M) = sup(0, lim sup ; ∑_{i=1}^{d} α_{i}→∞ −v(G\underline{α})/ ∑_{i=1}^{d} α_{i}) .

Having shown that the Poincaré-Katz rank may be calculated from the spectral norms of the elements of any adapted basis of Der_{v}(F/K), we pause in the two following subsections to give an explicit procedure for the calculation of the Poincaré-Katz rank in cyclic cases.

2.2 The classical one-variable case (trivially valued constant field).

2.2.1. Here (M, ∇) is a K((x))/K-differential module. Let

\underline{e} = (e, ∇(\frac{d}{dx})e …, ∇(\frac{d}{dx})^{μ−1}e)

be a cyclic basis of (M, ∇) and

(2.2.1.1)  Λ = (\frac{d}{dx})^{μ} − ∑_{i=0}^{μ−1} γ_{i}(x)(\frac{d}{dx})^{i} ,

be the monic differential operator such that Λe = 0. So, as a left K((x))[\frac{d}{dx}]- module, (M, ∇) ≅ K((x))[\frac{d}{dx}]/K((x))[\frac{d}{dx}]Λ, ∇(\frac{d}{dx})^{j}e ↔ (\frac{d}{dx})^{j}. For any m ≥ μ, let Λ_{m} be defined by

(2.2.1.2)  Λ_{m} = (\frac{d}{dx})^{m} − ∑_{j=0}^{μ−1} γ_{m,j}(x)(\frac{d}{dx})^{j} ∈ K((x))[\frac{d}{dx}]Λ .

We write (2.2.1.3)

γ_{j} = x^{−δ_{j}}λ_{j} + higher order terms, λ_{j} ∈ K ,

σ := max(1, max_{j=0,...,μ−1} \frac{δ_{j}}{μ − j}) ,

γ_{m,j} = x^{−σ(m−j)}λ_{m,j} + higher order terms (λ_{m,j} = 0 if σ(m − j) ∉ Z) .

Lemma 2.2.2. ([BS, Lemma 1]). Assume σ > 1 and let

(2.2.2.1)  Λ̃ = (\frac{d}{dx})^{μ} − ∑_{δ_{j}=σ(μ−j)^{0≤j≤μ−1}} λ_{j}(\frac{d}{dx})^{j} .

A basis of solutions of Λ̃ in K[[x]], is given by

(2.2.2.2)  v_{k}(x) = \frac{x^{k}}{k!} + \frac{λ_{μ,k}}{μ!} x^{μ} + \frac{λ_{μ+1,k}}{(μ + 1)!}x^{μ+1} + …,

for k = 0, …, μ − 1.

Proof. For σ > 1, one easily finds

(2.2.2.3)  λ_{m+1,j} = λ_{m,j−1} + λ_{μ,j}λ_{m,μ−1} ,

for m ≥ μ and j = 0, …, μ − 1. On the other hand,

(2.2.2.4)  Λ̃v_{k} = ∑_{i=0}^{∞}(λ_{μ+i,k} − λ_{μ,k−i} − ∑_{j} λ_{μ,μ−j}λ_{μ+i−j,k})\frac{x^{i}}{i!} .

It is easy to check that the two inductive formulas for λ_{μ+i,k} in terms of λ_{μ+h,l}, for h < i and l = 0, …, μ − 1:

λ_{μ+i,k} = λ_{μ,k−i} + ∑_{j} λ_{μ,μ−j}λ_{μ+i−j,k}

and

λ_{μ+i,k} = λ_{μ+i−1,k−1} + λ_{μ,k}λ_{μ+i−1,μ−1} ,

are compatible. Q.E.D.

2.2.3. We have from (2.2.1.2)

(2.2.3.1) (\frac{d}{dx})^{m}\underline{e} = \underline{e}Γ_{m} = \underline{e} ( \begin{matrix}γ_{m,0} & γ_{m+1,0} & … & γ_{m+μ−1,0} \\ γ_{m,1} & γ_{m+1,1} & … & γ_{m+μ−1,1} \\ ⋮ & & … & ⋮ \\ γ_{m,μ−1} & γ_{m+1,μ−1} & … & γ_{m+μ−1,μ−1} \end{matrix} ) ,

and from the previous lemma we deduce that, at least if σ > 1, for each column of Γ_{m}, one of its entries γ_{m+i,j} has minimal value −σ(m + i − j). Therefore, if σ > 1,

−σm − σ(μ − 1) ≤ T_{v,\underline{e}}((\frac{d}{dx})^{m}) ≤ −σm ,

that is, using (I.2.1.2.1,2),

(2.2.3.2)  −(σ − 1)m − σ(μ − 1) ≤ T_{v,\underline{e}}((x\frac{d}{dx})^{m}) ≤ −(σ − 1)m ,

and, now for any σ ≥ 1, and for any n ∈ N,

(2.2.3.3) −(σ − 1)n − σ(μ − 1) ≤ inf_{L∈Diff_{v}^{n}(F̂/K)} T_{v,\underline{e}}(∇(L)) ≤ −(σ − 1)n .

We conclude (cf. [De, II, Thm. 1.9 and Prop. 1.10]):

Proposition 2.2.4. Let v be the x-adic valuation on K((x)). For the left K((x))[\frac{d}{dx}]-module (M, ∇) ≅ K((x))[\frac{d}{dx}]/K((x))[\frac{d}{dx}]Λ, with Λ given by (2.2.1.1), we have

ρ_{v}(M) = −v_{sp}(∇(x\frac{d}{dx})) = max(0, max_{j=0,...,μ−1}(−\frac{v(γ_{j})}{μ − j} − 1)) .

In particular, ρ_{v}(M) is a non-negative rational number of denominator at most μ. Let \underline{e} be any K((x))-basis of M. The function

(2.2.4.1)  n ⟼ |T_{v,\underline{e}}(∇((x\frac{d}{dx})^{n})) + ρ_{v}(M) n| ,

is bounded.

Parallel results hold for function fields F of one variable over K.

Corollary 2.2.5. Let Λ ∈ K((x))[\frac{d}{dx}] and (M, ∇) be as in (2.2.4). Write

(2.2.5.1)  x^{μ}Λ = (x\frac{d}{dx})^{μ} − ∑_{i=0}^{μ−1} θ_{i}(x)(x\frac{d}{dx})^{i} .

Then

(2.2.5.2)  ρ_{v}(M) = max(0, max_{j=0,...,μ−1}(−\frac{v(θ_{j})}{μ − j})) .

Proof. We have in DiffF̂/K relations of the type (cf. (I.2.1.2.1,2))

(2.2.5.3)  \begin{aligned}(x\frac{d}{dx})^{i} & = x^{i}(\frac{d}{dx})^{i} + ∑_{j<i} a_{ij}x^{j}(\frac{d}{dx})^{j} \\ x^{i}(\frac{d}{dx})^{i} & = (x\frac{d}{dx})^{i} + ∑_{j<i} b_{ij}(x\frac{d}{dx})^{j} , \end{aligned}

with a_{ij}, b_{ij} ∈ Z. We rewrite (2.2.5.1) as

(2.2.5.4) x^{μ}Λ = (x\frac{d}{dx})^{μ} − ∑_{i=0}^{μ−1} θ_{i}(x)(x\frac{d}{dx})^{i} = x^{μ}(\frac{d}{dx})^{μ} − ∑_{i=0}^{μ−1} γ̃_{i}(x)x^{i}(\frac{d}{dx})^{i} ,

with γ̃_{i}(x) = x^{μ−i}γ_{i}(x). It follows from (2.2.5.3) that ∀i = 0, …, μ − 1

\begin{aligned}θ_{i} & = −b_{μi} + γ̃_{i} − ∑_{j>i} b_{ji}γ̃_{j} , \\ γ̃_{i} & = −a_{μi} + θ_{i} − ∑_{j>i} a_{ji}θ_{j} . \end{aligned}

Hence, ∀i = 0, …, μ − 1,

−v(θ_{i}) ≤ max(0, sup_{j≥i} −v(γ̃_{j})) ,
−v(γ̃_{i}) ≤ max(0, sup_{j≥i} −v(θ_{j})) .

A fortiori, ∀i = 0, …, μ − 1,

−\frac{v(θ_{i})}{μ − 1} ≤ max(0, sup_{j≥i} −\frac{v(γ̃_{j})}{μ − j}) ,
−\frac{v(γ̃_{i})}{μ − 1} ≤ max(0, sup_{j≥i} −\frac{v(θ_{j})}{μ − j}) .

Hence, ∀i = 0, …, μ − 1, (2.2.5.5)

max(0, sup_{j≥i} −\frac{v(θ_{j})}{μ − j}) = max(0, sup_{j≥i} −\frac{v(γ̃_{j})}{μ − j}) = max(0, sup_{j≥i} −\frac{v(γ_{j})}{μ − j} − 1) .

Formula 2.2.5.2 follows from proposition 2.2.4 and (2.2.5.5) with i = 0. Q.E.D.

2.2.6. The results of this subsection will apply in particular to the calculation of v_{sp}(D_{1}) in the case of a divisorially valued function field (F, v) over K (with an adapted R_{v}-basis (D_{1}, …, D_{d}) of Der_{v}(F/K)). For that application, one completes v-adically, takes x\frac{d}{dx} = D_{1}, x = x_{1}, and replaces K by the field of constants F^{D}.

2.3. One-variable case with non trivially valued field of constants.

2.3.1. Let (G, v) be a divisorially valued function field over K, and let (Ĝ, v) be its completion. We now consider the case when F̂ is a finite extension of the completion of Ĝ(x) with respect to the v-adic Gauss norm (v(∑_{i} a_{i}x^{i}) = min_{i} v(a_{i}), if ∑_{i} a_{i}x^{i} ∈ Ĝ[x]), and D is the Ĝ-linear derivation of F̂ such that Dx = 1. This situation arises for instance in the case of the completion of a divisorially valued function field (F, v) over K (with an adapted R_{v}-basis (D_{1}, …, D_{d}) of Der_{v}(F/K)), if we take D = D_{i}, i > 1, Ĝ = F̂^{D} (with the induced valuation v), and x = x_{i}.

We observe that in the present case for a F̂/Ĝ-differential module M, −v_{M,sp}(D) is (in valuative form) the v-adic “generic” radius of convergence of solutions of (M, ∇) [ABa]. This may be explained as follows. We consider a second copy (F̂^{(1)}, v^{(1)}) of (F̂, v), isomorphic via f ⟼ f^{(1)}, and the Taylor map T_{D} : f ⟼ ∑_{i}(D^{i}f)^{(1)}(T − x^{(1)})^{i}/i! of F̂ into F̂^{(1)}[[T − x^{(1)}]]. Let (Ω, v^{(1)}) be the completion of the algebraic closure of (F̂^{(1)}, v^{(1)}). Let B_{Ω}(Δ) denote the ring of bounded analytic Ω-valued functions on the “generic” disk Δ = {a ∈ Ω | v^{(1)}(a − x^{(1)}) > 0 }. The valuation v^{(1)} extends to a valuation on B_{Ω}(Δ), on setting v^{(1)}(f) = inf_{a∈Δ} v^{(1)}(f(a)).

It is clear that T_{D} takes values in B_{Ω}(Δ) and that it is an isometric embedding of (F̂, v) into (B_{Ω}(Δ), v^{(1)}), with T = T_{D}(x).

Then, −v_{M,sp}(D) is related to the radius of the maximum common disk of convergence of power series solutions of M ⊗F̂ B_{Ω}(Δ) in Ω[[x−x^{(1)}]]. In fact, if \underline{e} is an F̂-basis of M and D^{i}\underline{e} = \underline{e}G_{i}, with G_{i} ∈ M_{μ×μ}(F̂), a fundamental matrix solution ∑_{i} G_{i}^{(1)}(x − x^{(1)})^{i}/i!, converges “precisely” for v(x − x^{(1)}) bigger than

− lim inf_{i→∞} v^{(1)}(G_{i}^{(1)})/i = − lim inf_{i→∞} v(G_{i})/i = lim sup_{i→∞} −v(G_{i})/i .

By remark 2.1.3.2,

−v_{M,sp}(D) = max(0, − lim inf_{i→∞} v^{(1)}(G_{i}^{(1)})/i) .

Let

(2.3.1.1)  Λ = D^{μ} − ∑_{i=0}^{μ−1} γ_{i}D^{i} ∈ F̂[D] .

For any m ≥ μ, let Λ_{m} be defined, as in (2.2.2), by

(2.3.1.2)  Λ_{m} = D^{m} − ∑_{j=0}^{μ−1} γ_{m,j}D^{j} ∈ F̂[D]Λ .

We set, as in (2.2.3),

(2.3.1.3)  σ := max(0, max_{j=0,...,μ−1} −\frac{v(γ_{j})}{μ − j}) .

Proposition 2.3.2 ^{(1)}. We have

v(γ_{m,j}) ≥ −σ(m − j) .

Therefore, the solutions of Λ at the generic point converge for v(x) > −σ. If σ > 0 the generic radius of convergence of solutions of Λ is precisely −σ.

Proof. The first part of the statement follows recursively from the calculation

γ_{m+1,j} = γ_{m,j−1} + γ_{m,μ−1}γ_{j} + D(γ_{m,j}) .

(1) Proposition 2.3.2 may be regarded as an analogue of Young’s theorem [DGS, VI, Lemma 2.1] for an equicharacteristic valuation.

Let us now follow the reasoning of [DGS, loc.cit.], assuming σ > 0. For a cyclic F̂-basis \underline{e} of the left Diff(F̂/Ĝ)-module M = Diff(F̂/Ĝ)/Diff(F̂/Ĝ)Λ, we have D\underline{e} = \underline{e}G, with

G = ( \begin{matrix}0 & 0 & 0 & … & γ_{0} \\ 1 & 0 & 0 & … & γ_{1} \\ & … & & … \\ 0 & & & … & γ_{μ−2} \\ 0 & 0 & … & 1 & γ_{μ−1} \end{matrix} ) .

Let γ ∈ Ĝ be such that v(γ) = σ and let

H = ( \begin{matrix}1 & & & & & 0 \\ & 1/γ \\ & & . & . & . \\ 0 & & & & & 1/γ^{μ−1} \end{matrix} ) .

We replace the basis \underline{e} by \underline{e}′ = \underline{e}H. Then

D^{i}\underline{e}′ = \underline{e}′T_{i} ,

with

T_{1} =: γW = γ ( \begin{matrix}0 & 0 & 0 & … & γ_{0}/γ^{μ} \\ 1 & 0 & 0 & … & γ_{1}/γ^{μ−1} \\ & … & & … & … \\ 0 & & & 0 & γ_{μ−2}/γ^{2} \\ 0 & 0 & … & 1 & γ_{μ−1}/γ \end{matrix} ) .

We check that

T_{i} ≡ γ^{i}W^{i} mod γ^{i−1} ,

for every i ≥ 1. The eigenvalues of W are v-integral and at least one of them, say λ, is a unit. Let u⃗ ∈ F̂^{μ} be an eigenvector of eigenvalue λ such that v(u⃗) = 0. We then have

T_{i}u⃗ ≡ γ^{i}W^{i}u⃗ = γ^{i}λ^{i}u⃗ modγ^{i−1} ,

so that

v(T_{i}u⃗) = i v(γ)

which implies that the solution

∑_{i} \frac{T_{i}u⃗}{i!}x^{i}

can only converge for v(x) > −σ. Q.E.D.

Corollary 2.3.3. Let M be the F̂/Ĝ-differential module defined by M = Diff(F̂/Ĝ)/Diff(F̂/Ĝ)Λ, with Λ given by (2.3.1.1). Then

ρ_{v}(M) = −v_{sp}(∇(D)) = max(0, max_{j=0,...,μ−1} −\frac{v(γ_{j})}{μ − j}) .

In particular, ρ_{v}(M) is a non-negative rational number of denominator at most μ. Let \underline{e} be any F̂-basis of M. The function

n ⟼ |T_{v,\underline{e}}(∇(D^{n})) + ρ_{v}(M) n| ,

is bounded.

2.4 Several variables again.

We are ready to synthetize the estimates of the previous sections.

Proposition 2.4.1. Let (F, v) be a divisorially valued function field (or a field of formal functions) over K and M, M_{1}, M_{2} be F/K-differential modules. Then i) ρ_{v}(F, d_{F/K}) = 0; ii) ρ_{v}(M^{∨}) = ρ_{v}(M); iii) ρ_{v}(M_{1} ⊗_{F} M_{2}) ≤ max(ρ_{v}(M_{1}), ρ_{v}(M_{2})); iv) ρ_{v}(Hom_{F}(M_{1}, M_{2})) ≤ max(ρ_{v}(M_{1}), ρ_{v}(M_{2}));

v) if

0 ⟶ M_{1} ⟶ M ⟶ M_{2} ⟶ 0

is an exact sequence of differential modules, then

ρ_{v}(M) = max(ρ_{v}(M_{1}), ρ_{v}(M_{2})) .

vi) If ρ_{v}(M_{1}) ≠ ρ_{v}(M_{2}) we have equality in iii) and iv). If M is of dimension μ as a F-vector space, ρ_{v}(M) is a non-negative rational number of denominator at most μ. Let \underline{e} be any F-basis of M and (D_{1}, …, D_{d}) be an adapted basis of Der_{v}(F/K). The function

(2.4.1.1)  n ⟼ |ρ_{v}(M) n + min ; ∑ α_{i}=n T_{v,\underline{e}}(∇(\underline{D}\underline{α}))| ,

is bounded.

Proof. The statements i) to vi) follow directly from lemmata 1.3-1.6. The last part of the lemma is a consequence of (2.2.4), (2.3.3) and of remark 1.9.2. Q.E.D.

Remark 2.4.1.2. It will be proven in corollary 3.2 below that we may always take D = D_{1} in (2.4.1.1).

Corollary 2.4.2. M is regular at v if and only if ρ_{v}(M) = 0.

Proof. This is clear from the characterization (I.3.3.4) of regularity. Q.E.D.

Proposition 2.4.3. Let (F, v)/(G, w) be an extension of divisorially valued function fields (or fields of formal functions), of relative ramification e = e(v/w). Let N be a G/K-differential module. The Poincaré-Katz rank of the F/Kdifferential module N_{F} is given by ρ_{v}(N_{F}) = eρ_{w}(N).

Proof. By (I.2.4.ii), we may choose an adapted basis (D_{1}, …, D_{d}) of Der_{v}(F/K), with (D_{1|G}, …, D_{r|G}) an adapted basis of Der_{w}(G/K), while D_{i|G} = 0, for i = r + 1, …, d. We write _{G}N_{F} for _{G}(N_{F}); then we have v_{N_{F} ,sp}(∇_{F}(D_{i})) = e v_{_{G}N_{F} ,sp}(∇_{F}(D_{i})). Since N is a submodule of _{G}N_{F}, v_{_{G}N_{F} ,sp}(∇_{F}(D_{i})) ≤ v_{N,sp}(∇(D_{i|G})). On the other hand, ∇_{F}(D_{i}) is obtained from ∇(D_{i|G}) ⊗_{K} 1_{F} + 1_{N} ⊗_{K} D_{i} as in ii) of lemma 1.4, where the field G plays now the role played there by F. Then v_{_{G}N_{F} ,sp}(∇_{F}(D_{i})) ≥ min(v_{N,sp}(∇(D_{i|G})), v_{_{G}F,sp}(D_{i})). We note that v_{_{G}F,sp}(D_{i}) = 0. Therefore, for i ≤ r v_{N,sp}(∇(D_{i|G})) ≤ 0 implies that

v_{N_{F} ,sp}(∇_{F}(D_{i})) = ev_{N_{F} ,sp}(∇(D_{i|G})) ,

while for i > r, v_{N,sp}(∇(D_{i|G})) = v_{N,sp}(0) = +∞ implies

v_{_{G}N_{F} ,sp}(∇_{F}(D_{i})) = v_{N_{F} ,sp}(∇_{F}(D_{i})) = 0 .

Since

ρ_{v}(N_{F}) = max_{i=1,...,d} −v_{N_{F} ,sp}(∇_{F}(D_{i}))

and

ρ_{w}(N) = max_{i=1,...,r} −v_{N,sp}(∇(D_{i|G})) ,

this concludes the proof. Q.E.D.

Proposition 2.4.4. Assume that [F : G] is finite, and let M be a F/Kdifferential module. The Poincaré-Katz rank of the G/K-differential module GM is given by ρ_{w}(_{G}M) = max_{v|w} e(v/w)^{−1}ρ_{v}(M).

Proof. This follows from (I.3.1.3). Q.E.D.

2.5. The Poincaré-Katz rank of a connection at a divisor.

Let (X, Z) be a model of a divisorially valued function field (F, v). Let E be a F/K-differential module, and let (E, ∇) be a model of E on X \ Z, i.e. a O_{X\Z} -coherent module with integrable connection whose generic fiber gives rises to E.

Definition 2.5.1. We define the rank of (E, ∇) at Z as the number ρ_{Z}(∇) := ρ_{v}(E) .

Proposition 2.5.2. Assume that (x_{1}, …, x_{d}) is an adapted system of coordinates for (X, Z). Then ρ_{Z}(∇) = ρ if and only if there exists a coherent extension Ẽ ⊂ j_{∗}E of E on X and an integer c such that for any n, Diff_{X/K,Z}^{n} Ẽ ⊂

x_{1}^{−[nρ]−c} Ẽ.

Proof. From (2.4.1.1) , the proof is parallel to that of (I.3.4.2). We do not repeat it.

Proposition 2.5.3. Let (X, Z) be a model and (E, ∇),(E_{1}, ∇_{1}),(E_{2}, ∇_{2}) be coherent sheaves with integrable connections on X \ Z.

i) If E sits in a horizontal exact sequence 0 ⟶ E_{1} ⟶ E ⟶ E_{2} ⟶ 0 , then ρ_{Z}(∇) = max(ρ_{Z}(∇_{1}), ρ_{Z}(∇_{2})) ; ii) ρ_{Z}((E_{1}, ∇_{1}) ⊗ (E_{2}, ∇_{2})) ≤ max(ρ_{Z}(∇_{1}), ρ_{Z}(∇_{2})) and ρ_{Z}(\underline{Hom}((E_{1}, ∇_{1}), (E_{2}, ∇_{2}))) ≤ max(ρ_{Z}(∇_{1}), ρ_{Z}(∇_{2})) with equality if ρ_{Z}(∇_{1}) ≠ ρ_{Z}(∇_{2}).

Proof. This follows from (2.4.1).

Proposition 2.5.4. Let (X, Z)⟶^{f}(Y, W) be a morphism of models such that f(Z) ⊂ W. Let (E, ∇) be a coherent sheaf with integrable connection on Y \ W. Then ρ_{Z}(f^{∗}∇) ≤ e_{f}.ρ_{W} (∇), with equality if f is log-dominant.

Proof. Starting from (2.5.2), the proof of the inequality is parallel to that of (I.3.4.4). We do not repeat it. The equality for a log-dominant morphism is a translation of (2.4.3). Q.E.D.

Corollary 2.5.5. The Poincaré-Katz rank at Z is local at η_{Z} for the étale topology on X.

N.B. 2.5.5.1. In order to obtain a notion local at η_{Z} for the fppf topology on X, we should normalize the Poincaré-Katz rank at Z on any fppf neighborhood f : U ⟶ X of η_{Z}, by dividing it by e_{f}.

Proposition 2.5.6. Let X⟶^{f}Y be a finite morphism, and let W be a smooth connected divisor of Y. Let us assume that f^{−1}W = ∑ e_{i}Z_{i}, where Z_{i} are smooth connected divisors of X, and that f is étale on X \ ⋃ Z_{i}. Let (E, ∇) be a coherent sheaf with integrable connection on X \ ⋃ Z_{i}. Then ρ_{W} (f_{∗}∇) ≤ max e_{i}^{−1} ρ_{Z_{i}}(∇).

Proof. Let w be the valuation of κ(Y) attached to W. Since f is finite, each valuation v of κ(X) above w is attached to some Z_{i}. The result then follows from (2.4.4). Q.E.D.

2.5.7. Let X be a smooth connected variety over K, F = κ(X). Let j : X ⟶ X′ be a dominant open immersion of X in a normal K-variety X′, and let Z′ be a subvariety of ∂X = X′ \ X. If (E, ∇) is a coherent sheaf with integrable connection on X, we define the rank of (E, ∇) at Z′ as the number ρ_{Z′}(∇) := max(ρ_{v}(E_{η_{X}})) , where v runs among the divisorial valuations of F defined by the irreducible components of Z′ of codimension 1 in X′. If codim_{X′}Z′ ≥ 2 (in particular, if Z′ = ∅) ρ_{Z′}(∇) = 0.

The following result generalizes (I.5.4) in the irregular case, as well as (2.5.4) in the confluent situation.

Theorem 2.5.8. Let P be a closed point of X′ and assume that every irreducible component of ∂X contains P. Let \overline{C}⟶^{h}X′ be a morphism from a smooth curve \overline{C}, such that C = h^{−1}(X) is the complement of a point Q with h(Q) = P in \overline{C}. For any component Z′_{i} of ∂X of codimension one in X′, we denote by e_{i} the valuation of the image in O_{\overline{C},Q} of a local equation of Z′_{i} in O_{X′,P}. Then ρ_{Q}(h^{∗}∇) ≤ ∑_{i} e_{i} ρ_{Z′_{i}}(∇).

Proof. We may assume that h is a closed immersion. We may replace X′ by any affine neighborhood of P, and assume that ∂X ∩ \overline{C} = \{P \}. Let \{x_{1}, x_{2} …\} be a finite set of generators of the ideal of ∂X in O(X′) such that x_{i} ≠ 0 on C. We may and shall replace ∂X by the reduced divisor defined by the ideal √(Πx_{i}). There is a closed subset T ⊂ ∂X such that Z̃ = ∂X \ T is a disjoint union of smooth divisors Z_{i} in X′ \ T defined by the equation x_{i} = 0. We denote by j′ (resp. j̃, resp. k) the open immersion of X̃ = X′ \ T into X′ (resp. X into X̃, resp. C into \overline{C}), so that j = j′ ∘ j̃. If P ∉ T, the result follows from (2.5.4). In order to settle the case P ∈ T, we follow the method of remark I.5.6, using jets with logarithmic poles (cf. App. B.6). We have a map of functoriality

(2.5.8.1)  h^{∗}j′_{∗}P_{X̃/K,Z̃}^{n} ⟶ P_{\overline{C}/K,Q}^{n}.

Let Ẽ ⊂ k_{∗}E be a coherent extension of E on X̃ as in (2.5.2). Then for all n, the mapping ˜∇ gives rise to a composite mapping

h^{∗}j′_{∗} ˜∇ : h^{∗}j′_{∗}Ẽ ⟶ (Πx_{i}^{−[nρ_{Z′_{i}} (∇)]−c})h^{∗}j′_{∗}Ẽ ⊗_{O\overline{C}} h^{∗}j′_{∗}P_{X̃/K,Z̃}^{n}
⟶ (x^{−[n∑_{i} e_{i} .ρ_{Z′_{i}} (∇)]−c′})h^{∗}j′_{∗}Ẽ ⊗_{O\overline{C}} P_{\overline{C}/K,Q}^{n}

for some c′ independent of n. We conclude by (2.5.2) again. Q.E.D.

Remark 2.5.9. This theorem shows in particular that in the integrable case, no confluence occurs at the crossing of two singular divisors, if the connection is regular along each one of them.

§3 Some consequences of the Turrittin-Levelt theorem.

Let us recall the well-known Turrittin-Levelt theorem, in the form given in [B1] (see also [Le],[R]).

Theorem 3.1. Let (Ĝ, w) be a field of formal functions of d variables over K, (D_{1}, …, D_{d}) be an adapted R_{w}-basis of Der_{w}(Ĝ/K) and M be a Ĝ/Ĝ^{D_{1}}- differential module of rank μ over Ĝ. There exists a finite extension (F̂, v) of (Ĝ, w), with ramification index e = e(v/w) dividing μ!, and a well-determined finite subset \{\overline{P}_{1}, …, \overline{P}_{r}\} of F̂/(𝔪_{v} + Z), such that for every choice of representatives P_{i} ∈ F̂ of \overline{P}_{i}, we have the following decomposition of left F̂[D_{1}]-modules,

(3.1.1)  MF̂ = ⊕_{i=1}^{r}F̂ ⊗F̂^{D_{1}} Ker_{MF̂} (D_{1} − P_{i})^{μ}

(we still denote by D_{j} the unique extension of D_{j} ∈ Der_{w}(Ĝ/K) to a derivation in Der_{v}(F̂/K)).

If moreover M is a Ĝ/K-differential module, then the increasing filtration of MF̂ by the F̂[D_{1}]-submodules

(3.1.2)  F_{σ}(MF̂) = ⊕_{v(P_{i})≥−σ} F̂ ⊗F̂^{D_{1}} Ker_{MF̂} (D_{1} − P_{i})^{μ} ,

for σ ∈ Q_{≥0}, is in fact a filtration by F̂/K-differential submodules, and descends to a filtration (the slope filtration) F_{•}(M) of M by Ĝ/K-differential submodules, such that (F\frac{σ}{e} (M))F̂ = F_{σ}(MF̂).

Moreover, for any σ ∈ Q_{≥0}, F_{σ}(M) consists of those m ∈ M for which the monic operator of minimal order Γ_{m} = D_{1}^{n} − ∑_{j=0}^{n−1} γ_{j}D_{1}^{j}, with γ_{j} ∈ Ĝ such that Γ_{m}m = 0 satisfies

(3.1.3)  \frac{w(γ_{j})}{n − j} ≥ −σ .

Remark 3.1.3.1. In an estimate like (3.1.3), the number n indicates the degree of the monic polynomial involved. Note that the estimates (3.1.3) just mean that the Gauss valuation

inf_{v(y)=−σ} v(y^{n} − ∑_{k=0}^{n−1} γ_{k}y^{k})

on the commutative associated graded ring F̂[y] of F̂[D_{1}] (filtered by the order of differential operators in D_{1}) is ≥ −nσ. So, if two monic polynomials in F̂[D_{1}] satisfy that estimate, so does their product. Also, by the Gauss lemma, if that estimate holds for an operator Λ, it also holds for any monic factor of it.

Proof. We refer to [B1]. The only new point with respect to loc. cit. is the stability of F_{σ}(M) under any ∂ ∈ {D_{2}, …, D_{d}}. This is easily proven as follows.

Let m ∈ F_{σ}(M) and Γ_{m}m = 0 be as in (3.1.3). Then 0 = ∂Γ_{m}m = Γ_{m}∂m+ ∂(Γ_{m})m, where ∂(Γ_{m}) = − ∑_{j=0}^{n−1} ∂(γ_{j})D_{1}^{j} so that Γ_{m}∂m = −∂(Γ_{m})m := p ∈ F_{σ}(M) . Then Γ_{p}Γ_{m}∂m = 0 and Γ_{p}Γ_{m} still satisfies the estimate (3.1.3). Then so does Γ_{∂m} | Γ_{p}Γ_{m}, hence ∂m ∈ F_{σ}(M).

In the notation of (3.1.1), for ν = 0, 1, …, we set

K_{i}^{(ν)} = Ker_{MF̂} (D_{1} − P_{i})^{ν} (a finite dimensional κ(Z)−space),
M_{i}^{(ν)} = F̂ ⊗_{κ(Z)} K_{i}^{(ν)} .

Lemma 3.1.4. Any element m ∈ MF̂ such that (D_{1} − P_{i})m ∈ M_{i}^{(ν)} belongs to M_{i}^{(ν+1)}.

Proof. It is clear that m belongs to the factor F̂ ⊗_{κ(Z)} Ker_{MF̂} (D_{1} − P_{i})^{μ} of the Turrittin decomposition. There exists a F̂-basis \underline{e} = (\underline{e}^{(1)}, …, \underline{e}^{(r)}), \underline{e}^{(j)} = (e_{1}^{(j)}, …, e_{s(j)}^{(j)}), of this factor, such that

D_{1}e_{1}^{(j)} = P_{i}e_{1}^{(j)} and D_{1}e_{h}^{(j)} = P_{i}e_{h}^{(j)} + e_{h−1}^{(j)} ,

for every 1 < h ≤ s(j) (≤ μ). Let us write m = ∑_{h,j} a_{h,j}e_{h}^{(j)}, with a_{h,j} ∈

κ(Z)((t)), (D_{1} − P_{i})m = D_{1}(a_{s(j),j})e_{s(j)}^{(j)} + ∑_{h<s(j)}(a_{h+1,j} + D_{1}(a_{h,j}))e_{h}^{(j)}. Then by assumption, a_{h+1,j} + D_{1}(a_{h,j}) = 0 for h > ν. By descending induction, we get a_{h,j} = 0 for h > ν + 1 and a_{ν+1,j} ∈ κ(Z): indeed D_{1}(a_{h,j}) ∈ κ(Z) only if D_{1}(a_{h,j}) = 0 (D_{1} = \frac{t}{e} \frac{d}{dt}).

Corollary 3.2. In the notation of (3.1),

v_{M,sp}(D_{1}) = e(v/w)^{−1} inf_{i=1,...,r} v(P_{i}) .

Let us assume moreover that M is a Ĝ/K-differential module. We have v_{M,sp}(D_{1}) ≤ v_{M,sp}(D_{j}) , for j = 2, …, d.

Proof. In order to prove the first statement, it is enough (according to (1.6)) to establish the formula

v_{MF̂,sp}(D_{1}) = inf_{i=1,...,r} v(P_{i})

or even the analogous formula with MF̂ replaced by a single summand (in (3.1.1)) M_{i}^{(μ)} = F̂ ⊗F̂^{D_{1}} Ker_{MF̂} (D_{1} − P_{i})^{μ}. The estimate (3.1.3) holds for (D_{1} − P_{i}), hence for (D_{1} − P_{i})^{μ}; our formula then follows from (2.2.4) and (2.2.5).

As for the second statement, we again extend the scalars to F̂, so to have the decomposition (3.1.1), and to deal with a single factor MF̂ = M_{i}^{(μ)}, with basis as in the proof of (3.1.4). We fix a derivation ∂ ∈ {D_{2}, …, D_{d}} and, for e ∈ MF̂, we let Λ_{e} = ∂^{n} + ∑_{k=0}^{n−1} λ_{k}∂^{k}, with λ_{k} ∈ F̂, be the monic operator of minimal order such that Λ_{e}e = 0. Lemma 1.5 shows that

min(0, v_{MF̂,sp}(∂)) = min(0, inf_{j,h} v_{F̂[∂]e_{h}^{(j)},sp}(∂)) ,

where F̂[∂]e_{h}^{(j)} is the cyclic F̂[∂]-submodule of MF̂ generated by e_{h}^{(j)}. According to (2.3.2) it will be enough to prove, for k = 0, …, n − 1, the estimate

(3.2.1)  \frac{v(λ_{k})}{n − k} ≥ \{ \begin{matrix}v(P_{i}) & if v(P_{i}) < 0, \\ 0 & if v(P_{i}) ≥ 0, \end{matrix}

for Λ_{e}, for any basis element e = e_{h}^{(j)} ∈ M_{i}^{(μ)}.

Remark 3.1.3.1 applies also to estimates of the type (3.2.1). So, if two monic polynomials in F̂[∂] satisfy them, so does their product, and if the estimates hold for the operator Λ, they hold for any monic factor of it.

We start with e_{1}^{(j)} ∈ M_{i}^{(1)} and set Λ = Λ_{e_{1}^{(j)}}.

D_{1}Λ = ΛD_{1} + Λ′ , Λ′ = ∑_{k=0}^{n−1} D_{1}(λ_{k})∂^{k} ,

and

ΛP_{i} = P_{i}Λ + Λ̃ , Λ̃ = n∂(P_{i})∂^{n−1} + ….

We have

0 = D_{1}Λe = ΛD_{1}e + Λ′e = P_{i}Λe + Λ̃e + Λ′e = (Λ̃ + Λ′)e .

Therefore Λ′ = −Λ̃, that is

(3.2.2)  ∑_{k=0}^{n−1} D_{1}(λ_{k})∂^{k} = [P_{i}, ∂^{n} + ∑_{k=0}^{n−1} λ_{k}∂^{k}] .

This implies (3.2.1) for Λ.

We now consider the following statements: (∗)_{h}: Λ_{e_{1}^{(j)}}e_{h}^{(j)} ∈ M_{i}^{(h−1)}, for j = 1, …, r;

(∗∗)_{h}: for every non-zero basis element e ∈ M_{i}^{(h)} the polynomial Λ_{e} satisfies the estimate (3.2.1).

Note that (∗)_{1} is obvious, while (∗∗)_{1} was proved above. Let us assume by induction that (∗)_{<h} and (∗∗)_{<h}, for some h > 1, hold. For Λ = Λ_{e_{1}^{(j)}} and Λ′, Λ̃ as before, we compute

D_{1}Λe_{h}^{(j)} = ΛD_{1}e_{h}^{(j)} + Λ′e_{h}^{(j)} = Λ(P_{i}e_{h}^{(j)} + e_{h−1}^{(j)}) + Λ′e_{h}^{(j)} =
P_{i}Λe_{h}^{(j)} + Λ̃e_{h}^{(j)} + Λe_{h−1}^{(j)} + Λ′e_{h}^{(j)} = P_{i}Λe_{h}^{(j)} + Λe_{h−1}^{(j)} .

So, (D_{1} − P_{i})Λe_{h}^{(j)} ∈ M_{i}^{(h−2)}; by lemma 3.1.4, Λe_{h}^{(j)} ∈ M_{i}^{(h−1)}. This establishes (∗)_{h}.

To prove (∗∗)_{h} we observe that, since Λe_{h}^{(j)} ∈ M_{i}^{(h−1)}, the estimates (3.2.1) hold for Λ_{Λe_{h}^{(j)}}. Hence they hold for Λ_{Λe_{h}^{(j)}}Λ and for its monic factor Λ_{e_{h}^{(j)}}. Q.E.D.

Corollary 3.3. The slope filtration of any Ĝ/K-differential module M is independent of the auxiliary choice of an adapted R_{w}-basis of Der_{w}(Ĝ/K): F_{σ}(M) is the maximal Ĝ/K-differential submodule of M of Poincaré-Katz rank ≤ σ. One has ρ(F_{σ}(M)) = σ if and only if the graded piece Gr_{σ}(M) is not 0.

Proof. We may assume that Ĝ = F̂. It follows from (3.2) that ρ_{w}F_{σ}(M) = σ. On the other hand, it is clear that any submodule of F_{σ}(M) which projects nontrivially on some factor F̂ ⊗F̂^{D_{1}} Ker_{MF̂} (D_{1} − P_{i})^{μ} of the decomposition (3.1.2) with v(P_{i}) = −σ has Poincaré-Katz rank at least σ. Whence the result.Q.E.D.

Remark 3.4. The elements \overline{P}_{1}, …, \overline{P}_{r} of F̂/(𝔪_{v}+Z) introduced in (3.1) depend on D_{1}, as may already be seen with the connection in two variables with formal solution e^{y/x}.

The next result shows that not only the slope filtration is horizontal (with respect to every derivation), as we saw in (3.1), but the Turrittin-Levelt decomposition itself is horizontal.

Proposition 3.5. M_{i}^{(μ)} is stable under D_{1}, D_{2}, …, D_{d}.

Proof. We first prove that for all ν = 0, 1, … and ∂ ∈ {D_{1}, …, D_{d}}, one has

(ν)  ∂M_{i}^{(ν)} ⊆ M_{i}^{(ν+1)} .

This is obvious for D_{1}. Let ∂ ∈ {D_{2}, …, D_{d}}. The statement (ν) is trivial for ν = 0. We proceed by induction: assume (ν) holds for ν < ν_{0} (ν_{0} ≥ 1) and let m ∈ K_{i}^{(ν_{0})}. So, (D_{1} − P_{i})^{ν_{0}}m = 0 and (D_{1} − P_{i})m ∈ K_{i}^{(ν_{0}−1)}. Then (D_{1} − P_{i})∂m = ∂(D_{1} − P_{i})m + (∂P_{i})m ∈ M_{i}^{(ν_{0})}, by the induction assumption. Due to (3.1.4), this implies ∂M_{i}^{(ν_{0})} ⊆ M_{i}^{(ν_{0}+1)}, and this achieves the proof of (3.5) since M_{i}^{(ν)} = M_{i}^{(ν+1)} for ν ≥ μ.Q.E.D.

Remark 3.6. The elements \overline{P}_{i} ∈ F̂/(𝔪_{v} + Z) need not belong to Ĝ/(𝔪_{w} + Z), as is shown by the example of the direct image of the rank one connection with formal solution e^{y/x} on the plane deprived from the axes, under the finite etale morphism (x, y) ↦ (x, y^{2}); indeed, in this example t = x and \overline{P}_{i} ≡ ±√y \frac{1}{t}.

§4. Newton polygons.

4.1 Classical Newton polygons.

4.1.1. Let us revert to the situation of (2.2.1), and write γ_{i}(x) = − ∑ γ_{ij}x^{j} for i ≤ μ (γ_{μ}(x) = −1), so that Λ = ∑ γ_{ij} x^{j} \frac{d^{i}}{dx^{i}}.

We recall that the Newton polygon NP (Λ) of Λ is the convex hull of the quadrants {u ≤ i, v ≥ j − i + μ | γ_{ij} ≠ 0} ⊂ R^{2}. The vertices of NP (Λ) are among the points (i, ord_{x}(γ_{i}) − i + μ), 0 ≤ i ≤ μ. The vertical edge of NP (Λ) is {u = μ, v ≥ 0}. If we write x^{μ}Λ = (x\frac{d}{dx})^{μ} − ∑_{i=0}^{μ−1} θ_{i}(x)(x\frac{d}{dx})^{i}, θ_{μ}(x) = 1, as in (2.2.5.1), then NP (Λ) can also be described as the convex hull of the quadrants {u ≤ i, v ≥ ord_{x}(θ_{i})} , 0 ≤ i ≤ μ. The height of NP (Λ) (i.e. the maximal distance between the ordinates of the vertices) is called the irregularity ir(Λ) of Λ. All finite slopes of NP (Λ) are ≥ 0 and the maximal one is max(0, max_{j=0,...,μ−1}(−\frac{v(γ_{j})}{μ−j} −1)). The points (0, ir(Λ)) and (μ, 0) belong to the boundary of NP (Λ) and the latter is always a vertex.

Proposition 4.1.2. NP (M) := NP (Λ) is independent of the choice of the cyclic vector e. The finite slopes of NP (M) are the indices of the (Turrittin) slope filtration 3.1 (in particular, the maximal finite slope is ρ_{v}(M)).

Proof. See [Mal, III 1.5] or [B1, prop.3], and (3.1.3) (in particular (2.2.4)). It is clear that NP (Λ) does not depend on the choice of the uniformizing

parameter x either.

Proposition 4.1.3. i) Let 0 ⟶ M_{1} ⟶ M ⟶ M_{2} ⟶ 0 be an exact sequence of K((x))/K-differential modules. Then NP (M) is the algebraic sum NP (M_{1}) + NP (M_{2}). ii) NP (M^{∨}) = NP (M).

Proof. For i), see [Mal, III 1.6] or [B1, lemma 5]. For ii), notice that if M is the differential module associated with (x\frac{d}{dx})^{μ} − ∑_{i=0}^{μ−1} θ_{i}(x)(x\frac{d}{dx})^{i}, then M^{∨} is associated with (x\frac{d}{dx})^{μ} − ∑_{i=0}^{μ−1}(−1)^{μ−i}(x\frac{d}{dx})^{i} θ_{i}(x). Q.E.D.

4.1.4. Here the field K is not necessarily algebraically closed. Let Q be a closed point on a smooth connected curve C defined over K. Let (E, ∇) be a coherent O_{C\Q}-module with (automatically integrable) connection. We equip κ(C) with the valuation ord_{Q} and set NP_{Q}(∇) = NP (E_{η_{C}} ⊗ κ̂(C)).

The rationality of the point Q does not matter here: if \overline{Q} is any geometric point of C over Q, \overline{C} is the connected component of C containing \overline{Q}, and (\overline{E}, \overline{∇}) is the extension of (E, ∇) to \overline{C}, we have NP_{Q}(∇) = NP\overline{Q}(\overline{∇}).

4.2 Variation of the Newton polygon in a family of integral curves.

4.2.1. Let X, S be K-varieties with S normal, and let Z ⊂ X be a reduced divisor in X. We consider a diagram of the type (I.1.3.1)

(4.2.1.1)  \begin{matrix}X & ↩^{i} & Z \\ f ↓ & ↙ g \\ S \end{matrix}

where

i) i is the closed immersion of Z; ii) f is smooth of pure relative dimension 1; iii) g is quasi finite.

Let E be a locally free O_{X\Z}-module of finite rank μ and

∇_{(X\Z)/S} : E ⟶ Ω_{(X\Z)/S}^{1} ⊗ E

be a relative connection on E.

Proposition 4.2.2. In this situation, the function on Z

N_{f} = N_{f}(∇, •) : Q ⟼ N_{f}(Q) = NP_{Q}(∇_{|f^{−1}(g(Q))\g^{−1}(g(Q))})

is constructible. A fortiori, the numerical functions

ρ_{f} : Q ⟼ ρ_{f}(Q) = ρ_{Q}(∇_{|f^{−1}(g(Q))\g^{−1}(g(Q))})

on Z and

ρ : s ⟼ ρ_{Z_{s}}(∇_{|(X\Z)_{s}/κ(s)})

on S, are constructible.

Proof. To prove the first assertion, it is enough to show that, for any point Q in Z, the condition N(P) = N(Q), for P in the closure Q^{−} of Q in Z, is open in Q^{−}. We will show, equivalently, that there exists a neighborhood U of Q in Q^{−} such that N(P) = N(Q) for any P ∈ U. After replacing S by the smooth part of g(Q^{−}) ⊂ S and localizing around the components of Z, we are reduced to the case where X, S are smooth connected Z is connected and étale over S, Q = η_{Z} and g(Q) = η_{S}; then X_{η_{S}} is a smooth connected curve defined over κ(η_{S}) and κ(X_{η_{S}}) = κ(X). We may replace X by any affine connected neighborhood of η_{Z}. This allows us to assume that:

a) X, Z, S are smooth connected;

b) (cf. (I.1.3.1.2)) there is a transversal derivation ∂ = ∂_{t,f} ∈ Γ(X, \underline{Der}_{X/K}) such that f is a family of integral curves of ∂; in particular, by (I.1.4.1.2), we may assume that ∂ is the image of a section ∂_{s} ∈ Γ(X_{s}, \underline{Der}_{X_{s}/κ(s)}) that generates (\underline{Der}_{X_{s}/κ(s)})_{P} for any P ∈ X_{s}.

We consider the restriction (E, ∇)_{|X_{η_{S}} \{η_{S}}} =: (E′, ∇′), a (X_{η_{S}} \{η_{S}})/κ(S)- connection, and pick a generic cyclic vector m ∈ E′_{η_{X}}. Then there exists a divisor W ⊂ X, Z ⊄ W, such that m extends to a cyclic section m of E on X \ (Z ∪ W) w.r.t. ∂. Since η_{Z} ∉ Z ∩ W, we may replace X by X \ W, and we may also assume

c) there is a section m of E over X \ Z such that E is free with basis (m, ∇(∂)m, …, ∇(∂)^{μ−1}m).

Let us write the associated differential equation in the form ∂^{μ}m−∑_{j<μ} b_{j}∂^{j}m = 0, with b_{j} ∈ O(X \ Z). By further shrinking X around η_{Z}, we may assume that the functions t^{−ord_{Z}b_{j}}b_{j} extend to invertible functions on X. It is then clear that in this situation, the Newton polygon of ∇_{|f^{−1}(g(P))\g^{−1}(g(P))} is independent of P ∈ Z. The second assertion is obvious. As for the last assertion, we recall that it suffices to prove that ρ is constant on a dense open subset of S. We may then assume that Z⟶^{g}S is an étale covering, and we see that ρ(s) = sup_{g(Q)=s} ρ_{f}(Q). The last assertion follows from the fact that the supremum of a finite set of realvalued constructible functions is constructible. Q.E.D.

A function N with polygonal values on a topological space T lower semicontinuous if, for any given polygon N_{0}, {t ∈ T |N(t) ⊂ N_{0}} is closed in T. If T is a noetherian scheme and N is constructible, N is lower semicontinuous as soon as for any specialization Q ⟶ P in T, N(Q) ⊂ N(P).

Proposition 4.2.3. Assume moreover that Z⟶^{g}S is étale. Then the function N_{f} is lower semi-continuous. In particular, the function ρ_{f} : Z ⟶ (μ!)^{−1}Z_{≥0} is lower semi-continuous. If g is an étale covering, the function ρ is also lower semi-continuous.

Proof. The assertion about ρ follows from the corresponding assertion for ρ_{f}, due to the fact that the sup of (any set of) lower semi-continuous realvalued functions is lower semi-continuous. The assertion for ρ_{f} is an obvious consequence of the statement about N_{f}, considering that for all Newton polygons under consideration, (μ, 0) is a vertex and ρ_{f} is the last non infinite slope.

We already know that N_{f} is constructible. We must prove that if Q and P are points of Z, with P a specialization of Q, then N_{f}(P) ≤ N_{f}(Q). We may assume that the relative dimension of the specialization is 1. If the (reduced locally closed subscheme of S) g(Q^{−}) is normal, we may replace S by g(Q^{−}) and assume that Q = η_{Z}; then P is a smooth point of Z. On replacing S by an étale neighborhood of g(P), we may assume that (4.2.1.1) is a strict coordinatized tubular neighborhood of Q (I.1.3.1.2). The definition (4.1.4) shows that we may extend our picture to κ(g(P)), and then to the algebraic closure of κ(g(P)). This allows us to assume that g(P) = P is a closed point of the smooth curve S = Z (of equation t = 0) over K; f is a smooth retraction f : X ⟶ Z with geometrically connected fibers. As in the proof of (4.2.2), we let ∂ = ∂_{t,f} pick a cyclic section m of E on X \ (Z ∪ f^{−1}(P) ∪ W) w.r.t. ∂, for some divisor W of X \ (Z ∪ f^{−1}(P)). Let

Λ(m) = ∂^{μ}m − ∑_{j<μ} b_{j}∂^{j}m = 0

be the associated differential equation, with with b_{j} ∈ O(X \ (Z ∪ f^{−1}(P) ∪ W)). Let R = O_{Z,P}, x be a uniformizer of R and let η_{P} be the generic point of the fiber f^{−1}(P). Then (E_{η_{X}} , ∇_{η_{X}}) is a κ(X)/κ(S)-differential module and m is a cyclic vector for it. We equip κ(X) with the valuation v with valuation ring O_{X,η_{P}}, a discrete valuation of κ(X)/K extending the x-adic valuation of κ(S). Since ∇(∂)(E_{η_{P}}) ⊂ E_{η_{P}}, (2.3.2) shows that v(b_{j}) ≥ 0, for j = 0, …, μ, so that b_{j} ∈ O(X \ (Z ∪ W)). We replace X by X \ W and then by X_{P} = X ×_{K} R. We define a cyclic X_{P} /R-differential module F = O_{X_{P}} [∂]/O_{X_{P}} [∂]Λ, with the natural action ℵ(∂) of ∂. By construction, (E_{η_{X}} , ∇_{η_{X}}) ≅ (F_{η_{X_{P}}} , ℵ_{η_{X_{P}}}). By Gabber’s theorem [Ka4, 2.5.2], the semisimplifications of (F_{|f^{−1}(P)}, ℵ_{|f^{−1}(P)}) and (E_{|f^{−1}(P)}, ∇_{|f^{−1}(P)}) are also isomorphic. By (4.1.3), we conclude that N_{f}(P) = NP ((F_{|f^{−1}(P)}, ℵ_{|f^{−1}(P)})) = NP (Λ_{x=0}). The assertion is now clear in this particular case.

If g(Q^{−}) is not normal, we consider the natural morphism S′ ⟶ S from the normalization S′ of g(Q^{−}) to S, and let X′ = X ×_{S} S′. The consideration of the inverse image of ∇ on X′ \ Z′ reduces us to the former case. Q.E.D.

4.2.4 Remarks.

i) Example (I.3.4.8.1) shows that the rank can indeed drop.

ii) The condition that Z is étale over S (or, equivalently, that Z meets all the integral curves transversally) is essential in order to prevent confluence phenomena, as in the following examples.

4.2.5 Examples. We take X = the (x, y)-plane with the line x = 0 removed.

4.2.5.1.

Z = the divisor y = 0,

E = O_{X\Z} with the integrable connection ∇ whose formal solution is e^{1/xy}. Let us consider the family of lines C_{λ} of equation x−y = λ. Then ρ_{Z}((E, ∇)) = 1, ρ_{x=0}((E, ∇)_{|C_{λ}}) = 1 for λ ≠ 0, but ρ_{(0,0)}((E, ∇)_{|C_{0}}) = 2.

4.2.5.2. Here Z = the parabola y = x^{2}, E = O_{X\Z} with the integrable connection ∇ whose formal solution is e^{1/(y−x^{2})}.

For the family of horizontal lines C_{λ} : y = λ, ρ_{Z}((E, ∇)) = 1, ρ_{x=0}((E, ∇)_{|C_{λ}}) = 1 for λ ≠ 0, but ρ_{(0,0)}((E, ∇)_{|C_{0}}) = 2. (However C_{0} is not transversal to Z).

4.2.5.3. We fix two constants b, c, and take Z = the hyperbola xy = 1, L_{y} ∈ D := Diff((X \ Z)/S) = K[x, y, \frac{1}{x(1−xy)}, \frac{∂}{∂x}] is the differential oper-

ator

L_{y} := x(1 − xy)(\frac{∂}{∂x})^{2} + (c − (1/y + b + 1)xy)\frac{∂}{∂x} − b

attached to the hypergeometric function _{2}F_{1}(1/y, b, c; xy),

(E, ∇_{(X\Z)/S}) = the free O_{(X\Z)}-module of rank 2 with relative connection associated to the D-module D/DL_{y}. The restriction of ∇ to (X \ Z)_{y} is regular. But the differential operator

L_{0} = x(\frac{d}{dx})^{2} + (c − x)\frac{d}{dx} − b

attached to the Kummer function _{1}F_{1}(b, c; x) is irregular at infinity. This phenomenon could not occur for an integrable connection (E, ∇) on X \ Z (I.5.4). All this should be compared with [De, II, 1.23] which, as it stands, is false. (The mistake comes from the fact that his \overline{X}_{s} \ X_{s} is not necessarily reduced. The example also shows that the argument given in [loc.cit. 2.4, case 2], is wrong).

4.2.5.4. A simpler example of confluence is the elementary formula

e^{− \frac{1}{x}} = lim_{y→0}(1 − \frac{y}{x})^{\frac{1}{y}} = lim_{y→0}(x − y)^{\frac{1}{y}} x^{− \frac{1}{y}} .

In this example, L_{0} = x^{2} \frac{d}{dx} − 1 appears as a specialization of the generically regular relative differential operator L_{y} = x(x − y)\frac{d}{dx} − 1.

Proposition 4.2.6. In the situation of (4.2.1), assume moreover that S is smooth and g is étale (resp. an étale covering), so that both (X, Z) is a model. Assume moreover that ∇ comes from an (absolute) integrable (X \Z)/Kconnection. Then the maximal value of the function ρ_{f} (resp. ρ) is ρ_{Z}(∇).

Proof. The fact that ρ_{Z}(∇) is an upper bound comes from (2.5.8). The fact that ρ_{Z}(∇) = ρ_{η_{Z}} (∇_{|f^{−1}(g(η_{Z}))\g^{−1}(g(η_{Z}))}) follows from (3.2). Q.E.D.

Remark 4.2.7. The difference between the integrable and non-integrable cases is more striking if one considers the irregularity ir (i.e. the height of the Newton polygon) instead of the Poincaré-Katz rank ρ. Let us consider a diagram as in (4.2.1.1), with X, S, Z smooth and Z finite over S, and let E be again a locally free O_{X\Z}-module of finite rank μ endowed with a relative connection on ∇. Following [Me2, §4], let us consider the functions

Ψ : s ∈ S ⟼ ∑_{Q∈Z_{s}} ir_{Q}(∇_{(X\Z)_{s}/κ(s)}) and Φ : s ∈ S ⟼ Ψ(s) + μ · ♯Z_{s} .

P. Deligne proved (at least if dim S = 1) that Φ is lower semicontinuous (letter to N. Katz 1976). Examples like (4.2.5.3) (or rather its variant in which X = P^{1} × A^{1}, f = pr_{2}, Z = {0, ∞} × A^{1} ∪ hyperbola) show that Ψ may not be lower semicontinuous. If ∇ comes from an (absolute) integrable X/K-connection, it is known that Φ is lower semicontinuous and that Ψ is constructible [Me1]; moreover, B. Malgrange conjectures that Ψ is also lower semicontinuous (at least if dim S = 1). In this direction, Z. Mebkhout proved that if Φ is constant, then Z is étale over S [Me2, §4].

4.3 The case of several variables.

Definition 4.3.1. Let (F̂, v) be a field of formal functions over K, and let M̂ be a F̂/K-differential module of rank μ. The Newton polygon of M̂ is the convex polygon NP (M̂) ⊂ R^{2} characterized by the following properties:

i) NP (M̂) contains the quadrant {x ≤ μ, y ≥ 0}, and its extreme right vertex is (μ, 0), ii) for any σ > 0, the horizontal length of the edge of slope σ is dim Gr_{σ}(M̂), the dimension of the graded piece of index σ in the slope filtration of M̂ (3.3).

In particular, the slopes of NP (M̂) are the indices σ of the slope filtration for which Gr_{σ}(M̂) ≠ 0, and 0, ∞, and the maximal finite slope is ρ(M̂).

4.3.2. Due to (3.3), this is a well-posed definition, independent of the choice of any adapted basis of Der_{v}(F̂/K). However, if (D_{1}, …, D_{d}) is such a basis, let us set C = F̂^{D_{1}} so that F̂ = C((x)), for a parameter x of R̂_{v} such that D_{1}x = x; so D_{j}C ⊂ C and D_{j}x = 0, j = 2, …, d. Then NP (M̂) is the Newton polygon of the C((x))/C-differential module M_{/C} in the sense of (4.1.1). In particular, the vertices of NP (M̂) have integral coordinates.

Proposition 4.3.3. i) Let 0 ⟶ M̂_{1} ⟶ M̂ ⟶ M̂_{2} ⟶ 0 be an exact sequence of F̂/K-differential modules. Then NP (M̂) is the algebraic sum of NP (M̂_{1}) and NP (M̂_{2}). ii) NP (M̂^{∨}) = NP (M̂).

Proof. (4.3.2) reduces the statement to (4.1.3).

In the sequel, we denote by φ_{a,b} , for any a, b ∈ Q^{×}, the automorphism of R^{2} given by φ_{a,b}(x, y) = (ax, by).

Proposition 4.3.4. Let (F̂, v)/(Ĝ, w) be an extension of fields of formal functions, as in (1.1). Let N̂ be a Ĝ/K-differential module. The Newton polygon of the F̂/K-differential module N̂F̂ is given by NP (N̂F̂) = φ_{1,e_{v}}NP (N̂).

Proof. Thanks to (3.3), this follows from the application of (2.4.3) to F_{σ}(M̂).

Proposition 4.3.5. Assume moreover that d_{v} = [F̂ : Ĝ] is finite. Let M̂ be a F̂/K-differential module. The Newton polygon of the Ĝ/K-differential module ĜM̂ is given by NP (ĜM̂) = φ_{d_{v},d_{v}/e_{v}}NP (M̂).

Proof. Thanks to (3.3), this follows from the application of (2.4.4) to F_{σ}(M̂).

4.3.6. Let (F, v) be a divisorially valued function field with completion F̂. We define the Newton polygon of any F/K-differential module M at v to be NP_{v}(M) = NP (M ⊗_{F} F̂).

Proposition 4.3.7. Let G ↪ F be a finite extension of function fields, and let w be a divisorial valuation of G. For any valuation v of F, we denote by d_{v} = [F̂ : Ĝ] (resp. e_{v}) the local degree (resp. the ramification index). Let M be a F̂/K-differential module. Then NP_{w}(_{G}M) is the algebraic sum of the polygons φ_{d_{v},d_{v}/e_{v}}NP_{v}(M), v | w.

Proof. Thanks to the decomposition (_{G}M)ˆ ≅ ⊕_{v|w} Ĝ(M ^{ˆ (v)}) (I.3.1.3), this follows from (4.3.4) and (4.3.3.i).

§5 Stratification of the singular locus by Newton polygons.

5.1 Newton polygon of an integrable connection at a singular divisor.

5.1.1. Let (X, Z) be a model of a divisorially valued function field (F, v), and let (E, ∇) be a coherent sheaf with integrable connection on X \Z. We define the (principal) Newton polygon of ∇ at Z to be the Newton polygon of the generic fiber E at v: NP_{Z}(∇) = NP_{v}(E). The vertices of NP_{Z}(∇) have integral coordinates, and the maximal slope is ρ_{Z}(∇) (4.3.1),(4.3.2). The behaviour of this invariant under extensions, duality, log-dominant inverse images and finite étale direct images of connections is given by the results (4.3.3.i,ii), (4.3.4) and (4.3.7) respectively (compare also with (2.5)). In particular NP_{Z}(∇) does not change if one replaces ∇ by its inverse image on any étale neighborhood of Z on X. Recall that we denote by φ_{a,b} , for any a, b ∈ Q^{×}, the automorphism of R^{2} given by φ_{a,b}(x, y) = (ax, by).

Theorem 5.1.2. Let (X, Z)⟶^{f}(Y, W) be a morphism of models such that f(Z) ⊂ W. Let (E, ∇) be a coherent sheaf with integrable connection on Y \ W. Then NP_{Z}(f^{∗}∇) ⊂ φ_{1,e_{f}} NP_{W} (∇), with equality if f is log-dominant (with the notation of (4.3.4)).

Proof. The equality for a log-dominant morphism is a translation of (4.3.4), as was said above. Thanks to (4.3.2) and (4.2.2), we may assume that X is a curve. On the other hand, we may replace X by any étale neighborhood of Z. Factorizing f as in (I.1.2.8) then leaves us with the case when f is a closed embedding (with X a curve). So, X is, locally at the point Z = X ∩ W, a complete intersection of smooth hypersurfaces in Y transversal to W. Hence f as a product of closed embeddings of models of relative codimension 1. On repeating the previous argument of reduction to curves, we may finally assume that f is a closed embedding with X a curve and Y a surface. By (I.1.3.3), we may assume that X is a fiber in an affine coordinatized tubular neighborhood f : X ⟶ S of W in Y with S a curve (in particular, X is an integral curve for a derivation D_{1} with simple zero at W). We conclude by (4.2.3). Q.E.D.

5.2 Gauss maps.

5.2.1. Let X be a smooth connected K-variety. The scheme Inf^{1}X of infinitely near points of first order on X is the projective bundle of tangent directions, i.e. P(Ω_{X/K}^{1}). Let C ⊂ X be a locally closed smooth curve. The Gauss map γ_{C} : C ⟶ Inf^{1}X attaches to any point Q ∈ C its tangent direction in T_{X,Q}. It is a locally closed immersion. One can interpret this map as follows: Inf^{1}X is the exceptional divisor in the blow-up of the diagonal in X^{2}. The strict transform of C × X meets Inf^{1}X transversally and their intersection is canonically isomorphic via the projection on the first factor to C: γ_{C} is the inverse of that isomorphism.

5.2.2. Let C′ be another curve in X such that γ_{C}(C) and γ_{C′}(C′) meet at some point Q^{1} in Inf^{1}X. Let Q be the image of Q^{1} in X. Then the strict transforms of C and C′ meet at some point Q_{1} of the exceptional divisor in the blow-up of Q in X, as is easily seen in local coordinates.

5.2.3. We set Inf^{0}X = X. For any integer n ≥ 1, we construct the scheme Inf^{n}X of infinitely near points of order n on X inductively, as being Inf^{1}(Inf^{n−1}X). This iterative process gives us a higher Gauss map γ_{C}^{n} : C ⟶ Inf^{n}X. For n ≥ m, we denote by π^{n,m} : Inf^{n}X ⟶ Inf^{m}X the natural projection. For any locally closed subset Z ⊂ X,(Inf^{n}X)_{|Z} sits atop a tower of projective bundles over Z:

π_{Z}^{n} : (Inf^{n}X)_{|Z} ⟶ (Inf^{n−1}X)_{|Z} ⟶ … ⟶ Z .

5.2.4. Let us assume that Z is a smooth closed subvariety of pure codimension one, and that C meets Z transversally at some point Q. Then γ_{C}^{n}(C) meets the divisor (Inf^{n}X)_{|Z} transversally in Inf^{n}X at the point Q^{n} = γ_{C}^{n}(Q) above Q.

Proposition 5.2.5. Let (E, ∇) be a coherent sheaf with integrable connection on X \ Z, and let n be > ρ_{Z}(∇). Then for any locally closed curve C which meets Z transversally at some point Q, the polygon NP_{Q}(∇_{|(C\Q)}) depends only on the point γ_{C}^{n}(Q) ∈ (Inf^{n}X)_{|Z}.

Proof. We consider a coherent extension Ẽ of E in a neighborhood of Q, as in (2.5.2). Let C′ be another curve in X such that γ_{C}(C) and γ_{C′}^{n}(Q) = γ_{C}^{n}(Q). We set X_{0} = X, Z_{0} = Z, C_{0} = C, C′_{0} = C′, Q_{0} = Q. Using the remark (5.2.2), we construct inductively (X_{m}, Z_{m}, C_{m}, C′_{m}, Q_{m}), for m ≤ n, in the following way: X_{m} is the blow-up of X_{m−1} with center Q_{m−1} deprived of the strict transform of Z_{m−1}, Z_{m} is the exceptional divisor in X_{m}; C_{m}, C′_{m} are the strict transforms of C_{m−1}, C′_{m−1} respectively, and Q_{m} is the intersection point of C_{m} and C′_{m} in Z_{m}. We denote by π the natural morphism (of log-schemes, if one likes) (X_{n}, Z_{n}) ⟶ (X, Z). We may localize the situation around Q and thus assume that (X, Z) admits a global set of adapted coordinates (x_{1}, …, x_{d}) centered at Q. Then (X_{n}, Z_{n}) admits a global set of adapted coordinates (y_{1}, …, y_{d}) centered at Q_{n} such that π is given by x_{1} = y_{1}, x_{2} = y_{2}y_{1}^{n}, …, x_{d} = y_{d}y_{1}^{n}. Note that π identifies C_{n} \ Q_{n} and C′_{n} \ Q_{n} to C \ Q and C′ \ Q respectively, and that under this identification, ∇_{|(C\Q)} = (π^{∗}∇)_{|(C_{n}\Q_{n})}, ∇_{|(C′\Q)} = (π^{∗}∇)_{|(C′_{n}\Q_{n})}.

Let Ê_{n} denote the K[[y_{1}, …, y_{d}]][\frac{1}{y_{1}} ]-module Ẽ_{Q}⊗_{O_{X,Q}} Ô_{X_{n},Q_{n}}[\frac{1}{y_{1}} ]. We note that Ô_{X_{n},Q_{n}}[\frac{1}{y_{1}} ] is a noetherian local ring with maximal ideal (y_{2}, …, y_{d}) and Ê_{n} is a finitely generated Ô_{X_{n},Q_{n}}[\frac{1}{y_{1}} ]-module. Let | | be an absolute value on K((y_{1})) trivial on K such that 0 < |y_{1}| < 1. If, for m ∈ Ê_{n}, we set ||m|| = |y_{1}^{n}|, where n ∈ N is minimal with

y_{1}^{n}m ∈ E_{Q} ⊗_{O_{X,Q}} Ô_{X_{n},Q_{n}} ,

then (Ê_{n}, || ||) is also a Banach space over (K((y_{1})), | |). Due to the hypothesis n > ρ_{Z}(∇), it follows from (2.4.1) that there is a well-defined K((y_{1}))-linear operator

Π = ∑_{\underline{α}=(α_{2},...,α_{d})}(−1)\underline{α}(\underline{α}!)^{−1}\underline{x}\underline{α}∇(\frac{∂}{∂\underline{x}})\underline{α} : E_{Q} ⟶ Ê_{n} .

Notice that Π involves only the well-defined derivations \frac{∂}{∂x_{2}} = \frac{1}{y_{1}^{n}} \frac{∂}{∂y_{2}} , …,

\frac{∂}{∂x_{d}} = \frac{1}{y_{1}^{n}} \frac{∂}{∂y_{d}} , of Ô_{X_{n},Q_{n}}[\frac{1}{y_{1}} ]. We have the formula

Π(hm) = (h_{|x_{2}=...=x_{d}=0})Π(m)

which allows to extend Π to an element of L_{K((y_{1}))}(Ê_{n}). By the classical telescoping [Ka1, 8.9], Im(Π) is killed by ∇(\frac{∂}{∂x_{2}} ), …, ∇(\frac{∂}{∂x_{d}}). On the other hand, it follows from Nakayama’s lemma and the congruence Π(m) ≡ m mod (y_{2}, …, y_{d}) that the K((y_{1}))-vector space Im(Π) generates Ê_{n} over Ô_{X_{n},Q_{n}}[\frac{1}{y_{1}} ]. The argument of [loc. cit., 8.9.12-15] shows that

Ê_{n} ≅ Im(Π) ⊗_{K((y_{1}))} Ô_{X_{n},Q_{n}}[\frac{1}{y_{1}} ] .

In particular, the K((y_{1}))-vector space Im(Π) is of finite dimension equal to the rank of the free Ô_{X_{n},Q_{n}}[\frac{1}{y_{1}} ]-module Ê_{n}. Moreover, Im(Π) is canonically endowed with a structure of K((y_{1}))[\frac{∂}{∂y_{1}} ]-module. Let N be its Newton polygon. The composed map

\begin{aligned}Der_{cont} & Ô_{C,Q} → Ô_{C,Q} ⊗ Der_{cont}Ô_{X_{n},Q_{n}} → \\ & Ô_{C,Q} ⊗ Der_{cont}Ô_{X_{n},Q_{n}}/(\frac{∂}{∂y_{2}} , …, \frac{∂}{∂y_{d}}) ≅ K[[y_{1}]]\frac{∂}{∂y_{1}} \end{aligned}

is an isomorphism, and via this isomorphism, the K((y_{1}))[\frac{∂}{∂y_{1}} ]-module

(Ẽ_{|C})_{Q} ⊗_{O_{C,Q}} Frac Ô_{C,Q} = Ê_{n} ⊗_{Ô_{Xn,Qn} [ \frac{1}{y_{1}} ]} Frac Ô_{C,Q}

is isomorphic to Im(Π). Hence NP_{Q}(∇_{|(C\Q)}) = N. Similarly NP_{Q}(∇_{|(C′\Q)}) = N. Q.E.D.

5.3 The big cell.

Theorem 5.3.1. Let (X, Z) be a model of a divisorially valued function field, and let (E, ∇) be a coherent sheaf with integrable connection on X \ Z. There is an open dense subset Z_{0} ⊂ Z such that for any locally closed curve C which meets Z transversally at some point Q of Z_{0}, NP_{Q}(∇_{|(C\Q)}) = NP_{Z}(∇).

Proof. let n be > ρ_{Z}(∇). For any point P ∈ (Inf^{n}X)_{|Z} \(π^{n,1})^{−1}(Inf^{1}Z), there is a family of smooth locally closed curves transversal to Z: Y ↪ X × S parametrized by a smooth K-variety S, such that ⋃_{s∈S} γ_{Y_{s}}^{n}(Y_{s} ∩ Z) contains some neighborhood of P. In fact, by quasicompactness of (Inf^{n}X)_{|Z} \ (π^{n,1})^{−1}(Inf^{1}Z), we may even assume that ⋃_{s∈S} γ_{Y_{s}}^{n}(Y_{s} ∩ Z) = (Inf^{n}X)_{|Z} \ (π^{n,1})^{−1}(Inf^{1}Z) if we allow S to have several connected components. We set W = Y ∩ (Z × S), which is a divisor in Y etale over S (since every Y_{s} meets Z transversally). We are thus in the situation (4.2.1) of a tubular neighborhood of W, denoting by f (resp. g) the projection Y → S (resp. W → S). According to (4.2.2), the subset W_{1} = {Q ∈ W | N_{Q}(∇_{|Y_{g(Q)}\Q}) ≠ N_{Z}(∇)} is constructible, and so is its projection Z_{1} in Z. By construction of Y, for any locally closed curve C which meets Z transversally at some closed point Q of Z_{0}, there is a point s ∈ S such that γ_{Y_{s}}^{n}(Q) = γ_{C}^{n}(Q); if Q ∉ Z_{1}, we thus have, by (5.2.5), NP_{Q}(∇_{|(C\Q)}) = NP_{Z}(∇). What remains to show is that Z_{1} is not dense in Z. Since Z_{1} is constructible, this means that η_{Z} ∉ Z_{1}, which follows from the fact that for any s ∈ S such that Y_{s} meets Z transversally at η_{Z}, NP_{η_{Z}} (∇_{|(Y_{s}\η_{Z})}) = NP_{Z}(∇) (see 4.3.2). Q.E.D.

Remark 5.3.2. It already follows from (4.3.2) and (4.2.2) that for any tubular neighborhood f of Z in X, there is a (maximal) dense open subset Z_{f} of Z with the property that N_{Q}(∇_{|(C\Q)}) = NP_{Z}(∇) for the fibers C of f which meet Z in Z_{f}, and any locally closed curve can be considered locally as a fiber of f for a suitable tubular neighborhood f (I.1.3.3). The previous theorem can be roughly interpreted as saying that the intersection of the Z_{f} for all tubular neighborhoods f is non-empty.

We next complete our results of (I) concerning the regularity.

Corollary 5.3.3. Let X be a smooth connected K-variety, Z a smooth closed subvariety of pure codimension one, (E, ∇) be a coherent sheaf with integrable connection on X \ Z. Then ∇ is regular at Z if and only if there exists a dense subset of closed points P ∈ Z and, for each P, a locally closed smooth curve C in X which meets Z transversally at P and for which ∇_{|(C\C∩Z)} is regular at P.

Proof. This follows from (I.3.4.7) and (5.3.1) (by localizing around each component of Z and noting that the regularity means that the Newton polygon has only two edges (horizontal and vertical)). Q.E.D.

Remark 5.3.4. As Mebkhout pointed out to us, this criterion can be made more precise: it suffices that there exists one non-characteristic curve C which meets Z transversally at P and for which ∇_{|(C\C∩Z)} is regular at P.

Corollary 5.3.5. Let X be a smooth connected K-variety, and let j : X ⟶ \overline{X} be an open dominant immersion of X in a proper normal K-variety \overline{X}. Let (E, ∇) be a coherent sheaf with integrable connection on X. Then ∇ is regular if and only if for every irreducible component \overline{Z} of \overline{X} \ j(X) of codimension 1, there exists a dense subset of simple closed points P ∈ \overline{Z} and, for each P, a locally closed smooth curve \overline{C} in \overline{X} which meets \overline{Z} transversally at P and for which ∇_{|\overline{C}∩X} is regular.

Proof. This is a combination of (I.5.10) and (5.3.1). Q.E.D.

5.4 Stratification by Newton polygons.

5.4.1. Let X be a smooth connected K-variety, and Q be a point of X. We say some property P is shared by almost every smooth locally closed curve containing Q if there is a dense open subset U ⊂ Inf^{1}X such that every smooth locally closed curve in X containing Q and such that γ_{C}^{1}(Q) ∈ U satisfies P.

Theorem 5.4.2. Let Z = ∪Z_{(i)} ⊂ X be a union of smooth closed subvarieties of pure codimension one, and let (E, ∇) be a coherent sheaf with integrable connection on X\Z. There is a finite set N of polygons with integral vertices, and a unique finite partition of Z into non-empty constructible subsets Z_{N} indexed by the elements of N, with the following property: (∗) for every point Q ∈ Z_{N} and almost every smooth locally closed curve C which meets each relevant component of Z transversally at Q, NP_{Q}(∇_{|(C\Q)}) = N.

Proof. It is clear that the partition Z = ∐ Z_{N} is unique if it exists. We can show its existence locally on Z. By applying (5.3.1) to X \ ∪_{j≠i}Z_{j}, we obtain dense open subsets Z_{i,0} ⊂ Z_{i} such that NP_{Q}(∇_{|(C\Q)}) = NP_{Z_{i}}(∇) for any smooth locally closed curve C which meets Z_{i,0} transversally at some point Q. We then blow up Z \ ∪Z_{i,0} (which is of codimension ≥ 2 in X). By applying (5.3.1) to the smooth part of the components of the exceptional divisor, we obtain dense open subsets Z_{i,1} ⊂ (Z \ ∪Z_{i,0}) such that NP_{Q}(∇_{|(C\Q)}) is constant for every point Q ∈ Z_{i,1} and almost every smooth locally closed curve C which meets each relevant component of Z transversally at Q. We then blow up (Z \∪Z_{i,0})\∪Z_{i,1} (which is of codimension ≥ 3 in X), and iterate the process. The construction of the partition is complete in finitely many steps: at the end, we glue together the strata corresponding to the same Newton polygon, taking into account the fact that a finite union of constructible subsets is constructible. Q.E.D.

Remark 5.4.3. Even if Z is smooth and connected, one cannot remove the restriction “for almost every” in (∗), as is shown for instance by the case of the rank one connection ∇ with formal solution e^{\frac{y}{x^{2}}} : the curve y = 0 is exceptional for the stratum {0}, since the restriction of ∇ to it is regular, while the restriction of ∇ to almost every curve passing through the origin has Poincaré-Katz rank 1.

If Z is smooth and connected, the principal Newton polygon NP_{Z}(∇) is a privileged element of N. We call the other elements the secondary Newton polygons of ∇ at Z.

We now state the semicontinuity theorem:

Theorem 5.4.4. Assume that Z is smooth. Then ⋃_{N′⊂N} Z_{N′} is closed in Z. If moreover Z is connected, then every secondary Newton polygon lies above (i.e. is contained in) the principal Newton polygon NP_{Z}(∇).

Proof. We have to show that if P′ ∈ Z_{N′} lies in the Zariski closure of Z_{N}, then N′ ⊂ N. We may replace X by an affine neighborhood of P′. Let C′ be a smooth locally closed curve in X which meets Z transversally at P′, such that NP_{P′}(∇_{|(C′\P′)}) = N′, and let C be a smooth locally closed curve in X which meets Z transversally at some point P ∈ Z_{N}, such that NP_{P} (∇_{|(C\P)}) = N. We may and shall assume that P′ ∉ C, and P ∉ C′. We need the following generalization of (I.1.3.3):

Proposition 5.4.5. Let (C, P) ⟶ (X, Z), (C′, P′) ⟶ (X, Z) be a closed immersion of affine models, with P′ ∉ C, P ∉ C′. There exist an open common neighborhood U of P and P′ in X and a tubular neighborhood (U⟶^{f}S, t) of Z ∩ U, such that C ∩ U = f^{−1}(f(P)), C′ ∩ U = f^{−1}(f(P′)).

Proof. One can find elements λ_{1}, …, λ_{d−1} (resp. μ_{1}, …, μ_{d−1}) of I_{C} (resp. I_{C′}) which are local generators at P (resp. P′) and such that the λ_{i} (resp. μ_{i}) do not vanish at P′ (resp. P). Then ((λ_{i}, μ_{i})_{i=1,...,d−1}) defines a regular map f′ : U′ ⟶ (P^{1})^{d−1} on some open common neighborhood U′ of P and P′ in X, which induces a tubular neighborhood after shrinking U′. Q.E.D.

We apply this proposition to our previous situation. Let T be the normalization of some irreducible curve joining f(P) and f(P′) in f({Z_{N}}). Let X_{T} (resp. Z_{T}) be the pull-back of U (resp. Z ∩ U) on T, and let ∇_{X_{T}} be the pullback of ∇ on X_{T} \ Z_{T}. We have NP_{Z_{T}} (∇_{X_{T}}) = N. The result then follows from (5.1.2) applied to the closed immersion of C′ in X_{T}.

The second assertion follows from the first, and from the fact (5.3.1) that Z_{NP_{Z}(∇)} is dense. Q.E.D.

Remark 5.4.6. (5.4.4) is no longer true if one removes the assumption that Z is smooth, as is shown for instance by the case of the rank one connection with formal solution e^{\frac{1}{xy}}.

§6. Formal decomposition of an integrable connection at a singular divisor.

6.1. From now onwards, we assume that K is algebraically closed. Let (Ĝ, w) be a field of formal functions over K, (Ŷ , W) a smooth affine formal model of (Ĝ, w) (I.1.1.6), (x_{1} = t_{w}, x_{2}, …, x_{d}) étale coordinates on Ŷ adapted to W, and (D_{1}, …, D_{d}) the corresponding adapted O(Ŷ)-basis of Der_{Ŷ /K,W}. According to (I.1.4.5), the transversal derivation t_{w}^{−1}D_{1} determines a canonical isomorphism O(Ŷ) ≅ O(W)[[t_{w}]] and a family of (formal) integral curves {C_{P} }_{P ∈W} of t_{w}^{−1}D_{1}.

Let (M, ∇) be a projective O(Ŷ)[\frac{1}{t_{w}}]-module of finite rank μ with integrable connection. Then MĜ := M ⊗ Ĝ is naturally a Ĝ/K-differential module. According to (3.1), there exists a finite extension (F̂, v) of (Ĝ, w), with ramification index e = e(v/w) (the Turrittin index) dividing μ!, and a well-determined finite subset \{\overline{P}_{1}, …, \overline{P}_{r}\} of F̂/(𝔪_{v} + Z), such that for every choice of representatives P_{i} ∈ F̂ of \overline{P}_{i}, we have the following decomposition of left F̂[D_{1}]-modules,

(6.1.1)  MF̂ = ⊕_{i=1}^{r}F̂ ⊗F̂^{D_{1}} Ker_{MF̂} (D_{1} − P_{i})^{μ}.

We set t = (t_{w})^{\frac{1}{e}}, a generator of 𝔪_{v} integral over O(Ŷ). We denote by Z the normalization of W in F̂^{D_{1}}; this is an affine normal connected variety over K, finite over W. Then F̂ ≅ κ(Z)((t)), D_{1} = \frac{t}{e} \frac{d}{dt}, and O(Z)((t)) ≅ O(W)((t)) ⊗_{O(W)} O(Z) is finite over O(Ŷ)[\frac{1}{t_{w}}] ≅ O(W)((t_{w})). The previous decomposition can be rewritten as

(6.1.2)  M_{κ(Z)((t))} = ⊕_{i=1}^{r}κ(Z)((t)) ⊗_{κ(Z)} Ker_{M_{κ(Z)((t))}}(D_{1} − P_{i})^{μ} .

We denote by e_{i} the projection of M_{κ(Z)((t))} on the factor κ(Z)((t)) ⊗_{κ(Z)} Ker_{M_{κ(Z)((t))}}(D_{1} − P_{i})^{μ} in the previous decomposition, that we may rewrite

(6.1.3)  M_{κ(Z)((t))} = ⊕_{i=1}^{r}e_{i}(M_{κ(Z)((t))}) .

Note that M_{O(Z)((t))} ↪ M_{κ(Z)((t))} because M is projective. The actions of D_{2}, …, D_{d} on κ(Z) do not respect O(Z) in general. In fact, there is a natural injection Der_{K}O(Z) ⟶ O(Z) ⊗_{O(W)} Der_{K}O(W) which becomes an isomorphism after tensoring by κ(Z). Although Z may not be smooth, we shall say abusively that the commuting actions of D_{1} and Der_{K}O(Z) on M_{O(Z)((t))} define an integrable connection ∇ (in d variables).

Theorem 6.2. We make the following technical assumption

(∗) The horizontal side of the Newton polygon N_{P} (End(M_{|C_{P}})) does not depend on the point P ∈ W, except possibly on a subset of codimension 2 in W.

Then, under the canonical isomorphism F̂/(𝔪_{v} + Z) ≅ κ(Z)[\frac{1}{t}]/Z, the elements \overline{P}_{i} belong to O(Z)[\frac{1}{t}]/Z. For any choice of representatives P_{i} of \overline{P}_{i} in O(Z)((t)), the decomposition (6.1.3) descends to a decomposition of O(Z)((t))-modules with integrable connection (in d variables)

(6.2.1)  M_{O(Z)((t))} = ⊕_{i=1}^{r}e_{i}(M_{O(Z)((t))})

where

(6.2.2)  e_{i}(M_{O(Z)((t))}) = O(Z)((t)) ⊗_{O(Z)} Ker_{M_{O(Z)((t))}}(D_{1} − P_{i})^{μ} .

Moreover, for any ν = 0, 1, …,

(6.2.3)  Ker_{M_{κ(Z)((t))}}(D_{1} − P_{i})^{ν} = κ(Z) ⊗_{O(Z)} Ker_{M_{O(Z)((t))}}(D_{1} − P_{i})^{ν} .

Proof. We set E := End(M_{O(Z)((t))}), and note that the e_{i} form a complete set of orthogonal idempotents in E_{κ(Z)((t))} which does not depend on the choice of the representatives P_{i}. The existence of the decomposition (6.2.1) can be expressed as: e_{i} ∈ E^{∇}. It follows from (6.2.1) that

(6.2.4) e_{i}(M_{O(Z)((t))}) = M_{O(Z)((t))} ∩[κ(Z)((t))⊗_{κ(Z)} Ker_{M_{κ(Z)((t))}}(D_{1} −P_{i})^{μ}] .

This is much weaker than (6.2.2). However, (6.2.1) allows us to prove (6.2.2) under the further assumption that in decomposition (6.1.2) r = 1, i.e. when there is only one P_{i}. Assumption (∗) is used in the proof of (6.2.1).

We shall proceed in several steps.

i) First step: e_{i} ∈ (E_{κ(Z)((t))})^{∇}.

This follows from 3.5: if we set

K_{i}^{(ν)} = Ker_{M_{κ(Z)((t))}}(D_{1} − P_{i})^{ν} , M_{i}^{(ν)} = κ(Z)((t)) ⊗_{κ(Z)} K_{i}^{(ν)} (ν = 0, 1, …), 3.5 tells that M_{i}^{(μ)} is stable under D_{1} and Der_{K}O(Z).

6.3 Interlude. In various steps of the proof of (6.2), we will have to deal with the following situation. We will be given a dense affine open subset Z′ ⊂ Z, and, by further shrinking Z′ if necessary, we will be able to assume that Z \ Z′ = ∪ T_{h} is a finite union of irreducible divisors T_{h} ⊂ Z (with generic point η_{h}). Because Z is normal, we then have

(6.3.1)  O(Z) = O(Z′) ∩ ⋂_{h} O_{Z,η_{h}} ,

the intersection taking place in κ(Z). Hence, we also have

(6.3.2)  O(Z)((t)) = O(Z′)((t)) ∩ ⋂_{h} O_{Z,η_{h}}((t)) ,

the intersection taking place in κ(Z)((t)). A component T_{h} of Z \ Z′ corresponds to a discrete valuation v_{h} of κ(Z) with valuation ring O_{Z,η_{h}}. Let us fix one such h and let x be a uniformizer of O_{Z,η_{h}} and R_{v_{h}} = O_{Z,η_{h}}((t)). The quotient field F_{v_{h}} of R_{v_{h}} is contained in κ(Z)((t)), and R_{v_{h}} is the ring of integers of a discrete valuation of F_{v_{h}}/K extending v_{h}, and still denoted by v_{h}, with v_{h}(t) = 0. Let ∂ be a derivation of O(Z) generically transversal to T_{h} and such that tr deg (κ(Z)/κ(Z)^{∂}) = 1. On replacing Z by an open neighborhood of η_{h}, we may assume to have a family of integral curves of ∂ (I.1.4.3)

\begin{matrix}Z & ↩ & T_{h} \\ f ↓ & ↙ g \\ S \end{matrix}

The generic fiber C_{h} = Z_{η_{S}} of f is then a smooth curve over κ(η_{S}) = κ(S) = κ(Z)^{∂}, with a distinguished closed point Q = η_{h}, with residue field κ(Q) = κ(T_{h}). We may then assume ∂ = ∂_{x,f} as in (1.4.1). So, x is a local coordinate on C_{h} centered at Q, such that ∂_{|C_{h}} = \frac{d}{dx}. We have κ(C_{h}) = κ(Z) and O_{C,η_{h}} = O_{Z,η_{h}}. By shrinking C_{h} around η_{h}, we may and shall assume that Z′_{η_{S}} = C_{h}\Q. We then have O(Z′)((t)) ⊂ O(C_{h})[\frac{1}{x}]((t)). We sometimes replace C_{h} by its completion Ĉ_{h} at the point η_{h}, so that O(Ĉ_{h}) = κ(η_{h})[[x]]. We notice that O_{C,η_{h}}((t)) and κ(η_{h})[[x, t]][\frac{1}{t}] are euclidian, hence principal, noetherian rings ([A.C.VII,1. ex. 9]).

ii) Second step: e_{i} ∈ E^{∇} (under assumption (∗)).

Let F = Frac(O(Z)((t))) and let us pick a cyclic vector of the dual of E_{F} with respect to the derivation \frac{d}{dt}, and let

Λ = (\frac{d}{dt})^{μ} − ∑_{j<μ} b_{j}(\frac{d}{dt})^{j}

be the associated monic differential operator. Then (E_{F})^{∇( \frac{d}{dt})} ≅ Ker_{F} Λ (see also (III.4.4) for more detail on this standard construction). Since the dual of E_{F} is a F/K-differential module, the indicial polynomial of Λ at 0 is a product τφ where τ ∈ κ(Z) and φ is a polynomial with coefficients in K (I.6.1.3).

Lemma 6.4. Λ ∈ O(Z)((t))[\frac{d}{dt}] and, under assumption (∗), τ ∈ O(Z)^{×}.

Proof. It is clear that there is a dense affine open subset Z′ ⊂ Z such that Λ ∈ O(Z′)((t))[\frac{d}{dt}] and τ ∈ O(Z′)^{×}. The considerations of the “Interlude” apply, and we have to show that (for any chosen component T_{h} and v = v_{h}) Λ ∈ R_{v}[\frac{d}{dt}] and τ ∈ O_{Z,η_{h}}^{×}. Since ∇(\frac{d}{dt})E_{R_{v}} ⊂ E_{R_{v}}, (2.3.2) shows that v(b_{j}) ≥ 0 for j = 0, …, μ − 1, i.e. b_{j} ∈ R_{v}. Thus we can define a cyclic R_{v}/O_{Z,η_{h}}-differential module F = R_{v}[\frac{d}{dt}]/R_{v}[\frac{d}{dt}]Λ, with the natural action ℵ(\frac{d}{dt}) of \frac{d}{dt}. By construction, (E_{F_{v}}, ∇_{F_{v}}) ≅ (F_{F_{v}}, ℵ_{F_{v}}).

Sublemma 6.4.1. The semisimplifications of the reductions of (F, ℵ) and (E, ∇) modulo the maximal ideal of v are isomorphic.

Proof. The proof is the same as for Gabber’s theorem mentioned in (4.2.3). For convenience, we sketch the argument. We regard E_{R_{v}} and F_{R_{v}} as two R_{v}- lattices inside the F_{v}-vector space E_{F_{v}} = F_{F_{v}}. We introduce M(a, b) = x^{a}E_{R_{v}} + x^{b}F_{R_{v}}, a submodule of E_{F_{v}} = F_{F_{v}} horizontal w.r.t. \frac{d}{dt}. It is also an R_{v}-lattice (it is of finite rank as a quotient of E_{R_{v}} ⊕ F_{R_{v}} and has no x-torsion). For n >> 0, we have x^{n}E_{R_{v}} ⊂ F_{R_{v}}, x^{n}F_{R_{v}} ⊂ E_{R_{v}}, and therefore M(0, n) = E_{R_{v}}, M(n, 0) = F_{R_{v}}. By symmetry and iteration, it suffices to compare M(a, b) and M(a + 1, b). We have

xM(a + 1, b) ⊂ xM(a, b) = M(a + 1, b + 1) ⊂ M(a + 1, b) ⊂ M(a, b) ,

and associated exact sequences of R_{v}/xR_{v}-modules with K-linear connection

0 ⟶ M(a+1, b)/xM(a, b) ⟶ M(a, b)/xM(a, b) ⟶ M(a, b)/M(a+1, b) ⟶ 0 ,

\begin{matrix}0 & ⟶ xM(a, b)/xM(a + 1, b) ≅ M(a, b)/M(a + 1, b) ⟶ \\ M(a + 1, b)/xM(a + 1, b) ⟶ M(a + 1, b)/xM(a, b) ⟶ 0 , \end{matrix}

which show that M(a, b)/xM(a, b) and M(a + 1, b)/xM(a + 1, b) have the same semisimplification. Q.E.D.

Assumption (∗) implies that the Newton polygons of E_{F_{v}} and of E_{R_{v}}/xE_{R_{v}} have the same horizontal slope. From (6.4.1), we deduce that the Newton polygons of F_{F_{v}} and of F_{R_{v}}/xF_{R_{v}} also have the same horizontal slope. This implies that τ ∈ O_{Z,η_{h}}^{×} which proves (6.4). Q.E.D.

Since τ ∈ O(Z)^{×}, the coefficient a_{n} in the expansion ∑ a_{n}t^{n} of any element of Ker_{κ(Z)((t))}Λ, for n big enough, is a linear combination of the terms of lower order with coefficients in O(Z). Hence

(E_{κ(Z)((t))})^{∇( \frac{d}{dt})} ≅ Ker_{κ(Z)((t))}Λ
≅ Ker_{O(Z)((t))⊗κ(Z)}Λ ≅ (E_{O(Z)((t))⊗κ(Z)})^{∇( \frac{d}{dt})} ≅ E^{∇( \frac{∂}{∂t})} ⊗ κ(Z) .

Moreover, E^{∇( \frac{∂}{∂t})} is an O(Z)-module of finite type, equipped with an action of Der_{K}O(Z). It is torsion free, as a submodule of E. Let Z′ ⊂ Z be an open dense subvariety such that

e_{i} ∈ ∩_{D∈Der_{K}O(Z)}(E^{∇( \frac{∂}{∂t})} ⊗ O(Z′))^{∇(D)} .

Again, we use the notation of (6.3). Using the fact that e_{i} ∈ (E^{∇( \frac{∂}{∂t})}⊗κ(T_{h})((x)))^{∇( \frac{∂}{∂x})}, we see that it will be enough to show that (for any h) e_{i} ∈ E^{∇( \frac{∂}{∂t})} ⊗ Ô_{Z,η_{h}} = E^{∇( \frac{∂}{∂t})} ⊗ κ(T_{h})[[x]]. But E^{∇( \frac{∂}{∂t})} ⊗ κ(T_{h})[[x]] has no x-torsion, hence it is a projective κ(T_{h})[[x]]-module (of finite type) with a connection (for the κ(T_{h})-linear x-adically continuous derivations of κ(T_{h})[[x]]). So,

(E^{∇( \frac{∂}{∂t})} ⊗ κ(T_{h})((x)))^{∇( \frac{∂}{∂x})} = (E^{∇( \frac{∂}{∂t})} ⊗ κ(T_{h})[[x]])^{∇( \frac{∂}{∂x})} . Q.E.D.

iii) Third step: \overline{P}_{i} belongs to O(Z)[\frac{1}{t}]/Z.

By the second step, we may assume that only one P_{i} = P appears in decomposition (6.2.1). Clearly, there is a dense affine open subset Z′ ⊂ Z such that \overline{P} belongs to O(Z′)[\frac{1}{t}]/Z. We may and will assume that P ∈ O(Z′)[\frac{1}{t}]. We apply the considerations of the “Interlude” to the open subvariety Z′ of Z. Then we may assume that, for any fixed component T_{h} and any i, P ∈ O(C_{h})[\frac{1}{x}, \frac{1}{t}]: we have to prove that P ∈ O(C_{h})[\frac{1}{t}]. We may replace C_{h} by its completion Ĉ_{h} at the point η_{h}, and we will only have to prove that P ∈ κ(T_{h})[[x]][\frac{1}{t}]. So, after replacing K by the algebraic closure of κ(T_{h}), and taking into account the fact that κ(η_{h})[[x, t]][\frac{1}{t}] is principal, we may rephrase our problem as follows.

Proposition 6.5. Let (M, ∇) be a free K[[x, t]][\frac{1}{t}]-module of finite rank μ with a K[[x]]-linear connection (for the t-adically continuous K[[x]]-linear derivations of K[[x, t]]). Let P ∈ K((x))[\frac{1}{t}] be such that

(6.5.1) M_{K((x))((t))} = K((x))((t)) ⊗_{K((x))} Ker_{M_{K((x))((t))}}(∇(t\frac{∂}{∂t}) − P)^{μ} .

Then P ∈ K[[x]][\frac{1}{t}].

Proof. Let ρ denote the (generic) Katz-Poincaré rank of M along t = 0. We may write P = ∑_{j=0}^{ρ} a_{j}(x)t^{−j}, with a_{j}(x) = ∑_{h=n_{j}}^{∞} a_{jh}x^{h} ∈ K((x)), a_{jn_{j}} ≠ 0.

We regard K((x)) as a valued field, via some x-adic absolute value, trivial over K, | |_{x}, with 0 < |x|_{x} < 1. We use x-adic analysis and Dwork’s notion of an x-adic generic point ξ_{r} at distance r ∈ (0, 1] from 0. This is a point of the Berkovich x-adic analytic space A_{K((x))}^{1 an} associated to the affine line, with canonical coordinate t: it is an extension | |_{r} of the absolute value | |_{x} to K((x))(t), such that |t|_{r} = r. We denote by K_{r} the completion of K((x))(t) w.r.t. the absolute value | |_{r}, and indicate by t(ξ_{r}) = t_{r} the canonical image of t in K_{r}. Then on the extension A_{K_{r}}^{1 an} of A_{K((x))}^{1 an} over K_{r}, there is a canonical point ξ′_{r} lying over ξ_{r}: ξ′_{r} has a fundamental system of neighborhoods in the Berkovich topology isomorphic to standard disks defined over K_{r}. The (easy) x-adic analogue of a result of Clark ([Cl], [B1]), implies that the elements of Ker_{M_{K((x))((t))}}(∇(t\frac{∂}{∂t}) − P)^{μ} are defined over the field of x-adic meromorphic functions of t on A_{K((x))}^{1 an} at t = 0: there is a radius 0 < r_{0} ≤ 1, such that those elements are defined over the ring A_{r_{0}} of x-adic meromorphic functions, defined over K((x)), on the disk D_{r_{0}} of radius r_{0} centered at 0, with poles only at t = 0.

Let \underline{e} = (e_{1}, …, e_{μ}) be a basis of M, and let us write a full set of solutions of M at ξ_{r} as

\underline{y} = \underline{e}Y_{ξ_{r}}(x, t)

where Y_{ξ_{r}}(x, t) satifies Y_{ξ_{r}}(x, t_{r}) = 1_{μ} and the differential system

\frac{d}{dt}Y = GY , G = G(x, t) ∈ M_{μ}(K[[x, t]][\frac{1}{t}]) .

From the previous differential system, we compute the matrices G^{(i)} ∈ M_{μ}(K[[x, t]][\frac{1}{t}]) such that (\frac{d}{dt})^{i}Y = G^{(i)}Y. Then

Y_{ξ_{r}}(x, t) = ∑_{i≥0} G^{(i)}(x, t_{r})\frac{(t − t(ξ_{r}))^{i}}{i!} ∈ M_{μ}(K_{1}[[t − t(ξ_{r})]]) .

As a consequence of proposition 2.4.1, there is a constant C such that

t^{(ρ+1)i+C}G^{(i)}(x, t) ∈ M_{μ}(K[[x, t]]) .

So, |G^{(i)}(x, t_{r})|_{r} = |G^{(i)}(x, t)|_{r} ≤ (ρ + 1)ir + Cr. In particular, the solutions of M at ξ_{1} converge and are bounded by 1 in the open unit disk centered at ξ_{1} in A_{K_{1}}^{1 an}. More generally, the (maximal common) radius of (bounded) convergence R_{r} of solutions of M at the x-adic generic point ξ_{r} at distance r from 0 satisfies R_{r} ≥ r^{ρ+1}. For 0 ≤ j ≤ ρ let us denote by A_{r_{0}}e^{∫ a_{j}(x)t^{−j−1}dt} the projective module of rank 1 over A_{r_{0}}

A_{r_{0}}[\frac{∂}{∂t}]/A_{r_{0}}[\frac{∂}{∂t}](t\frac{∂}{∂t} − a_{j}(x)t^{−j}) ,

equipped with the natural connection (relative to K((x))) (for j = 0, we may

write A_{r_{0}}t^{a_{0}(x)} for A_{r_{0}}e^{∫ a_{0}(x)t^{−1}dt}). An easy calculation shows that, for any j > 0 and r < r_{0}, the radius of convergence of A_{r_{0}}e^{∫ a_{j}(x)t^{−j−1}dt} at ξ_{r} is at least r^{j+1}|x|_{x}^{−n_{j}}; besides, if n_{j} < 0 the previous estimate gives the exact radius of convergence at ξ_{r}. We start by considering the term a_{ρ}(x)t^{−ρ} of P to show that n_{ρ} ≥ 0. In fact, assuming the contrary, let us pick r < r_{0} such that all values r^{j+1}|x|_{x}^{−n_{j}} are distinct. Then the radius of convergence at ξ_{r} of

A_{r_{0}}e^{∫ P \frac{dt}{t}} := A_{r_{0}}[\frac{∂}{∂t}]/A_{r_{0}}[\frac{∂}{∂t}](t\frac{∂}{∂t} − P) ,

tensor product of the previous A_{r_{0}}e^{∫ a_{j}(x)t^{−j−1}dt}, is at most r^{ρ+1}|x|_{x}^{−n_{ρ}}. Under assumption 6.5.1, Ker_{M_{Ar_{0}}} (∇(t\frac{∂}{∂t}) − P) ≠ 0 and any of its non-zero elements

provides a solution (i.e. a full set of solutions) of A_{r_{0}}e^{∫ P \frac{dt}{t}} at ξ_{r} of radius of convergence R_{r}. By choosing a sufficiently small value of r, we get a contradiction. So, n_{ρ} ≥ 0. We may then replace M by its tensor product N with the inverse of

K[[x, t]][\frac{1}{t}]e^{∫ a_{ρ}(x)t^{−ρ−1}dt} := K[[x, t]][\frac{1}{t}][\frac{∂}{∂t}]/K[[x, t]][\frac{1}{t}][\frac{∂}{∂t}](t\frac{∂}{∂t} − a_{ρ}(x)t^{−ρ}) .

Since a_{ρ}(x) ∈ K[[x]], N has the same properties as M, for P replaced by P − a_{ρ}(x)t^{−ρ}, but its Poincaré-Katz rank is less than ρ. We then iterate our reasoning and eliminate step by step the possibility of a pole at x = 0 for any a_{j}(x), j = ρ, ρ − 1, …, 0. Q.E.D.

From now on, we may and shall assume that the representatives P_{i} of \overline{P}_{i} belong to O(Z)((t)).

iv) Fourth step: reduction of (6.2.2) to the case where Z is one-dimensional.

We may assume here again that there is only one P_{i} in the decomposition (6.1.2). We may also find Z′ ⊂ Z, as in the “Interlude”, such that

(6.6.1)  M_{O(Z′)((t))} = O(Z′)((t)) ⊗_{O(Z′)} Ker_{M_{O(Z′)((t))}}(D_{1} − P_{i})^{μ} .

We again use the notation of the “interlude”, assuming Z \ Z′ = ∪_{h}T_{h}. From (6.3.1) and (6.3.2), and because M_{O(Z)((t))} is flat over O(Z), we draw

(6.6.2)  M_{O(Z)((t))} = M_{O(Z′)((t))} ∩ ⋂_{h} M_{O_{Z,η_{h}} ((t))} ,

Ker_{M_{O(Z)((t))}}(D_{1}−P_{i})^{μ} = Ker_{M_{O(Z′)((t))}}(D_{1}−P_{i})^{μ}∩⋂_{h} Ker_{M_{O_{Z,η_{h}} ((t))}}(D_{1}−P_{i})^{μ} .

In this fourth step, we reduce the desired equality

(6.6.3)  M_{O(Z)((t))} = O(Z)((t)) ⊗_{O(Z)} Ker_{M_{O(Z)((t))}}(D_{1} − P_{i})^{μ}

to the following one:

(6.6.4)  M_{O_{Z,η_{h}} ((t))} = O_{Z,η_{h}}((t)) ⊗_{O_{Z,η_{h}}} Ker_{M_{O_{Z,η_{h}} ((t))}}(D_{1} − P_{i})^{μ}

for every h. We first note that the natural map

O_{Z,η_{h}}((t)) ⊗_{O_{Z,η_{h}}} Ker_{M_{O_{Z,η_{h}} ((t))}}(D_{1} − P_{i})^{μ} → M_{O_{Z,η_{h}} ((t))}

is injective, since O(Z)((t)) is flat over O(Z) and M_{i}^{(μ)} = κ(Z)((t)) ⊗_{κ(Z)} K_{i}^{(μ)} can be identified with its image in M_{κ(Z)((t))}. From (6.6.1) and (6.6.4), we deduce

M_{i}^{(μ)} = κ(Z)⊗_{O(Z)} Ker_{M_{O(Z′)}}(D_{1} −P_{i})^{μ} = κ(Z)⊗_{O(Z)} Ker_{M_{O_{Z,η_{h}} ((t))}}(D_{1} −P_{i})^{μ}

whence, by (6.6.2),

(6.6.5)  M_{i}^{(μ)} = κ(Z) ⊗_{O(Z)} Ker_{M_{O(Z)((t))}}(D_{1} − P_{i})^{μ} .

This implies in turn:

(6.6.6) Ker_{M_{O(Z′)((t))}}(D_{1} − P_{i})^{μ} = O(Z′) ⊗_{O(Z)} Ker_{M_{O(Z)((t))}}(D_{1} − P_{i})^{μ} ,

(6.6.7) Ker_{M_{O_{Z,η_{h}} ((t))}}(D_{1} − P_{i})^{μ} = O_{Z,η_{h}} ⊗_{O(Z)} Ker_{M_{O(Z)((t))}}(D_{1} − P_{i})^{μ} ,

whence

(6.6.8)  M_{O(Z′)((t))} = O(Z′)((t)) ⊗_{O(Z)} Ker_{M_{O(Z)((t))}}(D_{1} − P_{i})^{μ} ,

(6.6.9)  M_{O_{Z,η_{h}} ((t))} = O_{Z,η_{h}}((t)) ⊗_{O(Z)} Ker_{M_{O(Z)((t))}}(D_{1} − P_{i})^{μ} .

In order to conclude (6.6.3), it then suffices to combine (6.6.8), (6.6.9), (6.6.2), the remark following (6.6.4), and the fact that in M_{κ(Z)((t))} the images of M_{O(Z)((t))}⊗(O(Z′)((t))∩⋂_{h} O_{Z,η_{h}}((t))) and of (O(Z′)((t))⊗_{O(Z′)}Ker_{M_{O(Z′)((t))}}(D_{1}− P_{i})^{μ}) ∩ ⋂_{h} (O_{Z,η_{h}}((t)) ⊗_{O_{Z,η_{h}}} Ker_{M_{O_{Z,η_{h}} ((t))}}(D_{1} − P_{i})^{μ}) coincide ([A.C.I, 2.6]).

Taking into account the fact that O_{Z,η_{h}}((t)) is principal, as we noticed before, we have thus reduced our statement (6.6.3) to the following lemma.

Lemma 6.7. Let (M, ∇) be a free O_{C,η_{h}}((t))-module of finite rank μ with integrable connection (with respect to ∂ = ∂_{x,f} = ∂/∂x and ∂/∂t). Let P ∈ O_{C,η_{h}}((t)) be such that M_{κ(C)((t))} = κ(C)((t)) ⊗_{κ(C)} Ker_{M_{κ(C)((t))}}(∇(t\frac{∂}{∂t}) − P)^{μ} . Then M = O_{C,η_{h}}((t)) ⊗_{O_{C,η_{h}}} Ker_{M}(∇(t\frac{∂}{∂t}) − P)^{μ} .

v) Fifth step: proof of (6.7).

We are allowed to tensor M by the inverse of the projective O_{C,η_{h}}((t))-module of rank 1

O_{C,η_{h}}((t))e^{∫ P \frac{dt}{t}} := O_{C,η_{h}}((t))[\frac{∂}{∂t}]/O_{C,η_{h}}((t))[\frac{∂}{∂t}](t\frac{∂}{∂t} − P) ,

endowed with the canonical K-linear integrable connection. This reduces us to proving the following

Lemma 6.8. Let (M, ∇) be as in lemma 6.7 but assume that

(6.8.1)  M_{κ(C)((t))} = κ(C)((t)) ⊗_{κ(C)} Ker_{M_{κ(C)((t))}}(∇(t\frac{∂}{∂t}))^{μ} .

Then for all ν, Ker_{M}∇(t\frac{∂}{∂t})^{ν} is a free O_{C,η_{h}}-submodule of finite rank of M stable under \frac{∂}{∂x}. Moreover

(6.8.2)  Ker_{M_{κ(C)((t))}}∇(t\frac{∂}{∂t})^{ν} = κ(C) ⊗_{O_{C,η_{h}}} Ker_{M}∇(t\frac{∂}{∂t})^{ν}

and

(6.8.3)  M = O_{C,η_{h}}((t)) ⊗_{O_{C,η_{h}}} Ker_{M}∇(t\frac{∂}{∂t})^{μ} .

Proof. The first assertion is clear. Formula (6.8.3) is a consequence of (6.8.2). Indeed, let us provisionally indicate by N the r.h.s. of (6.8.3): N is a free O_{C,η_{h}}((t))-submodule of finite type of M stable by \frac{∂}{∂t} and \frac{∂}{∂x}. On the other hand, by (6.8.1) and (6.8.2), M/N is killed by a power of x. The usual argument based on the simplicity of the differential ring O_{C,η_{h}}((t)) shows that it must be zero. As for (6.8.2), we first observe that the formula for all M and all ν is equivalent to the same formula for all M and ν = 1 : in fact, assuming the formula for all modules as in (6.8) satisfying (6.8.1) and for ν = 1, we observe that, given any (M, ∇) and ν, we may endow M^{ν} = M ⊕ … ⊕ M of the connection

\begin{aligned}∇_{ν}(t\frac{∂}{∂t})(m_{1}, …, m_{ν}) & = (∇(t\frac{∂}{∂t})(m_{1}), ∇(t\frac{∂}{∂t})(m_{2}) − m_{1}, …, ∇(t\frac{∂}{∂t})(m_{ν}) − m_{ν−1}) \\ ∇_{ν}(\frac{∂}{∂x})(m_{1}, …, m_{ν}) & = (∇(\frac{∂}{∂x})(m_{1}), ∇(\frac{∂}{∂x})(m_{2}), …, ∇(\frac{∂}{∂x})(m_{ν})) . \end{aligned}

Obviously, m ⟼ (∇(t\frac{∂}{∂t})^{ν−1}(m), ∇(t\frac{∂}{∂t})^{ν−2}(m), …, ∇(t\frac{∂}{∂t})(m), m) establishes an isomorphism of O_{C,η_{h}}-modules between Ker_{M}∇(t\frac{∂}{∂t})^{ν} and Ker_{M^{ν}} ∇_{ν}(t\frac{∂}{∂t}).

For ν = 1, formula (6.8.2) is equivalent to

(6.8.5)  Ker_{M_{κ(C)((t))}}∇(t\frac{∂}{∂t}) = Ker_{M⊗Frac(O_{C,η_{h}})}∇(t\frac{∂}{∂t}) .

This is a statement on x-adic boundedness of solutions of a linear differential equation, which follows from the following general result (replacing K by the algebraic closure of κ(η_{h})):

Proposition 6.9. Let (M, ∇) be a free K[[x, t]][\frac{1}{t}]-module of finite rank μ with a K[[x]]-linear connection (for the t-adically continuous K[[x]]-linear derivations of K[[x, t]]). Let us assume that

M_{K((x))((t))} = K((x))((t)) ⊗_{K((x))} Ker_{M_{K((x))((t))}}(∇(t\frac{∂}{∂t}))^{μ} .

Then

(6.9.1)  Ker_{M_{K((x))((t))}}∇(t\frac{∂}{∂t}) = Ker_{M_{K[[x,t]][ \frac{1}{t} , \frac{1}{t}]}}∇(t\frac{∂}{∂t}) .

For the proof, we need the following result.

Lemma 6.9.2. In the situation of (6.9), there exists a basis \underline{e} = (e_{1}, …, e_{μ}) of M_{Frac(K[[x,t]])} in which ∇(t\frac{∂}{∂t})\underline{e} = \underline{e}G with G ∈ M_{μ}(K[[x, t]]). Moreover, the eigenvalues of G(t = 0) are rational integers.

Proof. Let us pick an element m of M which is generically cyclic with respect to ∇(t\frac{∂}{∂t}). In the basis \underline{e}′ = (m, …, ∇(t\frac{∂}{∂t})^{μ−1}(m)) of M⊗Frac(K[[x, t]]), let us write ∇(t\frac{∂}{∂t})\underline{e}′ = \underline{e}′G′. By regularity at t = 0, we then have G′ ∈ GL_{μ}(Frac(K[[x, t]])) ∩ M_{μ}(K((x))[[t]]) (we view Frac(K[[x, t]]) as a subfield of K((x))((t)) ). Let us choose a basis \underline{e}′′ of M, and let H ∈ GL_{μ}(Frac(K[[x, t]])) ∩ M_{μ}(K[[x, t]][\frac{1}{t}]) be the matrix transforming \underline{e}′′ into \underline{e}′. On modifying \underline{e}′′ by powers of t if necessary, we may and shall assume also that H ∈ M_{μ}(K[[x, t]]). If det H does not vanish at t = 0, it is clear that we can take \underline{e} = \underline{e}′′. If ord_{t=0} det H > 0, let (b_{1}, …, b_{μ}) be the coefficients of a dependence relation over K((x)) between the columns of H(t = 0). We may assume that the b_{i} belong to K[[x]] and that some b_{i_{0}} = 1. Let

H_{1} = ( \begin{matrix}1 & 0 & … & & & & 0 \\ . & . & . \\ 0 & … & 1 \\ −b_{1} & … & −b_{i_{0}−1}, & t, & −b_{i_{0}+1}, & …, & −b_{μ} \\ 0 & … & & & 1 \\ . & . & . \\ & & & & & & & 1 \end{matrix} ) .

We have

H_{1}^{−1} = ( \begin{matrix}1 & 0 & … & & & & 0 \\ . & . & . \\ 0 & … & 1 \\ \frac{b_{1}}{t} & … & \frac{b_{i_{0}−1}}{t} , & \frac{1}{t}, & \frac{b_{i_{0}+1}}{t} , & …, & \frac{b_{μ}}{t} \\ 0 & … & & & 1 \\ . & . & . \\ & & & & & & & 1 \end{matrix} ) ,

hence H_{1} ∈ GL_{μ}(K[[x, t]][\frac{1}{t}]). On the other hand, ord_{t=0} det H_{1}^{−1}H < ord_{t=0} det H. We see that after modifying the basis \underline{e}′′ by a finite product of matrices of type H_{1}, we may assume that ord_{t=0} det H = 0. The last assertion of the lemma is clear.

Let us return to the proof of (6.9). We write G = ∑_{j=0}^{+∞} G^{(j)}t^{j} , with G^{(j)} ∈ M_{μ}(R) and G^{(0)} = Δ + N, with Δ diagonal with coefficients in Z, N nilpotent, ΔN = NΔ. We first consider a solution m = \underline{e}y(t) of ∇(\frac{∂}{∂t})(m) = 0 of the form y(t) = ∑_{i=−N}^{∞} y_{i}t^{i} , with y_{i} ∈ C^{μ}, y_{−N} ≠ 0. We must prove that there exists an M such that x^{M}y_{i} ∈ K[[x]] for all i ≥ −N. From (t\frac{∂}{∂t} + G)y(t) = 0 we obtain

(6.9.3)  (Δ + i + N)y_{i} = − ∑_{j=0}^{i−1} G^{(i−j)}y_{j}, ∀ i > 0 .

On the other hand, there exists i_{0} such that the matrix Δ + i + N is invertible in M_{μ}(K[[x]]) for i > i_{0}. We conclude that |y_{i}| ≤ max{|y_{N}|, …, |y_{i_{0}}|}, for all i. The result follows. This achieves the proof of (6.7) and (6.2.2).

vi) Last step: proof of (6.2.3).

It follows from (6.2.1) and (6.2.2) that

(Ker_{M_{κ(Z)((t))}}(D_{1} − P_{i})^{μ}) = κ(Z) ⊗_{O(Z)} Ker_{M_{O(Z)((t))⊗κ(Z)}}(D_{1} − P_{i})^{μ},

and this implies that for any ν = 0, 1, …,(Ker_{M_{κ(Z)((t))}}(D_{1} − P_{i})^{ν}) ∩ M_{O(Z)((t))}

= Ker_{M_{O(Z)((t))⊗κ(Z)}}(D_{1} − P_{i})^{ν} = κ(Z) ⊗_{O(Z)} Ker_{M_{O(Z)((t))}}(D_{1} − P_{i})^{ν}.

Q.E.D.

Corollary 6.10. (Under assumption (*) of (6.2)). Let F_{•}(M) be the slope filtration of M = M ⊗ Ĝ as in (3.1), (3.3). Then F_{•}(M) := M ∩ F_{•}(M) is a filtration of M by horizontal O(Ŷ)[\frac{1}{t_{w}}]-submodules of M, and F_{•}(M) ⊗ Ĝ ≅ F_{•}(M). Moreover, F_{σ}(M) is the maximal O(Ŷ)[\frac{1}{t_{w}}]-submodule of M stable under ∇ of Poincaré-Katz rank ≤ σ.

Proof. Since Ĝ is flat over O(Ŷ)[\frac{1}{t_{w}}], F_{σ}(M) ⊗ Ĝ ⟶ M ⊗ Ĝ = M is injective so that

Im(F_{•}(M) ⊗ Ĝ ⟶ M) ≅ F_{•}(M) ⊗ Ĝ .

The fact that F_{•}(M) is stable under ∇ comes from (3.1). On the other hand, with the notation of (6.1), the image of (M∩F_{•}(M))⊗O(Z)((t)) in M ⊗_{O(Ŷ)[ \frac{1}{tw} ]} O(Z)((t)) ≅ MF̂ coincides with the intersection of the images of M ⊗ O(Z)((t)) and F_{•}(M) ⊗ O(Z)((t)) in MF̂ respectively ([A.C.I, 2.6]), i.e. with M_{O(Z)((t))} ∩ F_{e•}(MF̂). We get

Im(F_{•}(M) ⊗ O(Z)((t)) ⟶ MF̂) = M_{O(Z)((t))} ∩ F_{e•}(MF̂) .

It follows from the decomposition (6.2.1) that

(M_{O(Z)((t))} ∩ F_{e•}(MF̂)) ⊗_{O(Z)((t))} F̂ = F_{e•}(MF̂) .

Hence

Im(F_{•}(M)⊗O(Z)((t)) ⟶ M⊗_{O(Ŷ)[ \frac{1}{tw} ]}O(Z)((t)))⊗_{O(Z)((t))}F̂ = F_{e•}(MF̂) = F_{•}(M)⊗ĜF̂ .

We deduce that

\begin{aligned}Im(F_{•} & (M) ⊗ O(Z)((t)) ⟶ MF̂) ⊗ F̂ = Im((F_{•}(M) ⊗ F̂) → MF̂) \\ & = Im((F_{•}(M) ⊗ Ĝ) → M) ⊗Ĝ F̂ = (F_{•}(M) ⊗ Ĝ) ⊗Ĝ F̂ \end{aligned}

coincides with F_{•}(M) ⊗Ĝ F̂. Hence F_{•}(M) ⊗ Ĝ ≅ F_{•}(M).

Let us turn to the second assertion. We already know that F_{σ}(M) is a O(Ŷ)[\frac{1}{t_{w}}]-submodule of M stable under ∇ of Poincaré-Katz rank ≤ σ. Let N be another submodule with the same property. Then F_{σ}(M) + N has again the same property. In order to conclude that N ⊂ F_{σ}(M), it suffices to note that M/F_{σ}(M) ⊗ Ĝ ≅ M/F_{σ}(M), and that any non-zero submodule of M/F_{σ}(M) stable under ∇ has Poincaré-Katz rank > σ. Q.E.D.

Remark 6.11. A slight difficulty arises here from the fact that O(Z)((t)) is not necessarily flat over O(Ŷ)[\frac{1}{t_{w}}] (Z is not necessarily flat over W). In fact, as M. Chardin pointed out to us, Z is flat over W if and only if Z is Cohen-Macaulay. However, there is a closed subset T ⊂ W of codimension ≥ 2 such that Z_{|(W \T)} is flat over W \ T and smooth over K (it suffices to take T to be the image in W of the singular locus of Z).

Corollary 6.12. The P_{i} being taken in κ(Z)[\frac{1}{t}], assumption (*) is equivalent to the condition that for every pair (i, i′) with i ≠ i′, P_{i} − P_{i′} is a unit in O(Z)((t)) of negative t-order.

Indeed, it is clear that the latter condition implies (*), and the converse follows from (6.2).

Remark 6.13. It is likely that assumption (*) is not necessary for (6.2). This would for instance follow from a positive answer to the following

Open question 6.14. Let (M, ∇) be a free K[[x, t]][\frac{1}{t}]-module of finite rank with integrable connection (for the (x, t)-adically continuous K-linear derivations of K[[x, t]]). Is it true that (M_{K((x))((t))})^{∇} = K((x)) ⊗ M^{∇}?

§7 Cyclic vectors, indicial polynomials and tubular neighborhoods.

7.1 Exponents and indicial polynomials.

7.1.1. In the notation of (3.1), let t = t_{v} be a parameter of R_{v}, such that eD_{1}t = t and D_{j}t = 0, for j = 2, …, d. The elements eRes_{t=0}P_{i} \frac{dt}{t} ∈ k(v) (well determined modulo Z) are called the Turrittin exponents of (E, ∇) (along w). Robba’s result reproduced in (I App.A) asserts that if ∇ is integrable, the Turrittin exponents of (E, ∇) along w are constant, i.e. that they belong to the algebraic closure of K in k(v). Then F_{0}(E, ∇) is called the fuchsian part of (E, ∇). The Turrittin exponents along w of the fuchsian part of (E, ∇) are also called the Fuchs exponents of (E, ∇). The Fuchs exponents of (E, ∇) along w, can be computed modulo Z, in terms of a cyclic vector, as the roots of the indicial polynomial of (I.6.1.1) (cf. loc. cit., or [B1]).

Proposition 7.1.2. Let (Y, W) be a model of a divisorially valued function field, and let (E, ∇) be a coherent sheaf with integrable connection on Y \ W. Let ∂ be any transversal derivation admitting a system of integral curves

U ↩^{i} W ∩ U
f ↓ ↙ g
S

(with U open in Y). Let us assume that NP_{Q}(∇_{|f^{−1}(f(Q))}) does not depend on the point Q ∈ W. Then for any cyclic vector (if any) of E(U) with respect to ∇(∂), with associated (monic) differential polynomial Λ, the leading coefficient of the indicial polynomial of Λ at W is a unit in O(W).

Proof. Let us shrink U in such a way that Û_{W ∩U} ≅ SpfO(W ∩ U)[[t]] with ∂ = \frac{d}{dt} as in (I.1.4.5). Then the coefficients of Λ ∈ κ(W)((t))[∂] are in O(W ∩U), and they specialize at each Q ∈ W ∩ U, to the coefficients of the analogous polynomial for (E, ∇)_{|f^{−1}(f(Q))} at Q. Since the connection is integrable, the roots of the indicial polynomial of Λ are in K (I.6.1.3), hence the vanishing of the leading coefficient at some point Q of W ∩ U implies the vanishing of the indicial polynomial itself at Q. This implies in turn that the horizontal edge of the Newton polygon N_{f}(Λ, •) is not constant, contrary to our assumption. Q.E.D.

7.2 Cyclic vectors.

Up to now, we have used the lemma of the cyclic vector in its coarsest, birational form. We shall need refined variants in the sequel. Let us start with the existence of cyclic vectors in the neighborhood of a given ordinary point.

Katz’ lemma of the cyclic vector 7.2.1. Let R be a Q-algebra, t an element of R, ∂ a derivation of R such that ∂t = 1, R^{∂} the Q- subalgebra of ∂-constants. Let N be a R[∂]-module, and let us assume that N is a free R-module of finite rank μ, with fixed basis of vectors (e_{0}, …, e_{μ−1}). Let α_{0}, …, α_{μ(μ−1)} be elements in R^{∂} such that α_{i} −α_{j} is a unit whenever i ≠ j. Then locally on Spec R, one of the vectors

(7.2.1.1)  v_{α_{i}} = ∑_{j=0}^{μ−1} \frac{(t − α_{i})^{j}}{j!} ∑_{k=0}^{j}(−1)^{k}\binom{j}{k}∂^{k}(e_{j−k}) , 0 ≤ i ≤ μ(μ − 1),

is a cyclic vector with respect to ∂.

Proof. See [Ka3, th.2].

Corollary 7.2.2. Let K be a subfield of R^{∂} and let P ∈ SpecR. Then v_{α} is a cyclic vector at P for all α ∈ K except possibly μ(μ − 1) of them.

Corollary 7.2.3. Let C be a smooth affine curve over K, and let (N , ∇) be a coherent O_{C}-module with connection. For any proper closed subset T ⊂ C, there is a closed subset T′ ⊂ C disjoint from T and a global section v of N which is a cyclic vector over C \ T′.

Proof. Let μ be the rank of the projective O_{C}-module N. By Nakayama’s lemma, one can find global sections e_{0}, …, e_{μ−1} of N which are linearly independent at the points of T. One can then apply (7.2.2) to the valuation rings of C at these points.

7.2.4. We now turn to the more delicate existence problem of cyclic vectors in the neighborhood of a singularity. The natural question in this context is:

Question 7.2.5. Assume that f is a coordinatized tubular neighborhood, i.e. factors through an étale map ϕ : X ⟶ A_{S}^{1} (with coordinate x on A^{1}), and that E extends to a locally free O_{X}-module. For any point s ∈ S, does there exist an affine neighborhood U of the finite subset g^{−1}(s) of X and a cyclic vector v ∈ Γ(U \ (Z ∩ U), E) with respect to ∇(\frac{∂}{∂x})?

The answer is no in general, even in the integrable case. Indeed, let X be the (x, y)-plane, Z= the y-axis, E = O_{X\Z}e_{1}⊕O_{X\Z}e_{2}, where the first (resp. second) factor is endowed with the trivial connection (resp. the connection with formal solution e^{− \frac{y}{x}}). If v = ae_{1} + be_{2}, \frac{∂}{∂x}v = \frac{∂a}{∂x}e_{1} + (\frac{∂b}{∂x}b − \frac{by}{x^{2}} )e_{2}; by looking at the the of lowest x-order, we see that for any affine neighborhood U of the origin, the determinant a\frac{∂b}{∂x} − b\frac{∂a}{∂x} − \frac{aby}{x^{2}} cannot be a unit in O(U \ (Z ∩ U)).

However we shall give a qualified positive answer to the previous question.

7.2.6. We first consider the formal situation, as in (6.1): (Ĝ, w) is a field of formal functions over K, (Ŷ , W) a smooth affine formal model of (Ĝ, w), (D_{1}, …, D_{d}) an adapted O(Ŷ)-basis of Der_{Ŷ /K,W} and (M, ∇) is a projective O(Ŷ)[\frac{1}{t_{w}}]-module of finite rank μ with integrable connection. Let e|μ!, Z, and P_{i} ∈ κ(Z)[\frac{1}{t}] be as in (6.1).

Lemma 7.2.7. Assume that the Newton polygon of the κ(P)((t_{w}))[D_{1}]-module \underline{End}(M_{P}) does not depend on the point P ∈ W, except possibly on a closed subset of codimension 2 in W. Then any point Q ∈ Z outside some closed subset T ⊂ Z of codimension ≥ 2 has an affine smooth neighborhood Z_{Q} such that M_{O(Z)((t))⊗O(Z_{Q})} is cyclic with respect to D_{1}.

Proof. Since we allow to exclude a closed subset of codimension ≥ 2 from Z, we may restrict to the part of W where the Newton polygon of \underline{End}(M_{P}) is constant, and over which Z is etale. Also, by quasicompactness of Z, it is enough to consider points Q of codimension 1. The assumption that the Newton polygon of \underline{End}(M_{P}) is constant on W implies that the Newton polygon of (\underline{End}(M_{O(Z)((t))}))_{Q} is constant on Z, hence that P_{i} − P_{i′} is either zero or a unit in O(Z)((t)) of non-positive t-order (cf. (6.2), (6.12)). We saw in (6.2.3) that

Ker_{M_{κ(Z)((t))}}(D_{1} − P_{i})^{ν} = κ(Z) ⊗_{O(Z)} Ker_{M_{O(Z)((t))}}(D_{1} − P_{i})^{ν}.

Moreover Ker_{M_{O(Z)((t))}}(D_{1}−P_{i})^{ν}/Ker_{M_{O(Z)((t))}}(D_{1}−P_{i})^{ν−1} has no O(Z)-torsion and is finitely generated (as a subquotient of the noetherian module Ker_{M_{O(Z)((t))}}(D_{1}− P_{i})^{μ}). Hence by shrinking Z_{Q}, we may assume that there exist elements e_{ν,i,1}, …, e_{ν,i,s_{ν,i}} in Ker_{M_{O(Z)((t))⊗O(Z_{Q})}}(D_{1} − P_{i})^{ν} whose classes form a basis in the quotient

Ker_{M_{O(Z)((t))⊗O(Z_{Q})}}(D_{1} − P_{i})^{ν}/Ker_{M_{O(Z)((t))⊗O(Z_{Q})}}(D_{1} − P_{i})^{ν−1}.

Altogether, the elements e_{ν,i,k} then form a O(Z)((t))⊗O(Z_{Q})-basis of M_{O(Z)((t))⊗O(Z_{Q})}. We reorder them in lexicographical order, and rename them (m_{0}, …, m_{μ−1}). Let us remark that, as a D_{1}-module, M_{O(Z)((t))⊗O(Z_{Q})} is an iterated extension of rank-one differential modules: D_{1}m_{j} = Q_{j}m_{j} + ∑_{k>j} a_{kj}m_{k}, where Q_{j} is one of the P_{l}, and a_{kj} ∈ O(Z_{Q}). We set −s = min ord_{t} P_{i}, and choose a large integer n (e.g. n > sμ).

We claim that m := m_{0} + t^{−en}m_{1} + … + t^{−en(μ−1)}m_{μ−1} is a cyclic vector of M_{O(Z)((t))⊗O(Z_{Q})}. Let us set m_{j}^{0} = t^{−enj}m_{j}, m^{0} := m = m_{0} + ∑_{j=1}^{μ−1} m_{j}^{0}, Q_{j}^{0} = Q_{j} − nj, so that D_{1}m_{j}^{0} = Q_{j}^{0}m_{j}^{0} + ∑_{k>j} a_{kj}^{0}m_{k}^{0}, with ord_{t} a_{kj}^{0} > 0 and, for j ≠ k, Q_{j}^{0} − Q_{k}^{0} is a unit in O(Z)((t)) ⊗ O(Z_{Q}), ord_{t} (Q_{j}^{0} − Q_{k}^{0}) ≤ 0. We have m^{1} := (Q_{1}^{0} − Q_{0}^{0} + a_{1,0}^{0})^{−1}(D_{1} − Q_{0}^{0})m = m_{1}^{0} + ∑_{k>1} u_{k,1}m_{k}^{0}, where u_{k,1} is a unit in O(Z)((t)) ⊗ O(Z_{Q}), −s ≤ ord_{t}u_{k,1} ≤ s. If we set m_{j}^{1} = u_{j,1}m_{j}^{0}, m^{1} = m_{1}^{0} + ∑_{j=2}^{μ−1} m_{j}^{1}, Q_{j}^{1} =

Q_{j}^{0} + \frac{D_{1}(u_{j,1})}{u_{j,1}} for j > 1, we get D_{1}m_{j}^{1} = Q_{j}^{1}m_{j}^{1} + ∑_{k>j} a_{kj}^{1}m_{k}^{1}, with ord_{t} a_{kj}^{1} > 0 and, for j ≠ k, Q_{j}^{1} − Q_{k}^{1} is a unit in O(Z)((t)) ⊗ O(Z_{Q}), ord_{t} (Q_{j}^{1} − Q_{k}^{1}) ≤ 0. We then construct m^{2} := (Q_{2}^{1} − Q_{1}^{1} + a_{2,1}^{1})^{−1}(D_{1} − Q_{1}^{1})m^{1} = m_{2}^{1} + ∑_{k>2} u_{k,2}m_{k}^{1}, and iterate μ−1 times. Iteration step ν shows that ∑_{j≤ν} O(Z)((t))⊗O(Z_{Q})D_{1}^{j}m projects onto M_{O(Z)((t))⊗O(Z_{Q})}/ ∑_{k>ν} O(Z)((t))⊗O(Z_{Q}).m_{k}. Hence m is cyclic. Q.E.D.

7.2.8. We now consider a diagram of smooth K-varieties

(7.2.8.1)  \begin{aligned}Y & ↪ & \overline{Y} & ↩ & (∐_{i=0}^{r} W_{i}) \\ & ↘^{f} & ↓\overline{f} & ↙^{g} \\ & & S \end{aligned}

where i) Y is the complement of ∐_{i=0}^{r} W_{i} in \overline{Y}, ii) Y and S are affine, and each W_{i} is a divisor defined by one equation

t_{i} = 0,

iii) \overline{f} is smooth surjective with geometric fibers irreducible of dimension one,

iv) each W_{i} is an etale covering of S. Let ∂ be a non-zero section of Der_{\overline{Y} /K,(∐_{i=0}^{r} W_{i})} over Y.

On the other hand, let \overline{E} be a locally free O\overline{Y} -module of rank μ. We assume that the restriction E of \overline{E} to Y is endowed with an integrable connection ∇ such that the Turrittin index e_{i} of (E, ∇) at each singular divisor W_{i} is equal to 1 (this is stronger than requiring that the slopes are integral).

Proposition 7.2.9. Assume that the Newton polygon of the O(Y_{g(P)})[∂]-module \underline{End}(E_{g(P)}) at P ∈ W_{i} does not depend on the point P (nor on i). Then there exists a smooth K-variety S′ and a flat morphism h : S′ → S such that

i) S′ is finite over h(S′), ii) S \ h(S′) has codimension ≥ 2 in S, iii) for any Q ∈ S′, there is an affine neighborhood U of the finite subset g^{−1}(Q) of \overline{Y} ×_{S} S′ such that the inverse image of E on U ∩ (Y ×_{S} S′) is cyclic with respect to ∇(∂).

Roughly speaking, this result establishes a close link between the set of points s ∈ S for which question (7.2.5) has a negative answer, and the set of points s where the Newton polygon of \underline{End}(E_{s}) is strictly above the generic ones.

Proof. We first notice that we may replace S by any finite etale covering, hence assume that each W_{i} is the image of a section σ_{i} of \overline{f}. Then the completion of \overline{Y} at W_{i} is isomorphic to the affine t_{i}-adic formal scheme SpfO(S)[[t_{i}]].

We shall apply the previous lemma to M_{i} := E(Y) ⊗_{O(Y)} O(S)((t_{i})) (with the induced integrable connection). We notice that the derivation D_{1} considered in loc. cit. is a multiple of ∂ by a unit in O(S)[[t_{i}]], hence that the condition of being cyclic with respect to D_{1} or to ∂ are equivalent. Also, we can arrange that the normal variety Z is independent of i, the finite morphism Z → W_{i} being the composite of a finite (surjective) morphism Z → S and σ_{i}. By assumption, the Turrittin indices are 1, so that what was denoted by O(Z)((t)) in loc. cit. is now O(Z)((t_{i})) ≅ O(S)((t_{i})) ⊗_{O(S)} O(Z) (since Z is finite over S). There is a closed subset T ⊂ S of codimension ≥ 2 such that S′ := Z_{|(S\T)} is flat over S \ T and smooth over K (it suffices to take T to be the image in S of the singular locus of Z).

It then follows from (7.2.7) that one can adjust T in such a way that any Q ∈ S′ has an affine neighborhood S′_{Q} such that the O(S)((t_{i})) ⊗_{O(S)} O(S′_{Q})[∂]- module M_{i} ⊗_{O(S)} O(S′_{Q}) is cyclic. We denote by m_{i} a cyclic vector in M_{i} ⊗_{O(S)} O(S′_{Q}).

We have to find a cyclic vector m for E ⊗_{O_{S}} O_{S′_{Q}} in some affine neighborhood of g^{−1}(h(Q)) in Y′ := Y ×_{S} S′. The question is now Zariski-local on S \ T. In particular, we may replace S \T by an affine open subset S_{0} of S \h(S′ \S′_{Q}) such that \overline{E} is free in some affine neighborhood of (∐_{i=0}^{r} W_{i}) ×_{S} S_{0} in \overline{Y} ×_{S} S_{0}, and replace S′ by S′ ×_{S} S_{0}. After shrinking S in such a way, we have the following properties:

i) in some affine neighborhood V of ∐_{i=0}^{r} W_{i} in \overline{Y}, \overline{E}(V) = ⊕_{j=1}^{μ}O(V)e_{j}, ii) S′ is finite flat over S, iii) m_{i} is a cyclic vector in M_{i}⊗_{O(S)}O(S′) ≅ ⊕_{j=1}^{μ}O(S)((t_{i}))⊗_{O(S)}O(S′) e_{j}

with respect to ∂. We thus have m_{i} ∧ ∂m_{i} ∧ … ∧ ∂^{μ−1}m_{i} = u_{i} e_{1} ∧ e_{2} ∧ … ∧ e_{μ}, where u_{i} is a unit in O(S)((t_{i})) ⊗_{O(S)} O(S′). On the other hand, the vectors m_{i} can be simultaneously approximated to arbitrarily high (t_{i}-)order by elements of \overline{E}(V ×_{S} S′)[\frac{1}{t_{0}...t_{r}}] = ⊕_{j=1}^{μ}O(V ×_{S} S′)[\frac{1}{t_{0}...t_{r}}]e_{j}. For a good enough approximation m, we have m ∧ ∂m ∧ … ∧ ∂^{μ−1}m = u e_{1} ∧ e_{2} ∧ … ∧ e_{μ}, where u ∈ O(V ×_{S} S′) is a unit in O(S)((t_{i})) ⊗_{O(S)} O(S′)[\frac{1}{t_{0}...t_{r}}] for every i. This implies that for some affine neighborhood U of (∐_{i=0}^{r} W_{i}) ×_{S} S′ in Y′, u is a unit in O(U ∩ Y′), hence that m is a cyclic vector in E(S) ⊗_{O(S)} O(U ∩ Y′) with respect to ∂. Q.E.D.

Chapter III.

Direct images (the Gauss-Manin connection).

Table of contents:

§1 Elementary fibrations.

§2 Review of connections and De Rham cohomology.

§3 Dévissage.

§4 Generic finiteness of direct images.

§5 Generic base change for direct images.

§6 Coherence of the cokernel of a regular connection.

§7 Regularity and exponents of the cokernel of a regular connection.

§8 Proof of the main theorems: finiteness, regularity, monodromy, base change (in the regular case).

Appendix C: Berthelot’s comparison theorem on O_{X}- vs. D_{X}-linear duals.

Appendix D: Introduction to Dwork’s algebraic dual theory.

Introduction.

This chapter develops a new approach to the study of direct images of connections (i.e. De Rham cohomology with coefficients, endowed with the Gauss-Manin connection) with respect to a smooth, not necessarily proper, morphism of smooth algebraic varieties in characteristic 0.

We present elementary and purely algebraic proofs of the generic finiteness and base change theorems, as well as of the fundamental finiteness, regularity, monodromy and base change theorems for direct images of regular algebraic connections.

Our proofs use neither resolution of singularities (beyond the classical case of embedded resolution of curves in surfaces), nor the theory of holonomy. In fact, in contrast to the now standard methods which consist of trying to extend all objects to a good compactification and studying ramification at infinity, our strategy relies upon a dévissage inspired by Artin’s theory of elementary fibrations. This approach allows us to reduce problems to the simple case of an ordinary differential operator in one variable (but in a relative situation), which we handle directly.

Let us now describe in more detail the content of each section.

In the first section, we consider M. Artin’s technique of good neighborhoods (local fibrations of a smooth variety by curves). This stands beside the technique of Lefschetz pencils (global fibrations of a smooth quasi-projective variety by hypersurfaces) as a powerful tool in algebraic geometry, offering the possibility of reasoning by induction on the dimension.

We combine both techniques to unscrew smooth morphisms into towers of coordinatized elementary fibrations; the basic pieces of this dévissage are thus relative versions of the projective line minus a few points, which we call rational elementary fibrations (1.2,3,4). By means of such elementary surgery, we attach to any smooth morphism f of smooth varieties an open dense subset A(f) of the base, the Artin set (1.8), which will play in this chapter the role held in P. Deligne’s theory [De] by those open subsets of the base over which f admits a good compactification à la Hironaka.

Section 2 is a review, in the elementary style of [Ka1], of connections (E, ∇) and their direct images R_{DR}^{i}f_{∗}(E, ∇) := R^{i}f_{∗}DR_{X/S}(E, ∇) under a smooth mor-

phism X⟶^{f}S over an algebraically closed field K of characteristic 0.

Section 3 is fundamental to the whole chapter. We consider properties P of connections, which are local for the étale topology, and strongly exact (in the sense that P holds for the middle term of a short exact sequence if and only if P holds for the other terms). The main lemma on dévissage asserts that in order to establish that for any smooth morphism of smooth K-varieties X⟶^{f}S, for any (quasi-coherent) connection (E, ∇) on X satisfying P, and for any i ≥ 0, the restriction of R_{DR}^{i}f_{∗}(E, ∇) to the Artin set A(f) ⊂ S satisfies P, it suffices to consider rational elementary fibrations f (in which case A(f) = S), and i = 0, 1. This essentially reduces the problem to the study of the kernel and cokernel of a connection in one variable.

There are several variants of this lemma, and furthermore, the arguments of dévissage extend to some of the properties we are interested in, and which cannot be expressed simply as a property P as above (e.g. base change). The idea of this dévissage has already appeared in our work on G-functions [AB].(1)

In §4, we apply this technique to establish generic finiteness of direct images R_{DR}^{i}f_{∗}(E, ∇), reducing the question to the case of a rational elementary fibration

(1) It is interesting to note that Artin’s technique of elementary fibrations was also used, independently, as a basic tool by A. de Jong in his now famous alteration theorem, as we learned shortly after beginning this work.

X⟶^{f}S (4.1); we are left with the task of proving that the cokernel of any nonzero differential operator Λ ∈ O(S)[x, \frac{d}{dx}] becomes finitely generated over O(S) after replacing S by some open dense subset. The point of the proof is that, as a consequence of integrability, the indicial polynomials ϕ_{i} at the singularities of Λ have the form

ϕ_{i} = ( non-zero element τ_{i} of O(S)) × (polynomial with constant coefficients) ;

in order to obtain finiteness of CokerΛ, it then suffices to remove from S the zero-locus of the τ_{i}.

In §5, we use the same reduction technique to prove generic base change for R_{DR}^{i}f_{∗} (5.1). Base change for a cokernel is automatic, while “generic” base change for Kerf_{∗}∇ follows from “generic” flatness of Cokerf_{∗}∇ (a consequence of “generic” coherence proven in §4).

With the help of the extension with logarithmic poles, constructed in (I, §4), we again take up in §6 the question of finiteness of R_{DR}^{1}f_{∗}(E, ∇), in the case of a rational fibration f and of a regular connection ∇, without localizing on S.

In §7, we handle the question of the regularity and exponents of the Gauss-Manin connection on R_{DR}^{1}f_{∗}(E, ∇) by using a method of N. Katz [Ka2]. In this question, one can assume that the base S is a curve, so that X is a surface, and embedded resolution of singularities (upon which Katz’s method relies) is elementary in this case.

One is then ready to prove in §8, by dévissage, the first main results of the theory: the finiteness, regularity, monodromy and base change theorems for R_{DR}^{i}f_{∗}(E, ∇) (endowed with the Gauss-Manin connection ℵ), when f : X ⟶ S is any smooth morphism of smooth K-varieties with S = A(f), and (E, ∇) is any regular connection. (The so-called monodromy theorem, in the form given by J. Bernstein [Bn], states that the exponents of ℵ belong to the rational space generated by 1 and by the exponents of ∇.)

We remark that unlike what occurs in the usual approaches, there is no simplification at all in our proof if the morphism f happens to be proper.

Appendix C is completely due to P. Berthelot. It is a comparison result between O_{X}- vs. D_{X/S}-linear duals of coherent D_{X/S}-modules, for any smooth morphism f : X ⟶ S of smooth algebraic varieties. This is preliminary to our definition of algebraic De Rham cohomology with compact supports in Appendix D.

The second appendix to this chapter originally started as an introduction to Dwork’s favorite approach to differential modules, namely his dual theory. While Dwork’s main concern is to give sharp p-adic growth estimates for the elements (power series) of the dual theory, we decided to restrict ourselves, for simplicity, to the algebraic situation. With time, appendix D developed into a proposal of definition of relative algebraic De Rham cohomology with compact supports and relative algebraic Poincaré duality based on the duality for coherent sheaves suggested by Deligne in the appendix of [H1].

We define a functor f_{!} for coherent O_{X}-modules, and extend it to a triangulated category, intermediate between the bounded derived category of O_{X}- modules and that of abelian sheaves, whose objects are complexes of O_{X}-modules and the differentials are relative differential operators. This category, already considered by Herrera and lieberman [HL], seems to be the natural ambiance where De Rham complexes live, and its objects should presumably play the role of analytically constructible coefficients in an algebraic Riemann-Hilbert correspondence. On applying f_{!} to the relative De Rham complex, one gets the definition of relative algebraic De Rham cohomology with compact supports. Our main duality theorem D.2.17 is based on Berthelot’s comparison theorem C.2, and shows how our definition is related to the standard algebraic definition of algebraic direct image with compact supports of a D-module [Me5, I.5.2.4], [Bo, VI.4.0, eq.1].

After this general construction, we show how Dwork’s definition of his “algebraic dual theory”, coincides with the previous notion of algebraic De Rham cohomology with compact supports. The proof of the comparison theorem D.2.20 was part of L. Fiorot’s thesis work. In the next sections, we make explicit calculations for a rational elementary fibration and illustrate some results of Dwork’s on the Gauss hypergeometric function. The dual theory offers alternative proofs of most of the results we obtain in this book and poses interesting comparison problems.

It is a pleasure to acknowledge some useful conversations with P. Berthelot on this topic, and the help we received from M. Cailotto and L. Fiorot.

§1 Elementary fibrations.

Let K be an algebraically closed field of characteristic 0. A K-variety will always be meant to be a reduced separated K-scheme of finite type, not necessarily irreducible.

Definition 1.1. A morphism f : X ⟶ S of smooth algebraic K-varieties is called an elementary fibration if it can be embedded in a commutative diagram

(1.1.1)  \begin{aligned}X & ↪^{j} & \overline{X} & ↩ & Z \\ & ↘^{f} & ↓ \overline{f} & ↙^{g} \\ & & S \end{aligned}

where

(i) Z is a closed reduced subscheme of \overline{X} and j is the complementary open immersion; (ii) \overline{f} is projective and smooth with geometric fibers irreducible of dimension 1; (iii) g is an étale covering.

Lemma 1.1.2. An elementary fibration is a surjective affine morphism.

Proof. Proof. We may then assume that Z is a disjoint union of divisors isomorphic to S via g. For each point s ∈ S, Z_{s} = g^{−1}(s) is then a positive effective divisor on the projective curve \overline{X}_{s}/κ(s), hence it is ample [H3, Chap. IV, Cor. 3.3]. By [EGA III, 4.7.1], Z is then a relatively ample divisor in the relative projective curve \overline{X}/S. We conclude by [EGA II, 5.5.7] that f is affine. Q.E.D.

Definition 1.2. An elementary fibration f as in (1.1) is rational if \overline{X} = P_{S}^{1}, \overline{f} is the projection on S and Z is a disjoint union of images of sections σ_{i}, i = 1, …, r, ∞, of \overline{f}, one of which is the section σ_{∞} : S ⟶ Z_{∞} = \{∞\}×S ⊂ P_{S}^{1}. We thus have the following diagram

(1.2.1)  \begin{aligned}X & ↪ & A_{S}^{1} & ↩ & ∐_{i=1}^{r} σ_{i}(S) \\ & ↘^{f} & ↓_{S}^{pr} & ↙ \\ & & S \end{aligned}

If K = C, any rational elementary fibration is thus topologically locally trivial for the classical complex topology.

Definition 1.3. An elementary fibration, as in (1.1), is coordinatized if it can be embedded in a commutative diagram

(1.3.1)  \begin{aligned}X & ↪^{j} & \overline{X} & ↩ & Z \\ ↓ π & & ↓ \overline{π} & & ↓ π′ \\ Y & ↪^{j′} & P_{S}^{1} & ↩ & Z′ \\ & ↘^{f′} & ↓_{S}^{pr} & ↙^{g′} \\ & & S \end{aligned}

where

(i) \overline{π} is finite and π is an étale covering; (ii) f = f′ ∘ π, \overline{f} = pr_{S} ∘ \overline{π}, g = g′ ∘ π′; (iii) the lower part of the diagram is an elementary fibration with j′(Y) ⊂ A_{S}^{1}.

We point out that the two squares in (1.3.1) are necessarily cartesian and that π′ is an étale covering. A rational elementary fibration is ipso facto coordinatized. Conversely, given a coordinatized elementary fibration as in (1.3.1), there exists an étale covering S′ ⟶ S such that the lower part of the pull-back of (1.3.1) over S′, is a rational elementary fibration.

Let f : X ⟶ S be a smooth morphism of smooth K-varieties, of relative dimension d. Assume S is connected. It follows from M. Artin’s theorem [SGA 4, Exp. XI, prop. 3.3] on “good neighborhoods” that there exists an étale dominant morphism ϵ : S′ ⟶ S and a finite open (affine) cover {U_{α}} of X′ = X ×_{S} S′, such that each U_{α} ⟶ S′ is a tower of elementary fibrations

(1.3.2)  U_{α} = V_{α,d} ⟶ V_{α,d−1} ⟶ … ⟶ V_{α,0} = S′ .

The integer d ≥ 0 will be the height of the tower (a tower of height 0 is then an isomorphism U_{α}⟶^{∼}S′). This result will play a fundamental role in the present chapter, in the following more precise form.

Proposition 1.4. Let f : X ⟶ S be a smooth morphism of smooth Kvarieties, with S connected. Then there exists an étale dominant morphism ϵ : S′ ⟶ S and a finite open (affine) cover {U_{α}} of X′ = X ×_{S} S′ such that each U_{α} ⟶ S′ is a tower of coordinatized elementary fibrations.

Proof. We may assume that f is of pure relative dimension d ≥ 0. Standard thickening arguments allow us to replace S by its geometric generic point SpecK(S)^{alg}, and then to replace the ground field K by K(S)^{alg}. The cases d = 0, 1 are then trivial. By induction on d (d ≥ 2), it suffices to find, for an arbitrary closed point ξ ∈ X, an open neighborhood U of ξ in X and a coordinatized elementary fibration U ⟶ T.

Let us first recall, for convenience, Artin’s construction of an elementary fibration U ⟶ T, let alone the condition of being “coordinatized”. We may assume X affine and consider a normal projective closure \overline{X}. We pick a very ample line bundle L on \overline{X} and embed \overline{X} in a projective space P^{N} via L^{⊗2}. Then a “general” linear space L ⊂ P^{N} of codimension d − 1 passing through ξ cuts \overline{X} and \overline{X} \ X transversally and avoids the singular locus of \overline{X}. Let H ⊂ P^{N} be a “general” hyperplane (not passing through ξ and avoiding L ∩ (\overline{X} \ X)), and let us consider the blow-up P̃^{N} of P^{N} along L ∩ H, and the morphism p : P̃^{N} ⟶ P^{d−1} induced by the linear projection with center L ∩ H. Let ˜ \overline{X} be the strict transform of \overline{X} in P̃^{N}, and let Z ⊂ ˜ \overline{X} be the disjoint union of \overline{X} \ X (considered as a subscheme of ˜ \overline{X} of codimension 1) and of the exceptional divisor ≃ P^{d−1} × (X ∩ L ∩ H). Then ˜ \overline{X} \ Z identifies with an open subscheme of X containing ξ, and the pull-back of the diagram

(1.4.1)  \begin{aligned}˜\overline{X} \ Z & ↪ & ˜\overline{X} & ↩ & Z \\ & ↘ & ↓ p_{| ˜\overline{X}} & ↙ \\ & & P^{d−1} \end{aligned}

over a suitable (affine) neighborhood T of p(ξ) in P^{d−1} defines an elementary fibration. Moreover the fiber of p_{| ˜\overline{X}} above p(ξ) identifies with \overline{X} ∩ L. The proposition is then a consequence of the following lemma.

Lemma 1.5. Given an elementary fibration

(1.5.1)  \begin{aligned}X & ↪ & \overline{X} & ↩ & Z \\ & ↘^{f} & ↓ \overline{f} & ↙ \\ & & S \end{aligned}

and a closed point ξ ∈ X, with image ζ = f(ξ), there is an open neighborhood U of ξ (resp. T of ζ), such that f induces a coordinatized elementary fibration U ⟶ T.

Proof. We denote by {ξ_{1}, …, ξ_{r}} the finite set {ξ} ∪ Z_{ζ} of closed points of \overline{X}_{ζ}. Let us choose a closed S-embedding \overline{X} ⟶ P_{S}^{M}. The fiber \overline{X}_{ζ} is a closed smooth curve in P^{M} to which the theory of Lefschetz pencils applies [SGA 7, II, exp. XVII]. There exists an open dense subset U of the Grassmannian of lines in the dual projective space P̌^{M} such that for any D_{ζ} ∈ U: (a)_{ζ} D_{ζ} is a Lefschetz pencil; in particular, the axis of D_{ζ} (which has codimension 2 in P^{M}) does not cut \overline{X}_{ζ}, so that D_{ζ} gives rise to a finite morphism \overline{π}_{ζ} : \overline{X}_{ζ} ⟶ P^{1} ≃ D_{ζ}; (b)_{ζ} if i ≠ j, \overline{π}_{ζ}(ξ_{i}) ≠ \overline{π}_{ζ}(ξ_{j}); (c)_{ζ} for each i, \overline{π}_{ζ} is étale at each point of \overline{π}_{ζ}^{−1}\overline{π}_{ζ}(ξ_{i}). In order to justify (b)_{ζ} and (c)_{ζ}, it suffices to show that the generic pencil satisfies these conditions; but this follows from the fact [loc. cit. 3.2.8] that for any hyperplane H_{i} ⊂ P^{M} containing ξ_{i}, but not ξ_{j} for j ≠ i, and cutting \overline{X}_{ζ} transversally, there exists a Lefschetz pencil of hyperplanes containing H_{i}.

Let us fix D_{ζ} ∈ U. Up to changing the coordinate in P^{1} ≃ D_{ζ}, we may assume that (d)_{ζ} the hyperplane H_{∞} corresponding to ∞ ∈ D_{ζ} cuts \overline{X}_{ζ} transversally, and does not meet the points ξ_{i}.

Let us now extend D_{ζ} to a relative line D/S in the dual projective space P̌_{S}^{M} (D is defined by an S-point in a suitable Grassmannian). After replacing S by a suitable Zariski neighborhood T of ζ, we may assume that the axis of D does not cut \overline{X}, so that D gives rise to a finite T -morphism \overline{π} : \overline{X} ⟶ P_{T}^{1} . We may also assume that:

(b) \overline{π}(Z) is étale over T, and does not contain \overline{π}(ξ);

(c) \overline{π} is étale at any point of \overline{π}^{−1}\overline{π}(Z);

(d) \overline{π}^{−1}(∞ × T) does not meet \overline{π}^{−1}\overline{π}(Z) nor \{ξ\}, and the restriction of \overline{π} to \overline{π}^{−1}(∞ × T) is étale.

According to [loc. cit. 6.3], we may assume furthermore that:

(a) for any closed point ζ′ ∈ T, D_{ζ′} is a Lefschetz pencil, and the branch locus B ⊂ P_{T}^{1} of \overline{π} is étale over T.

Then there is at most one ramification point in each fiber of \overline{π}^{−1}(B), so that \overline{π}^{−1}(B) is étale over T (and does not meet \overline{π}^{−1}\overline{π}(Z) ∐ \overline{π}^{−1}(∞ × T) nor \{ξ\}). It is now clear that U = \overline{X} \ (\overline{π}^{−1}(B) ∐ \overline{π}^{−1}\overline{π}(Z) ∐ \overline{π}^{−1}(∞ × T)) is a coordinatized elementary fibration (via \overline{π}), and ξ ∈ U ⊂ X.

This concludes the proof of lemma 1.5, hence of proposition 1.4. Q.E.D.

1.6 Remark. In fact, proposition 1.4 holds for a separably closed ground field K of any characteristic. The proof is the same, except that one must replace K(S)^{alg} by the separable closure of K(S), and that in lemma 1.5, it may be necessary to replace the given embedding \overline{X} ↪ P_{S}^{N} by a Veronese multiple (in characteristic 2, one should also invoke [loc. cit. 6.4] and replace L^{⊗2} by L^{⊗3} in 1.4).

1.7 Notation. Let f : X ⟶ S be a smooth morphism of K-varieties. We denote by d the maximum among the dimensions of the fibers. Let {U_{α}}_{α=0,...,r} be a finite open cover of X. For \underline{α} = (α_{0}, …, α_{p}), 0 ≤ α_{0} < α_{1} < … < α_{p} ≤ r, |\underline{α}| = p + 1, we denote by U\underline{α} the open subset U_{α_{0}} ∩ … ∩ U_{α_{p}} of X, and by f\underline{α} the restriction of f to U\underline{α}. We now introduce a chain of subsets of S, the Artin sets of f, which will be well suited for the dévissage in section 3 below.

Definition 1.8. The Artin set of f of level i ≥ 0, denoted by A_{i}(f), is the union of images ϵ(S′) ⊂ S, for all étale morphisms ϵ : S′ ⟶ S, finite over their image, such that:

1) X_{S′} admits a finite open cover {U_{α}} such that the restriction of f_{S′} to each U_{α} is a tower of coordinatized elementary fibrations;

2) each U\underline{α}, |\underline{α}| > 1, admits a finite open cover \{U_{\underline{α},β}\} such that the restriction of f_{S′} to each U_{\underline{α},β} is a tower of coordinatized elementary fibrations;

3) each U_{\underline{α},\underline{β}}, |\underline{β}| > 1, admits a finite open cover \{U_{\underline{α},\underline{β},γ}\} such that the restriction of f_{S′} to each U_{\underline{α},\underline{β},γ} is a tower of coordinatized elementary fibrations; … and so on, for all (\underline{α}, \underline{β}, \underline{γ}, …) such that

(|\underline{α}| − 1) + (|\underline{β}| − 1) + (|\underline{γ}| − 1) + … ≤ i .

Obviously, A_{i}(f) ⊃ A_{i+1}(f) ∀ i, and, for any U\underline{α} appearing in the previous definition, A_{i−|\underline{α}|+1}(f\underline{α}) ⊃ A_{i}(f). We set A(f) = A_{d+dimX}(f) (this definition will be justified in 3.2.4). From proposition 1.4, we deduce:

Corollary 1.9. For any i ≥ 0, A_{i}(f) is a dense open subset of the image of the (open) morphism f in S.

Clearly, all Artin sets of a tower X ⟶ S of coordinatized elementary fibrations coincide with S itself. If f is étale, they coincide with the maximal open subset of S on which f is an étale covering.

§2 Review of connections and De Rham cohomology.

2.1. Let X be a smooth algebraic K-variety and E be an O_{X}-module. We recall that an integrable connection on E is an additive map

(2.1.1)  E ⟶ E ⊗_{O_{X}} Ω_{X}^{1}

satisfying Leibniz rule and such that the usual composite arrow (“curvature”)

(2.1.2)  E ⟶ E ⊗_{O_{X}} Ω_{X}^{1} ⟶ E ⊗_{O_{X}} Ω_{X}^{2}

is 0. This is equivalent to giving an additive morphism

∇ : \underline{Der}_{X} ⟶ \underline{End}_{K}E

compatible with the Lie bracket [ , ] and satisfying

\begin{aligned}∇(fD)(m) & = f∇(D)(m) \\ ∇(D)(fm) & = f∇(D)(m) + D(f)m \end{aligned}

for local sections f, D, m of O_{X}, Der_{X} and E, respectively. In these circumstances, one may construct the De Rham complex of (E, ∇)

(2.1.3)  DR(X/K, (E, ∇)) := [E ⟶ E ⊗_{O_{X}} Ω_{X}^{1} ⟶ E ⊗_{O_{X}} Ω_{X}^{2} ⟶ …]

We denote by MIC(X) the category of O_{X}-modules with integrable connection. In this general setting (no assumptions on the underlying O_{X}-module), MIC(X) is isomorphic to the category of left D_{X}-modules ([BO, §2], [Kas]). The point is that any action, compatible with Lie brackets, of differential operators of order ≤ 1, extends to an action of D_{X}.

The category MIC(X) is thus abelian and has enough injectives. It has an internal ⊗, and an internal \underline{Hom} [Ka1, 1.1]^{(1)}.

(1) We drop Katz’s general quasi-coherence assumptions: see also [H2, §4]; otherwise, the internal \underline{Hom} would be only partially defined

Given two objects (E, ∇), (E′, ∇′) ∈ Ob MIC(X), an O_{X}-linear mapping E ⟶ E′ is called horizontal if it comes from a morphism in MIC(X).

Let (E, ∇) ∈ Ob MIC(X). We say that (E, ∇) is simple, if it is a simple object of MIC(X). We say that it is quasi-coherent (resp. coherent, resp. free) if the underlying O_{X}-module is. As is well-known, if (E, ∇) is coherent, then E is automatically locally free [Ka1, 8.8]; this is then the usual (restricted) notion of connection on a vector bundle, which is actually our main object of interest.

If f : X ⟶ S is smooth and Der_{X/S} is free of rank one with basis ∂ (e.g. f = a rational elementary fibration cf. (1.2.1)), we say that (E, ∇) is cyclic with respect to f (or ∂) if it is free of finite rank and admits a basis of the form {v, ∇(∂)v, …, ∇(∂)^{μ−1}v}. We then say that v is a cyclic vector. The definition is easily seen to be independent of the choice of ∂.

2.2 Inverse image. Let f : X ⟶ S be a morphism of smooth K-varieties. There is a functor

\begin{matrix}f^{∗} : MIC(S) & ⟶ MIC(X) \\ (E, ∇) & ⟼ (f^{∗}E, f^{∗}∇) , \end{matrix}

where f^{∗}∇ is defined as follows: by sheaf-theoretic inverse image, ∇ defines a mapping

(2.2.1)  f^{−1}E ⟶ f^{−1}E ⊗_{f^{−1}O_{S}} f^{−1}Ω_{S}^{1}

sitting in a canonical diagram

(2.2.2)  \begin{matrix}f^{−1}E & ⟶ & f^{−1}E ⊗_{f^{−1}O_{S}} f^{−1}Ω_{S}^{1} \\ ↓ & & ↓ \\ f^{∗}E & & f^{∗}E ⊗_{O_{X}} Ω_{X}^{1} \end{matrix}

There is a unique integrable connection f^{∗}∇ on f^{∗}E such that

(2.2.3)  f^{∗}∇ : f^{∗}E ⟶ f^{∗}E ⊗_{O_{X}} Ω_{X}^{1}

completes (2.2.2) into a commutative square. It is not immediate that (2.2.1), which is not f^{−1}O_{S}-linear, extends to (2.2.3)^{(1)}. For this reason, f^{∗}∇ is better described using Atiyah’s linear description of connections (cf. [De, I, 2.3]): with the notation of [EGA IV, 16.7.1.2], ∇ corresponds to an O_{S}-linear section of

(2.2.4)  P_{S}^{1}(E) ⟶ E ,

which defines an O_{X}-linear section of

(2.2.5)  f^{∗}P_{S}^{1}(E) ⟶ f^{∗}E ;

(1) The same slight difficulty arises in defining the internal ⊗ and is solved in the same way. since (2.2.5) factors through

(2.2.6)  P_{X}^{1}(f^{∗}E) ⟶ f^{∗}E

[loc. cit., 16.7.9], we obtain an O_{X}-linear section of (2.2.6) which corresponds to f^{∗}∇. One checks easily that this connection is integrable. Viewed as a D_{X}- module, f^{∗}(E, ∇) corresponds to D_{X→S} ⊗_{f^{−1}D_{S}} f^{−1}E (cf. [Bo, VI, 4.2]).

The functor f^{∗} is right-exact.

2.3 Direct image. Assume moreover that f is smooth. There is a functor

R_{DR}^{0}f_{∗} : MIC(X) ⟶ MIC(S)
(E, ∇) ⟼ (f_{∗}E^{∇_{|\underline{Der}X/S}}, induced connection) ,

which is left-exact.

Proposition 2.3.1. Let f : X ⟶ S be a smooth morphism of smooth Kvarieties with connected geometric fibers. Then for any coherent (E, ∇) ∈ ObMIC(X), the natural morphism f^{∗}R_{DR}^{0}f_{∗}(E, ∇) ⟶ (E, ∇) is a monomorphism.

Proof. Let x be any closed point of X. We first show that O_{X,x} ⊗_{O_{S,f(x)}} E_{x}^{∇} ⟶ E_{x} is injective. Because the geometric fibers of f are connected, F rac(O_{X,x})/F rac(O_{S,f(x)}) is a regular extension and F rac(O_{X,x})⊗_{F rac(O_{S,f(x)})}F rac(Ô_{S,f(x)}) ⟶ F rac(Ô_{X,x}) is injective; hence, O_{X,x} ⊗_{O_{S,f(x)}} Ô_{S,f(x)} ⟶ Ô_{X,x} is itself injective. Because E_{x} is free, it embeds into its 𝔪_{x}-adic completion Ê_{x}, and E_{x}^{∇} ⊆ Ê_{x}^{∇}. We are thus reduced to show that Ô_{X,x} ⊗Ô_{S,f(x)} Ê_{x}^{∇} ⟶ Ê_{x} is injective. This is in fact an isomorphism by the argument of Taylor series in [Ka1, 8.9]. We have shown that O_{X} ⊗_{f^{−1}O_{S}} E^{∇} ⟶ E is injective, and we conclude by noticing that f^{−1}f_{∗}E^{∇} ⟶ E^{∇} is injective (because f^{−1}f_{∗}E ⟶ E is, E being locally free and X connected) and f is flat. Q.E.D.

2.4 Higher direct images. The q-th derived functor of R_{DR}^{0}f_{∗} is denoted by

R_{DR}^{q}f_{∗} : MIC(X) ⟶ MIC(S) .

For any (E, ∇) ∈ Ob MIC(X), the concrete description of R_{DR}^{j}f_{∗}(E, ∇) is given by the R^{q}f_{∗} (for complexes of abelian sheaves or, what amounts to the same, of f^{−1}O_{S}-modules on X) of the relative de Rham complex DR(X/S, (E, ∇)) of (E, ∇), endowed with the Gauss-Manin connection [Ka1, 3.0]. (One checks that one obtains in this way an effaceable δ-functor, and applies [H3, III, 1.3A and 1.4]).

By [EGA 0_{III}, 12.4.3] applied to the morphism of ringed spaces

f : (X, f^{−1}O_{S}) ⟶ (S, O_{S})

and K^{·} = DR(X/S, (E, ∇)), the sheaf underlying R_{DR}^{q}f_{∗}(E, ∇) is the sheaf associated to the presheaf

(2.4.1)  H^{q}(f, K^{·}) : U ⟼ H^{q}(f^{−1}(U), K_{|f^{−1}(U)}^{·})

on S. We see in particular that the formation of R_{DR}^{q}f_{∗} is compatible with Zariski localization on S. If S is a point, we also write H_{DR}^{q}(X, (E, ∇)) for the K-vector space R_{DR}^{q}f_{∗}(E, ∇).

Let us recall one of the constructions of the Gauss-Manin connection (cf. [KaO]). We filter the absolute De Rham complex DR(X/K, (E, ∇)) by the subcomplexes

F^{p} = image of f^{∗}(Ω_{S}^{p}) ⊗_{O_{X}} Ω_{X}^{·−p} ⊗_{O_{X}} E ,

so that

gr^{p} = F^{p}/F^{p+1} ≅ f^{∗}(Ω_{S}^{p}) ⊗_{O_{X}} Ω_{X/S}^{·−p} ⊗_{O_{X}} E ,
R^{q}f_{∗}(gr^{0}) ≅ R_{DR}^{q}f_{∗}(E, ∇) , R^{q+1}f_{∗}(gr^{1}) ≅ Ω_{S}^{1} ⊗_{O_{S}} R_{DR}^{q}f_{∗}(E, ∇) .

The Gauss-Manin connection is the coboundary map in the long cohomology sequence of the R^{q}f_{∗} arising from the short exact sequence

(2.4.2)  0 ⟶ gr^{1} ⟶ F^{0}/F^{2} ⟶ gr^{0} ⟶ 0 .

When f is a (smooth) morphism of pure relative dimension d, R_{DR}^{q}f_{∗} coincides with the notion of higher direct image h^{q−d}f_{+} of D_{X}-modules [Bo, VI, 5.0]^{(1)}.

2.5 Open covers. Let {U_{α}} be an open cover of X (not necessarily finite). In the following proposition, a variation of [EGA 0_{III}, 12.4.6], h^{q}(f, K^{·}) is the

(1) Although this fact only plays no role in the present chapter, we sketch the following argument. As in the proof of [Bo, VI, 5.3.2], what has to be proven is that, for d = dimX − dimS, DR(X/S, D_{X})[d] is a locally free left resolution of D_{S←X} in the category of right D_{X}-modules. Local freedom is obvious. For an étale map f = j : X ⟶ S, D_{S←X} = D_{X→S} = j^{∗}(D_{S}) = D_{X} = DR(X/S, D_{X}), by definition ([Bo, VI, 4.2] and [Bo, VI, 5.1 (4)]). For a projection f = p : A_{S}^{d} ⟶ S the assertion is proven in [Bo, VI, 5.3.1 (i)]. Consider the case of f = p ∘ j : X⟶^{j}X′ = A_{S}^{d}⟶^{p}S, composite of a projection and of an étale map j. Then, by [Bo, VI, 6.3 (2)], D_{S←X} = j^{−1}(D_{S←X′}) ⊗_{j^{−1}(D_{X′})} D_{X′←X} =

j^{−1}(D_{S←X′})⊗^{L}_{j^{−1}(D_{X′})}D_{X′←X}, and D_{X′←X} = D_{X}. So, D_{S←X} = j^{∗}(D_{S←X′}) =

Lj^{∗}(D_{S←X′}) = j^{∗}(DR(X′/S, D_{X′})[d]) = DR(X/S, D_{X})[d]. Since any smooth morphism f : X ⟶ S is locally for the Zariski topology on X of the type p ∘ j, the assertion follows. In fact, this argument accounts only for the coincidence of R_{DR}^{q}f_{∗}(E, ∇) and h^{q−d}f_{+}(E, ∇) as O_{S}-modules; for the comparison of the D_{S}- module structures, the crucial case is again that of a projection A^{n+d} → A^{n}, for which one can use the explicit computations of [Bo, V, 3.5]. presheaf U ⟼ R^{q}f_{|U∗}(K_{|U}^{·}), while h^{q}(f, K^{j}) is U ⟼ R^{q}f_{|U∗}(K_{|U}^{j}), both presheaves on X taking values in the category of sheaves of O_{S}-modules, and h^{q}(f, K^{·}) is the complex of presheaves obtained from term by term application of the latter. For a presheaf F (resp. for a complex of presheaves F^{·}) on X, with values in any abelian category, the notation C^{·}({U_{α}}, F) (resp. C^{·}({U_{α}}, F^{·})) refers to the usual Čech complex (resp. bicomplex). If F is a presheaf of abelian groups on X, we also use the notation C_{f}^{i}(\{U_{α}\}, F) = ⊕_{α_{0}<α_{1}<...<α_{i}}f_{\underline{α}∗}(F_{|U\underline{α}}), for the abelian presheaf of (alternating) Čech cochains.

Lemma 2.5.1. Let K^{·} be a complex of f^{−1}O_{S}-modules on X. There exist two regular spectral functors _{I}E_{·}^{·,·} and _{II}E_{·}^{·,·} in K^{·} taking values in the category of O_{S}-modules and converging to the relative hypercohomology R^{·}f_{∗}(K^{·}). The E_{1} terms are given by

\begin{aligned}_{I}E_{1}^{p,q} = (C^{·}(\{U_{α}\}, h^{q}(f, K^{·})))_{tot}^{p} = ⊕_{i+j=p} & C^{i}(\{U_{α}\}, h^{q}(f, K^{j})) \\ & = ⊕_{i+j=p,α_{0}<α_{1}<...<α_{i}}R^{q}f_{\underline{α}∗}(K_{|U\underline{α}}^{j}) , \end{aligned}

IIE_{1}^{p,q} = C^{p}(\{U_{α}\}, h^{q}(f, K^{·})) = ⊕_{α_{0}<α_{1}<...<α_{p}}R^{q}f_{\underline{α}∗}(K_{|U\underline{α}}^{·}) ,

respectively. If K^{·} is bounded from below, the previous spectral sequences are biregular.

Proof. Following [EGA 0_{III}, 12.4.6], we consider an injective Cartan-Eilenberg resolution L^{·,·} of K^{·} and the tricomplex of O_{S}-modules

C_{f}^{·}({U_{α}}, L^{·,·}) = ⊕_{i,j,k}C_{f}^{i}({U_{α}}, L^{j,k}) .

We first regard C_{f}^{·}({U_{α}}, L^{·,·}) as a bicomplex for the degrees (i, j + k). The second spectral sequence of this bicomplex is regular and degenerate, since H_{I}^{q} is the associated sheaf to the presheaf V ⟼ H^{q}({U_{α} ∩ f^{−1}(V)}, L^{j,k}) on S, and this is 0 for q > 0, L_{|f^{−1}(V)}^{j,k} being flasque. So, H^{n}(C_{f}^{·}({U_{α}}, L^{·,·})_{tot})⟶^{∼}R^{n}f_{∗}K^{·}. We then define _{I}E_{·}^{·,·} as the first spectral sequence of C_{f}^{·}({U_{α}}, L^{·,·}), viewed as a bicomplex for the degrees (i+j, k) (this is the spectral sequence in the statement of [loc. cit.]). Here H_{II}^{q} = ⊕_{i,j}H^{q}(C_{f}^{i}({U_{α}}, L^{j,·})) = ⊕_{i,j}C^{i}({U_{α}}, h^{q}(f, K^{j})), since L_{|U}^{j,·} is an injective resolution of K_{|U}^{j}. As _{II}E_{·}^{·,·} we take instead the first spectral sequence of C_{f}^{·}({U_{α}}, L^{·,·}), viewed as a bicomplex for the degrees (i, j + k). Q.E.D.

In the special case of K^{·} = DR(X/S, (E, ∇)), the spectral sequence _{II}E_{·}^{·,·} of the lemma is the Zariski spectral sequence of O_{S}-modules of [Ka1, 3.5.1.0]:

(2.5.2)  E_{1}^{p,q} = ⊕_{α_{0}<α_{1}<...<α_{p}}R_{DR}^{q}f_{\underline{α}∗}(E, ∇)_{|U\underline{α}} ⇒ R_{DR}^{p+q}f_{∗}(E, ∇) .

If we replace DR(X/S, (E, ∇)) by the short exact sequence (2.4.2), and use the fact that hypercohomology of a direct image is a cohomological functor, we obtain a coboundary morphism of spectral sequences

E_{1}^{p,q} ≅ E_{1}^{p,q}(gr^{0})  ⇒  R_{DR}^{p+q}f_{∗}(E, ∇)
↓  ↓
E_{1}^{p,q+1}(gr^{1}) ≅ Ω_{S/K}^{1} ⊗_{O_{S}} E_{1}^{p,q}(gr^{0}) ⇒ R^{p+q+1}f_{∗}(gr^{1}) ≅ Ω_{S/K}^{1} ⊗_{O_{S}} R_{DR}^{p+q}f_{∗}(E, ∇) ,

which expresses the fact that (2.5.2) “is” a spectral sequence in MIC(S). On the other hand, the E_{2}^{pq} term of the spectral sequence _{I}E_{·}^{·,·} of the proposition is

IE_{2}^{p,q} = total homology of degree p of the bicomplex C^{·}({U_{α}}, h^{q}(f, K^{·})) .

This implies, in particular, that if each U_{α} is affine, and if E is quasi-coherent, then R_{DR}^{j}f_{∗}(E, ∇) may be calculated “à la Čech”, as the j-th (total) homology O_{S}-module of the bicomplex of O_{S}-modules C_{f}^{·}({U_{α}}, Ω_{X/S}^{·} ⊗ E), of alternating Čech cochains on the nerve of the cover {U_{α}}.

It turns out that the Gauss-Manin connection comes, upon passage to homology, from a (not integrable) connection on this bicomplex: if ∂ is a vector field on S, and if ∂_{α} is a lifting of ∂ on U_{α}, the value at ∂ of this connection can be expressed as a sum of the Lie derivative L(∂_{α_{0}}) and the interior product I(∂_{α_{0}} − ∂_{α_{1}}) (up to sign), see [KO, §3].

The proof of loc. cit. depends on a local calculation. We present here an alternative argument, based on functoriality. Let us denote by d_{DR} the differential of the De Rham complex Ω_{X/S}^{·} ⊗ E or of C_{f}^{p}({U_{α}}, Ω_{X/S}^{·} ⊗ E), and by ď the Čech differential on C_{f}^{p}({U_{α}}, Ω_{X/S}^{·} ⊗ E). We follow the convention of [KO] (not the one in [EGA III]) by putting the total differential δ = d_{DR} + (−)^{q}ď on the term C_{f}^{p,q} = C_{f}^{p}({U_{α}}, Ω_{X/S}^{q} ⊗ E) of the bicomplex. We denote by C_{f}^{·} = C_{f}^{·}(E, ∇) the associated simple complex, so that

H^{i}(C_{f}^{·}(E, ∇)) ≅ R^{i}f_{∗}(Ω_{X/S}^{·} ⊗ E) ,

functorially in (E, ∇). For each vector field ∂ on S, we define an additive endomorphism ℵ̃(∂) of C_{f}^{i} by

ℵ̃(∂) = L(∂_{α_{0}}) + (−)^{q}I(∂_{α_{0}} − ∂_{α_{1}})

on C_{f}^{i−q,q}. This endomorphism satisfies the Leibniz rule, because the Lie derivative does; hence the ℵ̃(∂) define a connection ℵ̃ on C_{f}^{i}(E, ∇), which is functorial in (E, ∇) (because the Lie derivative L(∂_{α_{0}}) : Ω_{U_{α_{0}} /S}^{q} ⊗ E_{|U_{α_{0}}} ⟶ Ω_{U_{α_{0}} /S}^{q} ⊗ E_{|U_{α_{0}}} is). On the other hand, we have the formulas i) [L(∂_{α_{0}}), d_{DR}] = 0, ii) L(∂_{α_{0}}) − L(∂_{α_{1}}) = d_{DR} ∘ I(∂_{α_{0}} − ∂_{α_{1}}) + I(∂_{α_{0}} − ∂_{α_{1}}) ∘ d_{DR}, iii) I(∂_{α_{0}} − ∂_{α_{1}}) + I(∂_{α_{1}} − ∂_{α_{2}}) + I(∂_{α_{2}} − ∂_{α_{0}}) = 0. From ii) we deduce iv) [L(∂_{α_{0}}), (−)^{q}ď] + [(−)^{q}I(∂_{α_{0}} − ∂_{α_{1}}), d_{DR}] = 0, and iii) implies

v) [(−)^{q}I(∂_{α_{0}} − ∂_{α_{1}}), (−)^{q}ď] = 0.

Formulas i), iv), v) put together show that [ℵ̃(∂), δ] = 0, i.e. ℵ̃(∂) is an (additive) endomorphism of the complex C_{f}^{·}(E, ∇). We thus obtain functors

\begin{matrix}ℵ^{i} : MIC(X) & ⟶ MC(S) \\ (E, ∇) & ⟼ (H^{i}(C_{f}^{·}(E, ∇)) ≅ R^{i}f_{∗}(Ω_{X/S}^{·} ⊗ E), H^{i}(ℵ̃)) , \end{matrix}

where MC(S) stands for the abelian category of O_{S}-modules with (not necessarily integrable) connection. By functoriality of ℵ̃ with respect to (E, ∇) it is clear that the collection of the ℵ^{i} form a δ-functor. This δ-functor is effaceable (because any (E, ∇) imbeds into an injective (I, ∇) of MIC(X) and R^{i}f_{∗}(Ω_{X/S}^{·} ⊗ E) = 0, for i > 0), hence ℵ^{i} = R^{i}ℵ^{0}. Let ω : MIC(S) ⟶ MC(S) stand for the (exact) imbedding. We have ℵ^{0} = ω ∘ R_{DR}^{0}f_{∗}, because ℵ̃(∂) is none but the Gauss-Manin connection on H^{0}(C_{f}^{·}) ≅ f_{∗}E^{∇_{|\underline{Der}_{X/S}}}, being induced by the Lie derivative in the direction ∂ acting on E. We conclude that ℵ^{i} = R^{i}(ω ∘ R_{DR}^{0}f_{∗}) = ω ∘ R_{DR}^{i}f_{∗}.

2.6 Flat base change. Let

X^{♯} ⟶^{u^{♯}} X
f^{♯} ↓  ↓ f
S^{♯}  ⟶^{u} S

be a fibered square, u being a morphism of smooth K-varieties. We have u^{♯∗}Ω_{X/S}^{·} ≃ Ω_{X^{♯}/S^{♯}}^{·} and u^{♯∗}DR(X/S, (E, ∇)) ≃ DR(X^{♯}/S^{♯}, u^{♯∗}(E, ∇)). (Notice that despite the non-linearity of the relative De Rham complex, the complex u^{♯∗}DR(X/S, (E, ∇)) is well-defined because u^{♯∗}(−) may be interpreted as

f^{♯−1}(O_{S^{♯}}) ⊗_{u^{♯−1}f^{−1}(O_{S})} u^{♯−1}(−) ).

Moreover, if E is quasi-coherent, and if u is flat (e.g. an étale localization), we have u^{∗}R_{DR}^{q}f_{∗}(E, ∇)⟶^{∼}R_{DR}^{q}f_{∗}^{♯}(u^{♯∗}(E, ∇)), by flat base change for hypercohomology. One can also choose an affine open cover of X, use the first spectral sequence of lemma 2.5.1 (where _{I}E_{1}^{p,q} = 0 for q > 0), and flat base change for f_{\underline{α}∗}(Ω_{U\underline{α}/S}^{j}).

2.7 Vanishing. Assume E is quasi-coherent and let d be the maximum among the dimensions of the fibers of f. Then R_{DR}^{j}f_{∗}(E, ∇) = 0 for j ∉ [0, d + dimX], as can be seen from the Hodge-DeRham spectral sequence E_{1}^{p,q} = R^{q}f_{∗}(Ω_{X/S}^{p} ⊗ E) ⇒ R_{DR}^{p+q}f_{∗}(E, ∇).

Assume moreover that f is affine. Then R_{DR}^{j}f_{∗}(E, ∇) = 0 for j ∉ [0, d]. In this case, the first spectral sequence of hypercohomology [EGA 0_{III}, 12.4.1.1]

E_{2}^{p,q} = H^{p}(R^{q}f_{∗}DR(X/S, (E, ∇))) ⇒ R^{p+q}f_{∗}DR(X/S, (E, ∇))

degenerates at E_{2}, so that

R_{DR}^{j}f_{∗}(E, ∇) ≅ H^{j}(f_{∗}DR_{X/S}(E, ∇))

(in particular R_{DR}^{1}f_{∗}(E, ∇) ≅ Cokerf_{∗}(∇), if d = 1). If S is affine (hence so is X), R_{DR}^{j}f_{∗}(E, ∇) may thus be computed as the O_{S}-module attached to the j-th cohomology Γ(S, O_{S})-module of the global relative De Rham complex

E(X) ⟶ E(X) ⊗ Ω_{X/S}^{1}(X) ⟶ ….

More generally, if S is affine and {U_{α}} is an affine open cover of X, R_{DR}^{j}f_{∗}(E, ∇) is the O_{S}-module attached to the j-th total cohomology group of the Čech bicomplex C^{·}({U_{α}}, Ω_{X/S}^{·} ⊗ E).

§3 Dévissage.

3.1. In the sequel, we consider properties of O_{X}-modules with integrable connection, for arbitrary smooth K-varieties X; we write P((E, ∇)) to indicate that (E, ∇) satisfies property P.

We say that P is local for the étale topology (resp. stable under finite étale direct image) if for any X, any (E, ∇) ∈ Ob MIC(X) and any étale cover {V_{i}} of X (resp. any finite étale π : X ⟶ X′), one has

P((E, ∇)) ⇔ (P((E, ∇)_{|V_{i}}) , ∀ i)

(resp. P((E, ∇)) ⇒ P(π_{∗}(E, ∇)) ) (here we write simply π_{∗}(E, ∇) instead of R_{DR}^{0}π_{∗}(E, ∇), the underlying O_{X′}-module being π_{∗}E).

We say in general that a property P of objects of an abelian category A is strongly exact if P(0) and for any exact sequence

E_{1} ⟶ E ⟶ E_{2}

in A, one has ((P(E_{i}), for i = 1, 2) ⇒ P(E)). Strong exactness is equivalent to the conjunction of exactness in the sense of [EGA III, 3.1.1] and stability by taking subobjects.

Lemma 3.1.1. Let P be a property of modules with integrable connection on smooth K-varieties, which is strongly exact and local for the étale topology. Then P is stable under finite étale direct image.

Proof. Let (E, ∇) ∈ Ob MIC(X) and π : X ⟶ Z be an étale covering. By [SGA 1, Exp. V, §4 and §7] there exists τ : Y ⟶ X such that π ∘ τ is a Galois covering of Z of Galois group, say, G. Then [loc. cit., Lemme 3.3] τ is necessarily étale. Let σ : Z′ ⟶ Z be an étale covering such that Y_{Z′} ⟶ Z′ [loc. cit., Prop. 2.6, (ii bis)] is the trivial Galois covering Z′×G⟶^{pr_{1}}Z′. We have the commutative diagram with fibered squares

Y ⟶^{τ} X ⟶^{π} Z
↑  ↑ σ′  ↑ σ
Z′ × G = Y_{Z′}  ⟶^{τ′} X_{Z′}  ⟶^{π′} Z′

where all morphisms are étale, π and σ are étale coverings, and π′ ∘ τ′ = pr_{1}.

We have: P((E, ∇)) ⇔ P(σ^{′∗}(E, ∇)) and, since π′_{∗}σ^{′∗}(E, ∇) ≅ σ^{∗}π_{∗}(E, ∇), P(π_{∗}(E, ∇)) ⇔ P(π′_{∗}σ^{′∗}(E, ∇)). So we may assume that (E′, ∇′) is an object of MIC(X_{Z′}) and must prove that P((E′, ∇′)) ⇒ P(π′_{∗}(E′, ∇′)). Now, P((E′, ∇′)) ⇒ P(τ^{′∗}(E′, ∇′)). Since Z′ × G = ∐_{γ∈G} Z′ × {γ}, an object (M, ∇) of MIC(Z′ × G), consists of local components (M, ∇)_{γ} in MIC(Z′ × {γ}) = MIC(Z′), and P((M, ∇)) ⇔ (P((M, ∇)_{γ}) , ∀ γ ∈ G). Since G operates on Z′ × G by Z′-automorphisms, for any γ ∈ G

(pr_{1}^{∗} ∘ pr_{1∗}(M, ∇))_{γ} ≅ ⊕_{δ∈G} (M, ∇)_{δ} ,

and P((M, ∇)) ⇔ P(pr_{1}^{∗} ∘ pr_{1∗}(M, ∇)). In particular,

P((E′, ∇′)) ⇒ P(pr_{1}^{∗} ∘ pr_{1∗} ∘ τ^{′∗}(E′, ∇′)) ⇔ P(τ^{′∗} ∘ π^{′∗} ∘ π′_{∗} ∘ τ′_{∗} ∘ τ^{′∗}(E′, ∇′)) .

But (E′, ∇′) ↪ τ′_{∗}∘τ^{′∗}(E′, ∇′) is a monomorphism, and the functor τ^{′∗}∘π^{′∗}∘π′_{∗} = pr_{1}^{∗} ∘ π′_{∗} is left-exact, pr_{1}^{∗} being exact and π′_{∗} left-exact. The conclusion is that τ^{′∗} ∘ π^{′∗} ∘ π′_{∗}(E′, ∇′) is a sub-object of τ^{′∗} ∘ π^{′∗} ∘ π′_{∗} ∘ τ′_{∗} ∘ τ^{′∗}(E′, ∇′), so that P(τ^{′∗} ∘ π^{′∗} ∘ π′_{∗}(E′, ∇′)), i.e. P(pr_{1}^{∗}(π′_{∗}(E′, ∇′))), holds. But pr_{1} is an étale covering, hence P(π′_{∗}(E′, ∇′)) must hold. Q.E.D.

Lemma 3.2. (Lemma on dévissage, first form.) Let P be a property of modules with integrable connection on smooth K-varieties, which is strongly exact and local for the étale topology. Assume that (∗) for any rational elementary fibration f′ : X′ ⟶ S′ with S′ affine, étale over some affine K-space, and for any quasi-coherent (E′, ∇′) ∈ Ob MIC(X′),

P((E′, ∇′)) ⇒ P(R_{DR}^{j}f′_{∗}(E′, ∇′)) , j = 0, 1 .

Then for any i ≥ 0, any smooth morphism f : X ⟶ S of smooth K-varieties, and any quasi-coherent (E, ∇) ∈ Ob MIC(X),

P((E, ∇)) ⇒ P((R_{DR}^{i}f_{∗}(E, ∇))_{|A_{i}(f)}) .

Proof.

3.2.1. The case when f has relative dimension 0 is a simple consequence of the stability of P under finite étale direct image; we discard this trivial case in the sequel. Because P is local, we may assume that S is connected. On the other hand, if f = f_{1} ∐ f_{2} : X = X_{1} ∐ X_{2} ⟶ S, then R_{DR}^{j}f_{∗}(E, ∇) = R_{DR}^{j}f_{1∗}((E, ∇)_{|X_{1}}) ⊕ R_{DR}^{j}f_{2∗}((E, ∇)_{|X_{2}}); because P is strongly exact, we may thus assume that X is connected so that f has pure relative dimension d. We may also assume that 0 ≤ j ≤ d + dimX, otherwise R_{DR}^{j}f_{∗}(E, ∇) = 0, hence the result is trivial.

3.2.2. We may replace at once S by A_{i}(f), and then (because P is local for the étale topology and R_{DR}^{j}f_{∗} is compatible with étale localization for quasicoherent E (2.3)) by an affine connected étale neighborhood S′ (étale over some affine space if we wish) such that X_{S′} admits a finite open cover {U_{α}} with the properties listed in definition 1.8. Let us consider the Zariski spectral sequence in MIC(S) (2.2.2):

E_{1}^{p,i−p} = ⊕_{α_{0}<α_{1}<...<α_{p}}R_{DR}^{i−p}f_{\underline{α}∗}(E, ∇)_{|U\underline{α}} ⇒ R_{DR}^{i}f_{∗}(E, ∇) .

By induction on i, since A_{i−|\underline{α}|+1}(f\underline{α}) = A_{i}(f) = S, P(R_{DR}^{i−p}f_{\underline{α}∗}(E, ∇)_{|U\underline{α}}) for all \underline{α} such that |\underline{α}| = p + 1 > 1. Thus, by strong exactness of P, this spectral sequence reduces us to prove P(R_{DR}^{i}f_{α∗}(E, ∇)_{|U_{α}}); hence we may assume that f itself is a tower

X = X_{d} ⟶ X_{d−1} ⟶ … ⟶ X_{0} = S

of coordinatized elementary fibrations over an affine connected base S (étale over some affine space if we wish). Such a tower gives rise to Leray spectral sequences in MIC (cf. [Ka1, 3.3]); by strong exactness of P, we are reduced to proving P for each level of the tower, i.e. we may assume that f itself is a coordinatized elementary fibration.

3.2.3. In case f is a coordinatized elementary fibration,

(3.2.3.1)  \begin{aligned}X & ↪^{j} & \overline{X} & ↩ & Z \\ ↓ π & & ↓ \overline{π} & & ↓ \\ Y & ↪ & P_{S}^{1} & ↩ & Z′ \\ & ↘^{f′} & ↓ & ↙ \\ & & S \end{aligned}

(f = f′ ∘ π), we may assume that i = 0, 1: other cohomology sheaves are 0, since f is affine of relative dimension 1 (cf. 2.4). Because P is étale-local, we may replace S by an étale covering so that f′ becomes a rational elementary fibration. Since π is an étale covering, the Leray spectral sequence for f = f′ ∘ π degenerates: R_{DR}^{i}f_{∗}(E, ∇) = R_{DR}^{i}f′_{∗}(π_{∗}(E, ∇)). Then P((E, ∇)) ⇒ P(π_{∗}(E, ∇)) ⇒ P(R_{DR}^{i}f′_{∗}(π_{∗}(E, ∇))), by (∗). This proves the lemma.

3.2.4 Remarks.

i) In lemma 3.2, we may replace A_{i}(f) by A(f) = A_{d+dimX}(f) since

R_{DR}^{j}f_{∗}(E, ∇) = 0 for j > d + dimX .

ii) If in lemma 3.2, the assumption (∗) holds only for j = 0, the conclusion keeps holding for i = 0.

Lemma 3.3. (Lemma on dévissage, second form.) Let P be a property of modules with integrable connection on smooth K-varieties, which implies coherence, is strongly exact and local for the étale topology. Assume that (∗) for any rational elementary fibration f′ : X′ ⟶ S′ with S′ affine, étale over some affine K-space, and for any simple, cyclic (E′, ∇′) ∈ Ob MIC(X′), there exists an open dense subset U′ ⊂ S′, such that

P((E′, ∇′)) ⇒ P((R_{DR}^{j}f′_{∗}(E′, ∇′))_{|U′}) , j = 0, 1 .

Then for any smooth morphism f : X ⟶ S of smooth K-varieties, and any (E, ∇) ∈ Ob MIC(X), there exists a non-empty open subset U ⊂ S (depending upon (E, ∇)) such that, for any i ≥ 0,

P((E, ∇)) ⇒ P((R_{DR}^{i}f_{∗}(E, ∇))_{|U}) .

Proof.

3.3.1. As in 3.2.1, we reduce to the case when S is connected, f is equidimensional of relative dimension d ≥ 1, and 0 ≤ j ≤ d + dimX. By Artin’s theorem (i.e. proposition 1.4, without insisting on “coordinatized” elementary fibrations), there exists an étale map ϵ : S′ ⟶ S such that X_{S′} admits a finite open affine cover {U_{α}} such that the restriction of f_{S′} to each U_{α} is a tower of elementary fibrations. We may then replace S by S′, and assume from the beginning that X admits a finite open affine cover {U_{α}} such that the restriction of f to each U_{α} is a tower of elementary fibrations. Let us assume P((E, ∇)), and let us consider the Zariski spectral sequence in MIC(S):

E_{1}^{p,i−p} = ⊕_{α_{0}<α_{1}<...<α_{p}}R_{DR}^{i−p}f_{\underline{α}∗}(E, ∇)_{|U\underline{α}} ⇒ R_{DR}^{i}f_{∗}(E, ∇) .

By induction on i, we know that after replacing S by a dense affine open subset, R_{DR}^{i−p}f_{\underline{α}∗}((E, ∇)_{|U\underline{α}}) satisfies P, for all \underline{α} such that |\underline{α}| = p + 1 > 1.

Thus, by strong exactness of P, this spectral sequence reduces us to prove P(R_{DR}^{i}f_{α∗}(E, ∇)_{|U_{α}}), after replacing S by a dense affine open subset. Hence we may assume that f is a tower of elementary fibrations over an affine connected base S.

3.3.2. We argue by induction on the height d of the tower. The induction hypothesis will be:

(∗)_{d} For any tower of elementary fibrations of height d

X_{d} ⟶^{f_{d−1}} X_{d−1} ⟶ … ⟶^{f_{0}} X_{0} ,

with X_{0} affine connected, and any (E, ∇) ∈ Ob MIC(X_{d}), there exists a dense affine open subset X′_{0} of X_{0} such that for any (i_{0}, …, i_{d−1}) ∈ {0, 1}^{d}, P((E, ∇)) ⇒ P((R_{DR}^{i_{0}}f_{0∗} … R_{DR}^{i_{d−1}}f_{d−1∗}(E, ∇))_{|X′_{0}}).

We point out that via the Leray spectral sequences for f = f_{0} ∘ … ∘ f_{d−1}, it follows from (∗)_{d}, taking into account the strong exactness of P, that for any i ≥ 0, P((E, ∇)) ⇒ P((R_{DR}^{i}f_{∗}(E, ∇))_{|X′_{0}}).

3.3.3. Let us consider the case d = 1. It is clear that an elementary fibration may be coordinatized after replacing S by an open dense subset. The argument in (3.2.3) then shows that (∗)_{1} is equivalent to

(∗)′ For any rational elementary fibration f : X ⟶ S with S affine, étale over some affine K-space, and for any (E, ∇) ∈ Ob MIC(X), there exists an open dense subset U ⊂ S such that P(E, ∇) ⇒ P((R_{DR}^{i}f_{∗}(E, ∇))_{|U}), i = 0, 1.

Let us show that (∗) ⇒ (∗)′. Let (E, ∇) be an object of MIC(X) satisfying P. Because P implies coherence, (E, ∇) has finite length. By applying strong exactness of P to long exact cohomology sequences, we see that it is enough to establish (∗) for the simple subquotients of (E, ∇). Hence we may assume that (E, ∇) is simple.

We remark that we are free of replacing S by some open dense subset; because P is étale-local we may even replace S by an affine étale neighborhood. Hence we may even assume that the restriction of (E, ∇) to the geometric generic fiber of f is simple.

On the other hand, there exists a finite open affine cover {U_{α}}_{α=0,...,r} of X, such that, for every α, E_{|U_{α}} is free and (E, ∇)_{|U_{α}} admits a cyclic vector v_{α}. In fact, according to [Ka3] (cf. also II.7.2), starting from any basis e_{0}, …, e_{μ−1} of sections of E_{|U_{α}}, one can take r = μ(μ − 1) and

(3.3.3.1)  v_{α} = ∑_{j=0}^{μ−1} \frac{(x − α)^{j}}{j!} ∑_{k=0}^{j}(−1)^{k}\binom{j}{k}∇(\frac{d}{dx})^{k}(e_{j−k}) .

Then after replacing S by some affine étale neighborhood, we may assume that each U_{α}, and each U_{α} ∩ U_{β}, is a rational elementary fibration over S. The Čech spectral sequence for {U_{α}} then reduces us to the cyclic case, so that (∗)_{1} follows from (∗).

3.3.4. We now consider a tower of height d ≥ 2 as in (∗)_{d}, and (E, ∇) ∈ Ob MIC(X_{d}) satisfying P (in particular, E is torsion-free over O_{X_{d}}, (2.6)). We apply the induction hypothesis (∗)_{d−1} to X_{d} ⟶ … ⟶ X_{1}, to the effect that there exists a dense affine open subset X′_{1} ⊂ X_{1} such that the O_{X′_{1}}-module with integrable connection (R_{DR}^{i_{1}}f_{1∗} … R_{DR}^{i_{d−1}}f_{d−1∗}(E, ∇))_{|X′_{1}} satisfies P. By shrinking X_{0}, we may assume that the restriction of f_{0}, say f′_{0} : X′_{1} ⟶ X_{0}, is an elementary fibration and that X′_{1} is the complement of a relatively smooth divisor in X_{1}. We then apply (∗)_{1} to the latter connections, with respect to f′_{0}. Upon replacing X_{0} again by a dense affine open subset, we obtain

P(R_{DR}^{i_{0}}f′_{0∗}(R_{DR}^{i_{1}}f_{1∗} … R_{DR}^{i_{d−1}}f_{d−1∗}(E, ∇))_{|X′_{1}}) , for any (i_{0}, …, i_{d−1}) ∈ {0, 1}^{d} .

We summarize the geometric situation in the diagram

X′_{d} ⟶^{f′_{d−1}} X′_{d−1} ⟶ … ⟶^{f′_{1}} X′_{1} ⟶^{f′_{0}} X′_{0}
j_{d} ↓  j_{d−1} ↓  j_{1} ↓  =↓
X_{d} ⟶^{f_{d−1}} X_{d−1} ⟶ … ⟶^{f_{1}} X_{1} ⟶^{f_{0}} X_{0}

in which the horizontal arrows are elementary fibrations, the vertical arrows are affine open embeddings, and all squares not involving the 0-subscript are fibered.

From these properties, it follows that for any k ∈ {1, …, d − 1} and any quasi-coherent (N , ∇) ∈ Ob MIC(X_{k+1}),

\begin{aligned}j_{k∗}j_{k}^{∗}R_{DR}^{i_{k}}f_{k∗}(N , ∇) & = j_{k∗}R_{DR}^{i_{k}}f′_{k∗}j_{k+1}^{∗}(N , ∇) = \\ & = R_{DR}^{i_{k}}(j_{k} ∘ f′_{k})_{∗}j_{k+1}^{∗}(N , ∇) = R_{DR}^{i_{k}}f_{k∗}(j_{k+1∗}j_{k+1}^{∗}(N , ∇)) . \end{aligned}

By iteration, we obtain

(∗∗)_{k}  \begin{aligned}j_{k∗}j_{k}^{∗}(R_{DR}^{i_{k}}f_{k∗} & … R_{DR}^{i_{d−1}}f_{d−1∗}(E, ∇)) = \\ & = R_{DR}^{i_{k}}f_{k∗} … R_{DR}^{i_{d−1}}f_{d−1∗}(j_{d∗}j_{d}^{∗}(E, ∇)) . \end{aligned}

On the other hand, because E is torsion-free, there is an exact sequence

0 ⟶ (E, ∇) ⟶ j_{d∗}j_{d}^{∗}(E, ∇) ⟶ Coker ⟶ 0 .

We shall show, by descending induction on k ∈ {0, …, d − 1}, that the sequence of natural morphisms

(∗ ∗ ∗)_{k} \begin{aligned}0 & ⟶ R_{DR}^{i_{k}}f_{k∗} … R_{DR}^{i_{d−1}}f_{d−1∗}(E, ∇) ⟶ R_{DR}^{i_{k}}f_{k∗} … \\ & … R_{DR}^{i_{d−1}}f_{d−1∗}(j_{d∗}j_{d}^{∗}(E, ∇)) ⟶ R_{DR}^{i_{k}}f_{k∗} … R_{DR}^{i_{d−1}}f_{d−1∗}Coker ⟶ 0 \end{aligned}

is a short exact sequence.

By induction, and via the long exact sequence attached to R_{DR}^{·}f_{k∗}, this follows from

(∗ ∗ ∗)′_{k}  R_{DR}^{0}f_{k∗}(R_{DR}^{i_{k+1}}f_{k+1∗} … R_{DR}^{i_{d−1}}f_{d−1∗}Coker) = 0 .

By (∗∗)_{k+1} and (∗ ∗ ∗)_{>k},

R_{DR}^{i_{k+1}}f_{k+1∗} … R_{DR}^{i_{d−1}}f_{d−1∗}Coker =
Coker(R_{DR}^{i_{k+1}}f_{k+1∗} … R_{DR}^{i_{d−1}}f_{d−1∗}(E, ∇) ⟶ j_{k+1∗}j_{k+1}^{∗}R_{DR}^{i_{k+1}}f_{k+1∗} … R_{DR}^{i_{d−1}}f_{d−1∗}(E, ∇))

which is torsion, supported by X_{k+1} \ X′_{k+1}. This implies (∗ ∗ ∗)′_{k}. Indeed, let m ≠ 0, if any, be a section of R_{DR}^{i_{k+1}}f_{k+1∗} … R_{DR}^{i_{d−1}}f_{d−1∗}Coker supported on the component of X_{k+1} \X′_{k+1} of local equation t. Then t^{h}m = 0, for some h ∈ Z_{>0}, which we assume to be minimal. So, for the O_{X_{k}}-linear derivation \frac{d}{dt},

(3.3.4.1)  0 = ∇(\frac{d}{dt})(t^{h}m) = ht^{h−1}m + t^{h}∇(\frac{d}{dt})(m) .

This shows that ∇(\frac{d}{dt})(m) ≠ 0, hence the assertion (∗∗∗)′_{k}. We have now proved (∗ ∗ ∗)′_{0}, which shows that R_{DR}^{i_{0}}f_{0∗} … R_{DR}^{i_{d−1}}f_{d−1∗}(E, ∇) is a subobject of

\begin{aligned}R_{DR}^{i_{0}}f_{0∗} & … R_{DR}^{i_{d−1}}f_{d−1∗}(j_{d∗}j_{d}^{∗}(E, ∇)) = R_{DR}^{i_{0}}f_{0∗}j_{1∗}j_{1}^{∗}R_{DR}^{i_{1}}f_{1∗} … \\ & … R_{DR}^{i_{d−1}}f_{d−1∗}(E, ∇) = R_{DR}^{i_{0}}f′_{0∗}((R_{DR}^{i_{1}}f_{1∗} … R_{DR}^{i_{d−1}}f_{d−1∗}(E, ∇))_{|X′_{1}}) , \end{aligned}

which satisfies P. This achieves the proof of (∗)_{d} and of the lemma. Q.E.D.

Remark 3.3.5. This argument shows in particular that the natural map

R_{DR}^{i_{k}}f_{k∗} … R_{DR}^{i_{d−1}}f_{d−1∗}(E, ∇) ⟶ R_{DR}^{i_{k}}f_{k∗}j_{k+1∗}j_{k+1}^{∗} … R_{DR}^{i_{d−1}}f_{d−1∗}(E, ∇)

is an isomorphism if (i_{k}, i_{k+1} …, i_{d−1}) ≠ (1, 1, …, 1).

Remark 3.3.6. One can give an alternative proof of Lemma 3.3, under the extra assumption that for any closed immersion i : Y ⟶ X of smooth K-varieties, P((E, ∇)) ⇒ P(i^{∗}(E, ∇)). This second proof uses the language and results of [Bo], but avoids the use of proposition 1.4 (see [ABa] in a special case).

§4 Generic finiteness of direct images.

Theorem 4.1. For any smooth morphism f : X ⟶ S of smooth K-varieties, and any coherent O_{X}-module with integrable connection (E, ∇), there exists a dense open subset U ⊂ S such that for every j ≥ 0, R_{DR}^{j}f_{∗}(E, ∇)_{|U} is locally free of finite rank.

We may and shall assume that f is dominant.

4.2. We treat separately the case j = 0, for which one can in fact take U = A_{0}(f), the Artin set of level 0 (1.8), independently of (E, ∇).

Lemma 4.2.1. Let f : X ⟶ S be a smooth morphism with S = A_{0}(f). Then for any coherent (E, ∇) ∈ ObMIC(X), R_{DR}^{0}f_{∗}(E, ∇) is locally free of finite rank as an O_{S}-module.

Proof. It suffices to check the coherence of R_{DR}^{0}f_{∗}(E, ∇), and this is an étale-local condition on S. By definition of A_{0}(f), we thus may assume that f is Zariski-locally on X a tower of elementary fibrations, hence (separating the connected components of X) that f has non-empty connected geometric fibers. The coherence of R_{DR}^{0}f_{∗}(E, ∇) then follows from proposition 2.3.1 and faithfully flat descent [EGA IV, 2.5.2]. Q.E.D.

4.3. For j > 0, we use the lemma on dévissage 3.3, with P = coherence (i.e. (E, ∇) satisfies P if and only if E is O_{X}-coherent). Strong exactness and étalelocalness are fulfilled. Therefore we are left to show that for any rational elementary fibration f : X ⟶ S with S affine, étale over some affine K-space, and for any cyclic (E, ∇), there is a dense open subset U ⊂ S such that R_{DR}^{1}f_{∗}(E, ∇)_{|U} is O_{U}-coherent.

4.4. Because f and S are affine the O_{S}-module underlying R_{DR}^{1}f_{∗}(E, ∇) is nothing but the O_{S}-module attached to the cokernel of the relative De Rham complex Γ(X, E) ⟶ Γ(X, E)dx. We have to show that this cokernel is an O(S)-module of finite type. Let (v, ∇(\frac{d}{dx})v, …, ∇(\frac{d}{dx})^{μ−1}v) be a cyclic base of Γ(X, E). Notice that the dual connection (E^{∨}, ∇^{∨}) is also cyclic: a cyclic vector is for example w ∈ Γ(X, E^{∨}) = Hom_{O(X)}(Γ(X, E), O(X)), such that

< w, ∇(\frac{d}{dx})^{j}v >= δ_{j,μ−1} .

We easily see in fact, by ascending induction on h = 0, 1, …, μ − 1, that

< ∇^{∨}(\frac{d}{dx})^{h}w, ∇(\frac{d}{dx})^{j}v >= \{ \begin{matrix}0 & if j < μ − h − 1 \\ (−1)^{h} & if j = μ − h − 1 \end{matrix}

so that det(< ∇^{∨}(\frac{d}{dx})^{h}w, ∇(\frac{d}{dx})^{j}v >)_{0≤j,h≤μ−1} = (−1)^{\frac{μ(μ−1)}{2}} . In the basis

(w, ∇^{∨}(\frac{d}{dx})w, …, ∇^{∨}(\frac{d}{dx})^{μ−1}w)

of Γ(X, E^{∨}), ∇^{∨}(\frac{d}{dx}) is represented by a matrix of the form

A = ( \begin{matrix}0 & … & 0 & −γ_{0} \\ & & & −γ_{1} \\ & I_{μ−1} & & … \\ & & & −γ_{μ−1} \end{matrix} ) ,

with γ_{k} ∈ O(X).

Let us introduce the differential polynomial (associated to (E, ∇) via the cyclic vector w ∈ Γ(X, E^{∨})) Λ = (\frac{d}{dx})^{μ} + ∑_{k=0}^{μ−1} γ_{k}(\frac{d}{dx})^{k} ∈ O(X)[\frac{d}{dx}], so that ∇^{∨}(Λ)w = 0. Let (v_{0}, …, v_{μ−1}) denote the basis of Γ(X, E) dual to the basis (w, ∇^{∨}(\frac{d}{dx})w, …, ∇^{∨}(\frac{d}{dx})^{μ−1}w), i.e. such that

< ∇^{∨}(\frac{d}{dx})^{i}w, v_{j} >= δ_{i,j} .

One has

∇(\frac{d}{dx})v_{0} = γ_{0}v_{μ−1}

and

∇(\frac{d}{dx})v_{j} = γ_{j}v_{μ−1} − v_{j−1}

for j ≥ 1. There is a commutative diagram

(4.4.1)  \begin{matrix}O(X) & ⟶^{Λ} & O(X) \\ ↓ & & ↓ \\ Γ(X, E) = ⊕_{k=0}^{μ−1}O(X)v_{k} & ⟶^{∇( \frac{d}{dx})} & Γ(X, E) = ⊕_{k=0}^{μ−1}O(X)v_{k} \end{matrix} ,

where the vertical arrows are η ⟼ (η, …, (\frac{d}{dx})^{μ−1}η) and η ⟼ (0, …, 0, η), respectively. Diagram 4.4.1 induces an isomorphism between the cokernel (resp. kernel) of Λ and the cokernel (resp. kernel) of ∇(\frac{d}{dx}). Therefore

Coker(Γ(X, E)⟶^{∇}Γ(X, E)dx) ≃ O(X)/ΛO(X) .

Let P (x) = ∏_{i=1}^{r}(x−θ_{i})^{s_{i}} ∈ O(S)[x] be such that Λ′ = P (x)Λ = ∑_{k=0}^{μ} γ′_{k}(\frac{d}{dx})^{k} ∈ O(S)[x, \frac{d}{dx}], with minimal s_{i} ≥ 0. We also have

Coker(Γ(X, E)⟶^{∇}Γ(X, E)dx) ≃ O(X)/Λ′O(X) .

4.5. Let us write again the diagram pertaining to a rational elementary fibration

(4.5.1)  \begin{aligned}X & ↪ & P_{S}^{1} & ↩ & (∐_{i=1}^{r} σ_{i}(S)) ∐ σ_{∞}(S) \\ & ↘^{f} & ↓^{pr} & ↙ \\ & & S \end{aligned}

where σ_{∞}(S) = ∞ × S, σ_{i} : ζ ⟼ (θ_{i}(ζ), ζ), θ_{i} ∈ O(S), θ_{i} − θ_{j} ∈ O(S)^{×}. We have a Mittag-Leffler decomposition

(4.5.2)  O(X) = O(S)[x] ⊕ ⨁_{i=1}^{r} \frac{1}{x − θ_{i}} O(S)[\frac{1}{x − θ_{i}}] .

4.6. We recall (see (I.6.1.1)) that the indicial polynomial of Λ′ at θ_{i}, denoted by ϕ_{θ_{i}}(s) (resp. ϕ_{∞}(s)), is the element of O(S)[s] defined by the condition that for any m ∈ Z,

Λ′ \frac{1}{(x − θ_{i})^{m}} = ϕ_{θ_{i}}(−m)\frac{1}{(x − θ_{i})^{m+r_{i}}} + lower order terms in \frac{1}{x − θ_{i}} ,

(resp.

Λ′x^{m} = ϕ_{∞}(−m)x^{m+r_{∞}} + l.o.t. in x ),

where r_{i} = max(k − ord_{θ_{i}}γ′_{k}) ≥ 0 (resp. r_{∞} = max(deg_{x}γ′_{k} − k) ≥ 0). According to (I.6.1.4), there exist τ_{i}, τ_{∞} ∈ O(S) \ \{0\}, such that \frac{ϕ_{θ_{i}} (s)}{τ_{i}}, \frac{ϕ_{∞}(s)}{τ_{∞}} ∈ K[s]. We claim that, after replacing S by its dense open subset SpecO(S)[_{\frac{1}{τ_{∞} ∏_{i} τ_{i}}}],

O(X)/Λ′O(X) becomes finitely generated over O(S). Let M ∈ Z_{≥0}, M ≥ r_{i} ∀i, be such that ϕ_{∞}(−n + r_{∞}) ∏_{j=1,...,r} ϕ_{θ_{j}}(−n + r_{j}) ≠ 0 for any n ∈ Z_{>M}. Then

\frac{1}{(x − θ_{i})^{n}} ∈ Λ′ \frac{σ_{i,n}}{(x − θ_{i})^{n−r_{i}}} + \frac{1}{(x − θ_{i})^{n−1}} O(S)[x] , σ_{i,n} ∈ O(S)^{×} ,

and

x^{n} ∈ Λ′σ_{∞,n}x^{n−r_{∞}} + ⨁_{k=0}^{n−1} x^{k}O(S) , σ_{∞,n} ∈ O(S)^{×} .

It follows from induction and the Mittag-Leffler decomposition that O(X)/Λ′O(X) is generated by the images of 1, x, …, x^{M}, \frac{1}{(x−θ_{1})}, …, \frac{1}{(x−θ_{1})^{M}} , …, \frac{1}{(x−θ_{r})}, … …, \frac{1}{(x−θ_{r})^{M}}. Q.E.D.

4.7 Remarks.

i) In particular, we have proved that the if the leading coefficient of each indicial polynomial of Λ′ is a unit, then R_{DR}^{1}f_{∗}(E, ∇) is finitely generated over the whole of S. As a special case, we see that R_{DR}^{1}f_{∗}(O_{X}, d) is finitely generated over O_{S}.

ii) The indicial polynomials of Λ and Λ′ at θ_{i} differ by multiplication by a unit in O(S).

iii) The proof does not use the full hypothesis that v is a cyclic vector on the whole of X (i.e. that one may take P (x) = ∏_{i=1}^{r}(x − θ_{i})^{s_{i}}). It would suffice to assume that v is a cyclic vector outside some divisor D whose Zariski closure in P_{S}^{1} is contained in X.

4.8 Counterexample. We show that, unlike the case of R_{DR}^{0}f_{∗}(E, ∇), it is not possible to choose an open subset U ⊂ S of coherence for R_{DR}^{1}f_{∗}(E, ∇), which is independent of (E, ∇). Indeed, let X = A_{K}^{2} = SpecK[x, y] ⟶ S = A_{K}^{1} be the second projection, and let us consider, for y_{0} ∈ K, the integrable O_{X}-connection of rank 1 (E = O_{X}e, ∇_{y_{0}}), defined by

\begin{aligned}∇_{y_{0}}(\frac{∂}{∂x})e & = (y − y_{0})e \\ ∇_{y_{0}}(\frac{∂}{∂y})e & = xe . \end{aligned}

The indicial polynomial ϕ_{∞}(s) is the “constant” y − y_{0}, and it is easy to see that the maximal open subset of coherence of E/∇_{y_{0}}(\frac{∂}{∂x})E is S \ \{y_{0}\}, which depends upon y_{0}. In fact, as a K[y − y_{0}]-module, Γ(E/∇_{y_{0}}(\frac{∂}{∂x})E) is isomorphic to K((y − y_{0}))/(y − y_{0})K[[y − y_{0}]] via x^{k} ↦ (y_{0} − y)^{−k}.

We complement the previous analysis by the following purity statement.

Lemma 4.9. Let f : X ⟶ S be a rational elementary fibration, and let (E, ∇) be a coherent O_{X}-module with integrable connection. Let us assume that there is a closed subset T ⊂ S of codimension ≥ 2 such that R_{DR}^{1}f_{∗}(E, ∇) is locally free of finite rank over S \T. Then R_{DR}^{1}f_{∗}(E, ∇) is locally free of finite rank over S.

Proof. Let us construct the natural commutative square

X′ = X_{S\T} ↪^{j′} X
f′ ↓  ↓ f
S \ T  ↪^{j} S.

By using the long exact sequence of De Rham cohomology associated to a short exact sequence of connections, we may reduce to the case where (E, ∇) is simple. Because T ⊂ S is of codimension ≥ 2, we have

(E, ∇) ≅ j′_{∗}j^{′∗}(E, ∇)

(taking into account the fact that E is locally free of finite rank),

j_{∗}R_{DR}^{0}f′_{∗}(j^{′∗}(E, ∇)) ≅ j_{∗}j^{∗}R_{DR}^{0}f_{∗}(E, ∇) ≅ R_{DR}^{0}f_{∗}(E, ∇)

and

f^{∗}j_{∗}R_{DR}^{0}f′_{∗}(j^{′∗}(E, ∇)) ≅ j′_{∗}f^{′∗}R_{DR}^{0}f′_{∗}(j^{′∗}(E, ∇))

is a subobject of (E, ∇) (taking into account (4.2.1)).

i) If this subobject is (E, ∇) itself, then

R_{DR}^{1}f_{∗}(E, ∇) ≅ R_{DR}^{1}f_{∗}(O_{X}, d) ⊗_{O_{S}} (j_{∗}R_{DR}^{0}f′_{∗}(j^{′∗}(E, ∇))) ,

which is coherent by (4.7.i). ii) Otherwise, we have

f^{∗}j_{∗}R_{DR}^{0}f′_{∗}(j^{′∗}(E, ∇)) = 0 ,

hence R_{DR}^{0}f′_{∗}(j^{′∗}(E, ∇)) = 0. We have Leray spectral sequences converging to

R_{DR}^{•}(jf′)_{∗}(j^{′∗}(E, ∇)) = R_{DR}^{•}(fj′)_{∗}(j^{′∗}(E, ∇))

and with E_{2}-terms

E_{2}^{ab} = R_{DR}^{a}j_{∗}R_{DR}^{b}f′_{∗}(j^{′∗}(E, ∇)) ,

and

′E_{2}^{ba} = R_{DR}^{b}f_{∗}R_{DR}^{a}j′_{∗}(j^{′∗}(E, ∇)) ,

respectively. Let H^{1} := R_{DR}^{1}(jf′)_{∗}(j^{′∗}(E, ∇)). In the present case, the first spectral sequence degenerates at the level E_{2} in degree one, so that H^{1} ≅ E_{2}^{0,1} = j_{∗}R_{DR}^{1}f′_{∗}(j^{′∗}(E, ∇)) = j_{∗}j^{∗}R_{DR}^{1}f_{∗}(E, ∇), by flat base change. Since T ⊂ S is of codimension ≥ 2, H^{1} is a coherent O_{S}-module. On the other hand, for the second spectral sequence we have the exact sequence [CE, Chap. XV, Prop. 5.5]

0 ⟶ ′E_{2}^{1,0} ⟶ H^{1} ⟶ ′E_{2}^{0,1} ⟶ 0 ,

showing that ′E_{2}^{1,0} = R_{DR}^{1}f_{∗}(E, ∇) is a (quasi-coherent) submodule of H^{1}. Since S is noetherian, R_{DR}^{1}f_{∗}(E, ∇) is coherent, too. Q.E.D.

§5 Generic base change for direct images.

Theorem 5.1. Let f : X ⟶ S be a morphism of smooth K-varieties, and let (E, ∇) be a coherent object of MIC(X). There is a dense open subset U ⊂ S with the following property. For any smooth K-variety S^{♯} and any morphism u : S^{♯} ⟶ S, let us construct the fibered diagram

X^{♯} ⟶^{f^{♯}} S^{♯} ↩ U^{♯}
u^{♯} ↓ ↓ u ↓
X ⟶^{f} S ↩ U

Then, for any i ≥ 0, the restriction to U^{♯} of the base change morphism

φ^{i} : u^{∗}R_{DR}^{i}f_{∗}(E, ∇) ⟶ R_{DR}^{i}f_{∗}^{♯}u^{♯∗}(E, ∇)

is an isomorphism in MIC(U^{♯}).

Notation 5.1.1. In the sequel we set (E^{♯}, ∇^{♯}) = u^{♯∗}(E, ∇).

5.2 Remarks.

i) U^{♯} is an open subset of S^{♯}, possibly empty (see 5.7 below).

The following points are straightforward consequences of the compatibility of the formation of R_{DR}^{i}f_{∗} with flat base change (2.3).

ii) It would suffice to consider the case of a closed immersion u (write u as a closed immersion given by its graph, followed by a projection).

iii) The statement is equivalent to the following. For any u : S^{♯} ⟶ S such that Im(u) ⊂ U, φ^{i} is an isomorphism.

iv) In order to prove 5.1, one may replace S by any S′ étale and dominant over S.

5.3. Replacing S by affine étale neighborhoods of its connected components, we may assume that X admits a finite open affine cover {U_{α}} such that the restriction f_{α} of f to U_{α} is a tower of elementary fibrations. Considering an affine cover of S^{♯}, or using remark 5.2, ii, we may assume that u^{♯−1}(U_{α}) are affine. In the notation 1.7, there is a natural morphism of Čech spectral sequences in MIC(S^{♯})

⊕_{α_{0}<...<α_{p}}u^{∗}R_{DR}^{i−p}f_{\underline{α}∗}(E, ∇)_{|U\underline{α}} ⇒ u^{∗}R_{DR}^{i}f_{∗}(E, ∇)
↓ φ_{i−p}  ↓ φ_{i}
⊕_{α_{0}<...<α_{p}}R_{DR}^{i−p}f_{\underline{α}∗}^{♯}(E^{♯}, ∇^{♯})_{|u^{♯−1}U\underline{α}} ⇒ R_{DR}^{i}f_{∗}^{♯}(E^{♯}, ∇^{♯})

Arguing by induction on i, we may assume that f itself is a tower of elementary fibrations.

5.4. We argue by induction on the height d of the tower, as in 3.3.2-4. The induction hypothesis will be:

(∗)_{d} For any tower of elementary fibrations of height d

X_{d} ⟶^{f_{d−1}} X_{d−1} ⟶ … ⟶^{f_{0}} X_{0} ,

with X_{0} affine connected, and any coherent (E, ∇) ∈ Ob MIC(X_{d}), there exists a dense affine open subset X′_{0} of X_{0} with the following property: for any smooth K-variety X_{0}^{♯} and any morphism u_{0} : X_{0}^{♯} ⟶ X_{0}, let us construct the fibered diagram

X_{d}^{♯} ⟶^{f_{d−1}^{♯}} X_{d−1}^{♯} ⟶ … ⟶^{f_{0}^{♯}} X_{0}^{♯}
u_{d} ↓  u_{d−1} ↓  u_{0} ↓
X_{d} ⟶^{f_{d−1}} X_{d−1} ⟶ … ⟶^{f_{0}} X_{0}

Then for any (i_{0}, …, i_{d−1}) ∈ {0, 1}^{d},

(R_{DR}^{i_{0}}f_{0∗}R_{DR}^{i_{1}}f_{1∗} … R_{DR}^{i_{d−1}}f_{d−1∗}(E, ∇))_{|X′_{0}}

is coherent, and the restriction to u_{0}^{−1}(X′_{0}) of the base change morphism

ψ : u_{0}^{∗}(R_{DR}^{i_{0}}f_{0∗}R_{DR}^{i_{1}}f_{1∗} … R_{DR}^{i_{d−1}}f_{d−1∗}E, ∇)) ⟶ R_{DR}^{i_{0}}f_{0∗}^{♯}R_{DR}^{i_{1}}f_{1∗}^{♯} … R_{DR}^{i_{d−1}}f_{d−1∗}^{♯}u_{d}^{∗}(E, ∇)

is an isomorphism in MIC(u_{0}^{−1}(X′_{0})).

We point out that via base change for the Leray spectral sequences of the composed morphism f = f_{0} ∘ … ∘ f_{d−1}, 5.1 (in the case of towers of elementary fibrations of height ≤ d) follows from (∗)_{≤d}.

Remark 5.4.1. If i_{0} = i_{1} = … = i_{d−1} = 1, then the base change morphism ψ is an isomorphism; this follows from the fact that for f affine smooth of relative dimension one the formation of R_{DR}^{1}f_{∗} commutes with arbitrary base change, being a cokernel.

5.5. Let us consider the case d = 1, i.e. f is an elementary fibration X_{1} = X ⟶ S = X_{0}. In this case, we can choose for X′_{0} = U any dense open subset such that R_{DR}^{1}f_{∗}(E, ∇)_{|U} is coherent (§4). According to 4.2, we may at once replace S by such an (affine) open subset. We note that only i = 0 or 1 occur. Since f is affine and flat, we have the exact sequence (cf. (2.4))

0 ⟶ R_{DR}^{0}f_{∗}(E, ∇) = Kerf_{∗}(∇) ⟶ f_{∗}E ⟶^{f_{∗}(∇)}f_{∗}(E ⊗_{O_{X}} Ω_{X/S}^{1}) ⟶
Cokerf_{∗}(∇) ≅ R_{DR}^{1}f_{∗}(E, ∇) ⟶ 0

in which all terms are flat over S; a fortiori, Imf_{∗}(∇) is flat. On the other hand, u^{∗}f_{∗}E = f_{∗}^{♯}E^{♯}, and u^{∗}f_{∗}(E ⊗_{O_{X}} Ω_{X/S}^{1}) = f_{∗}^{♯}(E^{♯} ⊗_{O_{X^{♯}}} Ω_{X^{♯}/S^{♯}}^{1}) [EGA I, 9.3.3]. Hence the sequences

0 ⟶ u^{∗}Imf_{∗}(∇) ⟶ u^{∗}f_{∗}(E⊗_{O_{X}} Ω_{X/S}^{1}) = f_{∗}^{♯}(E^{♯}⊗_{O_{X^{♯}}} Ω_{X^{♯}/S^{♯}}^{1}) ⟶ u^{∗}Cokerf_{∗}(∇) ⟶ 0

0 ⟶ u^{∗}Kerf_{∗}(∇) ⟶ u^{∗}f_{∗}E = f_{∗}^{♯}E^{♯} ⟶^{u^{∗}f_{∗}(∇)=f_{∗}^{♯}(∇^{♯})}u^{∗}Imf_{∗}(∇) ⟶ 0

are exact, from which we conclude that

u^{∗}Imf_{∗}(∇) = Imf_{∗}^{♯}(∇^{♯}) ,
u^{∗}Cokerf_{∗}(∇) = Cokerf_{∗}^{♯}(∇^{♯}) ≅ R_{DR}^{1}f_{∗}^{♯}(E^{♯}, ∇^{♯}) ,
u^{∗}Kerf_{∗}(∇) = Kerf_{∗}^{♯}(∇^{♯}) ≅ R_{DR}^{0}f_{∗}^{♯}(E^{♯}, ∇^{♯}) ,

hence φ^{0} and φ^{1} are isomorphisms in MIC(S^{♯}), which establishes (∗)_{1}.

5.6. We now consider a tower of height d ≥ 2 as in (∗)_{d}. We apply the induction hypothesis (∗)_{d−1} to X_{d} ⟶ … ⟶ X_{1}, to the effect that there exists a dense affine open subset X′_{1} ⊂ X_{1} such that (R_{DR}^{i_{1}}f_{1∗} … R_{DR}^{i_{d−1}}f_{d−1∗}(E, ∇))_{|X′_{1}} is coherent and the natural morphism

φ_{(1)} : (u_{1}^{∗}(R_{DR}^{i_{1}}f_{1∗} … R_{DR}^{i_{d−1}}f_{d−1∗}(E, ∇)))_{|u_{1}^{−1}(X′_{1})} ⟶ (R_{DR}^{i_{1}}f_{1∗}^{♯} … R_{DR}^{i_{d−1}}f_{d−1∗}^{♯}u_{d}^{∗}(E, ∇))_{|u_{1}^{−1}(X′_{1})}

is an isomorphism in MIC(u_{1}^{−1}(X′_{1})).

Let us consider the fibered prism

X_{1}^{′♯}  −− ⟶^{j_{1}^{♯}}  X_{1}^{♯}
| ↘^{f_{0}^{′♯}}  ↙^{f_{0}^{♯}}  |
u′_{1} |  X_{0}^{♯}  | u_{1}
↓  |  u_{0}  ↓
X′_{1} −−^{j_{1}}  |  ⟶ X_{1}
↘^{f′_{0}}  ↓  ↙^{f_{0}}
X_{0}

By shrinking X_{0}, we may assume that f′_{0} is an elementary fibration, and (by applying (∗)_{1} to f′_{0}) that

(5.6.1) R_{DR}^{i_{0}}f′_{0∗}((R_{DR}^{i_{1}}f_{1∗} … R_{DR}^{i_{d−1}}f_{d−1∗}(E, ∇))_{|X′_{1}}) is coherent over X_{0}.

(5.6.2) The natural morphism

\begin{aligned}& u_{0}^{∗}R_{DR}^{i_{0}}f′_{0∗}(R_{DR}^{i_{1}}f_{1∗} … R_{DR}^{i_{d−1}}f_{d−1∗}(E, ∇))_{|X′_{1}} ⟶ \\ R_{DR}^{i_{0}}f_{0∗}^{′♯}u_{1}^{′∗} & (R_{DR}^{i_{1}}f_{1∗} … R_{DR}^{i_{d−1}}f_{d−1∗}(E, ∇))_{|X′_{1}} ≅ R_{DR}^{i_{0}}f_{0∗}^{′♯}j_{1}^{♯∗}u_{1}^{∗}R_{DR}^{i_{1}}f_{1∗} … R_{DR}^{i_{d−1}}f_{d−1∗}(E, ∇) \end{aligned}

is an isomorphism in MIC(X_{0}^{♯}).

On combining with φ_{(1)}, we get a natural isomorphism in MIC(X_{0}^{♯})

(5.6.3)  \begin{aligned}φ : u_{0}^{∗}R_{DR}^{i_{0}}f_{0∗}j_{1∗} & j_{1}^{∗}R_{DR}^{i_{1}}f_{1∗} … R_{DR}^{i_{d−1}}f_{d−1∗}(E, ∇)⟶^{∼} \\ & R_{DR}^{i_{0}}f_{0∗}^{♯}j_{1∗}^{♯}j_{1}^{♯∗}R_{DR}^{i_{1}}f_{1∗}^{♯} … R_{DR}^{i_{d−1}}f_{d−1∗}^{♯}u_{d}^{∗}(E, ∇) . \end{aligned}

According to remarks 5.4.1 and 3.3.5, we may assume that one of the i_{j} is 0, to the effect that the natural maps

R_{DR}^{i_{0}}f_{0∗} … R_{DR}^{i_{d−1}}f_{d−1∗}(E, ∇) ⟶ R_{DR}^{i_{0}}f_{0∗}j_{1∗}j_{1}^{∗}R_{DR}^{i_{1}}f_{1∗} … R_{DR}^{i_{d−1}}f_{d−1∗}(E, ∇)

R_{DR}^{i_{0}}f_{0∗}^{♯} … R_{DR}^{i_{d−1}}f_{d−1∗}^{♯}u_{d}^{∗}(E, ∇) ⟶ R_{DR}^{i_{0}}f_{0∗}^{♯}j_{1∗}^{♯}j_{1}^{♯∗}R_{DR}^{i_{1}}f_{1∗}^{♯} … R_{DR}^{i_{d−1}}f_{d−1∗}^{♯}u_{d}^{∗}(E, ∇)

are isomorphisms. In particular, R_{DR}^{i_{0}}f_{0∗} … R_{DR}^{i_{d−1}}f_{d−1∗}(E, ∇) is coherent (5.6.1). Moreover, these isomorphisms fit in a commutative square

\begin{aligned}u_{0}^{∗}R_{DR}^{i_{0}}f_{0∗} … R_{DR}^{i_{d−1}}f_{d−1∗}(E, ∇) & ⟶^{∼}u_{0}^{∗}R_{DR}^{i_{0}}f_{0∗}j_{1∗}j_{1}^{∗}R_{DR}^{i_{1}}f_{1∗} … R_{DR}^{i_{d−1}}f_{d−1∗}(E, ∇) \\ ψ ↓ & φ ↓∼ (5.6.3) \\ R_{DR}^{i_{0}}f_{0∗}^{♯} … R_{DR}^{i_{d−1}}f_{d−1∗}^{♯}u_{d}^{∗}(E, ∇) & ⟶^{∼}R_{DR}^{i_{0}}f_{0∗}^{♯}j_{1∗}^{♯}j_{1}^{♯∗}R_{DR}^{i_{1}}f_{1∗}^{♯} … R_{DR}^{i_{d−1}}f_{d−1∗}^{♯}u_{d}^{∗}(E, ∇) \end{aligned}

We conclude that ψ (the base change map appearing in (∗)_{d} of 5.4) is an isomorphism, and this achieves the inductive proof of 5.1.

5.7 Counterexample. Let us consider the fibered square

SpecK[x]  ⟶^{u^{♯}} SpecK[x, y] = X
f^{♯} ↓  ↓ f
{y_{0}} = SpecK ⟶^{u} SpecK[y] = S

in which f is the projection and u is the embedding of the point defined by y = y_{0}. Let us consider the integrable connection on X of rank 1 (E, ∇_{y_{0}}) defined in 4.8. It is clear that R_{DR}^{0}f_{∗}(E, ∇_{y_{0}}) = 0, while R_{DR}^{0}f_{∗}^{♯}u^{♯∗}(E, ∇_{y_{0}}) ≅ K, because u^{♯∗}(E, ∇_{y_{0}}) ≅ (K[x], d). This shows the necessity of localizing in 5.1, when u is not flat. This example is also interesting in connection with the base change theorem of [Bo, VI, 8.4].

§6 Coherence of the cokernel of a regular connection.

6.1. In this section, we establish that for any rational elementary fibration f : X ⟶ S as in (4.5.1) and any locally free O_{X}-module of finite rank μ with regular integrable connection (E, ∇), R_{DR}^{1}f_{∗}(E, ∇) is locally free of finite rank (over the whole of S).

We may assume that S is affine and connected. The statement amounts to the finite generation of Γ(X, E)/∇(\frac{d}{dx})Γ(X, E) over O(S).

6.2. Let us first remark that two special cases already follow from 4.6, namely:

1) the case when S is reduced to a point (which allows us to assume dimS ≥ 1 in the sequel), and 2) the case when (E, ∇) is cyclic (because for regular cyclic (E, ∇), the leading coefficients of the indicial polynomials are units, cf. remark 4.6.1).

However, unlike the situation of (3.3.3), the reduction to the cyclic case is problematic: it is not clear that one can cover S by étale neighborhoods S′ with the property that X_{S′} admits a Zariski open covering {U_{α}} such that U_{α} ⟶ S′ and U_{α} ∩ U_{β} ⟶ S′ are rational elementary fibrations and such that (E, ∇)_{|U_{α}} is cyclic outside some divisor whose Zariski closure in P_{S′}^{1} is contained in |U_{α}, see (II, question 4.2.9). We shall take another approach, via τ-extensions.

6.3. For notational convenience, we shall assume, by translating the “vertical” variable x, that one of the θ_{1}, …, θ_{r}, say θ_{1}, is 0 (if r ≠ 0). Let us consider the τ-extension (Ẽ, ˜∇) of (E, ∇) on P_{S}^{1}, as constructed in (I.4.9): Ẽ is a locally free O_{P_{S}^{1}}-module, endowed with a τ-prepared connection ˜∇ : Der_{Z}P_{S}^{1} ⟶ End_{O_{S}} Ẽ with logarithmic poles along P_{S}^{1} \ X = Z = ∐_{ν=1,...r,∞} Z_{ν}.

Lemma 6.3.1. Let F be a locally free O_{P_{S}^{1}}-module of finite rank. Then there exists a locally free O_{S}-module M such that F_{|A_{S}^{1}} ≅ pr^{∗}M, where pr stands for the projection A_{S}^{1} ⟶ S.

Proof. The graded O(S)[x_{0}, x_{1}]-module F attached to F is projective of finite rank (since so are its localizations with respect to the open covering U = {U_{0}, U_{1}} of P_{S}^{1} with U_{0} := A_{S}^{1}, U_{1} := P_{S}^{1} \ ({0} × S) and

F ⊗_{O(S)[x_{0},x_{1}]} O(S)[x_{0}, x_{1}]_{x_{i}} ≅ Γ(U_{i}, F) ⊗_{O(U_{i})} O(S)[x_{0}, x_{1}]_{x_{i}} ,

for i = 0, 1). According to [A.C. X, p.144], there is a projective O(S)-module M such that F ≅ M ⊗_{O(S)} O(S)[x_{0}, x_{1}]. One may then take for M the locally free O_{S}-module attached to M. Q.E.D.

Remark 6.3.2. Lemma 6.3.1 is a special, elementary case of a result of H. Lindel [Li] extending the work of Quillen-Suslin on the Serre conjecture, and which says that in fact, for affine smooth S/K, any locally free O_{A_{S}^{1}}-module of finite rank has the form pr^{∗}M, for some locally free O_{S}-module M.

We apply (6.3.1) to F = Ẽ, to the effect that there are projective O(S)- modules M and M′ of rank μ, such that Γ(A_{S}^{1}, Ẽ) = M[x], Γ(P_{S}^{1} \({0}×S), Ẽ) = M′[\frac{1}{x}], and M[x, \frac{1}{x}] = M′[x, \frac{1}{x}]. Since M and M′ are finitely generated, there is a positive integer N such that

x^{N}M′[x] ⊂ M[x] ⊂ x^{−N}M′[x] .

6.4. We have ˜∇(∏_{i=1}^{r}(x − θ_{i})\frac{d}{dx})M[x] ⊂ M[x], and, for any i = 1, …, r, we are given an endomorphism Res_{θ_{i}} ˜∇ of M which enjoys the following properties (cf. (I.4)):

(6.4.1) the characteristic polynomial of Res_{θ_{i}} ˜∇ has coefficients in K and roots in the image of τ (in particular, none of the roots is a non-zero integer);

(6.4.2) for any m ∈ M, identified with m(x − θ_{i})^{0} ∈ M[x − θ_{i}] = M[x], one has

˜∇((x − θ_{i})\frac{d}{dx})m − (Res_{θ_{i}} ˜∇)m ∈ (x − θ_{i})M[x, \frac{1}{∏_{k≠i}(x − θ_{k})}] .

Let us set γ_{i} = ∏_{k≠i}(θ_{i} − θ_{k}), a unit in O(S). It follows from (6.4.2) that, for any m ∈ M ≅ M · 1 ⊂ M[x, \frac{1}{x−θ_{i}}], (6.4.3)

˜∇(∏_{k=1}^{r}(x−θ_{k})\frac{d}{dx})(m(x−θ_{i})^{−n})−γ_{i}(Res_{θ_{i}} ˜∇−n1_{M})(m)(x−θ_{i})^{−n} ∈ (x−θ_{i})^{−n+1}M[x] .

On the other hand, (6.4.1) implies that Res_{θ_{i}} ˜∇ − n1_{M} is an invertible endomorphism of M for any n > 0 (by Cayley-Hamilton, the endomorphism (Res_{θ_{i}} ˜∇ − n1_{M})^{−1} of M ⊗_{O(S)} κ(S) is a polynomial in Res_{θ_{i}} ˜∇ − n1_{M} with coefficients in K, hence induces an endomorphism of M). This shows that

m(x − θ_{i})^{−n} ∈ ˜∇(∏_{k=1}^{r}(x − θ_{k})\frac{d}{dx})(M(x − θ_{i})^{−n}) + (x − θ_{i})^{−n+1}M[x] .

In virtue of the Mittag-Leffler decomposition

M[x, \frac{1}{∏_{i=1}^{r}(x − θ_{i})}] = M⊗(O(S)[x, \frac{1}{∏_{i=1}^{r}(x − θ_{i})}]) = M[x]⊕⨁_{i=1}^{r} \frac{1}{x − θ_{i}} M[\frac{1}{x − θ_{i}}] ,

one concludes that

(6.4.4) M[x, \frac{1}{∏_{i=1}^{r}(x − θ_{i})}] = ˜∇(∏_{i=1}^{r}(x − θ_{i})\frac{d}{dx})M[x, \frac{1}{∏_{i=1}^{r}(x − θ_{i})}] + M[x] .

At this point the proof splits into two alternative arguments (6.5 and 6.6).

6.5. From

˜∇(\frac{1}{x} ∏_{i=2}^{r}(\frac{1}{x} − \frac{1}{θ_{i}})\frac{d}{d\frac{1}{x}})M′[\frac{1}{x}] ⊂ M′[\frac{1}{x}] ,

we deduce

˜∇(x^{1−r} ∏_{i=1}^{r}(x − θ_{i})\frac{d}{dx})M′ ⊂ ⨁_{j=0}^{l} \frac{1}{x^{j}} M′ ,

for sufficiently big l ≥ 1. We consider the endomorphism Res_{∞} ˜∇ of M′, which satisfies (6.4.1) above, and

(6.5.1)  ˜∇(x\frac{d}{dx})m + (Res_{∞} ˜∇)m ∈ \frac{1}{x}M′[[\frac{1}{x}]] ,

or, equivalently

(6.5.2)  ˜∇(x^{1−r} ∏_{i=1}^{r}(x − θ_{i})\frac{d}{dx})m + (Res_{∞} ˜∇)m ∈ ⨁_{j=1}^{l} \frac{1}{x^{j}} M′ ,

for any m ∈ M′. So, for any n ≥ 0 and m ∈ M′, (6.5.3)

˜∇(x^{1−r} ∏_{i=1}^{r}(x − θ_{i})\frac{d}{dx})(x^{n+l}m) + x^{n+l}(Res_{∞} ˜∇ − (n + l)1_{M′})m ∈ ⨁_{j=0}^{n−1} x^{j}M′ .

From the invertibility of Res_{∞} ˜∇ − (n + l)1_{M′}, we deduce, that

M′[x]/ ˜∇(x^{1−r} ∏_{i=1}^{r}(x − θ_{i})\frac{d}{dx})x^{l}M′[x]

is generated by ⨁_{j=0}^{l−1} x^{j}M′. So, for any h ≥ 0,

x^{r−h−1}M′[x]/ ˜∇(∏_{i=1}^{r}(x − θ_{i})\frac{d}{dx})x^{l}M′[x]

is generated by ⨁_{j=r−h−1}^{r+l−2} x^{j}M′. We now pick l, h sufficiently big so that

x^{l}M′[x] ⊂ M[x] ⊂ x^{r−h−1}M′[x] .

This proves that M[x]/ ˜∇(∏_{i=1}^{r}(x − θ_{i})\frac{d}{dx})M[x] is finitely generated over O(S). A fortiori Γ(X, E)/∇(\frac{d}{dx})Γ(X, E) is finitely generated over O(S).

6.6. Alternative conclusion of proof.

We come back to (6.4.4), and derive the coherence of Coker_{Γ(X,E)}∇(\frac{d}{dx}) using another argument which will be of importance in the next section.

Let us first remark that

(6.6.1) M[x] ∩ ˜∇(∏_{i=1}^{r}(x − θ_{i})\frac{d}{dx})M[x, \frac{1}{∏_{i=1}^{r}(x − θ_{i})}] = ˜∇(∏_{i=1}^{r}(x − θ_{i})\frac{d}{dx})M[x] .

Indeed, if an element of M[x, _{\frac{1}{∏_{i=1}^{r}(x−θ_{i})}}] has a pole of order n ≥ 0 at θ_{i}, with

Mittag-Leffler component m(x−θ_{i})^{−n}, and is sent to M[x] by ˜∇(∏_{i=1}^{r}(x−θ_{i})\frac{d}{dx}), then m ∈ Ker(Res_{θ_{i}} ˜∇ − n1_{M}), in view of (6.4.3); hence n ∈ Imτ ∩ Z = {0}. This argument also shows that

(6.6.2)  Ker_{M[x]} ˜∇ = Ker_{M[x,_{\frac{1}{∏{r \atop i=1}(x−θ_{i})}} ]}∇ .

Putting (6.4.4) and (6.6.1) together, we see that multiplication by ∏_{i=1}^{r}(x−θ_{i})^{−1} induces an isomorphism

(6.6.3)  \begin{matrix}M[x]/ ˜∇( & ∏_{i=1}^{r}(x − θ_{i})\frac{d}{dx})M[x] ≅ \\ M[x, \frac{1}{∏_{i=1}^{r}(x − θ_{i})}]/∇(\frac{d}{dx})M[x, \frac{1}{∏_{i=1}^{r}(x − θ_{i})}] . \end{matrix}

By symmetry x ⟼ \frac{1}{x}, we also have

(6.6.2)′  Ker_{M′[ \frac{1}{x}]} ˜∇ = Ker_{M′[x,_{\frac{1}{∏{r \atop i=1}(x−θ_{i})}} ]}∇ .

(6.6.3)′  \begin{aligned}M′[\frac{1}{x}]/ ˜∇(\frac{1}{x} & ∏_{i=2}^{r}(\frac{1}{x} − \frac{1}{θ_{i}})\frac{d}{d\frac{1}{x}})M′[\frac{1}{x}] ≅ \\ & M[x, \frac{1}{∏_{i=1}^{r}(x − θ_{i})}]/∇(\frac{d}{dx})M[x, \frac{1}{∏_{i=1}^{r}(x − θ_{i})}] . \end{aligned}

This can be summarized in the following diagram of isomorphisms

(Co)Ker_{M[x]} ˜∇(∏_{i=1}^{r}(x − θ_{i})\frac{d}{dx}) (Co)Ker_{M′[ \frac{1}{x}]} ˜∇(\frac{1}{x} ∏_{i=2}^{r}(\frac{1}{x} − \frac{1}{θ_{i}})\frac{d}{d \frac{1}{x}})

↘ ↙ ·(−1)^{r}x^{r−1} ∏_{i=2}^{r} θ_{i}
(Co)Ker_{M[x, \frac{1}{x}]=M′[x, \frac{1}{x}]} ˜∇(∏_{i=1}^{r}(x − θ_{i})\frac{d}{dx})
↓ · ∏_{i=1}^{r}(x − θ_{i})^{−1}
(Co)Ker_{M[x,_{\frac{1}{∏{r \atop i=1}(x−θ_{i})}} ]}∇(\frac{d}{dx})

Diagram 6.6.4.

In terms of the open covering U = {U_{0}, U_{1}} of P_{S}^{1} with U_{0} := A_{S}^{1}, U_{1} := P_{S}^{1} \ ({0} × S), (X ⊂ U_{01} := A_{S}^{1} \ ({0} × S)), the previous diagram may be rewritten as follows. For i = 0, 1, the maps

\overline{f}_{|U_{0}∗}\underline{h}^{i}(Ω_{U_{0}/S}^{·}(log(U_{0} \ X)) ⊗ Ẽ_{|U_{0}}) \overline{f}_{|U_{1}∗}\underline{h}^{i}(Ω_{U_{1}/S}^{·}(log(U_{1} \ X)) ⊗ Ẽ_{|U_{1}})

↘ ↙

\overline{f}_{|U_{01}∗}\underline{h}^{i}(Ω_{U_{01}/S}^{·}(log(U_{01} \ X)) ⊗ Ẽ_{|U_{01}})
↓

f_{∗}\underline{h}^{i}(Ω_{X/S}^{·} ⊗ E)

↓≅

R_{DR}^{i}f_{∗}(E, ∇)

Diagram 6.6.5. are isomorphisms of O_{S}-modules (in fact, isomorphisms in MIC(S)). Let us now compute R^{i}\overline{f}_{∗}(Ω·_{P_{S}^{1}/S}(log Z) ⊗ Ẽ), using the (2 by 2) Čech bicomplex C^{·}(U, Ω_{P_{S}^{1}/S}^{·}(log Z)⊗Ẽ) of O_{S}-modules. The differentials d_{2}^{0,0} and d_{2}^{1,0} are given

by ˜∇, while d_{1}^{0,0} (resp. d_{1}^{0,1}) is induced by (j_{0}^{∗}, −j_{1}^{∗}) (resp. (−j_{0}^{∗}, j_{1}^{∗})), where j_{k} stands for the inclusion U_{01} ⊂ U_{k}. For k = 0, 1, (6.6.4) identifies Kerd_{2}^{k,0} (resp. Cokerd_{2}^{k,0}) with 2 − k copies of R_{DR}^{0}f_{∗}(E, ∇) (resp. R_{DR}^{1}f_{∗}(E, ∇)), and for the first spectral sequence of the bicomplex, we find I_{2}^{0,0} ≅ R_{DR}^{0}f_{∗}(E, ∇), I_{2}^{0,1} ≅ R_{DR}^{1}f_{∗}(E, ∇), I_{2}^{1,0} = I_{2}^{1,1} = 0. Applying [CE, chap. XV, sect. 6, case 4], we conclude:

Proposition 6.6.6. The morphism of O_{S}-modules

R^{i}\overline{f}_{∗}(Ω_{P_{S}^{1}/S}^{·}(log Z) ⊗ Ẽ) ⟶ R^{i}\overline{f}_{∗}(j_{∗}(Ω_{X/S}^{·} ⊗ E)) ≅ R_{DR}^{i}f_{∗}(E, ∇) ,

induced by the open S-embedding j : X ↪ P_{S}^{1}, is an isomorphism.

Since R^{i}\overline{f}_{∗}(Ω_{P_{S}^{1}/S}^{·}(log Z) ⊗ Ẽ) is the abutment of a spectral sequence with

E_{1}^{pq} = R^{q}\overline{f}_{∗}(Ω_{P_{S}^{1}/S}^{p}(log Z) ⊗ Ẽ), the coherence of R_{DR}^{i}f_{∗}(E, ∇) now follows from

the stability of coherence under direct image by the projective morphism \overline{f}. Q.E.D.

6.7. Base change. On combining 6.1 and 5.5, we obtain that for any fibered square of smooth K-varieties

X^{♯} ⟶^{u^{♯}} X
f^{♯} ↓  ↓ f
S^{♯}  ⟶^{u} S

with f a rational elementary fibration, and for any regular object (E, ∇) of MIC(X), the base change morphism

u^{∗}R_{DR}^{i}f_{∗}(E, ∇) ⟶ R_{DR}^{i}f_{∗}^{♯}u^{♯∗}(E, ∇)

is an isomorphism for i = 0, 1.

In fact it is immediate to generalize 6.1 (coherence) and this base change property to the case where f is a coordinatized elementary fibration.

§7 Regularity and exponents of the cokernel of a regular connection.

7.1. In this section, we establish that for any rational elementary fibration

(7.1.1)  \begin{aligned}X & & ↪^{j} & & \overline{X} \\ & ↘^{f} & & ↙^{\overline{f}} \\ & & S \end{aligned}

and for any locally free O_{X}-module of finite rank with regular integrable connection (E, ∇), the Gauss-Manin connection on R_{DR}^{i}f_{∗}(E, ∇) is regular for i = 0, 1; moreover, if the exponents of (E, ∇) belong to a certain Q-space Δ (Q ⊂ Δ ⊂ K), so do the exponents of R_{DR}^{i}f_{∗}(E, ∇).

We notice that the case i = 0 is an easy consequence of (4.1.2).

7.2. By theorem (I.5.7) and by base change (6.7), we may assume that S is the complement of a finite set of points Σ in a smooth projective curve \overline{S}. By the classical theory of embedded resolution of curves (Zariski), there is a smooth projective surface \overline{\overline{X}} containing \overline{X}, such that \overline{Z} := \overline{\overline{X}} \ \overline{X} is a union ⋃_{i=1}^{s} \overline{Z}_{j} of smooth connected curves with normal crossings, and a projective morphism \overline{\overline{f}} : \overline{\overline{X}} ⟶ \overline{S} extending \overline{f}. By rearranging the \overline{Z}_{j}, we may assume that \overline{Z}_{j} ∩\overline{X} ≅ Z_{j},

for i ≤ r, while \overline{\overline{f}}^{−1}(Σ) = ∑_{j=r+1}^{s} e_{j}\overline{Z}_{j}, for some positive integers e_{j}. Following Katz [Ka2], we shall construct a locally free extension of R_{DR}^{i}f_{∗}(E, ∇) on \overline{S} and an extension of the Gauss-Manin connection with logarithmic poles along Σ.

7.3. Let us set

Ω_{\overline{\overline{X}}/\overline{S}}^{p}(log \overline{Z}/Σ) = ∧_{O\overline{\overline{X}}}^{p} (Ω1 ; \overline{\overline{X}}(log \overline{Z}))/\overline{\overline{f}}^{∗}Ω1 ; \overline{S}(log Σ)) .

Let (˜Ẽ, ˜ ˜∇) be a locally free extension of (E, ∇) on \overline{\overline{X}} with logarithmic poles along \overline{Z} (§8). Let us filter the logarithmic De Rham complex Ω· ; \overline{\overline{X}}(log \overline{Z}) ⊗_{O\overline{\overline{X}}} ˜ Ẽ by

F^{p} = image of \overline{\overline{f}}^{∗}Ωp ; \overline{S}(log Σ) ⊗ Ω·−p ; \overline{\overline{X}}(log \overline{Z}) ⊗_{O\overline{\overline{X}}} ˜ Ẽ ,

so that

gr^{p} = F^{p}/F^{p+1} = \overline{\overline{f}}^{∗}Ωp ; \overline{S}(log Σ) ⊗_{O\overline{\overline{X}}} Ω_{\overline{\overline{X}}/\overline{S}}^{·−p}(log \overline{Z}/Σ) ⊗_{O\overline{\overline{X}}} ˜ Ẽ .

We define

R_{DR}^{i}\overline{\overline{f}}_{∗}(˜Ẽ, ˜ ˜∇) = R^{i}\overline{\overline{f}}_{∗}(Ω_{\overline{\overline{X}}/\overline{S}}^{·}(log \overline{Z}/Σ) ⊗_{O\overline{\overline{X}}} ˜ Ẽ)/\{O\overline{S} − torsion \} ,

which is a locally free O\overline{S}-module of finite rank because \overline{\overline{f}} is projective and \overline{S} is a curve. Moreover,

R^{i}\overline{\overline{f}}_{∗}(Ω_{\overline{\overline{X}}/\overline{S}}^{·}(log \overline{Z}/Σ)⊗_{O\overline{\overline{X}}} ˜ Ẽ)_{|S} ≅ R^{i}\overline{f}_{∗}(Ω_{\overline{X}/\overline{S}}^{·}(log Z)⊗_{O\overline{X}} ˜ Ẽ_{|\overline{X}}) ≅ R_{DR}^{i}f_{∗}(E, ∇) ,

according to 6.6.6, so that R_{DR}^{i}\overline{\overline{f}}_{∗}(˜Ẽ, ˜ ˜∇) is a locally free extension of R_{DR}^{i}f_{∗}(E, ∇). On the other hand, an extension of the Gauss-Manin connection

R^{i}\overline{\overline{f}}_{∗}(Ω_{\overline{\overline{X}}/\overline{S}}^{·}(log \overline{Z}/Σ) ⊗_{O\overline{\overline{X}}} ˜ Ẽ) ⟶ Ω1 ; \overline{S}(log Σ) ⊗_{O\overline{S}} R^{i}\overline{\overline{f}}_{∗}(Ω_{\overline{\overline{X}}/\overline{S}}^{·}(log \overline{Z}/Σ) ⊗_{O\overline{\overline{X}}} ˜ Ẽ)

is provided by the coboundary map of the long exact sequence

0 ⟶ gr^{1} ⟶ F^{0}/F^{2} ⟶ gr^{0} ⟶ 0

(since Ω1 ; \overline{S}(log Σ)⊗_{O\overline{S}}R^{i}\overline{\overline{f}}_{∗}(Ω_{\overline{\overline{X}}/\overline{S}}^{·}(log \overline{Z}/Σ)⊗_{O\overline{\overline{X}}} ˜ Ẽ) may be identified with R^{i+1}\overline{\overline{f}}_{∗}(gr^{1})).

That map factors through torsion, and induces an extension of the Gauss-Manin connection on R_{DR}^{i}\overline{\overline{f}}_{∗}(˜Ẽ, ˜ ˜∇), with logarithmic poles along Σ.

7.4. Katz’s argument [Ka2, 7] applies without change to our situation and shows that the indicial polynomial of R_{DR}^{i}\overline{\overline{f}}_{∗}(˜Ẽ, ˜ ˜∇) at s ∈ Σ divides

(7.4.1)  ∏_{j∈J} ∏_{f_{j}=0}^{e_{j}−1} P_{j}(e_{j}X − f_{j}) ,

where P_{j} is the indicial polynomial of (˜Ẽ, ˜ ˜∇) at \overline{Z}_{j}, \overline{\overline{f}}^{−1}(s) = ∑_{j∈J} e_{j}\overline{Z}_{j}. This completes the proof of the statement in 7.1.

7.5 Remarks.

1) A closer examination of Katz’s argument allows to replace (7.4.1) by the least common multiple of the P_{j}(e_{j}X − f_{j}), for j ∈ J and f_{j} ∈ {0, 1, …, e_{j} − 1}.

2) If K ⊂ C and τ is the canonical section K/Z ⟶ K, defined by Re(τ(z)) ∈ [0, 1[, and if (˜Ẽ, ˜ ˜∇) is the τ-extension of (E, ∇) on \overline{\overline{X}}, then 7.4 shows that R_{DR}^{i}\overline{\overline{f}}_{∗}(˜Ẽ, ˜ ˜∇) is the τ-extension of R_{DR}^{i}f_{∗}(E, ∇) on \overline{S}; indeed Re(z) ∈ [0, 1[⇒ Re(\frac{z+f_{j}}{e_{j}}) ∈ [0, 1[.

3) The arguments 7.2,3,4 work as well for any elementary fibration, not necessarily rational, over a curve.

§8 Proof of the main theorems: finiteness, regularity, monodromy, base change (in the regular case).

Let i be a non-negative integer. Let f : X ⟶ S be a smooth morphism of smooth K-varieties, such that the Artin subset A_{i}(f) is S itself (cf. (1.8)). Let E be a coherent O_{X}-module endowed with an integrable regular connection ∇.

Theorem 8.1. (Finiteness theorem). R_{DR}^{i}f_{∗}(E, ∇) is a coherent O_{S}-module.

Theorem 8.2. (Regularity theorem). The Gauss-Manin connection ℵ on R_{DR}^{i}f_{∗}(E, ∇) is regular.

Theorem 8.3. (Monodromy theorem). If the exponents of ∇ all belong to some Q-subspace Δ of K (Δ ⊃ Q), so do the exponents of ℵ.

Theorem 8.4. (Base change theorem). For any smooth K-variety S^{♯} and any morphism u : S^{♯} ⟶ S, the base change morphism

u^{∗}R_{DR}^{i}f_{∗}(E, ∇) ⟶ R_{DR}^{i}f_{∗}^{♯}u^{♯∗}(E, ∇)

attached to the fibered square

X^{♯} ⟶^{u^{♯}} X
f^{♯} ↓  ↓ f
S^{♯}  ⟶^{u} S

is an isomorphism in MIC(S^{♯}).

Proofs. Let us say that (E, ∇) satisfies property P_{Δ} if and only if E is coherent, ∇ is regular, and all exponents of ∇ belong to Δ. Then P_{Δ} is a strongly exact property (I.5.2,I.6.5.3), local for the étale topology (I.5.3,I.6.5.4). By the lemma on dévissage 3.2, in order to prove theorems 8.1,2,3, it suffices to consider the case when f is a rational elementary fibration and i = 0 or 1, which was settled in sections 6 and 7.

As for the base change theorem, one reduces as in (5.8.2) to the case when f is a tower of coordinatized elementary fibrations. The associated Leray spectral sequences allow us to further reduce to the case when f is a single coordinatized elementary fibration, which was settled in 6.7. Q.E.D.

Appendix C: Berthelot’s comparison theorem on O_{X}- vs. D_{X}- linear duals.

We reproduce in this section the proof of a very powerful theorem of Berthelot, in its relative version^{(1)}. Let f : X ⟶ S be a smooth morphism of smooth K-varieties of relative dimension d, and let ω_{X/S} := Ω_{X/S}^{d}.

We recall a few basic definitions from the theory of D-modules.

D_{X/S} (resp. D_{X/S,n}) = the sheaf of differential operators (resp. of order ≤ n) on X relative to S, with its two structures of O_{X}-module, via left and right multiplication. For any O_{X}-module E, in the tensor product D_{X/S} ⊗_{O_{X}} E (resp. E ⊗_{O_{X}} D_{X/S}) the right (resp. left) structure of O_{X}-module of D_{X/S} is used.

For any sheaf of rings A on X, let D^{b}(A) be the bounded derived category of left A-modules.

D_{c}^{b}(D_{X/S}) = the full triangulated subcategory of D^{b}(D_{X/S}), consisting of objects with D_{X/S}-coherent cohomology. It is well-known that in this framework the relative De Rham functor may be viewed as a derived functor

\begin{matrix}DR_{X/S} = R\underline{Hom}_{D_{X/S}}(O_{X}, −) : D^{b}(D_{X/S}) & ⟶ D^{b}(f^{−1}O_{S}) \\ E & ⟼ Ω_{X/S}^{·} ⊗ E . \end{matrix}

In the next lemma, for a left D_{X/S}-module E, we form the two tensor products D_{X/S} ⊗_{O_{X}} E and E ⊗_{O_{X}} D_{X/S}. On the first, we have a left (resp. right) D_{X/S}-module structure given by

Q(P ⊗ e) = QP ⊗ e ,

(resp.

(P ⊗ e)D = P D ⊗ e − P ⊗ De ) ,

for sections P, Q of D_{X/S}, e of E and D of the relative tangent space. On the second tensor product, we have instead a left (resp. right) D_{X/S}-module structure given by

D(e ⊗ P) = De ⊗ P + e ⊗ DP ,

(resp.

(e ⊗ P)Q = e ⊗ P Q ) ,

One checks that both D_{X/S} ⊗_{O_{X}} E and E ⊗_{O_{X}} D_{X/S} are D_{X/S}-bi-modules. Berthelot [Be2, 1.3.1] noticed the existence of the following transposition isomorphism.

(1) We are indebted to M. Cailotto and L. Fiorot for providing us very good notes of 2 lectures held by Berthelot at Padova University in September 1998. P. Berthelot kindly gave us his permission to include them in this book.

Lemma C.1. Let E be a left D_{X/S}-module. There is a unique isomorphism of D_{X/S}-bi-modules

D_{X/S} ⊗_{O_{X}} E⟶^{≅}E ⊗_{O_{X}} D_{X/S}

sending 1 ⊗ e to e ⊗ 1, for any section e of E.

Theorem C.2. Let F ∈ D_{c}^{b}(D_{X}).

(i) There is a canonical morphism in D^{b}(D_{X/S})

F^{∨} := R\underline{Hom}_{O_{X}} (F, O_{X}) ⟶ R\underline{Hom}_{D_{X/S}} (F, D_{X/S} ⊗ ω_{X/S}^{−1}[d]) =: F_{X/S}^{∗}

(ii) If F has O_{X}-coherent cohomology, the previous morphism is an isomor-

phism. (iii) In the general case, the morphism in (i) induces an isomorphism

DR_{X/S} (F^{∨}) ⟶^{≅} DR_{X/S} (F_{X/S}^{∗}) .

Proof. We make use of the isomorphism, valid for E, F, G in D^{b}(D_{X}),

R\underline{Hom}_{D_{X/S}} (E ⊗_{O_{X}}^{L} F, G) ≅ R\underline{Hom}_{D_{X/S}} (E, R\underline{Hom}_{O_{X}} (F, G))

in the special case of E = O_{X} and G = D_{X/S} ⊗ ω_{X/S}^{−1}[d] so as to obtain

\begin{aligned}F_{X/S}^{∗} = R\underline{Hom}_{D_{X/S}} & (F, D_{X/S} ⊗ ω_{X/S}^{−1}[d]) \\ & ≅ R\underline{Hom}_{D_{X/S}} (O_{X}, R\underline{Hom}_{O_{X}} (F, D_{X/S} ⊗ ω_{X/S}^{−1}[d])) . \end{aligned}

We have a canonical morphism

(C.2.1)  R\underline{Hom}_{O_{X}} (F, O_{X}) ⊗_{O_{X}} (D_{X/S} ⊗ ω_{X/S}^{−1}[d]) ; ↓ ; R\underline{Hom}_{O_{X}} (F, D_{X/S} ⊗ ω_{X/S}^{−1}[d]) .

This in turn induces another canonical morphism

(C.2.2) R\underline{Hom}_{D_{X/S}} (O_{X}, R\underline{Hom}_{O_{X}} (F, O_{X}) ⊗_{O_{X}} (D_{X/S} ⊗ ω_{X/S}^{−1}[d])) ; ↓ ; R\underline{Hom}_{D_{X/S}} (O_{X}, R\underline{Hom}_{O_{X}} (F, D_{X/S} ⊗ ω_{X/S}^{−1}[d])) ≅ F_{X/S}^{∗} .

By lemma C.1 applied to the left D_{X/S}-module E = R\underline{Hom}_{O_{X}} (F, O_{X}), the top termof (C.2.2) may be rewritten as

R\underline{Hom}_{D_{X/S}} (O_{X}, D_{X/S} ⊗ R\underline{Hom}_{O_{X}} (F, O_{X}) ⊗_{O_{X}} ω_{X/S}^{−1}[d])

but, O_{X} being D_{X/S}-coherent, this equals

\begin{matrix}R\underline{Hom}_{D_{X/S}} (O_{X}, D_{X/S}) ⊗ R\underline{Hom}_{O_{X}} (F, O_{X}) ⊗_{O_{X}} ω_{X/S}^{−1}[d] \\ ≅ DR_{X/S}(D_{X/S}) ⊗ F^{∨} ⊗ ω_{X/S}^{−1}[d] ≅ F^{∨} \end{matrix}

since DR_{X/S}(D_{X/S}) ≅ ω_{X/S}[−d]. All in all, we obtain the canonical morphism (i), which is an isomorphism in the case of coherent O_{X}-modules (because then (C.2.1) is an isomorphism).

From (i), we get the induced map of relative De Rham complexes appearing in (iii). To prove that this map is an isomorphism, we may work locally. We may then assume that F is free of finite rank over D_{X/S}, and we are reduced to the case of F = D_{X/S}.

We then have

D_{X/S}^{∨} = R\underline{Hom}_{O_{X}} (D_{X/S}, O_{X})
= R \underleftarrow{lim} R\underline{Hom}_{O_{X}} (D_{X/S,n}, O_{X})
= \underleftarrow{lim} P_{X/S}^{n} =: P_{X/S}^{∞}

where the next to last equality holds true because the projective system P_{X/S}^{n} = R\underline{Hom}_{O_{X}} (D_{X/S,n}, O_{X}) has surjective transition morphisms. On the other side of (i) we get

R\underline{Hom}_{O_{X}} (D_{X/S}, D_{X/S} ⊗ ω_{X/S}^{−1}[d]) ≅ R \underleftarrow{lim} R\underline{Hom}_{O_{X}} (D_{X/S,n}, D_{X/S} ⊗ ω_{X/S}^{−1}[d])
≅ R \underleftarrow{lim} (R\underline{Hom}_{O_{X}} (D_{X/S,n}, O_{X}) ⊗_{O_{X}} (D_{X/S} ⊗ ω_{X/S}^{−1}[d]))
≅ \underleftarrow{lim} (P_{X/S}^{n} ⊗_{O_{X}} (D_{X/S} ⊗ ω_{X/S}^{−1}[d]))

The map (C.2.1) may then be spelled out as the natural morphism

P_{X/S}^{∞} ⊗_{O_{X}} D_{X/S} ⊗ ω_{X/S}^{−1}[d] ⟶ \underleftarrow{lim} (P_{X/S}^{n} ⊗_{O_{X}} (D_{X/S} ⊗ ω_{X/S}^{−1}[d]))

On applying the functor R\underline{Hom}_{D_{X/S}}(O_{X}, −) = DR_{X/S}(−) = Ω_{X/S}^{·} ⊗ − we get the morphism (i) of the two duals

D_{X/S}^{∨}  (D_{X/S})_{X/S}^{∗}
∥  ∥
Ω_{X/S}^{·} ⊗ P_{X/S}^{∞} ⊗_{O_{X}} D_{X/S} ⊗ ω_{X/S}^{−1}[d] ⟶ Ω_{X/S}^{·} ⊗ \underleftarrow{lim} (P_{X}^{n} ⊗_{O_{X}} (D_{X/S} ⊗ ω_{X/S}^{−1}[d]))

(in particular, those duals are interpreted as relative De Rham complexes).

Until now, only the left structure of D_{X/S} ⊗ ω_{X/S}^{−1} as a D_{X/S}-module has

been used. We now use the right structure for calculating the relative De Rham functor

DR_{X/S} (D_{X/S}^{∨})  DR_{X/S} ((D_{X/S})_{X/S}^{∗})
∥  ∥
Ω_{X/S}^{·} ⊗ P_{X/S}^{∞} ⊗_{O_{X}} D_{X/S} ⊗ ω_{X/S}^{−1} ⊗ Ω_{X/S}^{·} ⟶ Ω_{X/S}^{·} ⊗ \underleftarrow{lim} (P_{X/S}^{n} ⊗_{O_{X}} (D_{X/S} ⊗ ω_{X/S}^{−1})) ⊗ Ω_{X/S}^{·}

The above is a map L^{··} ⟶ K^{··} of bicomplexes whose cohomology may be calculated from either one of the two spectral sequences. The second spectral sequence immediately gives the identification

IIE_{2}^{p,d}(L^{··}) = Ω_{X/S}^{p} ⊗ P_{X/S}^{∞}

and _{II}E_{2}^{p,q}(L^{··}) = 0 for q ≠ d, since D_{X/S} ⊗ ω_{X/S}^{−1} ⊗ Ω_{X/S}^{·} ≅ O_{X}[−d]. The same

fact, after we express \underleftarrow{lim} (P_{X/S}^{n} ⊗_{O_{X}} (D_{X/S} ⊗ ω_{X/S}^{−1})) as \underline{Hom}_{O_{X}} (D_{X/S}, D_{X/S} ⊗ ω_{X/S}^{−1})

to write \underleftarrow{lim} (P_{X/S}^{n} ⊗_{O_{X}} (D_{X/S} ⊗ ω_{X/S}^{−1}))⊗Ω_{X/S}^{·} = \underline{Hom}_{O_{X}} (D_{X/S}, D_{X/S} ⊗ ω_{X/S}^{−1} ⊗ Ω_{X/S}^{·}), shows that

IIE_{2}^{p,d}(K^{··}) = Ω_{X/S}^{p} ⊗ \underline{Hom}_{O_{X}} (D_{X/S}, O_{X})

while _{II}E_{2}^{p,q}(K^{··}) = 0 for q ≠ d, and our conclusion follows. Q.E.D.

Appendix D: Introduction to Dwork’s algebraic dual theory.

D.1. Serre duality for an elementary fibration. We start with an elementary fibration as in (1.1.1). We denote by J = J_{Z} the invertible sheaf of O\overline{X}-ideals of Z in \overline{X}. The dualizing sheaf for the morphism \overline{f} is the O\overline{X}-module ω = Ω_{\overline{X}/S}^{1}. For any scheme Y, let Ab(Y) (resp. Mod(Y), resp. Coh(Y)) be the category of abelian sheaves on Y (resp. of O_{Y} -modules, resp. of coherent O_{Y} -modules, resp. of quasi-coherent O_{Y} -modules). We also set D_{c}^{b}(Y) := D_{Coh(Y)}^{b}(Mod(Y)), D^{b}(Y) := D^{b}(Mod(Y)).

We consider the exact functor

(D.1.1)  \begin{matrix}Mod(\overline{X}) & ⟶ Mod(\overline{X}) \\ E & ⟼ E(∗Z) = lim_{s^{⟶}} J_{Z}^{−s} ⊗_{O\overline{X}} E . \end{matrix}

For E coherent E(∗Z) = j_{∗}(j^{−1}(E)). For any E in Coh(\overline{X}),

Ê = lim_{N^{⟵}} E/J_{Z}^{N}E

is a coherent Ô_{\overline{X}|Z}-module, which we rather regard as an object of Mod(\overline{X}), in general not even quasi-coherent. So again we have an exact functor

(D.1.2)  \begin{matrix}ˆ : Coh(\overline{X}) & ⟶ Mod(\overline{X}) \\ E & ⟼ Ê = lim_{N^{⟵}} E/J_{Z}^{N}E . \end{matrix}

We assume from now on in this section that E is a coherent and locally free O\overline{X}-module. Grothendieck-Serre duality [H1, VII, Cor. 4.3] gives, for any E as before, an isomorphism

R\overline{f}_{∗}(ω ⊗ E^{∨})[1]⟶^{≅}R\underline{Hom}_{S}(R\overline{f}_{∗}E, O_{S}) .

We apply [CE, XV, Thm. 5.11] with r = 2, q = −1, q′ = 0 to the spectral sequence

E_{2}^{u,v} := \underline{Ext}_{S}^{u}(R^{−v}\overline{f}_{∗}E, O_{S}) ⇒ R^{u+v+1}\overline{f}_{∗}(ω ⊗ E^{∨})

and deduce exact sequences

(D.1.3)  E_{2}^{−2,0} = 0 ⟶ E_{2}^{0,−1}⟶^{∼}H^{−1} ⟶ E_{2}^{−1,0} = 0 ,

(D.1.4)  E_{2}^{−1,0} = 0 ⟶ E_{2}^{1,−1} ⟶ H^{0} ⟶ E_{2}^{0,0} ⟶ E_{2}^{2,−1} ⟶ H^{1} = 0 .

So we get exact sequences:

(D.1.5)  0 ⟶ \underline{Hom}_{S}(R^{1}\overline{f}_{∗}E, O_{S})⟶^{∼}R^{0}\overline{f}_{∗}(ω ⊗ E^{∨}) ⟶ 0 ,

and

(D.1.6)  \begin{matrix}0 ⟶ & \underline{Ext}_{S}^{1}(R^{1}\overline{f}_{∗}E, O_{S}) ⟶ R^{1}\overline{f}_{∗}(ω ⊗ E^{∨}) \\ ⟶ \underline{Hom}_{S}(R^{0}\overline{f}_{∗}E, O_{S}) ⟶ \underline{Ext}_{S}^{2}(R^{1}\overline{f}_{∗}E, O_{S}) ⟶ 0 . \end{matrix}

On interchanging E and ω ⊗ E^{∨}, we obtain the exact sequences

(D.1.7)  0 ⟶ \underline{Hom}_{S}(R^{1}\overline{f}_{∗}(ω ⊗ E^{∨}), O_{S})⟶^{∼}R^{0}\overline{f}_{∗}E ⟶ 0 ,

and (D.1.8)

\begin{matrix}0 ⟶ & \underline{Ext}_{S}^{1}(R^{1}\overline{f}_{∗}(ω ⊗ E^{∨}), O_{S}) ⟶ R^{1}\overline{f}_{∗}E \\ ⟶ \underline{Hom}_{S}(R^{0}\overline{f}_{∗}(ω ⊗ E^{∨}), O_{S}) ⟶ \underline{Ext}_{S}^{2}(R^{1}\overline{f}_{∗}(ω ⊗ E^{∨}), O_{S}) ⟶ 0 . \end{matrix}

Let us explain in concrete terms the pairing

(D.1.9)  R^{0}\overline{f}_{∗}(ω ⊗ E^{∨}) × R^{1}\overline{f}_{∗}(E) ⟶ O_{S} .

That pairing factors through cup-product and an isomorphism R^{1}\overline{f}_{∗}(ω)⟶^{Tr}O_{S} that may be described as follows. After finite étale base-change S′ ⟶ S, we may assume that Z is a disjoint union of components Z_{i}, for i = 1, …, r, ∞, isomorphic to S via g. In terms of an affine covering \{U_{j}\}_{j} of \overline{X}, we may represent a global section η of the O_{S}-module R^{1}\overline{f}_{∗}(ω) as the class of an (alternating) cocycle η_{j,k} ∈ Γ(U_{j,k}, ω). We may write η_{j,k} = η_{k} − η_{j}, for η_{j} ∈ Γ(U_{j}, j_{∗}(ω_{|X})). Then

Tr(η) = ∑_{i,j} Res_{Z_{i}}η_{j} .

We now replace in the previous pairing E by E ⊗ J_{Z}^{N} for sufficiently big N, so that R^{0}\overline{f}_{∗}(E ⊗ J_{Z}^{N}) = 0 and R^{1}\overline{f}_{∗}(ω ⊗ E^{∨} ⊗ J_{Z}^{−N}) = 0. The sequence D.1.8 then gives an isomorphism

(D.1.10) 0 ⟶ R^{1}\overline{f}_{∗}(E ⊗ J_{Z}^{N})⟶^{∼}\underline{Hom}_{S}(R^{0}\overline{f}_{∗}(ω ⊗ E^{∨} ⊗ J_{Z}^{−N}), O_{S}) ⟶ 0 .

Taking into account (D.1.5) that now gives

(D.1.11) 0 ⟶ \underline{Hom}_{S}(R^{1}\overline{f}_{∗}(E ⊗ J_{Z}^{N}), O_{S})⟶^{∼}R^{0}\overline{f}_{∗}(ω ⊗ E^{∨} ⊗ J_{Z}^{−N}) ⟶ 0 ,

we obtain, for sufficiently big N, perfect pairings

(D.1.12)  < , >_{N}: R^{0}\overline{f}_{∗}(ω ⊗ E^{∨} ⊗ J_{Z}^{−N}) × R^{1}\overline{f}_{∗}(E ⊗ J_{Z}^{N}) ⟶ O_{S} .

In order to take limits for N → ∞ in (D.1.12), we consider the inverse system of exact 6-terms rows (S_{N}, φ_{M,N})_{M≥N} (D.1.13) S_{N} = (0 ⟶ R^{0}\overline{f}_{∗}E ⟶ R^{0}\overline{f}_{∗}(E/J_{Z}^{N}E) ⟶ R^{1}\overline{f}_{∗}(J_{Z}^{N}E) ⟶ R^{1}\overline{f}_{∗}E ⟶ 0) .

Since when S = SpecA is affine the projective system of A-modules (Γ(S, R^{0}\overline{f}_{∗}(E/J_{Z}^{N}E)))_{N} satisfies the Mittag-Leffler condition, on taking the projective limit of (S_{N}, φ_{M,N})_{M≥N}, we obtain an exact row

(D.1.14) 0 ⟶ R^{0}\overline{f}_{∗}E ⟶ R^{0}\overline{f}_{∗}(Ê) ⟶ lim_{N^{⟵}} R^{1}\overline{f}_{∗}(J_{Z}^{N}E) ⟶ R^{1}\overline{f}_{∗}E ⟶ 0 .

Similarly, we obtain a commutative diagram with exact rows (D.1.15)

0 ⟶  R^{0}\overline{f}_{∗}E  ⟶  R^{0}\overline{f}_{∗}Ê  ⟶ lim_{N^{⟵}} R^{1}\overline{f}_{∗}(J_{Z}^{N}E) ⟶ R^{1}\overline{f}_{∗}E ⟶ 0
↓  ↓  ↓  ↓
0 ⟶ R^{0}\overline{f}_{∗}(E(∗Z)) ⟶ R^{0}\overline{f}_{∗}(Ê(∗Z)) ⟶ lim_{N^{⟵}} R^{1}\overline{f}_{∗}(J_{Z}^{N}E) ⟶  0  ⟶ 0

Therefore, (D.1.16)

R^{0}\overline{f}_{∗}(Ê(∗Z))/R^{0}\overline{f}_{∗}(E(∗Z))⟶^{∼} lim_{N^{⟵}} R^{1}\overline{f}_{∗}(J_{Z}^{N}E)⟶^{∼}\underline{Hom}_{S}(R^{0}\overline{f}_{∗}(ω⊗E^{∨}(∗Z)), O_{S}) .

If R^{1}\overline{f}_{∗}E = 0, we also get

(D.1.17)  R^{0}\overline{f}_{∗}Ê/R^{0}\overline{f}_{∗}E⟶^{∼}R^{0}\overline{f}_{∗}(Ê(∗Z))/R^{0}\overline{f}_{∗}(E(∗Z)) .

We say that an O_{S}-linear pairing E × F ⟶ O_{S} of O_{S}-modules is left-perfect if the corresponding O_{S}-linear map F ⟶ \underline{Hom}_{S}(E, O_{S}) is an isomorphism. We again prefer to regard the composite isomorphism in (D.1.16) as a left-perfect pairing

(D.1.18)  < , >: R^{0}\overline{f}_{∗}(ω ⊗ E^{∨}(∗Z)) × R^{0}\overline{f}_{∗}Ê(∗Z)/R^{0}\overline{f}_{∗}E(∗Z) ⟶ O_{S} .

We point out that in general the sheaf of O_{S}-modules R^{0}\overline{f}_{∗}Ê is not quasicoherent. We now assume that E is endowed with an integrable relative connection

(D.1.19)  ∇ : D_{\overline{X}/S} ⟶ \underline{End}_{\overline{f}^{−1}(O_{S})}(E(∗Z))

with poles along Z, which makes E(∗Z) into a left D_{\overline{X}/S}-module. From this we deduce a similar structure on

Coker(E(∗Z) → Ê(∗Z)) ≅ Coker(E → Ê) .

From the dual connection (E^{∨} = \underline{Hom}_{O\overline{X}} (E, O\overline{X}), ∇^{∨}) (with poles along Z), we get a structure of left D_{\overline{X}/S}-module on E^{∨}(∗Z), and from the natural right structure of ω, we deduce a structure of right D_{\overline{X}/S}-module on ω ⊗ E^{∨}(∗Z). For the previous structures and for sections L of R^{0}\overline{f}_{∗}D_{\overline{X}/S}, η of R^{0}\overline{f}_{∗}(ω ⊗E^{∨}(∗Z)), and h of R^{0}\overline{f}_{∗}Ê(∗Z), we have

(D.1.20)  < ηL, h >=< η, Lh > ,

as can be immediately checked via the previous residue description of Serre duality.

Proposition D.1.21. (Dwork duality principle for an elementary fibration.) Let E be a coherent and locally free O\overline{X}-module and assume E(∗Z) carries a structure of a left D_{\overline{X}/S}-module. Let

\begin{aligned}DR(\overline{X}/S, E(∗Z)) := [E & (∗Z) ⟶ ω ⊗ E(∗Z)] \\ & 0 1 \end{aligned}

be the De Rham complex of the D_{\overline{X}/S}-module E(∗Z) and let

\begin{matrix}DW (\overline{X}/S, E^{∨}) := [R^{0}\overline{f}_{∗}(Ê^{∨}(∗Z)) & /R^{0}\overline{f}_{∗}(E^{∨}(∗Z)) ⟶ R^{0}\overline{f}_{∗}(ω ⊗ Ê^{∨}(∗Z))/R^{0}\overline{f}_{∗}(ω ⊗ E^{∨}(∗Z))] \\ 0 1 \end{matrix}

be the complex of Dwork’s algebraic dual theory (for E), where E^{∨} denotes the dual O\overline{X}-module of E with the dual connection. There is a canonical isomorphism

(D.1.21.1)  \begin{aligned}DW & (\overline{X}/S, E^{∨})⟶^{∼}\underline{Hom}_{S}(\overline{f}_{∗}DR(\overline{X}/S, E(∗Z))[1], O_{S}) \\ & = R\underline{Hom}_{O_{S}}(Rf_{∗}DR(X/S, E_{|X})[1], O_{S}) . \end{aligned}

In particular, we have a canonical isomorphism

H^{0}(DW (\overline{X}/S, E^{∨}))⟶^{∼}\underline{Hom}_{S}(H_{DR}^{1}(X/S, (E, ∇)_{|X}), O_{S}) ,

and an exact sequence

0 ⟶ \underline{Ext}_{S}^{1}(H_{DR}^{1}(X/S, (E, ∇)_{|X}), O_{S}) ⟶ H^{1}(DW (\overline{X}/S, E^{∨}))

⟶ \underline{Hom}_{S}(H_{DR}^{0}(X/S, (E, ∇)_{|X}), O_{S}) ⟶ \underline{Ext}_{S}^{2}(H_{DR}^{1}(X/S, (E, ∇)_{|X}), O_{S})

⟶ 0 .

If, in particular, the O_{S}-module H_{DR}^{1}(X/S, (E, ∇)_{|X}) is locally free, we have a canonical isomorphism

H^{1}(DW (\overline{X}/S, E^{∨}))⟶^{∼}\underline{Hom}_{S}(H_{DR}^{0}(X/S, (E, ∇)_{|X}), O_{S}) .

Proof. The main point that still requires a proof is the last equality in (D.1.21.1). Now Rf_{∗}DR(X/S, E_{|X}) = f_{∗}DR(X/S, E_{|X}) because f is affine, and \underline{Hom}_{O_{S}}(f_{∗}DR(X/S, E_{|X}), O_{S}) = R\underline{Hom}_{O_{S}}(Rf_{∗}DR(X/S, E_{|X}), O_{S}) because the terms of DR(X/S, E_{|X}) are O_{S}-locally free. Granting that, the last two formulas follow by application of [CE, XV, Thm. 5.11] with r = 2, q = −1, q′ = 0 to the spectral sequence

E_{2}^{u,v} := \underline{Ext}_{S}^{u}(H_{DR}^{−v}(X/S, (E, ∇)_{|X}), O_{S}) ⇒ H^{u+v+1}(DW (\overline{X}/S, E^{∨})) .

We deduce exact sequences (D.1.3)

\begin{matrix}E_{2}^{−2,0} = 0 ⟶ & E_{2}^{0,−1} = \underline{Hom}_{S}(H_{DR}^{1}(X/S, (E, ∇)_{|X}), O_{S}) \\ ⟶^{∼}H^{0}(DW (\overline{X}/S, E^{∨})) ⟶ E_{2}^{−1,0} = 0 , \end{matrix}

and (D.1.4)

E_{2}^{−1,0} = 0 ⟶ E_{2}^{1,−1} = \underline{Ext}_{S}^{1}(H_{DR}^{1}(X/S, (E, ∇)_{|X}), O_{S}) ⟶ H^{0} = H^{1}(DW (\overline{X}/S, E^{∨}))
⟶E_{2}^{0,0} = \underline{Hom}_{S}(H_{DR}^{0}(X/S, (E, ∇)_{|X}), O_{S}) ⟶ E_{2}^{2,−1} = \underline{Ext}_{S}^{2}(H_{DR}^{1}(X/S, (E, ∇)_{|X}), O_{S})
⟶ H^{1} = H^{2}(DW (\overline{X}/S, E^{∨})) = 0 .

The conclusion follows. Q.E.D.

D.2 Cohomology with compact supports. It is tempting to interpret the results of the previous section in terms of Deligne’s theory of direct images with proper supports for coherent O_{X}-modules [H1, Appendix]. In this section we relax the conditions on the diagram of K-varieties

(D.2.0)  \begin{aligned}X & ↪^{j} & \overline{X} & ↩ & Z \\ & ↘^{f} & ↓ \overline{f} & ↙ \\ & & S \end{aligned}

by requiring only that f is smooth of relative dimension d, that \overline{f} is proper, and that j is an open immersion of complement Z = \overline{X} \ j(X), the closed subscheme defined by the ideal sheaf J_{Z}.

For any category C, let P ro C denote the category of (Artin-Rees) pro-objects of C [G2]. Objects of P ro C are projective systems indexed by Z of objects of C, and morphisms are defined by

Hom_{P ro C}((A_{n})_{n∈Z}, (B_{n})_{n∈Z}) = \underrightarrow{lim}_{k∈Z} Hom((A_{n})_{k}, (B_{n}))

where (A_{n})_{k} denotes the pro-object obtained from (A_{n}) by shifting k places to the right (i.e. its n-th component is A_{n+k}), and the second Hom means morphisms of diagrams. We point out that, for any C, the category P ro P ro C is equivalent to P ro C.

Deligne [loc. cit.] defines an exact functor that he denotes j_{!}, and we here j_{!}^{Del},

(D.2.1)  \begin{matrix}j_{!}^{Del} : Coh(X) & ⟶ P ro Coh(\overline{X}) \\ E & ⟼ “ lim_{N^{←}} ”J_{Z}^{N} \overline{E} , \end{matrix}

independent of the coherent extension \overline{E} of E to \overline{X}. We prefer to define our functor “direct image with compact supports” (already in derived categories) as the composition of R lim_{←} and Deligne’s j_{!}^{Del}, that is we set

(D.2.2)  \begin{matrix}j_{!} : D_{c}^{b}(X) & ⟶ D^{b}(\overline{X}) \\ E & ⟼ R lim_{N^{←}}(J_{Z}^{N} \overline{E}) , \end{matrix}

where for a bounded complex … ⟶ E_{i} ⟶ E_{i+1} ⟶ … of coherent O_{X}- modules that represents E, \overline{E} is represented an arbitrary extension to a bounded complex … ⟶ \overline{E}_{i} ⟶ \overline{E}_{i+1} ⟶ … of coherent O\overline{X}-modules and O\overline{X}-linear maps. The exact functor “extension by zero” of [H1, II, Ex. 1.19] will be denoted by j_{!}^{top} : Ab(X) ⟶ Ab(\overline{X}).

When \overline{X} is smooth, we may deduce from j_{!}, a triangulated functor

(D.2.3)  j_{!} : D_{Coh(X)}^{b}(D_{X}) ⟶ D^{b}(D\overline{X}) ,

where D_{Coh(X)}^{b}(D_{X}) denotes the bounded derived category of D_{X}-modules with O_{X}-coherent cohomology, and D^{b}(D\overline{X}) the bounded derived category of D\overline{X}- modules. If E is an object of D_{Coh(X)}^{b}(D_{X}) and \overline{E} denotes any extension of E

to an object of D_{Coh(\overline{X})}^{b}(D\overline{X}), the completion ˆ \overline{E} of \overline{E} along Z is a well defined

object of D^{b}(D\overline{X}). We may calculate (\overline{X} smooth) j_{!}E from the exact triangle in D^{b}(D\overline{X})

(D.2.3.2)  \overline{E} ; ↗  ↘ ; j_{!}(E)  ←^{[+1]}  ˆ  \overline{E}

In general, we set Rf_{!} = R\overline{f}_{∗} ∘ j_{!} : D_{c}^{b}(X) ⟶ D^{b}(S). The interpretation of the functor lim_{←} as a direct image of a morphism of topoi (cf. [BO, proof of 7.20], e.g.) and [SGA 4, Exp. V, Prop. 5.4] show that

Rf_{!}(E) = R\overline{f}_{∗}(R lim_{←}(J_{Z}^{N} \overline{E})) = R lim_{←}(R\overline{f}_{∗}(J_{Z}^{N} \overline{E})) .

The definition of \overline{f}^{!}F, for F ∈ D_{c}^{b}(S), is given in [H1, VII.3.4]. We are not too interested in what it is, except later when \overline{f} will be assumed to be smooth and \overline{f}^{!}F = \overline{f}^{−1}F ⊗ Ω_{\overline{X}/S}^{d}[d]. In any case, f^{!}F = (\overline{f}^{!}F)_{|X} = \overline{f}^{−1}F ⊗ Ω_{X/S}^{d}[d].

Lemma D.2.4. ∀E ∈ D_{c}^{b}(X), F ∈ D_{c}^{b}(\overline{X}), G ∈ D_{c}^{b}(S) we have in D^{b}(\overline{X}) the canonical isomorphism

j_{!}R\underline{Hom}_{X}(E, j^{∗}F)⟶^{∼}R\underline{Hom}\overline{X}(Rj_{∗}E, F) ,

and in D^{b}(S) the canonical isomorphism

Rf_{!}R\underline{Hom}_{X}(E, f^{!}G)⟶^{∼}R\underline{Hom}_{S}(Rf_{∗}E, G) .

Proof. The second assertion follows from the first, by taking F = \overline{f}^{!}G and applying the functor Rf_{∗}. In fact, by [H1, VII, Cor. 4.3]

R\overline{f}_{∗}R\underline{Hom}\overline{X}(Rj_{∗}E, \overline{f}^{!}G)⟶^{∼}R\underline{Hom}_{S}(Rf_{∗}E, G) .

So we only have to prove the first assertion. Now, let \overline{E} be a coherent extension of E and E^{∨} = R\underline{Hom}(E, O_{X}), so that \overline{E}^{∨} = R\underline{Hom}(\overline{E}, O\overline{X}) is a coherent extension of E^{∨}. From Deligne’s adjunction formula [H1, Appendix, Prop. 4] we deduce

\underline{Hom}(j_{!}^{Del}E^{∨}, O\overline{X}) = j_{∗}\underline{Hom}(E^{∨}, O_{X}) ,

and, (since for any injective O\overline{X}-module I, and any O\overline{X}-module H \underline{Hom}\overline{X}(H, I) is flasque [SGA 4, Exp. V, Prop. 4.10], while j_{!}^{Del} is exact)

R\underline{Hom}(j_{!}^{Del}E^{∨}, O\overline{X}) = Rj_{∗}R\underline{Hom}(E^{∨}, O_{X}) = Rj_{∗}E .

We now apply to the previous formula the functor R\underline{Hom}(−, F) to obtain

R\underline{Hom}(Rj_{∗}E, F) = R \underleftarrow{lim}_{N} R\underline{Hom}((J_{Z}^{N} \overline{E}^{∨})^{∨}, F) = R \underleftarrow{lim}_{N} J_{Z}^{N} \overline{E}^{∨}⊗^{L}F = j_{!}R\underline{Hom}_{X}(E, j^{∗}F) .

Q.E.D.

Corollary D.2.5. ∀E ∈ D_{c}^{b}(X),

j_{!}E^{∨} = (Rj_{∗}E)^{∨} .

Let E be a coherent O_{X}-module and \overline{E} be an arbitrary coherent extension of it to \overline{X}. The exact sequence of projective systems

0 ⟶ \{J_{Z}^{N} \overline{E}\}_{N} ⟶ \overline{E} ⟶ \{\overline{E}/J_{Z}^{N} \overline{E}\}_{N} ⟶ 0

shows that

(D.2.6)  H^{0}(j_{!}E) = j_{!}^{top}(E) , H^{1}(j_{!}E) = Coker(\overline{E} ⟶ ˆ \overline{E}) .

We are now interested in introducing, for any base K-scheme S and any smooth S-scheme Y, a triangulated category D^{b}(Coh(Y), diff.op.), intermediate between D_{c}^{b}(Y) and D^{b}(Ab(Y)), where De Rham complexes relative to S of D_{Y} -modules naturally live, and with the further property that j_{!} make sense as a triangulated functor from D^{b}(Coh(X), diff.op.) to, say, D^{b}(Ab(\overline{X})). Our construction is similar to the one proposed by Herrera and Lieberman [HL], but replaces their use of a Yoneda pairing of hyperext functors by a wonderful duality result in the theory of D-modules, recently obtained by Berthelot.

Definition D.2.7. Let S be a K-scheme. For any smooth S-scheme Y, let C be any of the categories Mod(Y), Coh(Y), P ro Coh(Y). Let C^{b}(C, diff.op.) be the (abelian) category of bounded complexes of objects of C, with O_{S}-linear differential operators as differentials. Morphisms of two such complexes are supposed to be O_{Y} -linear maps commuting with the differentials.

Definition D.2.8. Let Y ⟶ S and C be as in (D.2.7).

i) Two morphisms f, g : F^{·} ⟶ G^{·} in C^{b}(C, diff.op.) are homotopic if there exist O_{Y} -linear maps s^{i} : F^{i} ⟶ G^{i−1} such that f^{i} −g^{i} = s^{i+1} ∘d_{F}^{i} +d_{G}^{i−1} ∘s^{i}, ∀i ∈ Z. ii) We define an additive category K^{b}(C, diff.op.) whose objects are the objects of C^{b}(C, diff.op.), and whose morphisms the homotopy classes of morphisms of C^{b}(C, diff.op.). iii) K^{b}(C, diff.op.) is a triangulated category, where a triangle is distinguished if it comes from the mapping cone of a morphism in C^{b}(C, diff.op.).

We now recall Grothendieck’s linearization of differential operators [G2]. Let Y ⟶ S and C be as in (D.2.7). We denote by P_{Y/S}^{ν}(i) the i-th infinitesimal neighborhood of the diagonal Y ⟶ Y ×_{S} … ×_{S} Y (ν + 1 factors) and by P_{Y/S}^{ν}(i) the O_{Y} -module p_{1∗}O_{P_{Y /S}^{ν}(i)}, where p_{k} : P_{Y/S}^{ν}(i) ⟶ Y denotes the projection on the k-th factor. Besides its canonical O_{Y} -module structure (the “extreme left structure”), P_{Y/S}^{ν}(i) has ν more such structures, associated to each of other projections. We only need to use the canonical and the “extreme right structure”, associated to p_{ν+1}. For any object E of C, we set P_{Y/S}^{ν}(i)(E) = p_{1∗}p_{ν+1}^{∗}E = P_{Y/S}^{ν}(i) ⊗_{O_{Y}} E, using the extreme right structure to take the tensor product. A tensor product of the form E ⊗_{O_{Y}} P_{Y/S}^{ν}(i), would instead mean that the extreme right structure on P_{Y/S}^{ν}(i) is used. Given an object E^{·} of C^{b}(C, diff.op.), Grothendieck constructs functorially a bicomplex

Q^{∗}(E^{·}) = Q_{Y/S}^{∗}(E^{·}) = (P_{Y/S}^{ν}(i)(E^{j}))

indexed by ν and j in the category P ro(C) (the index i refers to inverse systems indexed by i ∈ N), such that the bicomplex lim_{i^{←}} Q^{∗}(E^{·})(i) is a resolution of E^{·} in

Ab(Y) (resp. P ro Ab(Y)), if C = Mod(Y) or Coh(Y) (resp. P ro Coh(Y)).

Remark D.2.9. If f, g ∈ Hom_{C^{b}(C,diff.op.)}(F^{·}, G^{·}) are homotopic via s^{i} : F^{i} ⟶ G^{i−1}, the maps Q^{0}(f) and Q^{0}(g) ∈ Hom_{P ro C}(Q^{0}(F^{·}), Q^{0}(G^{·})) are homotopic via Q^{0}(s^{i}) : Q^{0}(F^{i}) ⟶ Q^{0}(G^{i−1}). So, Q^{0} induces a functor K^{b}(C, diff.op.) ⟶ K^{b}(P ro C).

Definition D.2.10^{(1)}. Let Y ⟶ S and C be as in (D.2.7). Let S be the family of morphisms f of K^{b}(C, diff.op.) such that Q^{0}(f) (hence Q^{ν}(f), ∀ν) is a quasi-isomorphism in K^{b}(P ro C).

(1) We are indebted to P. Berthelot for suggesting localization of K^{b}(C, diff.op.) with respect to this class of morphisms.

It is easy to show that S is a multiplicative system in K^{b}(C, diff.op.).

Definition D.2.11. Let Y ⟶ S and C be as in (D.2.7). Let S be as in (D.2.10). We define the triangulated category D^{b}(C, diff.op.) as the localization of K^{b}(C, diff.op.) with respect to the multiplicative system S.

It is easily seen that D^{b}(C) is a full triangulated subcategory of D^{b}(C, diff.op.). Since Grothendieck’s construction gives a resolution of objects of C^{b}(C, diff.op.) as abelian sheaves, there is a canonical triangulated functor D^{b}(C, diff.op.) ⟶ D^{b}(Ab(Y)) (resp. D^{b}(C, diff.op.) ⟶ D^{b}(P ro Ab(Y))) if C = Mod(Y) or Coh(Y) (resp. P ro Coh(Y)).

Lemma D.2.12. Let E, F be coherent O_{X}-modules and E⟶^{L}F be an O_{S}- linear differential operator. Let \overline{F} be a coherent extension of F to \overline{X}, contained in j_{∗}(F). Then there exists a coherent extension \overline{E} of E to \overline{X}, contained in j_{∗}(E)

and an O_{S}-linear differential operator \overline{E}⟶^{\overline{L}}\overline{F} extending L.

Proof. Certainly j_{∗}(L) : j_{∗}(E) ⟶ j_{∗}(F) is an O_{S}-linear differential operator. Let \overline{E}′ be any coherent extension of E to \overline{X}, contained in j_{∗}(E). We may take \overline{E} = J_{Z}^{N} \overline{E}′ for sufficiently big N, and \overline{L} = j_{∗}(L)_{|\overline{E}}. Q.E.D.

The previous lemma shows that, when \overline{f} : \overline{X} ⟶ S is smooth, given a

bounded complex E^{·} = … ⟶ E^{i}⟶^{L^{i}}E^{i+1} … of coherent O_{X}-modules and O_{S}-linear differential operators, there exists a bounded complex \overline{E}^{·} = … ⟶

\overline{E}^{i}⟶^{\overline{L}^{i}}\overline{E}^{i+1} … in C^{b}(Coh(\overline{X}), diff.op.) extending E^{·}. Given homotopic maps φ, ψ : E^{·} ⟶ F^{·} in C^{b}(Coh(X), diff.op.), we may extend E^{·} and F^{·} to objects of C^{b}(Coh(\overline{X}), diff.op.) and φ, ψ to homotopic maps \overline{φ}, \overline{ψ} : \overline{E}^{·} ⟶ \overline{F}^{·} in C^{b}(Coh(\overline{X}), diff.op.).

We note that a differential operator of order d between O\overline{X}-modules \overline{E}⟶^{\overline{L}}\overline{F}, induces a differential operator of projective systems \{J_{Z}^{N} \overline{E}\}_{N} ⟶ \{J_{Z}^{N−d}\overline{F}\}_{N}. For E^{·} in C^{b}(Coh(X), diff.op.) and any extension \overline{E}^{·} of E^{·} in C^{b}(Coh(\overline{X}), diff.op.), “ lim_{N^{←}} ”J_{Z}^{N} \overline{E}^{·} is an object of C^{b}(P ro Coh(\overline{X}), diff.op.), independent of the chosen

extension \overline{E}^{·} of E^{·}. So, we get a functor

j_{!}^{Del} : C^{b}(Coh(X), diff.op.) ⟶ C^{b}(P ro Coh(\overline{X}), diff.op.) .

It is easily seen that j_{!}^{Del}E^{·} up to homotopy only depends upon the homotopy class of E^{·}, so that

j_{!}^{Del} : K^{b}(Coh(X), diff.op.) ⟶ K^{b}(P ro Coh(\overline{X}), diff.op.) .

Lemma D.2.13. Assume \overline{f} : \overline{X} ⟶ S is smooth. Let φ : E^{·} ⟶ F^{·} be a morphism in C^{b}(Coh(X), diff.op.), representing an element in the multiplicative system S of morphisms of K^{b}(Coh(X), diff.op.). Then

j_{!}^{Del}(φ) : j_{!}^{Del}(E^{·}) ⟶ j_{!}^{Del}(F^{·})

is a quasi-isomorphism in K^{b}(P ro Coh(\overline{X}), diff.op.). We conclude that j_{!}^{Del} defines a triangulated functor

j_{!}^{Del} : D^{b}(Coh(X), diff.op.) ⟶ D^{b}(P ro Coh(\overline{X}), diff.op.) .

Proof. Let \overline{φ} : \overline{E}^{·} ⟶ \overline{F}^{·} be an extension of φ to a morphism of C^{b}(Coh(\overline{X}), diff.op.). By application of Grothendieck’s linearization functor, we get a morphism of bicomplexes

(D.2.13.1)  Q_{\overline{X}/S}^{0}(\overline{φ}) : Q_{\overline{X}/S}^{0}(\overline{E}^{·}) ⟶ Q_{\overline{X}/S}^{0}(\overline{F}^{·})

extending

Q_{X/S}^{0}(φ) : Q_{X/S}^{0}(E^{·}) ⟶ Q_{X/S}^{0}(F^{·}) .

The functor “ lim_{N^{←}} ”J_{Z}^{N}(−) also applies to the morphism (D.2.13.1): we obtain

j_{!}^{Del}(Q_{X/S}^{0}(φ)) : j_{!}^{Del}(Q_{X/S}^{0}(E^{·})) ⟶ j_{!}^{Del}(Q_{X/S}^{0}(F^{·}))

where, for example, j_{!}^{Del}(Q_{X/S}^{0}(E^{·})) is (J_{Z}^{N}Q_{\overline{X}/S}^{0}(\overline{E}^{·})) = (J_{Z}^{N} ⊗ P_{\overline{X}/S}^{1}(i)(\overline{E}^{j})),

a complex of pro-objects (each indexed by (N, i) ∈ N × N) in Coh(\overline{X}). The difference between the left and the right structure of P_{\overline{X}/S}^{1}(i) gives

J_{Z}^{N} ⊗ P_{\overline{X}/S}^{1}(i)(\overline{E}^{j}) = P_{\overline{X}/S}^{1}(i)(\overline{E}^{j}) ⊗ J_{Z}^{N+i} = P_{\overline{X}/S}^{1}(i)(J_{Z}^{N+i}\overline{E}^{j}) ,

so that we may write Q_{\overline{X}/S}^{0}(j_{!}^{Del}E^{·}) = j_{!}^{Del}Q_{X/S}^{0}(E^{·}). More precisely,

j_{!}^{Del}(Q_{X/S}^{0}(φ)) : j_{!}^{Del}(Q_{X/S}^{0}(E^{·})) ⟶ j_{!}^{Del}(Q_{X/S}^{0}(F^{·}))

identifies with

Q_{\overline{X}/S}^{0}(j_{!}^{Del}(φ)) : Q_{\overline{X}/S}^{0}(j_{!}^{Del}E^{·}) ⟶ Q_{\overline{X}/S}^{0}(j_{!}^{Del}F^{·}) .

We recall that Deligne’s functor j_{!}^{Del} : Coh(X) ⟶ Coh(\overline{X}) (hence j_{!}^{Del} : P ro Coh(X) ⟶ P ro Coh(\overline{X})) is exact. So, if Q_{X/S}^{0}(φ) is a quasi-isomorphism in K^{b}(P ro Coh(X)), then j_{!}^{Del}(Q_{X/S}^{0}(φ)) = Q_{\overline{X}/S}^{0}(j_{!}^{Del}(φ)) is a quasi-isomorphism

in K^{b}(P ro Coh(\overline{X})), and the assertion is proven. Q.E.D.

Definition D.2.14. We define

Rj_{!} : D^{b}(Coh(X), diff.op.) ⟶ D^{b}(Ab(\overline{X}))

as the composition of

D^{b}(Coh(X), diff.op.)⟶^{j_{!}^{Del}}D^{b}(P ro Coh(\overline{X}), diff.op.) ⟶ D^{b}(P roAb(\overline{X}))⟶^{R lim_{←}}D^{b}(Ab(\overline{X})) .

We also define

Rf_{!} = R\overline{f}_{∗} ∘ Rj_{!} : D^{b}(Coh(X), diff.op.) ⟶ D^{b}(Ab(S)) .

Obviously, Rj_{!} takes its values in the derived category of the category of sheaves of \overline{f}^{−1}O_{S}-modules on \overline{X}, a triangulated subcategory of D^{b}(Ab(\overline{X})). As a consequence, Rf_{!} may be viewed as a triangulated functor

Rf_{!} = R\overline{f}_{∗} ∘ Rj_{!} : D^{b}(Coh(X), diff.op.) ⟶ D^{b}(S) .
Let \overline{f} be smooth and E^{·} be an object of C^{b}(Coh(X), diff.op.), and let \overline{E}^{·}

be an extension of E^{·} to an object of C^{b}(Coh(\overline{X}), diff.op.). We may compute Rj_{!}(E^{·}) from the exact triangle in D^{b}(Ab(\overline{X}))

(D.2.15)  \begin{aligned}\overline{E} & · \\ & ↗ & & ↘ \\ Rj_{!}(E^{·}) & & ←^{[+1]} & & ˆ\overline{E}^{·} \end{aligned}

We note that if E is an object of D_{Coh(X)}^{b}(D_{X}) (D.2.3),

(D.2.15.1)  DR_{\overline{X}/S}(j_{!}E) = Rj_{!}DR_{X/S}(E) .

Definition D.2.16. Let S (and therefore X) be smooth. For E in D_{Coh(X)}^{b}(D_{X}) we define the direct image with compact supports by f : X ⟶ S as f_{!}E := Rf_{!}DR_{X/S}(E)[d] ∈ D^{b}(D_{S}). For a coherent object (E, ∇) of MIC(X), we define the q-th De Rham cohomology sheaf with compact supports of (E, ∇) relative to S, denoted R_{DR}^{q}f_{!}(E, ∇) or H_{DR,c}^{q}(X/S, (E, ∇)), as the object H^{q}(Rf_{!}DR(X/S, (E, ∇))) of MIC(S).

The following is our main duality statement. It is a corollary of Berthelot’s duality theorem proven in Appendix C, combined with the previous definition. It allows to regard Dwork’s duality principle for elementary fibrations as a version in algebraic coherent cohomology of the familiar Poincaré duality.

We recall that, f : X ⟶ S being smooth, the direct image by f of D_{X}- modules may be expressed via the classical Gauss-Manin connection on relative De Rham cohomology:

\begin{matrix}f_{+} : D^{b}(D_{X}) & ⟶ D^{b}(D_{S}) \\ E & ⟼ Rf_{∗}DR_{X/S}(E)[d] . \end{matrix}

Theorem D.2.17. (Main Duality Theorem) Let f : X ⟶ S be a smooth map of smooth algebraic K-varieties. Assume that f can be embedded in a diagram (D.2.0), with \overline{f} smooth and proper. Then for any object E of D_{Coh(X)}^{b}(D_{X}), we have a canonical isomorphism in D^{b}(D_{S})

f_{!}E ⟶^{≅}(f_{+}E_{X/S}^{∗})^{∨} .

Proof. By (D.2.5),

j_{!}E^{∨} = (Rj_{∗}E)^{∨} .

By Berthelot’s theorem C.1, we have a natural morphism in D^{b}(D\overline{X})

(Rj_{∗}E)^{∨} ⟶ (Rj_{∗}E)_{\overline{X}/S}^{∗} ,

and therefore a morphism

j_{!}E^{∨} ⟶ (Rj_{∗}E)_{\overline{X}/S}^{∗} ,

inducing an isomorphism of De Rham complexes in D^{b}(D_{S})

Rj_{!}DR_{\overline{X}/S} (E^{∨}) = DR_{\overline{X}/S} (j_{!}E^{∨}) ⟶^{≅} DR_{\overline{X}/S} ((Rj_{∗}E)_{\overline{X}/S}^{∗}) .

We now apply to the previous isomorphism the functor R\overline{f}_{∗} and obtain to the r.h.s.

R\overline{f}_{∗}DR_{\overline{X}/S} ((Rj_{∗}E)_{\overline{X}/S}^{∗}) ≅ \overline{f}_{+} ((Rj_{∗}E)_{\overline{X}/S}^{∗}) ≅ (\overline{f}_{+}Rj_{∗}E)_{S/S}^{∗} ≅ (f_{+}E)^{∨} .

Q.E.D.

We point out that the formula obtained in our duality theorem has been taken as the definition of the algebraic direct image with compact support of a D-module by previous authors [Me5, I.5.2.4], [Bo, VI.4.0, eq. 1]. Our construction of f_{!} has the advantage of allowing explicit computations of De Rham relative cohomology groups, especially in the case of an elementary fibration, and should offer an alternative proof of the results in section 8. We start by showing how Dwork’s duality principle for elementary fibrations may be restated as a version in coherent cohomology of the familiar Poincaré duality. Some explicit computations due to Dwork, in the framework of his dual theory, will be given as examples in the following sections.

Proposition D.2.18. (Dwork-Poincaré duality for an elementary fibration.) Let f : X ⟶ S be an elementary fibration. Let (E, ∇) be a coherent object of MIC(X) and (E^{∨}, ∇^{∨}) be its dual. There is a canonical isomorphism

Rf_{!}DR(X/S, (E^{∨}, ∇^{∨}))[2]⟶^{∼}R\underline{Hom}_{S}(Rf_{∗}DR(X/S, (E, ∇)), O_{S}) .

In particular, we have a canonical isomorphism in MIC(S)

H_{DR,c}^{1}(X/S, (E^{∨}, ∇^{∨}))⟶^{∼}\underline{Hom}_{O_{S}}(H_{DR}^{1}(X/S, (E, ∇)), O_{S}) ,

and an exact sequence in MIC(S)

0 ⟶ \underline{Ext}_{S}^{1}(H_{DR}^{1}(X/S, (E, ∇)), O_{S}) ⟶ H_{DR,c}^{2}(X/S, (E^{∨}, ∇^{∨}))
⟶ \underline{Hom}_{S}(H_{DR}^{0}(X/S, (E, ∇)), O_{S}) ⟶ \underline{Ext}_{S}^{2}(H_{DR}^{1}(X/S, (E, ∇)), O_{S})
⟶ 0 .

Corollary D.2.19.

(i) If \underline{Ext}_{S}^{1}(H_{DR}^{1}(X/S, (E, ∇)), O_{S}) = 0, then H_{DR,c}^{2}(X/S, (E^{∨}, ∇^{∨})) is coherent (hence locally free). (ii) If \underline{Ext}_{S}^{1}(H_{DR}^{1}(X/S, (E, ∇)), O_{S}) = 0 = \underline{Ext}_{S}^{2}(H_{DR}^{1}(X/S, (E, ∇)), O_{S}) (e.g., if H_{DR}^{1}(X/S, (E, ∇)) is coherent) we also have a canonical isomorphism of coherent (4.2.1) objects of MIC(S)

H_{DR,c}^{2}(X/S, (E^{∨}, ∇^{∨}))⟶^{∼}\underline{Hom}_{O_{S}}(H_{DR}^{0}(X/S, (E, ∇)), O_{S}) .

(iii) If \underline{Ext}_{S}^{1}(H_{DR}^{1}(X/S, (E, ∇)), O_{S}) = 0 = \underline{Ext}_{S}^{2}(H_{DR}^{1}(X/S, (E, ∇)), O_{S}) and H_{DR,c}^{1}(X/S, (E^{∨}, ∇^{∨})) is coherent, then H_{DR}^{1}(X/S, (E, ∇)) is coherent, too.

We now show that the discussion of this section in fact interprets and generalizes Dwork’s duality principle (D.1.21).

Theorem D.2.20. In the situation of (D.2.18), let \overline{E} be any coherent extension of E to \overline{X}, contained in j_{∗}E, and let ω = Ω_{\overline{X}/S}^{1}. Then Rf_{!}DR(X/S, (E, ∇))[1] is quasi-isomorphic to

\begin{matrix}DW (\overline{X}/S, \overline{E}) = [R^{0}\overline{f}_{∗}(ˆ\overline{E}(∗Z))/ & R^{0}\overline{f}_{∗}(\overline{E}(∗Z)) ⟶ R^{0}\overline{f}_{∗}(ω ⊗ ˆ \overline{E}(∗Z))/R^{0}\overline{f}_{∗}(ω ⊗ \overline{E}(∗Z))] . \\ 0 & 1 \end{matrix}

Proof. (L. Fiorot) We may extend E^{·} := DR(X/S, (E, ∇)) to \overline{X} by

\begin{matrix}\overline{E}^{·} := [ & \overline{E} ⟶ ω ⊗ \overline{E} ⊗ J_{Z}^{−m}] \\ 0 1 \end{matrix}

for a sufficiently high m. By (D.2.15) j_{!}(E^{·}) is the mapping fiber of the morphism

\overline{E}^{·} ⟶ ˆ \overline{E}^{·}, independent of the choice of \overline{E} and m. So,

\begin{matrix}j_{!}(E^{·})[1] = [ & \overline{E} ⟶ ˆ \overline{E} ⊕ (ω ⊗ \overline{E} ⊗ J_{Z}^{−m}) ⟶ ω ⊗ ˆ \overline{E} ⊗ J_{Z}^{−m}] \\ − & 1  0  1 \end{matrix}

with the obvious differential. Since j_{∗}(E) = \overline{E}(∗Z) = lim_{→} \overline{E}_{i} where the direct

limit runs over the directed set of extensions \overline{E}_{i} of E to \overline{X}, that are coherent submodules of j_{∗}(E), considering extensions of E^{·} as above with \overline{E} replaced by \overline{E}_{i} (and suitable m = m_{i}) and taking direct limits, we obtain

\begin{matrix}j_{!}(E^{·})[1] = [ & \overline{E}(∗Z) ⟶ ˆ \overline{E}(∗Z) ⊕ (ω ⊗ \overline{E}(∗Z)) ⟶ ω ⊗ ˆ \overline{E}(∗Z)] \\ − 1  0  1 \end{matrix}

So,

Rf_{!}(E^{·})[1] = R\overline{f}_{∗}(j_{!}(E^{·}))[1] = R\overline{f}_{∗}[\overline{E}(∗Z) ⟶ ˆ \overline{E}(∗Z)⊕(ω⊗\overline{E}(∗Z)) ⟶ ω⊗ˆ\overline{E}(∗Z)] .

Since the objects of [\overline{E}(∗Z) ⟶ ˆ \overline{E}(∗Z)⊕(ω⊗\overline{E}(∗Z)) ⟶ ω⊗ˆ\overline{E}(∗Z)] are f_{∗}-acyclic, we obtain

Rf_{!}(E^{·})[1] = [\overline{f}_{∗}\overline{E}(∗Z) ⟶ \overline{f}_{∗}(ˆ\overline{E}(∗Z)) ⊕ \overline{f}_{∗}(ω ⊗ \overline{E}(∗Z)) ⟶ \overline{f}_{∗}(ω ⊗ ˆ \overline{E}(∗Z))] .

It is immediate to check that the map of complexes

[\overline{f}_{∗}\overline{E}(∗Z) ⟶ \overline{f}_{∗}(ˆ\overline{E}(∗Z))  ⊕  \overline{f}_{∗}(ω ⊗ \overline{E}(∗Z)) ⟶ \overline{f}_{∗}(ω ⊗ ˆ \overline{E}(∗Z))]
↓ 0  can ↘  ↙ 0  can ↓
[ 0  ⟶  \frac{\overline{f}_{∗}(ˆ\overline{E}(∗Z))}{\overline{f}_{∗}(\overline{E}(∗Z))}  ⟶ \frac{\overline{f}_{∗}(ω⊗ˆ\overline{E}(∗Z))}{\overline{f}_{∗}(ω⊗\overline{E}(∗Z))}]

is a quasi-isomorphism. Q.E.D.

D.3 Duality for rational elementary fibrations. We now come back to the situation of section D.1 and consider the case of a rational elementary fibration as in (1.2.1) and (4.5.1). So now \overline{X} = P_{S}^{1}, \overline{f} = pr, and we have a canonical relative affine coordinate x. We let (O\overline{X}, d) play the role of the (E, ∇) of (D.1). We set

(D.3.1)  R(f) := R^{0}\overline{f}_{∗}(Ô\overline{X})/R^{0}\overline{f}_{∗}(O\overline{X}) = R^{0}\overline{f}_{∗}(Ô\overline{X})/O_{S} ,
(D.3.2)  R′(f) := lim_{s→∞} R^{0}\overline{f}_{∗}(Ĵ_{Z}^{−s}) = R^{0}\overline{f}_{∗}(Ô\overline{X} ⊗_{O\overline{X}} j_{∗}O_{X}) ,
L(f) = R^{0}f_{∗}(O_{X})
ω(f) := R^{0}f_{∗}(ω_{|X}) = L(f)dx ,
ω′(f) := R^{0}\overline{f}_{∗}(Ô\overline{X} ⊗_{O\overline{X}} j_{∗}(ω_{|X})) = R′(f)dx ,
D(f) := R^{0}f_{∗}(D_{X/S}) ,
D̂(f) := R^{0}\overline{f}_{∗}(Ô\overline{X} ⊗_{O\overline{X}} D_{\overline{X}/S}) .

According to section D.1, there exists an exact sequence of left D(f)-modules (which defines the structure of left D(f)-module of R(f))

(D.3.3)  0 ⟶ L(f) ⟶ R′(f) ⟶^{γ_{−}(f)}R(f) ⟶ 0 .

There are also natural embeddings ω̂ ↪ ω′(f), of right D̂(f)-modules and ω(f) ↪ ω′(f) of right D(f)-modules.

Proposition D.3.4. There is a unique O_{S}-linear pairing

(D.3.4.1)  < , >: ω(f) × R(f) ⟶ O_{S}

compatible with the Serre pairings

< , >_{N}: R^{0}\overline{f}_{∗}(ω ⊗ J_{Z}^{−N}) × R^{1}\overline{f}_{∗}(J_{Z}^{N}) ⟶ O_{S}

and the natural morphisms

R(f) ⟶ R^{0}\overline{f}_{∗}(O\overline{X}/J_{Z}^{N}) ⟶ R^{1}\overline{f}_{∗}(J_{Z}^{N})

and

R^{0}\overline{f}_{∗}(ω ⊗ J_{Z}^{−N}) ⟶ ω(f) .

We have, for any point s ∈ S:

i) let h ∈ R(f)_{s}; if < η, h >= 0 in O_{S,s} for any η ∈ ω(f)_{s} then h = 0; ii) let η ∈ ω(f)_{s}; if < η, h >= 0 in O_{S,s} for any h ∈ R(f)_{s} then η = 0; iii) < , > establishes an isomorphism

R(f)⟶^{∼}ω(f)^{∨} = \underline{Hom}_{O_{S}}(ω(f), O_{S}) ;

iv) for any sections η of ω(f), h of R(f), L of D(f), we have

< ηL, h >=< η, Lh > .

Proof. It is a restatement of Dwork’s duality principle for elementary fibrations, in the special case of E = O_{P_{S}^{1}}, when R^{1}\overline{f}_{∗}O_{P_{S}^{1}} = 0. Q.E.D.

There is a natural embedding of O_{S}-modules

(D.3.5)  R^{0}\overline{f}_{∗}(Ô\overline{X}) ↪ R^{0}\overline{f}_{∗}(Ô\overline{X}(∗Z)) = R′(f) ,

that “almost splits” the exact sequence (D.3.3) in the sense that R^{0}f_{∗}(O_{X}) ∩ R^{0}\overline{f}_{∗}(Ô\overline{X}) = R^{0}\overline{f}_{∗}(O\overline{X}).

We define an O_{S}-linear pairing

(D.3.6)  \begin{matrix}< , >′: ω′(f) × R′(f) & ⟶ O_{S} \\ ((η_{i})_{i}, (ξ_{i})_{i}) & ⟼ ∑_{i=1}^{∞} Res_{Z_{i}}ξ_{i}η_{i} . \end{matrix}

where, for some open subset U ⊂ S and some N = 0, 1, …, η_{i} (resp. ξ_{i}), for i = 1, …, r, ∞, denotes a section of Ĵ^{−N}ω (resp. Ĵ^{−N}) over g^{−1}(U) ∩ Z_{i}. For any section L of D_{X/S} over f^{−1}(U), we then have

(D.3.6.1)  < (η_{i}L)_{i}, (ξ_{i})_{i} >′=< (η_{i})_{i}, (Lξ_{i})_{i} >′ .

Under the pairing < , >′, ω̂ ⊥ R^{0}\overline{f}_{∗}(Ô\overline{X}) and ω(f) ⊥ L(f). Therefore Proposition D.3.7. The pairing < , > of (D.3.4) is induced by the pairing < , >′ of (D.3.6), in the sense that for any section η (resp. ξ) of ω(f) (resp. R′(f))

< η, γ_{−}(f)(ξ) >=< η, ξ >′ .

In order to follow faithfully Dwork’s calculations, we now assume that one of the Z_{i}, say Z_{1}, has equation x = 0. We will identify ω′(f) with R′(f) via

ω′(f) = R′(f)\frac{dx}{x} , ω(f) = L(f)\frac{dx}{x}

To be completely explicit we recall Dwork’s definition of the (logarithmic) adjoint operator L^{∗} of L, with respect to the relative coordinate x, namely:

(D.3.8)  (∑_{i} a_{i}(x\frac{d}{dx})^{i})^{∗} = ∑_{i} (−x\frac{d}{dx})^{i}a_{i}

characterized by the property^{(1)} that for any section h\frac{dx}{x} = η of ω_{η_{X}},

(D.3.9)  (f \frac{dx}{x})L = (L^{∗}f)\frac{dx}{x} .

D.4 Explicit calculations. Let the assumptions be as in (D.3); we assume moreover that S = SpecA is affine, and compute global sections of the O_{S}- modules introduced above. We have: R^{0}f_{∗}(O_{X})(S) = L = A[x, _{\frac{1}{∏_{i}(x−θ_{i})}}]

D(f)(S) = D = L[\frac{d}{dx}] R′(f)(S) = R′ = ⊕R′_{i}, with

(D.4.1)  R′_{i} = \{ \begin{matrix}A((x − θ_{i})) & if i = 1, …, r, (θ_{1} = 0), \\ A((\frac{1}{x})) & if i = ∞. \end{matrix}

Note that L is embedded in R′ by the “diagonal” embedding h ⟼ (h, h, …, h). ω′(f)(S) = R′(f)(S)dx ω(f)(S) = Ldx

(1) The classical definition differs from ours in that it uses the basis dx of ω_{η_{X}}, so that, classically,

(∑_{i} a_{i}(\frac{d}{dx})^{i})^{∗} = ∑_{i} (−\frac{d}{dx})^{i}a_{i} .

R^{0}\overline{f}_{∗}(Ô\overline{X})(S) = ⊕_{i}R^{0}\overline{f}_{∗}(Ô_{\overline{X}|σ_{i}(S)})(S) where

(D.4.2)  R^{0}\overline{f}_{∗}(Ô_{\overline{X}|σ_{i}(S)})(S) = \{ \begin{matrix}A[[x − θ_{i}]] & if i = 1, …, r, \\ A[[\frac{1}{x}]] & if i = ∞. \end{matrix}

To give a useful (non-canonical) description of R(f)(S) = R, we split (D.3.3) (γ_{−}(f) =: γ_{−}), by a map

(D.4.3)  R ↪ R′

taking ξ ∈ R to the unique (ξ′)_{i} ∈ R^{0}f_{∗}(Ô\overline{X}) ∩ γ_{−}^{−1}(ξ) such that ξ_{∞} ∈ \frac{1}{x}A[[\frac{1}{x}]]. We will then (non-canonically) identify R with the A-submodule of R′ direct sum ⊕R_{i} where

(D.4.4)  R_{i} = \{ \begin{matrix}A[[x − θ_{i}]] & if i = 1, …, r, \\ \frac{1}{x}A[[\frac{1}{x}]] & if i = ∞. \end{matrix}

The map

(D.4.5)  γ_{+} : R′ ⟶ L

associated to this splitting of (D.3.3) takes (ξ_{i})_{i} to ∑_{i} P_{i}(ξ_{i}) ∈ L, where P_{i} : A((x − θ_{i})) ⟶ A[[x − θ_{i}]], is the usual “principal part” at θ_{i}, for i = 1, …, r, while P_{∞} : A((\frac{1}{x})) ⟶ \frac{1}{x}A[[\frac{1}{x}]] takes ∑_{i} a_{i}x^{i} to ∑_{i≤0} a_{i}x^{i}. This is easily seen because γ_{+}(R) = 0 while γ_{+|L} = id_{L}. So, as endomorphisms of R′,

(D.4.6)  γ_{+} + γ_{−} = id_{R′}

So we now have a decomposition

(D.4.7)  R′ = R ⊕ L ,

where the first projection is the map γ_{−}, which is a morphism of left D-modules, while the second projection is the A-linear map γ_{+}. On the other hand the (diagonal) embedding L ↪ R′ is an embedding of left D-modules. The pairing < , >′: (⊕_{i}R′_{i}) × (⊕_{i}R′_{i}) ⟶ A, ((η_{i})_{i}, (ξ_{i})_{i}) ⟼ ∑_{i} Res_{θ_{i}}η_{i}ξ_{i} \frac{dx}{x} of (D.3.6) then satisfies

(D.4.8)  < (Lη_{i})_{i}, (ξ_{i})_{i} >′=< (η_{i})_{i}, (L^{∗}ξ_{i})_{i} >′ ,

for any (η_{i})_{i}, (ξ_{i})_{i} in R′.

The subspaces L and R of R′ are isotropic for < , >′. Hence < , >′ induces a pairing (that of proposition D.3.4)

(D.4.9)  < , >: L × R ⟶ A

which identifies R with Hom_{A}(L, A). More precisely, for any (ξ_{i})_{i} in R′ and η in L, we have

< η, (ξ_{i})_{i} >′=< η, γ_{−}((ξ_{i})_{i}) > ,

and for any (ξ_{i})_{i} in R′ and (η_{i})_{i} in R, we have

< (ξ_{i})_{i}, (η_{i})_{i} >′=< γ_{+}((ξ_{i})_{i}), (η_{i})_{i} > .

Lemma D.4.10. Let T be an A-linear endomorphism of R′, stable on L. Let T^{∗} be an A-linear endomorphism of R′ such that

(D.4.10.1)  < T (η_{i})_{i}, (ξ_{i})_{i} >′=< (η_{i})_{i}, T^{∗}(ξ_{i})_{i} >′ ,

for any (η_{i})_{i}, (ξ_{i})_{i} in R′. Then the (unique) endomorphism of R, adjoint of T_{|L} under the pairing < , >, is

(D.4.10.2)  γ_{−} ∘ T^{∗} .

We can now make completely explicit the duality of proposition D.2.8, in the special case of a rational elementary fibration. Let \underline{λ} = (λ_{1}, …, λ_{d}) be étale coordinates on S, so that (x, λ_{1}, …, λ_{d}) are étale coordinates on X, and assume we are given an integrable connection ∇ on E = O_{X}^{μ}

(D.4.11)  ∇ = d_{X/K} − G_{1,\underline{0}}d_{X/K}x − ∑_{i=1}^{d} G_{0,\underline{1}_{i}}d_{X/K}λ_{i} ,

with G_{1,\underline{0}}, G_{0,\underline{1}_{i}} ∈ M_{μ×μ}(L), for i = 1, …, d. The de Rham complex relative to S is

(D.4.12)  \begin{matrix}DR(X/S, (E, ∇)) = \\ = [O_{X}^{μ} ⟶^{d_{X/S}−G_{1,\underline{0}}d_{X/S}x}(Ω_{X/S}^{1})^{μ}] ≅ \\ ≅ [O_{X}^{μ} ⟶^{\frac{∂}{∂x} −G_{1,\underline{0}}}O_{X}^{μ}] . \end{matrix}

To be completely clear we specify that the first-order differential operator \frac{∂}{∂x} − G_{1,\underline{0}} acts on O_{X}^{μ} as

( \begin{matrix}y_{1} \\ . \\ . \\ y_{μ} \end{matrix} ) ⟼ ( \frac{∂y_{1}}{∂x} ; . ; . ; \frac{∂y_{μ}}{∂x} ) − G_{1,\underline{0}} ( \begin{matrix}y_{1} \\ . \\ . \\ y_{μ} \end{matrix} )

We are interested in the two dual (up to a suitable shift) complexes of Amodules

(D.4.13)  \begin{matrix}(Rf_{∗}DR(X/S, (E, ∇)))(S) = Γ(X, DR(X/S, (E, ∇))) = \\ = [L^{μ}  ⟶^{d_{X/S}−G_{1,\underline{0}}d_{X/S}x}L^{μ}dx] ≅ \\ ≅ [L^{μ} ⟶^{\frac{∂}{∂x} −G_{1,\underline{0}}}L^{μ}] , \end{matrix}

and (D.4.14)

Γ(S, Rf_{!}DR(X/S, (E^{∨}, ∇^{∨})))[1] =

= [(\overline{f}_{∗}(Ô_{\overline{X}|Z}(∗Z))/\overline{f}_{∗}(O\overline{X}(∗Z)))^{μ}⟶^{∇^{∨}}(\overline{f}_{∗}(Ω̂_{\overline{X}/S}^{1}(∗Z))/\overline{f}_{∗}(Ω_{\overline{X}/S}^{1}(∗Z)))^{μ}] ≅

≅ [R^{μ} ⟶^{γ_{−}∘( \frac{∂}{∂x} +G_{1,\underline{0}}^{t})}R^{μ}] ≅ [(R′/L)^{μ} ⟶^{\frac{∂}{∂x} +G_{1,\underline{0}}^{t}}(R′/L)^{μ}] ,

where G_{1,\underline{0}}^{t} is the transposed matrix of G_{1,\underline{0}} and again \frac{∂}{∂x} + G_{1,\underline{0}}^{t} acts on R^{′μ} as

( \begin{matrix}y_{1} \\ . \\ . \\ y_{μ} \end{matrix} ) ⟼ ( \frac{∂y_{1}}{∂x} ; . ; . ; \frac{∂y_{μ}}{∂x} ) + G_{1,\underline{0}}^{t} ( \begin{matrix}y_{1} \\ . \\ . \\ y_{μ} \end{matrix} ) ,

while γ_{−} : R^{′μ} ⟶ R^{μ} operates componentwise.

We deduce the following down-to-earth description of De Rham cohomology groups.

Proposition D.4.15. Let (E, ∇) be an object of MIC(X), with E ≅ O_{X}^{μ} and ∇ as in (D.4.11). Then

(D.4.15.1)  H_{DR}^{0}(X/S, (E, ∇))(S) ≅ Ker_{L^{μ}}(\frac{∂}{∂x} − G_{1,\underline{0}}) ,

(D.4.15.2)  H_{DR}^{1}(X/S, (E, ∇))(S) ≅ L^{μ}/(\frac{∂}{∂x} − G_{1,\underline{0}})L^{μ} ,

(D.4.15.3)  H_{DR,c}^{1}(X/S, (E^{∨}, ∇^{∨}))(S) ≅ Ker_{R^{μ}}γ_{−} ∘ (\frac{∂}{∂x} + G_{1,\underline{0}}^{t}) ,

(D.4.15.4)  \begin{matrix}H_{DR,c}^{2}(X/S, & (E^{∨}, ∇^{∨}))(S) ≅ R^{μ}/(γ_{−} ∘ (\frac{∂}{∂x} + G_{1,\underline{0}}^{t})R^{μ}) \\ ≅ R^{′μ}/(R^{μ} + (\frac{∂}{∂x} + G_{1,\underline{0}}^{t})R^{′μ}) . \end{matrix}

The A-bilinear pairings of A/K-differential modules

(D.4.15.5) < , >: H_{DR}^{1}(X/S, (E, ∇))(S) × H_{DR,c}^{1}(X/S, (E^{∨}, ∇^{∨}))(S) ⟶ A

and

(D.4.15.6) < , >: H_{DR}^{0}(X/S, (E, ∇))(S) × H_{DR,c}^{2}(X/S, (E^{∨}, ∇^{∨}))(S) ⟶ A

are induced by the pairing (D.4.9).

Corollary D.4.16. If H_{DR,c}^{0}(X/S, (E^{∨}, ∇^{∨}))(S) is of finite type and H_{DR}^{1}(X/S, (E, ∇))(S) is projective, then (D.4.15.5) and (D.4.15.6) are perfect pairings of projective Amodules of finite type.

Proof. The O_{S}-module H_{DR,c}^{1}(X/S, (E^{∨}, ∇^{∨})) is not quasi-coherent in gen-

eral. Still, if H_{DR,c}^{1}(X/S, (E^{∨}, ∇^{∨}))(S) is an A-module of finite type, the same holds for H_{DR}^{1}(X/S, (E, ∇))(S). Since H_{DR}^{1}(X/S, (E, ∇)) is quasi-coherent, this implies that it is in fact coherent, and therefore locally free. So, H_{DR,c}^{1}(X/S, (E^{∨}, ∇^{∨})) is coherent and locally free, too, and (D.4.15.5) is perfect. On the other hand, the local freeness of H_{DR}^{1}(X/S, (E, ∇))(S) implies that (D.4.15.6) is left-perfect. But H_{DR}^{0}(X/S, (E, ∇)) is always coherent by (4.2.1), hence (D.4.15.6) is perfect. Q.E.D.

The A-modules H_{DR}^{j}(X/S, (E, ∇)) are endowed with the S/K-connection

ℵ^{j} induced by

(D.4.17)  ℵ(\frac{∂}{∂λ_{i}}) = \frac{∂}{∂λ_{i}} − G_{0,\underline{1}_{i}} , i = 1, …, d ,

on the quotient space (this makes sense because of the integrability of ∇). Similarly, the A-modules H_{DR,c}^{j}(X/S, (E^{∨}, ∇^{∨})) are endowed with an integrable S/Kconnection induced by

(D.4.18)  ℵ^{∨}(\frac{∂}{∂λ_{i}}) = \frac{∂}{∂λ_{i}} + G_{0,\underline{1}_{i}}^{t} , i = 1, …, d .

D.5 Example: Gauss hypergeometric functions. We are here in the situation of the previous section, but we also fix a, b, c ∈ K and assume

A = K[λ, \frac{1}{λ(1 − λ)}] ,

S = SpecA ,

θ_{1} = 0 , θ_{2} = 1 , θ_{3} = \frac{1}{λ} .

We prefer to reindex our θ_{i}’s on setting θ_{i} = i, for i = 0, 1, \frac{1}{λ}, ∞.

L = A[x, \frac{1}{x(1 − x)(1 − xλ)}] ,

h_{a,b,c}(λ, x) = x^{b}(1 − x)^{c−b}(1 − xλ)^{−a} ,

(in the sense of differential algebra)

E = O_{X} , ∇ = ∇_{a,b,c} = h_{a,b,c}^{−1} ∘ d_{X/K} ∘ h_{a,b,c} .

We also reindex our spaces as

R = R_{0} ⊕ R_{1} ⊕ R_{\frac{1}{λ}} ⊕ R_{∞} ,

where

R_{0} = A[[x]] , R_{1} = A[[1 − x]] , R_{\frac{1}{λ}} = A[[1 − xλ]] , R_{∞} = \frac{1}{x}A[[\frac{1}{x}]] ,

and similarly for R′. We also set

T_{0} = x , T_{1} = 1 − x , T_{\frac{1}{λ}} = 1 − λx , T_{∞} = \frac{1}{x} ,

and

t_{0} = λ , t_{1} = \frac{λ}{λ − 1} , t_{\frac{1}{λ}} = \frac{1}{1 − λ} , t_{∞} = \frac{1}{λ} .

Using the identification

\begin{matrix}L & ⟶ ω(f) \\ η & ⟼ η \frac{dx}{x} , \end{matrix}

the De Rham complex of (O_{X}, ∇_{a,b,c}) is identified with the complex

\begin{matrix}[ & O_{X} ⟶^{L_{a,b,c}}O_{X}] \\ 0 1 . \end{matrix}

where

L_{a,b,c} = x\frac{∂}{∂x} + c − a + \frac{b − c}{1 − x} + \frac{a}{1 − λx} .

Therefore the relative De Rham cohomology groups H^{i}(X/S, (O_{X}, ∇_{a,b,c})), are the associated sheaves to the A-modules obtained as cohomology groups of the complex of A-modules

(D.5.1)  \begin{matrix}[A[x, & \frac{1}{x(1 − x)(1 − xλ)}] ⟶^{L_{a,b,c}}A[x, \frac{1}{x(1 − x)(1 − xλ)}]] \\ 0  1 . \end{matrix}

One can show by direct computation [Dw, Ch.1] that, if

(D.5.2)  c − a , c − b , b , a ∉ Z ,
(D.5.3)  H_{DR}^{i}(X/S, (O_{X}, ∇_{a,b,c})) = \{ \begin{matrix}0 & if i ≠ 1, \\ A cl(1) ⊕ A cl(\frac{1}{1−x}) & if i = 1, \end{matrix}

where cl(η) denotes the class of η ∈ L modulo L_{a,b,c}L.

The dual theory of ∇_{a,b,c} involves the calculation of the dual connection

∇_{a,b,c}^{∨} = h_{a,b,c} ∘ d_{X/K} ∘ h_{a,b,c}^{−1} = ∇_{−a,−b,−c}

on E^{∨} = O_{X} (so that L_{a,b,c}^{∗} = L_{−a,−b,−c}) and of the corresponding De Rham complex with compact supports, dual of (D.5.1),

(D.5.4)  \begin{matrix}[ & R ⟶^{γ_{−}∘L_{−a,−b,−c}}R] \\ 0  1 . \end{matrix}

So,

(D.5.5) H_{DR,c}^{i}(X/S, (O_{X}, ∇_{−a,−b,−c})) = \{ \begin{matrix}0 & if i ≠ 1, \\ Ker_{R}γ_{−} ∘ L_{−a,−b,−c} & if i = 1. \end{matrix}

The dual A-module H_{DR,c}^{1}(X/S, (O_{X}, ∇_{−a,−b,−c})) of H_{DR}^{1}(X/S, (O_{X}, ∇_{a,b,c})), may be explicitly computed in the following way. We examine the equation

(D.5.6)  γ_{−} ∘ L_{−a,−b,−c}ξ = 0 ,

for ξ = (ξ_{0}, ξ_{1}, ξ_{\frac{1}{λ}} , ξ_{∞}) ∈ R. Equation D.5.6 is equivalent to

(D.5.7)  L_{−a,−b,−c}ξ_{j} = γ_{+}((L_{−a,−b,−c}ξ_{i})_{i}) = ∑_{i} P_{i}(L_{−a,−b,−c}ξ_{i}) ∀j .

Since ξ is going to be an element of R, so that each ξ_{i} is formally holomorphic at i (at ∞, it even has a zero) consideration of the poles of L_{−a,−b,−c} shows that the r.h.s. of (D.5.7) is (D.5.8)

(c−b)\frac{ξ_{1}(1)}{1 − x} −a\frac{ξ_{\frac{1}{λ}} (\frac{1}{λ})}{1 − xλ} = (c−b) < \frac{1}{1 − x}, ξ > \frac{1}{1 − x} +aλ < \frac{1}{1 − xλ}, ξ > \frac{1}{1 − xλ} .

Equation D.5.7 then permits explicit calculations [Dw, 2.3.5] and provides vadic growth estimates for the power series ξ_{i} for any discrete valuation v of the quotient field K(λ) of A. We easily obtain:

Proposition D.5.9. ([Dw, 2.3]) Assume v is of one of the following two types:

a) v is a discrete valuation of K(λ) trivial on K

b) v induces the p-adic valuation on Q, K is algebraic over Q, v(a), v(b), v(c) ≥ 0, v(∑_{j} a_{j}λ^{j}) = max v(a_{j}) + jv(λ) for any polynomial ∑_{j} a_{j}λ^{j} ∈ K[λ]. Then, let ξ ∈ H_{DR,c}^{1}(X/S, (O_{X}, ∇_{−a,−b,−c})); for any i = 0, 1, \frac{1}{λ}, ∞, ξ_{i} is a power series in T_{i} with coefficients in K(λ) converging for v(T_{i}) > max(0, −v(t_{i})).

From a result of this type, one could deduce comparison theorems of algebraic versus p-adic analytic De Rham cohomology.

Chapter IV.

Complex and p-adic comparison theorems.

Table of contents:

§1 Review of analytic connections and De Rham cohomology.

§2 Abstract comparison criteria.

§3 Comparison theorem for algebraic vs. complex-analytic cohomology.

§4 Comparison theorem for algebraic vs. rigid-analytic cohomology (regular coefficients).

§5 Rigid-analytic comparison theorem in relative dimension one.

§6 Comparison theorem for algebraic vs. rigid-analytic cohomology (irregular coefficients).

§7 The relative non-archimedean Turrittin theorem.

Appendix E: Riemann’s existence problem: an elementary approach.

Introduction.

In this chapter, we show how to adapt Artin’s strategy of proof of his comparison theorem for étale cohomology [SGA IV] in the De Rham context. The result is a particularly simple proof of the Grothendieck-Deligne comparison theorem (algebraic versus complex-analytic De Rham cohomology with regular coefficients [G1], [De]). As a corollary, we obtain an elementary proof of Riemann’s existence theorem for coverings, in higher dimensions.

Not only does this proof not rely on resolution of singularities, but it also does not make use of moderate growth conditions, nor even of the notion of monodromy. This enables us to translate almost literally our argument into the p-adic setting, which yields a simple proof of the Kiehl-Baldassarri theorem (algebraic versus rigid-analytic De Rham cohomology with regular coefficients [Ki2], [B3]). We then use our study (II) of irregularity in several variables to prove that, in the p-adic setting, the comparison theorem extends to irregular connections (with the usual proviso on the field of definition).

Let us now describe in more detail the content of each section.

In the first section, we review complex-analytic and rigid-analytic connections and De Rham cohomology.

Section 2 generalizes and formalizes, in the abstract setting of differential algebra (2.1,2.4), the basic structure of some “comparison theorems” for De Rham cohomologies in one variable which one may find scattered in the literature (especially the literature on p-adic analysis).

Section 3, which relies on the technique of dévissage of (III), contains our proof of the Grothendieck-Deligne comparison theorem between algebraic and complex-analytic De Rham cohomologies with regular coefficients (3.1). For a discussion of the original proofs and of Mebkhout’s contribution [Me3], we refer to the general introduction of this book.

In all these approaches (as well as their p-adic avatars [Ki2], [B2], [B3]), the central point remains the comparison between two analytic De Rham complexes, one meromorphic and one with essential singularities. This point does not arise (at least explicitly) in our proof. Indeed, by dévissage, we are able to reduce the question to a global comparison problem in one variable (1), so simple that it may be settled by the formal criteria of §2, together with Frobenius’ convergence theorem of formal solutions of Fuchsian differential equations.

This proof is therefore analytical in nature (free from monodromy considerations, cf. (3.5)), and may be transferred into the p-adic setting. It also avoids any result of “GAGA type”.

In section 4, we prove in this way the rigid-analytic analogue of the Grothendieck-Deligne comparison theorem, under the assumption that the group generated by the exponents contains no p-adic Liouville number (4.1).

In the subsequent sections, we consider the case of irregular connections (in the rigid-analytic setting). Our main theorem is the following statement (cf. (6.1)), which was conjectured by the second author [B2]:

Let X be a smooth \overline{Q}-variety and let (E, ∇) be a coherent O_{X}-module with integrable connection. Then for any i ≥ 0,

(H_{DR}^{i}(X, (E, ∇))) ⊗\overline{Q} C_{p} ≅ H_{DR}^{i}(X_{C_{p}}^{an}, (E_{C_{p}}^{an}, ∇_{C_{p}}^{an})).

We first treat the relative-dimension-one case (5.2) (a generalization of [B2] and [C]); here again, the statement is reduced to Clark’s non-archimedean convergence theorem of formal solutions of possibly irregular differential equations, but the method of reduction is different from those used in [B2] or [C]. The strategy of proof of the theorem in arbitrary dimension uses refinements of our dévissages and our previous study of irregularity in several variables (II).

(1) let us mention that Mebkhout’ strategy also uses an argument of projection of relative dimension 1, but in a local, analytic setting.

As a corollary, we obtain that over a smooth algebraic variety defined over \overline{Q} ⊂ C_{p}, the rigid GAGA functor

(algebraic connections) ⟶ (analytic connections)

is fully faithful.

We close the chapter with a p-adic version of the Turrittin theorem in the relative case.

§1 Review of analytic connections and De Rham cohomology.

1.1 Complex-analytic connections.

1.1.1. Let X be a smooth complex analytic variety. The notion of an integrable connection on a O_{X} -module E is defined as in the algebraic situation (III.2.1.1) (without finiteness condition). It is equivalent to the notion of D_{X} -module. We denote by MIC(X) the category of O_{X} -modules with integrable connection. If E is coherent, it is then automatically locally free. 1.1.2. Inverse images are constructed as in the algebraic case. For a smooth morphism of smooth complex analytic varieties f : X ⟶ S, one constructs the direct image and the higher direct images R_{DR}^{q}f_{∗}(E, ∇) ∈ Ob MIC(S) of any (E, ∇) ∈ Ob MIC(X), as in the algebraic situation (III.2.4). The formation of R_{DR}^{q}f_{∗} is compatible with localization on S. 1.1.3. One associates functorially to any (smooth) algebraic C-variety X a (smooth) complex analytic variety X^{an}. The GAGA functor

\begin{matrix}MIC(X) & ⟶ MIC(X^{an}) \\ (E, ∇) & ⟼ (E^{an}, ∇^{an}) \end{matrix}

commutes with inverse images (essentially because O_{X^{an}} is flat over O_{X}), but not with direct images in general. In the sequel, we limit ourselves to the case of analytic connections on algebraic complex analytic varieties (i.e. objects of MIC(X^{an}) for smooth algebraic varieties X over C). 1.1.4. The discussion of Čech spectral sequences in (III.2.5) carries over for analytic connections, except that “quasi-coherent” should be replaced by “coherent” (and using the fact that affine varieties are Stein, and Cartan’s theorem B). 1.1.5. Flat base change (III.2.6) is more delicate in the analytic context: the interpretation of (u^{♯ an})^{∗} as (f^{♯ an})^{−1}O_{S^{♯ an}} ⊗_{(u^{♯ an})^{−1}(f^{an})^{−1}O_{S^{an}}} (u^{♯ an})^{−1} can only hold for a finite base change morphism u. For finite flat u, the argument of (III.2.6) (flat base change for _{I}E_{1}^{p,q}, using Čech complexes for affine covers as usual) carries over, and shows in particular that the formation of R_{DR}^{q}f_{∗}^{an} is compatible with étale localization (by factoring an étale map as a composition of open immersions and finite flat morphisms^{(1)}). 1.1.6. The computation (III.2.7) R_{DR}^{j}f_{∗}^{an}(E, ∇) ≅ H^{j}(f_{∗}^{an}DR_{X^{an}/S^{an}}(E, ∇)) for affine f (and vanishing for j > d), also holds. (However, for affine (or Stein) S, the computation of that cohomology sheaf in terms of the j-th cohomology module of the global relative De Rham complex seems to require the a priori knowledge of its coherence; using Cartan’s theorem

(1) A singular variety may appear in such a factorization. This does not make trouble since the O_{S}-module R^{q}f_{∗}DR_{X/S}(E, ∇) is well defined even if S is singular, provided f is smooth.

B, one then shows that the presheaves of cohomology of f_{∗}^{an}DR_{X^{an}/S^{an}}(E, ∇) are sheaves.) 1.1.7. The first lemma of dévissage (III.3.2) also holds for properties P of modules with integrable connection on algebraic complex-analytic (with the same proof), if one replaces “quasi-coherent” by “coherent”. For instance, let P = (analytic) coherence, i.e. (E, ∇) on X^{an} satisfies P if and only if E is O_{X^{an}}- coherent. This property is clearly strongly exact and local for the étale topology. Moreover, since any rational elementary fibration f′ : X′ ⟶ S′ is (locally on S′) topologically trivial for the classical topology (III.1.2), and since for any ζ ∈ S^{′an} and any coherent (E′, ∇′) ∈ Ob MIC(X^{′an}), H^{j}(X′_{ζ}, E_{|X′_{ζ}}^{′∇′}) is finite-dimensional

(j = 0, 1), it follows that R_{DR}^{j}f_{∗}^{′an}(E′, ∇′) ≃ R^{j}f_{∗}^{′an}(E^{′∇′_{|Der_{cont}X^{an}/S^{an}}})⊗_{C}O_{S^{′an}} is O_{S^{′an}}-coherent [De, I, 2.28].

One concludes that for any smooth morphism f : X ⟶ S of smooth complex algebraic varieties of pure relative dimension d, for any j ≥ 0, and any coherent (E, ∇) ∈ Ob MIC(X^{an}), R_{DR}^{j}f_{∗}^{an}(E, ∇)_{|U^{an}} is coherent, with U = A_{j}(f) if j ≤ d + dimX, and U = S otherwise.

1.1.8. However, we do not know whether the second lemma of dévissage (III.3.3) holds in the analytic category. While trying to adapt the given algebraic proof (or its mentioned alternative), one is confronted to the delicate question of comparing a De Rham complex with essential singularities along a divisor, with a meromorphic one. 1.1.9. Just as for the finiteness question (cf. (1.1.7)), the analytic situation shows a simpler behaviour than the algebraic situation with respect to base change, namely:

Let f : X ⟶ S be a smooth morphism of smooth complex algebraic varieties, and let (E, ∇) be a coherent object of MIC(X^{an}). Then, for any smooth complex variety S^{♯}, any morphism u : S^{♯} ⟶ S, and any i ≥ 0, the restriction to (u^{−1}A_{i}(f))^{an} of the base change morphism

u^{an∗}R_{DR}^{i}f_{∗}^{an}(E, ∇) ⟶ R_{DR}^{i}f_{∗}^{♯an}(u^{♯an∗}(E, ∇))

is an isomorphism in MIC((u^{−1}A_{i}(f))^{an}).

Indeed, replacing S by affine étale neighborhoods of the connected components of A_{i}(f), we may assume that X admits a finite open affine cover {U_{α}} with the properties listed in definition (III.1.8). There is a morphism of Čech spectral sequences in MIC(S^{♯an}) analogous to the one considered in (III.5.3), which allows to reduce ourselves to the case when f is a tower of (coordinatized) elementary fibrations. The result is then an easy consequence of the topological local triviality of the fibration f, which implies that

u^{an∗}R^{i}f_{∗}^{an}E^{∇_{|Der_{cont}X^{an}/S^{an}}} ⟶^{∼}R^{i}f_{∗}^{♯an}E^{♯∇_{|Der_{cont}X^{♯an}/S^{♯an}}^{♯}} ,

is an isomorphism.

1.2 Rigid analytic connections.

1.2.1. Similar comments apply to the rigid-analytic situation, over an algebraically closed field K of characteristic 0, complete with respect to a nonarchimedean absolute value | | (e.g. K = C_{p}) ([Be1], [BGR]). Notions and comments (1.1.1) to (1.1.4) may be repeated in this context. One uses the fact that affine covers of X give rise to admissible covers of X^{an}, that affine varieties are quasi-Stein, and Kiehl’s analogue of theorem B [Ki1].

1.2.2. The discussion (III.2.5) also carries over if {U_{α}} is chosen to be an admissible affinoid cover of X^{an}. This remark allows us to prove finite flat base change (using the first spectral sequence of (III.2.5.1) for such a cover, and flat base change for f_{\underline{α}∗}(Ω_{U\underline{α}}^{j}/S) with affinoid f\underline{α}). Therefore, the formation of R_{DR}^{i}f_{∗}^{an} is again compatible with étale localization. The previous discussion of R_{DR}^{i}f_{∗}^{an} for affine f carries over in the rigid-analytic context (on replacing “Stein” by “quasi-Stein”, and Stein covers by admissible affinoid covers), but the question of coherence of R_{DR}^{i}f_{∗}^{an}(E, ∇) is more delicate.

In fact, since Dwork’s early studies, it is well-known that one cannot expect generic finiteness of R_{DR}^{1}f_{∗}^{an}(E^{an}, ∇^{an}) in general, due to “Liouville phenomena”. As we shall see later, in the case of an elementary fibration, the problem disappears when the roots α_{ij} of the indicial polynomials ϕ_{i} are non-Liouville (in the sense that the series ∑_{n≠−α_{ij}^{n≥0}} \frac{x^{n}}{α_{ij}+n} converge).

§2 Abstract comparison criteria.

Let (C, ∂) be a differential Q-algebra, A, B be differential sub-algebras, such that A^{∂} = C^{∂} (equality of rings of constants), A is faithfully flat over A^{∂}, and ∂_{|A} is surjective onto A.

Proposition 2.1. Let E be an (A ∩ B)[∂]-module, projective of finite rank μ as a module over A ∩ B. Assume that E is solvable in A, i.e.

(E ⊗ A)^{∂} ⊗_{A^{∂}} A⟶^{∼}E ⊗ A .

Then φ_{0} : Ker_{E}∂ ⟶ Ker_{E⊗B}∂ is an isomorphism and φ_{1} : Coker_{E}∂ ⟶ Coker_{E⊗B}∂ is injective.

Proof.

(i) Injectivity of φ_{0}. This follows trivially from the fact that, since E is flat over A ∩ B, E injects into E ⊗ B. (Similarly, we have a diagram of inclusions

\begin{aligned}& & E ⊗ A \\ & ↗ & & ↘ \\ E & & & & E ⊗ C \\ & ↘ & & ↗ \\ & & E ⊗ B \end{aligned} ;

since E is locally free (A ∩ B)-module, (E ⊗ A) ∩ (E ⊗ B) = E). (ii) Injectivity of φ_{1}. Since E is solvable in A, which is faithfully flat over A^{∂}, the A^{∂}-module (E ⊗ A)^{∂} is projective (of rank μ), and, locally with respect to the Zariski topology on SpecA^{∂}, the A[∂]-module E ⊗ A is a sum of copies of A. Since A^{∂} = C^{∂}, this implies in turn

Lemma 2.2. Under the previous assumptions, (E ⊗ A)^{∂} = (E ⊗ C)^{∂}.

In order to complete the proof of (ii), we need the following

Lemma 2.3. Under the previous assumptions, ∂ : E⊗A ⟶ E⊗A is surjective.

Proof. By assumption, ∂ : A ⟶ A is surjective. Since Zariski-locally on A^{∂} , E⊗A ≅ A^{μ} as A[∂]-modules, it follows that ∂ : E⊗A ⟶ E⊗A is Zariski-locally (on A^{∂}) surjective, hence surjective. Q.E.D.

We now deduce the injectivity of φ_{1} from lemma 2.3. Let e ∈ E ⊗B be such that ∂e ∈ E. By lemma 2.3, there exists e′ ∈ E ⊗A satisfying ∂e′ = ∂e. We have e − e′ ∈ (E ⊗ C)^{∂} = (E ⊗ A)^{∂}, by lemma 2.2. Hence e ∈ (E ⊗ A) ∩ (E ⊗ B) = E. (iii) Surjectivity of φ_{0}. It is a special case of the last argument, for ∂e = 0. Q.E.D.

Proposition 2.4. In addition to the previous assumptions, we assume that C (resp. A) is a product of differential Q-algebras without zero divisors C_{i} (resp. A_{i} ⊂ C_{i}), i = 0, 1, …, r containing a differential subalgebra C′_{i} (resp. A′_{i} := A_{i} ∩ C′_{i}). We also assume the existence for each i of a differential A′_{i}- subalgebra T_{i} of C_{i} subject to the following conditions:

a) ∂ : C_{i} ⟶ C_{i} is surjective and C_{i} is faithfully flat over C_{i}^{∂};

b) the composed homomorphism of differential algebras B ⟶ C⟶^{pr_{i}}C_{i} factors through an injection B ↪ C′_{i};

c) there is a decomposition C_{i} = T_{i} ⊗_{A′_{i}} C_{i}^{log}, a tensor product of differential algebras, where C_{i}^{log} := ⊕_{j≥0}C′_{i}∂_{i}^{−j}1, ∂_{i} = u_{i}∂ for some unit u_{i} in A′_{i}, and (1 = ∂_{i}^{0}1, ∂_{i}^{−1}1, …) is a sequence of elements of A_{i} such that ∂_{i}(∂_{i}^{−j}1) = ∂_{i}^{−j+1}1;

d) C′_{0} ⊂ A′_{0} + pr_{0}B, and, for i > 0, C′_{i} ⊂ A′_{i} + ⋂_{j<i} pr_{i}(B ∩ A_{j}) (we use b) to make sense of this intersection). Then φ_{1} : Coker_{E}∂ ⟶ Coker_{E⊗B}∂ is surjective.

Proof. Since C_{i} is a domain, A_{i}^{log} := A_{i} ∩ C_{i}^{log} = ⊕_{j≥0}A′_{i}∂_{i}^{−j}1; therefore A_{i} = T_{i} ⊗_{A′_{i}} A_{i}^{log}.

We remark that (on applying pr_{i}) E is solvable in A_{i}, which is faithfully flat over A_{i}^{∂}, and A_{i}^{∂} = C_{i}^{∂}. A fortiori, E is solvable in C_{i}. On the other hand,

b) implies that pr_{i|B} induces an injection A ∩ B ↪ A′_{i}. We have a diagram of inclusions

E ⊗ A′_{i} ⟶ E ⊗ A_{i}
↗
E ↓ ↓
↘
E ⊗ C′_{i} ⟶ E ⊗ C_{i}

and (E ⊗ A_{i}) ∩ (E ⊗ C′_{i}) = E ⊗ A′_{i} (see the argument in (2.1.i)). Moreover B ∩ A_{i} = B ∩ A′_{i}, which we shall view either as a subring of B or of C_{i}.

We also note that E_{T_{i}} := E ⊗ T_{i} is a T_{i}[∂]-module projective of rank μ over T_{i}. The results (2.1), (2.2), (2.3) above apply on replacing E by E_{T_{i}} and the differential algebras C, A, B by C_{i}, A_{i} and T_{i}, respectively.

Let us fix an element e = e_{−1} of E ⊗ B. We look for an element f ∈ E and an element e′ ∈ E ⊗ B such that e = f + ∂e′. In view of this, we shall construct recursively, for i = 0, …, r, an element e′_{i} ∈ E ⊗ (⋂_{j<i} B ∩ A_{j}), such that e_{i} := e_{i−1} − ∂e′_{i} belongs to E ⊗ (⋂_{j≤i} B ∩ A_{j}). It will then suffice to take e′ := ∑_{i=0}^{r} e′_{i}, and f := e_{r} (which belongs to E ⊗ (⋂_{j=0}^{r} B ∩ A_{j}) = E).

Let us start with e_{i−1} ∈ E ⊗(⋂_{j<i} B ∩A_{j}) ⊂ E ⊗C′_{i} (if i = 0 e_{−1} ∈ E ⊗B ⊂ E ⊗ C′_{i}). We draw the existence of an element f_{i} ∈ E ⊗ C_{i} = E_{T_{i}} ⊗_{A′_{i}} C_{i}^{log} such that e_{i−1} = ∂f_{i}. According to c), we may write f_{i} = ∑_{j=0}^{N} f_{i,j}∂_{i}^{−j}1, with f_{i,j} ∈ E_{T_{i}} ⊗_{A′_{i}} C′_{i}. Equation e_{i−1} = ∂f_{i} decomposes into the following set of equations

∂f_{i,N} = 0
∂f_{i,N−1} = −u_{i}^{−1}f_{i,N}
·
·
∂f_{i,1} = −u_{i}^{−1}f_{i,2}
∂f_{i,0} = −u_{i}^{−1}f_{i,1} + e_{i−1}

from which we deduce, step by step, using lemmata 2.2 and 2.3 for E_{T_{i}}, that f_{i,N} ∈ (E_{T_{i}} ⊗_{A′_{i}} C′_{i})^{∂} = (E_{T_{i}})^{∂}, f_{i,N−1}, …, f_{i,1} ∈ E_{T_{i}}. According to condition d), f_{i,0} = e′_{i} + f′_{i}, with e′_{i} ∈ E ⊗ (⋂_{j<i} B ∩ A_{j}), f′_{i} ∈ E ⊗ A′_{i}. One then observes that the element e_{i} := e_{i−1} − ∂e′_{i} of E ⊗ (⋂_{j<i} B ∩ A_{j}) may also be written u_{i}^{−1}f_{i,1} + ∂f′_{i}, an element of E_{T_{i}}. Therefore e_{i} ∈ E ⊗ (⋂_{j≤i} B ∩ A_{j}). Q.E.D.

§3 Comparison theorem for algebraic vs. complex-analytic cohomology.

Theorem 3.1. ([De, II, 6.13]). Let i be a non-negative integer, and let f : X ⟶ S be a smooth morphism of smooth complex algebraic varieties, with A_{i}(f) = S (cf. definition III.1.8). Let (E, ∇) be a coherent O_{X}-module endowed with an integrable regular connection. Then the canonical morphism in MIC(S^{an}):

φ_{i} : (R_{DR}^{i}f_{∗}(E, ∇))^{an} ⟶ (R_{DR}^{i}f_{∗}^{an}(E^{an}, ∇^{an}))

is an isomorphism.

In the special case S = SpecC and (E, ∇) = (O_{X}, d_{X/S}), this gives (taking into account the analytic Poincaré lemma):

Corollary 3.2. ([G1]). For any smooth complex algebraic variety X, and any i ≥ 0, H_{DR}^{i}(X) ≅ H_{DR}^{i}(X^{an}) ≅ H^{i}(X(C), C).

3.3 Reduction to the case of a rational elementary fibration.

3.3.1. If f is of relative dimension 0, the statement amounts to the fact that for étale finite f, the canonical morphism (f_{∗}E)^{an} ⟶ f_{∗}^{an}E^{an} is an isomorphism, which is an elementary result of GAGA type (one may reduce, as in (III.3.1.1), to the case when f is a Galois covering, and then remark that

(f^{∗}f_{∗}E)^{an}⟶^{∼}f^{an∗}f_{∗}^{an}E^{an}).

3.3.2. As in (III.3.2.1), one then reduces to the case when f has pure relative dimension d ≥ 1, S = A_{i}(f) is affine connected, and 0 ≤ i ≤ d + dimX. Moreover, since the statement is local on S^{an}, and since the formation of algebraic or analytic De Rham cohomology commutes with étale localization on S ((III.2.3), (1.1.5)), we may replace S, as in (III.3.2.2), by an affine connected étale neighborhood S′ such that X_{S′} admits a finite open cover {U_{α}} with the properties listed in definition (III.1.8).

There is a natural morphism of Čech spectral sequences in MIC(S^{an}),

⊕_{α_{0}<...<α_{p}}(R_{DR}^{i−p}f_{\underline{α}∗}(E, ∇)_{|U\underline{α}})^{an} ⇒ (R_{DR}^{i}f_{∗}(E, ∇))^{an}
↓ φ_{i−p}  ↓ φ_{i}
⊕_{α_{0}<...<α_{p}}R_{DR}^{i−p}f_{\underline{α}∗}^{an}(E^{an}, ∇^{an})_{|U an ; \underline{α}} ⇒ R_{DR}^{i}f_{∗}^{an}(E^{an}, ∇^{an})

We argue by induction on i: since A_{i−|\underline{α}|+1}(f\underline{α}) = A_{i}(f) = S, φ_{i−p} is an isomorphism for p > 0; hence we may assume that f itself is a tower of coordinatized elementary fibrations.

3.3.3. For f a tower of coordinatized elementary fibrations, we have a natural morphism of Leray spectral sequences, by which we are reduced to proving the comparison theorem for each level of the tower. Therefore we may assume that f itself is a coordinatized elementary fibration, taking into account the fact that regularity is preserved under direct images by coordinatized elementary fibrations (III.7). In fact, since we are free to replace S by an étale covering, we may even assume that f = f′ ∘ π, where f′ is a rational elementary fibration and π is an étale covering. We then have R_{DR}^{i}f_{∗}(E, ∇) = R_{DR}^{i}f′_{∗} ∘ π_{∗}(E, ∇) (resp. R_{DR}^{i}f_{∗}^{an}(E^{an}, ∇^{an}) = R_{DR}^{i}f_{∗}^{′an} ∘π_{∗}^{an}(E^{an}, ∇^{an}) ), and we are reduced to proving the comparison theorem for a rational elementary fibration, and i = 0, 1.

At this point the proof splits into two alternative arguments.

3.4 First argument: reduction to an ordinary linear differential system. We use the fact that both the source and the target of φ_{i} are locally free O_{S^{an}}-modules of finite rank ((III.6) and (1.1.7), resp.). Therefore it suffices to check that the fibre (φ_{i})_{s} is an isomorphism for any s ∈ S(C). By base change ((III.5.5), (1.1.9)), we are finally reduced to the case of an ordinary linear (regular) differential system, i.e. S = SpecC, X = A^{1} \ {θ_{1}, …, θ_{r}}, and it is enough to deal with global sections. Hence, we have reduced theorem 3.1 to the following special case.

Proposition 3.4.1. Let X = SpecC[x, _{\frac{1}{∏_{i=1}^{r}(x−θ_{i})}}], and let (E, ∇) be a projec-

tive O(X)-module of finite rank μ, with a regular connection. Then

φ_{0} : Ker_{E}∇(\frac{d}{dx}) ⟶ Ker_{E⊗O(X^{an})}∇(\frac{d}{dx})

and

φ_{1} : Coker_{E}∇(\frac{d}{dx}) ⟶ Coker_{E⊗O(X^{an})}∇(\frac{d}{dx})

are isomorphisms.

Notice that, since C[x, _{\frac{1}{∏_{i=1}^{r}(x−θ_{i})}}] is a principal Dedekind ring, the projec-

tive module E is necessarily free. A proof of the proposition can then be found in [Ad, Thm.2 and §5, Remark 2]. We prefer to deduce proposition 3.4.1 from the abstract comparison criteria of section 2.

3.4.2. We set

X = SpecC[x, _{\frac{1}{∏_{i=1}^{r}(x−θ_{i})}}],

B := O(X^{an}),

and for i = 1, …, r,

C′_{i} := ⋃_{ϵ>0} O(D(θ_{i}, ϵ) \ {θ_{i}}),

M_{θ_{i}} := the field of germs of meromorphic functions at θ_{i},

A_{i} := M_{θ_{i}}[log(x − θ_{i}), (x − θ_{i})^{α}]_{α∈C} (with the usual identification

(x − θ_{i})^{α}(x − θ_{i})^{β} = (x − θ_{i})^{α+β} ) ,

C_{i} = A_{i}C′_{i}.

We adopt similar definitions for i = 0, upon replacing θ_{i} by ∞, and set C = ∏_{i=0}^{r} C_{i}, A = ∏_{i=0}^{r} A_{i}, ∂ = \frac{d}{dx}. We have A ∩ B = (∏_{i=0}^{r} M_{θ_{i}}) ∩ B = O(X) = C[x, _{\frac{1}{∏_{i=1}^{r}(x−θ_{i})}}] (diagonally embedded in A), A_{i}^{∂} = C_{i}^{∂} = C, A′_{i} := A_{i} ∩ C′_{i} =

M_{θ_{i}}.

3.4.3. We now exploit the regularity of the (A∩B)[∂]-module E in the guise that E is solvable in A, or, what amounts to the same, in each A_{i} (Fuchs-Frobenius, the point being that formal solution series of a differential operator at a regular singular point, are convergent). It remains to check that all assumptions in section 2 are met. Both A and C are clearly faithfully flat over A^{∂} = C^{∂} ≅ C^{r+1}. Using the formula

x^{α−1} \frac{log^{k} x}{k!} = ∂(\frac{x^{α}}{α} ∑_{i+j=k}(−\frac{1}{α})^{i} \frac{log^{j} x}{j!}) ,

for α ≠ 0, and

x^{−1} \frac{log^{k} x}{k!} = ∂ \frac{log^{k+1} x}{(k + 1)!} ,

one sees that ∂ acts surjectively both on A and on C. This proves the requirements for (2.1), and condition a) of (2.4) in addition. Condition b) is clear (by analytic continuation). Condition c) holds for i ≥ 1 if we set u_{i} = x − θ_{i}, ∂_{i}^{−j}1 = \frac{log^{j}(x−θ_{i})}{j!}, T_{i} = M_{θ_{i}}[(x − θ_{i})^{α}]_{α∈C}. For i = 0, we adopt similar definitions, replacing θ_{i} by ∞.

Finally, d) holds in a stronger form. For any i ∈ {0, …, r}, C′_{i} = A′_{i} + ⋂_{j≠i}(B ∩ A_{j}), which reflects the decomposition of the Laurent expansion of an element of C′_{i}, into its positive and negative parts, respectively. This achieves the proof of (3.4.1) and of the comparison theorem. Q.E.D.

3.5 Alternative end of proof. The proofs (1.1.7), (1.1.9) of coherence and base change for R_{DR}^{1}f_{∗}^{an}(E^{an}, ∇^{an}), which are used in (3.4), rely on topological arguments (monodromy). We can avoid such arguments by handling directly the case of a rational elementary fibration in the following way.

3.5.1. For S compact in S(C) and ϵ > 0, we introduce the compact tube in S^{an} × (A^{1})^{an}

T_{S,i,ϵ} = {(s, x) , s ∈ S , |x − θ_{i}(s)| ≤ ϵ } ,

and

T_{S,i,ϵ}^{∗} = {(s, x) , s ∈ S , 0 < |x − θ_{i}(s)| ≤ ϵ } .

Let Δ be a Q-subspace of C containing Q (in the sequel, Δ will be the Q-space generated by 1 and the exponents of the canonical extension Ẽ of E, taking into account theorem (III.8.3) in the course of dévissage).

We define

C′_{S,i} = ⋃_{ϵ>0} O(T_{S,i,ϵ}^{∗}),

A_{S,i} = ⋃_{ϵ>0} O(T_{S,i,ϵ})[(x−θ_{i})^{α}, log(x−θ_{i})]_{α∈Δ} (with the usual identification

(x − θ_{i})^{α}(x − θ_{i})^{β} = (x − θ_{i})^{α+β} ) ,

C_{S,i} = C′_{S,i}A_{S,i},

A′_{S,i} := C′_{S,i} ∩ A_{S,i} = ⋃_{ϵ>0} O(T_{S,i,ϵ})[\frac{1}{x−θ_{i}}].

We adopt similar definitions for i = 0, upon replacing θ_{i} by ∞, and set

C_{S} = ∏_{i=0}^{r} C_{S,i} , A_{S} = ∏_{i=0}^{r} A_{S,i} , ∂ = \frac{d}{dx} .

Finally, we introduce the inverse image X of S in X^{an}, and set B_{S} = O(X), diagonally embedded into C_{S}.

One then checks as in (3.4.3) that A_{S} is faithfully flat over A_{S}^{∂} = C_{S}^{∂} = O(S)^{r+1}, that ∂_{|A_{S}} is surjective onto A_{S}, and that conditions a), …, d) of (2.4) are fulfilled (with

\begin{matrix}u_{i} & = x − θ_{i} , ∂_{i}^{−j}1 = \frac{log^{j}(x − θ_{i})}{j!} , \\ T_{i} = & ⊕_{α∈Δ∩\{Imτ\\{0\}\}} C′_{S,i}(x − θ_{i})^{α} ) . \end{matrix}

Note that A_{S} ∩ B_{S} = O(S)[x, \frac{1}{∏(x−θ_{i})}].

3.5.2. Because Ẽ is locally free, there exists a Zariski covering {U} of S such that Ẽ_{|Z_{i}∩f^{−1}(U)} is free over O_{Z_{i}∩f^{−1}(U)} for every i, where Z_{i} = div(x−θ_{i}) ⊂ P_{S}^{1} (Z_{0} is the divisor at infinity). Then for every compact S ⊂ U(C), there exists ϵ_{S} > 0 such that Ẽ ⊗ O(T_{S,i,ϵ}) is free over O(T_{S,i,ϵ}) if ϵ ≤ ϵ_{S}. We denote by E_{S} the O(S)[x, \frac{1}{∏(x−θ_{i})}, \frac{d}{dx}]-module Γ(E) ⊗_{O(S)} O(S).

Proposition 3.5.3. E_{S} is solvable in A_{S}.

Proof. We have to show that E_{S} is solvable in A_{S,i}, for every i = 0, …, r. Let \underline{e} be a basis of Ẽ ⊗ O(T_{S,i,ϵ}) in which the connection may be written ∇((x − θ_{i})\frac{d}{dx})\underline{e} = \underline{e}G, where G has entries analytic in T_{S,i,ϵ}, and G_{|Z_{i}} is a constant matrix with eigenvalues in Δ: in fact because T_{S,i,ϵ} is a compact tube, there exists a compact neighborhood S′ of S such that G ∈ M_{μ}(O(T_{S′,i,ϵ})).

It suffices to show that the differential system ((x − θ_{i})\frac{d}{dx} + G)Y = 0 has a solution of the form Y = W · (x − θ_{i})^{G_{|Z_{i}}}, where W ∈ GL_{μ}(O(T_{S,i,ϵ′})), for some 0 < ϵ′ < ϵ. We follow the classical method used in the case S = one point, cf. e.g. [DGS, III, 8.5 and Appendix II]: a formal computation expresses the coefficients of the Taylor expansion of W at x = θ_{i} by a recursive formula, under the extra condition: W_{|Z_{i}} = I; one then estimates the growth of these coefficients (with respect to the sup-norm on S′) as in loc. cit., app. II, and one concludes that W ∈ M_{μ}(O(T_{S,i,ϵ′′})), 0 < ϵ′′ < ϵ. Since W_{|Z_{i}} = I, it is clear that there exists ϵ′, 0 < ϵ′ ≤ ϵ′′, such that W ∈ GL_{μ}(O(T_{S,i,ϵ′})), as wanted. This concludes the proof of (3.5.3).

3.5.4. We can apply the criteria (2.1) and (2.4), and conclude that

(Co)Ker_{E_{S}} ∇(\frac{d}{dx}) ≅ (Co)Ker_{Γ(X ,E^{an})}∇(\frac{d}{dx}) .

Since O(S) is flat over O(S), this may be rewritten

(Co)Ker_{Γ(X,E)}∇(\frac{d}{dx}) ⊗_{O(S)} O(S) ≅ (Co)Ker_{Γ(X ,E^{an})}∇(\frac{d}{dx}) ,

which holds for any small enough compact neighborhood S of any point of S(C). Therefore the natural morphism

(R_{DR}^{i}f_{∗}(E, ∇))^{an} ⟶ R_{DR}^{i}f_{∗}^{an}(E^{an}, ∇^{an})

is an isomorphism for i = 0, 1, and this achieves the alternative proof of 3.1.

Corollary 3.6. Let X be a smooth complex algebraic variety. The functor

{regular connections on X} ⟶ {local systems on X(C)}
(E, ∇) ⟼ (E^{an})^{∇^{an}}

is fully faithful.

Proof. Let (E, ∇), (E′, ∇′) be two regular objects in MIC(X). Then (the internal Hom) \underline{Hom}((E, ∇), (E′, ∇′)) is a regular object in MIC(X) (7.5). By 3.1, with i = 0, we have

Hom((E, ∇), (E′, ∇′)) = H_{DR}^{0}(X, \underline{Hom}((E, ∇), (E′, ∇′))) ≅
H_{DR}^{0}(X^{an}, \underline{Hom}((E^{an}, ∇^{an}), (E^{′an}, ∇^{′an}))) ≅
H^{0}(X(C), \underline{Hom}_{loc. syst.}((E^{an})^{∇^{an}}, (E^{′an})^{∇^{′an}})) =
Hom_{loc. syst.}((E^{an})^{∇^{an}}, (E^{′an})^{∇^{′an}}) .

Q.E.D.

Corollary 3.7. Let X and (E, ∇), (E′, ∇′) be as before. Then the natural map

Ext_{MIC}^{1}((E, ∇), (E′, ∇′)) ⟶ Ext_{loc. syst.}^{1}((E^{an})^{∇^{an}}, (E^{′an})^{∇^{′an}})

is a bijection.

Proof. Same argument as for (3.6), but with i = 1.

§4 Comparison theorems for algebraic vs. rigid-analytic cohomology.

In this section, K is an algebraically closed field of characteristic 0, complete with respect to a non-archimedean absolute value.

Theorem 4.1. Let i be a non-negative integer, and let f : X ⟶ S be a smooth morphism of smooth K-varieties, with A_{i}(f) = S. Let (E, ∇) be a coherent O_{X}- module endowed with an integrable regular connection, such that the additive subgroup of K generated by 1 and the exponents of ∇ contains no Liouville number. Then the canonical morphism

φ_{i} : (R_{DR}^{i}f_{∗}(E, ∇))^{an} ⟶ R_{DR}^{i}f_{∗}^{an}(E^{an}, ∇^{an})

is an isomorphism.

In the special case S = SpecK and (E, ∇) = (O_{X}, d_{X/S}), this gives:

Corollary 4.2. ([Ki2]). For any smooth algebraic K-variety X, and any i ≥ 0, H_{DR}^{i}(X) ≅ H_{DR}^{i}(X^{an}).

In the case S = SpecK, theorem 4.1 was already proven in [B3] using resolution of singularities.

4.3. Our proof of (4.1) is very close to the proof in the complex case (in its second form, cf. (3.5)). We reduce to the case of a rational elementary fibration, i = 0, 1, and apply the abstract criteria of section 2. We introduce S, X, T_{S,i,ϵ}, A_{S,i}, C′_{S,i}, A_{S}, C_{S}, E_{S}, as in (3.5) (on replacing the word “compact” by “affinoid”). The only difference is that for the solvability of E_{S} in A_{S} - more specifically, in the non-archimedean estimates replacing [DGS, Appendix II] -, as well as for the stability of A_{S,i} and C′_{S,i} by integration, we have to use the fact that Δ (the Q-vector space generated by 1 and the exponents) does not contain Liouville numbers, which follows from the assumption in (4.1).

We conclude again that the natural morphism

(Co)Ker_{Γ(X,E)}∇(\frac{d}{dx}) ⊗_{O(S)} O(S) ⟶ (Co)Ker_{Γ(X ,E^{an})}∇(\frac{d}{dx}) ,

is an isomorphism for any affinoid subspace S of U^{an}, where {U} is a finite Zariski open covering of S as in (3.5.2).

4.4. Since (Co)Ker_{Γ(X,E)}∇(\frac{d}{dx}) is finitely generated over O(S) (III.6), we see that S ⟼ Ker_{Γ(X ,E^{an})}∇(\frac{d}{dx}) (resp. S ⟼ Coker_{Γ(X ,E^{an})}∇(\frac{d}{dx})) is a coherent sheaf on U^{an}, which coincides with R_{DR}^{0}f_{|f^{an−1}U^{an}∗}^{an}(E^{an}, ∇^{an})_{|f^{an−1}U^{an}} ≅ (R_{DR}^{0}f_{∗}^{an}(E^{an}, ∇^{an}))_{|U^{an}} (resp. (R_{DR}^{1}f_{∗}^{an}(E^{an}, ∇^{an}))_{|U^{an}}). By pasting, we conclude that φ_{i} is an isomorphism for i = 0, 1 in our special situation. This achieves the proof of (4.1). Q.E.D.

§5 Rigid-analytic comparison theorem in relative dimension one.

5.1 On the coherence of the cokernel of a connection in the analytic situation. Let us consider a rational elementary fibration

(5.1.0)  \begin{aligned}X & ↪ & P_{S}^{1} & ↩ & Z = (∐_{i=1}^{r} σ_{i}(S)) ∐ σ_{∞}(S) \\ & ↘^{f} & ↓^{pr} & ↙ \\ & & S \end{aligned}

where σ_{∞}(S) = ∞ × S, σ_{i} : ζ ⟼ (θ_{i}(ζ), ζ), θ_{i} ∈ O(S), θ_{i} − θ_{j} ∈ O(S)^{×}. We assume that the base S is affine smooth and consider a cyclic (E, ∇), with associated differential polynomial Λ ∈ O(X)[\frac{d}{dx}], as in (III.4.4); in particular, we assume that S is localized in such a way that the leading coefficients of the indicial polynomials ϕ_{i} at x = θ_{i} are units in O(X). We denote by f^{an} : X^{an} ⟶ S^{an} and (E^{an}, ∇^{an}) ∈ ObMIC(X^{an}) the corresponding analytic objects. Let also S ⊂ S^{an} be an affinoid subspace, and let X ⟶ S denote the (quasi-Stein) morphism obtained from f^{an} by base change.

Proposition 5.1.1. If the roots α_{ij} of the indicial polynomials ϕ_{i} are non-Liouville (i.e. if the power series ∑_{n≠α_{ij}^{n≥0}} \frac{x^{n}}{α_{ij}+n} has a non-zero radius of conver-

gence), then the natural map

Coker(Γ(X, E)⟶^{∇}Γ(X, E)dx) ⊗ O(S) ⟶ Coker(Γ(X , E^{an})⟶^{∇^{an}}Γ(X , E^{an})dx)

is surjective.

The rest of this section is devoted to the proof of this assertion.

5.1.2. There is a commutative diagram (induced by (III.4.4.1))

(5.1.2.1) \begin{matrix}O(X) & ⟶^{Λ} & O(X) \\ ↓ & & ↓ \\ Γ(X , E^{an}) = ⨁_{k=0}^{μ−1} O(X)v_{k} & ⟶^{∇^{an}( \frac{d}{dx})} & Γ(X , E^{an}) = ⨁_{k=0}^{μ−1} O(X)v_{k} \end{matrix}

which gives rise to isomorphisms:

(Co)Ker(Γ(X , E^{an})⟶^{∇}Γ(X , E^{an})dx) ≅ (Co)Ker_{O(X)}Λ ≅ (Co)Ker_{O(X)}Λ′

where Λ′ = P (x)Λ ∈ O(S)[x, \frac{d}{dx}] as in (III.4.4).

5.1.3. We have a Mittag-Leffler decomposition (induced by (III.4.5.2))

(5.1.3.1)  O(X) = O(A_{S}^{1 an}) ⊕ ⨁_{i=1}^{r} \frac{1}{x − θ_{i}} O(P_{S}^{1 an} \ σ_{i}(S)) ,

where

O(A_{S}^{1 an}) = {∑_{n≥0} a_{n,∞}x^{n} ∈ O(S)[[x]] , lim_{n→∞} ||a_{n,∞}||^{1/n} = 0 }

O(P_{S}^{1 an} \ σ_{i}(S)) = \{∑_{n≥0} \frac{a_{n,i}}{(x − θ_{i})^{n}} ∈ O(S)[[\frac{1}{x − θ_{i}}]] , lim_{n→∞} ||a_{n,i}||^{1/n} = 0 \}

for some (any) Banach norm || || on the Tate K-algebra O(S).

5.1.4. Let us restate more precisely the induction process in (III.4.6), assuming that the leading coefficients τ_{∞}, τ_{i} of the indicial polynomials ϕ_{∞}, ϕ_{θ_{i}} (i = 1, …, r) of Λ′ are invertible in O(S). Let us write

P (x)γ_{k} = ∑_{l} b_{k,l,i}(x − θ_{i})^{l} = ∑_{l} b_{k,l}x^{l} ,

with b_{k,l,i}, b_{k,l} ∈ O(S), and set

κ_{i} = max(1, max_{k,l} ||k!τ_{i}^{−1}b_{k,l,i}||) , κ_{∞} = max(1, max_{k,l} ||k!τ_{∞}^{−1}b_{k,l}||) .

We contend that for n > 0 (resp. n ≥ 0)

(∗)_{n,i}  \frac{1}{(x − θ_{i})^{n}} = Λ′( ∑_{l=1}^{n−r_{i}} γ_{n,l,i} \frac{1}{(x − θ_{i})^{l}}) + ∑_{k=1}^{M} δ_{n,k,i} \frac{1}{(x − θ_{i})^{k}} ,

with

||γ_{n,l,i}|| , ||δ_{n,l,i}|| ≤ κ_{i}^{max(0,n−M)} ∏_{m=M+1}^{n} max(1, |\frac{τ_{i}}{ϕ_{θ_{i}}(−m + r_{i})}|)

(resp.

(∗)_{n,∞}  x^{n} = Λ′( ∑_{l=0}^{n−r_{∞}} γ_{n,l,∞}x^{l}) + ∑_{k=0}^{M} δ_{n,k,∞}x^{k} ,

with

||γ_{n,l,∞}|| , ||δ_{n,l,∞}|| ≤ κ_{∞}^{max(0,n−M)} ∏_{m=M+1}^{n} max(1, |\frac{τ_{∞}}{ϕ_{∞}(−m + r_{∞})}|) ) .

Indeed this is trivial for n ≤ M. For n > M, we write

\begin{aligned}\frac{1}{(x − θ_{i})^{n}} = & Λ′(\frac{1}{τ_{i}} · \frac{τ_{i}}{ϕ_{θ_{i}}(−n + r_{i})} · \frac{1}{(x − θ_{i})^{n−r_{i}}}) + \\ & ∑_{k=n−r_{i}−r_{∞}}^{max(M,n−r_{i}−r_{∞})} β_{n,k,i} \frac{1}{(x − θ_{i})^{k}} + ∑_{k=max(M,n−r_{i}−r_{∞})+1}^{n−1} β_{n,k,i} \frac{1}{(x − θ_{i})^{k}} , \end{aligned}

with |β_{n,k,i}| ≤ κ_{i} max(1, |\frac{τ_{i}}{ϕ_{θ_{i}} (−n+r_{i})}|), and (∗)_{n,i} follows by induction, on applying (∗)_{<n,i} to the terms of the last sum (resp.

x^{n} = Λ′(τ_{∞}^{−1} · \frac{τ_{∞}}{ϕ_{∞}(−n + r_{∞})} · x^{n−r_{∞}}) + ∑_{k=max(0,n−r_{∞}−μ)}^{max(M,n−r_{∞}−μ)} β_{n,k,∞}x^{k}+

∑_{k=max(M,n−r_{∞}−μ)+1}^{n−1} β_{n,k,∞}x^{k} ,

with |β_{n,k,∞}| ≤ κ_{∞} max(1, |\frac{τ_{∞}}{ϕ_{∞}(−n+r_{∞})}|), …). On the other hand,

∏_{m=M+1}^{n} max(1, |\frac{τ_{i}}{ϕ_{θ_{i}}(−m + r_{i})}|)

grows at most exponentially with n, because the roots of the ϕ_{θ_{i}} are assumed to be non-Liouville (cf. [Ad, §2]). Hence ||γ_{n,l,i}||, ||δ_{n,l,i}|| ≤ κ^{′n}, for some κ′ ≥ 1.

Therefore, any ∑_{n≥0} a_{n,∞}x^{n} + ∑_{i} ∑_{n≥1} \frac{a_{n,i}}{(x−θ_{i})^{n}} ∈ O(X) may be written formally Λ′u + v, where

\begin{matrix}u = & ∑_{l≥0}( ∑_{n≥l+r_{∞}} a_{n,∞}γ_{n,l,∞})x^{l} + ∑_{i} ∑_{l≥1}( ∑_{n≥l+r_{i}} a_{n,i}γ_{n,l,i})\frac{1}{(x − θ_{i})^{l}} \\ v = & ∑_{k=0}^{M}(∑_{n≥0} a_{n,∞}δ_{n,k,∞})x^{k} + ∑_{i} ∑_{k=1}^{M}(∑_{n≥1} a_{n,i}δ_{n,k,i})\frac{1}{(x − θ_{i})^{k}} , \end{matrix}

and the previous estimates show that u ∈ O(X), v ∈ O(S)[x, _{\frac{1}{∏_{i}(x−θ_{i})}}].

This shows that the natural map

(Coker_{O(X)}Λ′) ⊗_{O(S)} O(S) ⟶ Coker_{O(X)}Λ′

is surjective, hence so is

(Coker(Γ(X, E)⟶^{∇}Γ(X, E)dx)⊗O(S)) ⟶ (Coker(Γ(X , E^{an})⟶^{∇^{an}}Γ(X , E^{an})dx)) .

Q.E.D.

Remark 5.1.5. As was already pointed out in the algebraic situation (III.4.7.iii), the proof does not use the full hypothesis that v is a cyclic vector on the whole of X (i.e. that one may take P (x) = ∏_{i=1}^{r}(x − θ_{i})^{s_{i}}). It would suffice to assume that v is a cyclic vector outside some divisor D whose Zariski closure in P_{S}^{1} is contained in X.

5.2 Comparison theorem for morphisms of relative dimension one.

We now give a relative version of the main theorem of [B2]. This result, bound to rigid-analytic comparison in relative dimension one, will be generalized to arbitrary dimensions in the next section.

Proposition 5.2.1. Let f : X ⟶ S be a smooth morphism of smooth algebraic K-varieties of relative dimension one, and let (E, ∇) be a coherent O_{X}-module endowed with an integrable connection. We assume that f and (E, ∇) may be defined over some algebraically closed subfield K_{0} ⊂ K which contains no Liouville number (e.g. K_{0} = \overline{Q}). Then there is a dense open subset U ⊂ S such that for any i ≥ 0, the restriction to U^{an} of the canonical morphism

φ_{i} : (R_{DR}^{i}f_{∗}(E, ∇))^{an} ⟶ R_{DR}^{i}f_{∗}^{an}(E^{an}, ∇^{an})

is an isomorphism in MIC(U^{an}).

Proof.

5.2.2. We may replace S by any dense open subset, and then by any finite flat S-scheme S′ (indeed, the formation of algebraic and analytic De Rham cohomologies commutes with finite flat base change (III.2.6), (1.2.2), O_{S^{′an}} is faithfully flat over O_{S^{an}}, and the functor “inverse image of O_{S^{an}}-modules” on S^{an} is nothing but − ⊗_{O_{S^{an}}} O_{S^{′an}}). As we saw in (III.3.3.1), this allows us to assume that X admits a finite open cover \{U_{α}\} such that all f\underline{α} := f_{|U\underline{α}=U_{α_{0}} ∩...∩U_{αp}} are coordinate elementary fibrations for p ≤ 1 + dimX.

Thanks to the morphism of Čech spectral sequences in MIC(S^{an}),

⊕_{α_{0}<...<α_{p}}(R_{DR}^{i−p}f_{\underline{α}∗}(E, ∇)_{|U\underline{α}})^{an} ⇒ (R_{DR}^{i}f_{∗}(E, ∇))^{an}
↓ φ_{i−p}  ↓ φ_{i}
⊕_{α_{0}<...<α_{p}}R_{DR}^{i−p}f_{\underline{α}∗}^{an}(E^{an}, ∇^{an})_{|U an ; \underline{α}} ⇒ R_{DR}^{i}f_{∗}^{an}(E^{an}, ∇^{an})

we may assume that f itself is a coordinatized elementary fibration.

On replacing S by an étale covering and arguing as in (3.3.3), we reduce at last to the case when f is a rational elementary fibration, and i = 0, 1.

5.2.3. If S = SpecK (as in the situation of [B2]), we are left with an ordinary differential system in one variable. Hence the comparison theorem

H_{DR}^{i}(X, (E, ∇)) ≅ H_{DR}^{i}(X^{an}, (E^{an}, ∇^{an}))

for a curve is equivalent to the following statement (whose proof only relies on the non-archimedean Turrittin theorem of [B1]).

Proposition 5.2.4. Let X = SpecK[x, _{\frac{1}{∏_{i=1}^{r}(x−θ_{i})}}], and let (E, ∇) be a projec-

tive O(X)-module of finite rank μ, with a connection (not necessarily regular), such that the Q-vector subspace Δ ⊂ K generated by all Turrittin exponents (at the θ_{i} and at ∞) contains no Liouville numbers. Let X^{an} denote the rigid analytic space associated to the K-scheme X. Then

φ_{0} : Ker_{E}∇(\frac{d}{dx}) ⟶ Ker_{E⊗O(X^{an})}∇(\frac{d}{dx})

and

φ_{1} : Coker_{E}∇(\frac{d}{dx}) ⟶ Coker_{E⊗O(X^{an})}∇(\frac{d}{dx})

are isomorphisms.

Proof. This is obtained by choosing B, C′_{i} as in (3.4.2),

T_{i} = M_{θ_{i}}[(x − θ_{i})^{α}, exp(a(x − θ_{i})^{− \frac{k}{μ!}} )]_{α∈Δ,k∈Z_{>0},a∈K} ,

with the usual identification

(x−θ_{i})^{α} exp(a(x−θ_{i})^{− \frac{k}{μ!}} )·(x−θ_{i})^{β} exp(b(x−θ_{i})^{− \frac{k}{μ!}} ) = (x−θ_{i})^{α+β} exp((a+b)(x−θ_{i})^{− \frac{k}{μ!}} ) ,

A_{i} = T_{i}[log(x − θ_{i})] ,

and C_{i} = A_{i}C′_{i} (for i = 0 we are tacitly replacing θ_{i} by ∞, as in (3.4.2)). We also set C = ∏_{i=0}^{r} C_{i}, A = ∏_{i=0}^{r} A_{i}, ∂ = \frac{d}{dx}.

Lemma 5.2.4.1. ∂ : A ⟶ A is surjective.

Proof. We tacitly replace A by A_{i}, and assume θ_{i} = 0. We also set z = x^{− \frac{1}{μ^{2}!}} , an element of A. We may work with d/dz instead of ∂. Any element of A is a finite sum of terms of the form u z^{α} \frac{log^{m} z}{m!} e^{a z^{−k}}, where u ∈ M_{0}[z] is a p-adic meromorphic function of z. We have to show that ∫ u z^{α} \frac{log^{m} z}{m!} e^{a z^{−k}}dz is of the same form. We first note that if k = 0 or a = 0 , the result is an easy consequence of the following formulas

z^{α−1} \frac{log^{m} z}{m!} = ∂(\frac{z^{α}}{α} ∑_{i+j=m}(−\frac{1}{α})^{i} \frac{log^{j} z}{j!} ) ,

for α ≠ 0, and

z^{−1} \frac{log^{m} z}{m!} = ∂ \frac{log^{m+1} z}{(m + 1)!} .

Let us then assume that k > 0, a ≠ 0. By integration by parts

∫ u z^{α} \frac{log^{m} z}{m!} e^{a z^{−k}}dz = \frac{log^{m} z}{m!} ∫ u z^{α} e^{a z^{−k}}dz−∫ z^{−1} \frac{log^{m−1} z}{(m − 1)!} (∫ u z^{α} e^{a z^{−k}}dz) dz

and descending induction on m, we reduce to the case where m = 0. We then have to show that, for α ∈ Δ and a ∈ K, there is a formal integral y = u^{−1} z^{−α} e^{−a z^{−k}} ∫ u z^{α} e^{a z^{−k}}dz which is a p-adic meromorphic function of z. We notice that y satisfies the inhomogenous differential equation

dy/dz = (−u^{−1}du/dz − α/z + kaz^{−k−1}) y + 1

hence the homogenous differential equation of second order

d^{2}y/dz^{2}+(u^{−1}du/dz+α/z−kaz^{−k−1})dy/dz+(d/dz(u^{−1}du/dz)−α/z^{2}+k(k+1)az^{−k−2})y = 0 .

Since u is meromorphic, d/dz(u^{−1}du/dz) has at worst a double pole at 0, and since k > 0, we see that the indicial polynomial of this differential equation at 0 is

ϕ_{0}(T) = ak(T − k − 1) .

As a simple instance of Clark’s theorem [Cl], we conclude that the differential equation possesses a meromorphic solution y. This achieves the proof of (5.2.4.1). Q.E.D.

To apply proposition 2.4, we need to show that

(5.2.4.2)  A′_{i} := A_{i} ∩ C′_{i} = M_{θ_{i}} .

We first show that

(5.2.4.3)  A_{i} ∩ C′_{i} ⊂ M_{θ_{i}}[(x − θ_{i})^{α}, log(x − θ_{i})]_{α∈Δ} .

In fact, any element y ∈ A_{i} may be expanded in convergent power series in x−ζ_{i}, for ζ_{i} ≠ θ_{i} sufficiently close to θ_{i}:

y = y_{0}(x − ζ_{i}) + ∑_{k=1}^{r} y_{k}(x − ζ_{i}) exp(a((x − ζ_{i})^{− \frac{k}{μ!}} − (ζ_{i} − θ_{i})^{− \frac{k}{μ!}} ) ,

where y_{k}(x − ζ_{i}) is the expansion at ζ_{i} of an element y_{k} of M_{θ_{i}}[(x − θ_{i})^{α}, log(x − θ_{i})]_{α∈Δ}. So, the radius of convergence of y_{k}(x − ζ_{i}) is of the form C|ζ_{i} − θ_{i}|, for a positive constant C depending on y but not on |ζ_{i} −θ_{i}| [B1, Lemma 6]. On the other hand, the radius of convergence of exp(a((x − ζ_{i})^{− \frac{k}{μ!}} − (ζ_{i} − θ_{i})^{− \frac{k}{μ!}} ) is of the form C|ζ_{i} − θ_{i}|^{1+k}. So, for sufficiently small value of |ζ_{i} − θ_{i}|, these radii are all distinct, and the radius of convergence of y is the infimum of them. Formula (5.2.4.3) follows. We now prove that

(5.2.4.4)  M_{θ_{i}}[(x − θ_{i})^{α}, log(x − θ_{i})]_{α∈Δ} ∩ C′_{i} ⊂ M_{θ_{i}} .

This follows directly from Dwork’s lemma [Ad, Lemma 1].

From (5.2.4.2) we deduce A∩B = (∏_{i=0}^{r} M_{θ_{i}})∩B = O(X) and A_{i}^{∂} = C_{i}^{∂} = C. As for the solvability of (E, ∇) in A, it follows from the non-archimedean Turrittin theorem of [B1]. Q.E.D.

5.2.5. However we cannot reduce the case of a rational elementary fibration to the case of “P^{1} minus a few points” as in (3.4), for lack of having yet proven coherence and base change for R_{DR}^{i}f_{∗}^{an}(E^{an}, ∇^{an}). Nevertheless, since we are free to replace S by a dense affine open subset, the argument of (III.3.3.3) together with the use of Čech spectral sequences as above allows us to assume that (E, ∇) is cyclic of rank μ, with associated differential operator Λ ∈ O(X)[\frac{d}{dx}], and that the leading coefficients of the indicial polynomials of Λ are invertible in S. In particular, Ker_{Γ(X,E)}∇(\frac{d}{dx}) and Coker_{Γ(X,E)}∇(\frac{d}{dx}) are projective O(S)-modules of finite rank. We notice that all previous reductions could be carried over over K_{0}, and in particular the roots of the indicial polynomials belong to K_{0}. More generally, by Robba’s result reproduced in (I, App. A), all Turrittin exponents (at θ_{i} and at ∞) belong to K_{0}; we denote by Δ the Q-subspace of K_{0} which they generate (together with 1).

5.2.6. Let S ⊂ S^{an} be an affinoid subspace, and let X ⟶ S be the morphism obtained from f^{an} by base change. Since Δ does not contain any Liouville number, we are in the situation of (5.1.1). On the other hand, by the non-archimedean Turrittin theorem [B1], there is a Banach field K′ which is a finite extension of the completion of the fraction field of O(S) with respect to the spectral norm, and which enjoys the following property. If we set E = Γ(X, E) ⊗_{O(S)} O(S), B = O(X), and choose A ⊂ C as in (5.2.4), but with K replaced by K′ (so that A ∩ B ≅ O(X) ⊗_{O(S)} O(S) ⊂ O(X), as follows combining the Mittag-Leffler decomposition (5.1.3.1) and A ∩ O((SpecK′[x, \frac{1}{∏(x−θ_{i})}])^{an}) = K′[x, \frac{1}{∏(x−θ_{i})}]), then E is solvable in A.

Proposition 2.1 (and flatness of O(S) over O(S)) shows that the natural morphism

Ker_{Γ(X,E)}∇(\frac{d}{dx}) ⊗_{O(S)} O(S) ⟶ Ker_{Γ(X ,E^{an})}∇(\frac{d}{dx}) ,

(resp.

Coker_{Γ(X,E)}∇(\frac{d}{dx}) ⊗_{O(S)} O(S) ⟶ Coker_{Γ(X ,E^{an})}∇(\frac{d}{dx}) ) ,

is an isomorphism (resp. is injective). On the other hand, we know by (5.1.1) that the latter morphism is surjective. As in (3.5.4), we conclude that φ_{i} is an isomorphism for i = 0, 1 in this situation, and this achieves the proof of (5.2.1). Q.E.D.

Remarks 5.2.7. i) In proving the surjectivity of φ_{1}, we cannot apply (2.4), because the last condition d) of (2.4) is not fulfilled. ii) In the situation of (5.2.1), assume moreover that S is affine, f is a rational elementary fibration, (E, ∇) is cyclic outside some divisor which does not meet Z, and that the leading coefficients of the indicial polynomials at the branches of Z are units in O(S). Then our argument shows that the conclusion of (5.2.1) holds with U = S (cf. (5.1.5)).

5.3 Purity. We complement the previous analysis by the following purity statement.

Lemma 5.3.1. Let f : X ⟶ S be a rational elementary fibration, and let (E, ∇) be a coherent O_{X}-module endowed with an integrable connection. We assume that there is an open subset U ⊂ S such that S \ U is of codimension ≥ 2 in S and for any i ≥ 0, the restriction to U^{an} of the canonical morphism

φ_{i} : (R_{DR}^{i}f_{∗}(E, ∇))^{an} ⟶ R_{DR}^{i}f_{∗}^{an}(E^{an}, ∇^{an})

is an isomorphism of coherent objects in MIC(U^{an}). Then φ_{i} is in fact an isomorphism of coherent objects in MIC(S^{an}).

Proof. We construct the natural commutative square

X′ = X_{U} ↪^{j′} X
f′ ↓  ↓ f
U  ↪^{j} S

Because S\U is of codimension ≥ 2 and E is locally free of finite rank, Lütkebohmert’s theorem [Lü1] applies and we have

(E^{an}, ∇^{an}) ≅ j_{∗}^{′an}j^{′an∗}(E^{an}, ∇^{an}) .

(By Zariski localization, this amounts to the fact that j_{∗}^{′an}O_{X^{′an}} ≅ O_{X^{an}}). For i ≠ 0, 1, the cohomology modules vanish. The case i = 0 is easy:

R_{DR}^{0}f_{∗}^{′an}(j^{′an∗}(E^{an}, ∇^{an})) ≅ (R_{DR}^{0}f′_{∗}(j^{′∗}(E, ∇)))^{an}

and compatibility of the formation of De Rham cohomology with localization (III.2.6), (1.2.2), imply the isomorphism of coherent objects in MIC(S^{an})

\begin{matrix}R_{DR}^{0}f_{∗}^{an}(E^{an}, & ∇^{an}) ≅ R_{DR}^{0}f_{∗}^{an}(j_{∗}^{′an}j^{′an∗}(E^{an}, ∇^{an})) ≅ j_{∗}^{an}R_{DR}^{0}f_{∗}^{′an}(j^{′an∗}(E^{an}, ∇^{an})) \\ ≅ (j_{∗}R_{DR}^{0}f′_{∗}(j^{′∗}(E, ∇)))^{an} ≅ (R_{DR}^{0}f_{∗}(E, ∇))^{an} . \end{matrix}

Let us now consider the case i = 1. We first prove the coherence of R_{DR}^{1}f_{∗}^{an}(E^{an}, ∇^{an}). For it, the same argument as the one used in the algebraic context (III.4.9) works: we may reduce to the case where (E, ∇) is simple, and we have the alternative that

f^{an∗}j_{∗}^{an}R_{DR}^{0}f_{∗}^{′an}(j^{′an∗}(E^{an}, ∇^{an})) ≅ (f^{∗}j_{∗}R_{DR}^{0}f′_{∗}(j^{′∗}(E, ∇)))^{an}

is either (E^{an}, ∇^{an}) or 0. In both cases, the analytic version of the argument of loc. cit. applies, and shows in particular that

R_{DR}^{1}f_{∗}^{an}(E^{an}, ∇^{an}) ≅ j_{∗}^{an}j^{an∗}R_{DR}^{1}f_{∗}^{an}(E^{an}, ∇^{an}).

The composition of this isomorphism with φ_{1} can also be decomposed as

(R_{DR}^{1}f_{∗}(E, ∇))^{an} → (j_{∗}j^{∗}R_{DR}^{1}f_{∗}(E, ∇))^{an} → j_{∗}^{an}(j^{∗}R_{DR}^{1}f_{∗}(E, ∇))^{an} →

j_{∗}^{an}(R_{DR}^{1}f′_{∗}j^{′∗}((E, ∇)))^{an} → j_{∗}^{an}R_{DR}^{1}f_{∗}^{′an}j^{′an∗}(E^{an}, ∇^{an}) → j_{∗}^{an}j^{an∗}R_{DR}^{1}f_{∗}^{an}(E^{an}, ∇^{an}) .

Using our assumption, we see that all maps of this sequence are isomorphisms. Hence φ_{1} is an isomorphism. Q.E.D.

§6 Comparison theorem for algebraic vs. rigid-analytic cohomology (irregular coefficients).

In this section, we tackle the problem of comparing algebraic and rigidanalytic De Rham cohomologies with possibly irregular coefficients, i.e. of extending (5.2.1) to the case of higher dimensional morphisms. We prove the following statement, which was conjectured in [B2]:

Theorem 6.1. Let X be a smooth K_{0}-variety and let (E, ∇) be a coherent O_{X}- module with integrable connection (as before, K_{0} denotes an algebraically closed subfield of the non-archimedean complete algebraically closed field K which does not contain Liouville numbers, e.g. K_{0} = \overline{Q}). Then for any i ≥ 0,

H_{DR}^{i}(X, (E, ∇)) ⊗_{K_{0}} K ≅ H_{DR}^{i}(X_{K}^{an}, (E^{an}, ∇^{an})) .

Our proof extends the strategy used before in the regular case: combining (5.2.1) and dévissage (III.3).

6.2. Let f : X ⟶ S be a coordinatized elementary fibration, and let (E, ∇) be a coherent object of MIC(X). We saw in (III.4) that R_{DR}^{0}f_{∗}(E, ∇) is locally free of finite rank, and that there exists a dense open subset U ⊂ S, depending on (E, ∇), such that R_{DR}^{1}f_{∗}(E, ∇)_{|U} is locally free of finite rank (the other cohomology modules R_{DR}^{i}f_{∗}(E, ∇)_{|U} vanish). Since U ≠ S in general for irregular connections, this prevents us from using the first form of dévissage (III.3.2) in the study of R_{DR}^{i}f_{∗}(E, ∇)_{|U} for higher dimensional f. On the other hand, as was mentioned in (1.1.8), the analytic version of the second form of dévissage (III.3.3) is problematic. In order to tackle the problem of comparing algebraic and rigid-analytic De Rham cohomologies with irregular coefficients by an extension of our strategy, we thus need suitable refinements of the coherence results of (III.4).

More precisely, the proof of (6.1) relies upon the following two propositions. The first one is purely algebraic, and makes essential use of the results of (II) about irregularity in several variables:

Proposition 6.3. Let X be a smooth K_{0}-variety. Let (E, ∇) be a coherent object of MIC(X). Then there exists a finite open affine cover (U_{α}) of X and for each α, a coordinatized elementary fibration f_{α} : U_{α} ⟶ S_{α} such that for j = 0, 1, R_{DR}^{j}f_{α∗}((E, ∇)_{|U_{α}}) is a coherent O_{S_{α}}-module.

The second proposition is a variant of (5.2.1), (5.2.7.ii):

Proposition 6.4. In the situation of (6.3), one can choose moreover f_{α} in such a way that for j = 0, 1, the canonical morphism

R_{DR}^{j}f_{α∗}((E, ∇)_{|U_{α}})^{an} ⟶ R_{DR}^{j}f_{α∗}^{an}((E^{an}, ∇^{an})_{|U_{α}^{an}})

is an isomorphism in MIC(S_{α}^{an}).

6.5. Let us deduce (6.1) from (6.3) and (6.4). We consider a cover (U_{α}) as in (6.3) and (6.4). For \underline{α} = (α_{0}, …, α_{p}), α_{0} < … < α_{p}, we set U\underline{α} = U_{α_{0}} ∩…∩U_{α_{p}}. There is a natural morphism of Čech spectral sequences,

⊕_{α_{0}<...<α_{p}}H_{DR}^{i−p}(U\underline{α}, (E, ∇)) ⊗ K ⇒ H_{DR}^{i}(X, (E, ∇)) ⊗ K
↓ φ_{i−p}  ↓ φ_{i}
⊕_{α_{0}<...<α_{p}}H_{DR}^{i−p}(U_{\underline{α}K}^{an}, (E^{an}, ∇^{an})) ⇒ H_{DR}^{i}(X_{K}^{an}, (E^{an}, ∇^{an})).

By induction on i, we are left with proving the theorem for X = U_{α}. There is a natural morphism of Leray spectral sequences,

⊕_{0≤j≤i} H_{DR}^{i−j}(S_{α}, R_{DR}^{j}f_{α∗}((E, ∇)_{|U_{α}})) ⊗ K ⇒ H_{DR}^{i}(X, (E, ∇)) ⊗ K
↓ φ′_{i−j}
⊕_{0≤j≤i} H_{DR}^{i−j}(S_{αK}^{an}, (R_{DR}^{j}f_{α∗}((E, ∇)_{|U_{α}})^{an})  ↓ φ_{i}
↓ φ′′_{j}
⊕_{0≤j≤i} H_{DR}^{i−j}(S_{αK}^{an}, R_{DR}^{j}f_{α∗}^{an}((E^{an}, ∇^{an})_{|U_{α}^{an}}) ⇒ H_{DR}^{i}(X_{K}^{an}, (E^{an}, ∇^{an})).

By (6.4), we know that φ′′_{j} is an isomorphism. By (6.3) and induction on the dimension, we may assume that φ′_{i−j} is an isomorphism. Hence the right vertical arrow φ_{i} is an isomorphism. Q.E.D.

6.6 Proof of 6.3. By quasi-compactness, it suffices to find an open affine neighborhood V of an arbitrary closed point ξ ∈ X, and a coordinatized elementary fibration f : V ⟶ S such that R_{DR}^{j}f_{∗}((E, ∇)_{|V}) is a coherent O_{S}-module. We may assume that j = 1 (the coherence of R_{DR}^{0}f_{∗}((E, ∇)_{|V}) follows from (III.4.2.1)), and that X is affine of pure dimension d ≥ 2 (the cases d = 0, 1 are trivial). If we take any coordinatized elementary fibration as being constructed in (III.1.4), and try to apply the method of (III.4), we encounter two major obstacles:

i) in general, there is no cyclic vector in the neighborhood of the singular divisors, as was pointed out in (II.7.2.5),

ii) even if one happens to find such a cyclic vector, the leading coefficients of the indicial polynomials will not be units in O(S) in general. We shall overcome these difficulties by selecting V very carefully, and applying the method of (III.1.4) to a suitable modification of f, making full use of our study (II) of irregularity in several variables.

6.6.1. Let us consider again in detail the construction of a coordinatized elementary fibration as in (III.1.4, 1.5). It depends on the choice of:

i) a projective embedding \overline{X} ⟶ P^{N} of a projective normal closure of X, ii) a general linear space L ⊂ P^{N} of codimension d − 1 passing through ξ, iii) a general hyperplane H ⊂ P^{N}

and is obtained as a suitable restriction of the linear projection p : X ⊂ P^{N} ⇀ P^{d−1} with center L ∩ H. More precisely, let ϵ : ˜ \overline{X} → \overline{X} denote the blowing up with center L ∩ H ∩ \overline{X}, with exceptional divisor E, and let D denote the divisor \overline{X} \ X. Then according to loc. cit., there is an affine open neighborhood S of ζ := p(ξ) in P^{d−1}, a divisor F in ˜ \overline{X}_{|S} and a finite morphism \overline{π} : ˜ \overline{X}_{|S} ⟶ P_{S}^{1} such that the induced diagram

(6.6.1.1)  \begin{aligned}U = ˜ \overline{X}_{|S} \ Z & ↪^{j} & ˜\overline{X}_{|S} & ↩ & Z = (D ∐ E)_{|S} ∐ F \\ ↓ π & & ↓ \overline{π} & & ↓ π′ \\ U′ & ↪^{j′} & P_{S}^{1} & ↩ & Z′ \\ & ↘^{f′} & ↓^{pr=\overline{f}′} & ↙^{g′} \\ & & S \end{aligned}

pertains to a coordinatized elementary fibration (III.1.3), with f := p_{|U} = f′ ∘ π. Moreover the fiber of p_{| ˜\overline{X}} above ζ identifies with \overline{X} ∩ L. Note that D is independent of the choice of (L, H). Note also that \overline{π} is finite étale above π′(D ∐ E)_{|S} and that (\overline{π}^{−1}π′(D ∐ E)_{|S}) ∩ Z = (D ∐ E)_{|S} according to (III.1.5.c).

6.6.2 Locally free extensions. There exists a closed subset T ⊂ D of codimension ≥ 1 such that

i) D \ T = ∐ D_{i} , a disjoint union of smooth divisors contained in the smooth part of \overline{X},

ii) the O_{X}-module E extends to a locally free module over \overline{X} \ T. Since T has codimension ≥ 2 in \overline{X} and since L has codimension d − 1 in P^{N} and is sufficiently general (among those linear spaces passing through ξ), we have L ∩ T = ∅. This implies, after shrinking S if necessary, that D_{|S} = ∐ D_{i}_{|S}, and that E_{|U} extends to a locally free O_{˜\overline{X}|S}-module.

6.6.3 Constancy of Newton polygons. It follows from (II.5.3.1) that we might have chosen the closed subset T ⊂ D of codimension ≥ 1 in such a way that, moreover, for an affine neighborhood U′ of ζ avoiding π(T) and U = π^{−1}(U′),

iii) for any locally closed curve C which meets D transversally at some point Q ∈ D_{i}, there are equalities of Newton polygons:

NP_{Q}((E, ∇)_{|(C\{Q})}) = NP_{D_{i}}((E, ∇)_{|U}) ,
NP_{Q}(\underline{End}((E, ∇))_{|(C\\{Q\})}) = NP_{D_{i}}(\underline{End}(E, ∇)_{|U}) .

We remark, besides, that ∇_{|U} has no singularity at all along E_{|S} nor along F. We thus have the following property:

(6.6.3.1) for any component Z_{i} of the divisor Z and any point Q ∈ Z_{i}, there are equalities of Newton polygons:

NP_{Q}(E, ∇)_{f^{−1}(f(Q))} = NP_{Z_{i}}(E, ∇)_{|U}),
NP_{Q}(\underline{End}(E, ∇))_{f^{−1}(f(Q))} = NP_{Z_{i}}(\underline{End}(E, ∇)_{|U})).

We claim that this implies, in turn:

(6.6.3.2) for any component Z′_{i} of the divisor Z′ and any point Q′ ∈ Z′_{i}, there are equalities of Newton polygons:

NP_{Q′}(π_{∗}(E, ∇)_{f^{′−1}(f′(Q′))}) = NP_{Z′_{i}}(π_{∗}(E, ∇)_{|U′}),
NP_{Q′}(\underline{End}(π_{∗}(E, ∇))_{f^{′−1}(f′(Q′))}) = NP_{Z′_{i}}(\underline{End}(π_{∗}(E, ∇))_{|U′}).

This is clear if Q′ ∉ \overline{π}(D), because in this case Z′_{i} is a regular singularity for π_{∗}(E, ∇). We may thus assume that \overline{π}^{−1}(Q′) = \{Q_{1}, …, Q_{δ}\}, with δ = deg \overline{π}, Q_{1} ∈ Z_{i} ⊂ Z, Q_{j} ∉ Z if j > 1. We set P = \overline{f}(Q_{j}) = \overline{f′}(Q′). In order to justify the claim, we may also replace S by an étale covering, assume that the Z_{i} (resp. Z′_{i}) are images of disjoint sections of \overline{f} (resp. \overline{f}′), and that π^{−1}(Z′_{i}) = ∪_{h=1,...,δ}Z_{ih}, with Z_{i1} = Z_{i}, Z_{ih} ∩ Z = ∅ if h > 1. For j > 1, we then have NP_{Q_{j}}(E, ∇)_{f^{−1}(P)} = LQ_{μ}, the left quadrant with vertex (μ, 0) (or else the Newton polygon of the trivial connection of rank μ). We denote provisionally by M the generic fiber of the O_{f^{−1}(P)}-module E_{f^{−1}(P)} (viewed as a differential module over F = κ(f^{−1}(P))). We introduce the subfield G = K_{0}(x), endowed with the valuation w induced by Q′, and consider the extensions v_{j} (induced by Q_{j}) of w to F. We have (_{G}M)ˆ ≅ ⊕_{j} Ĝ(M ^{ˆ (v_{j})}) (I.3.1.3) and (\underline{End}(_{G}M))ˆ ≅ \underline{End}(⊕_{j} Ĝ(M ^{ˆ (v_{j})})) ≅ ⊕_{i,j}\underline{Hom}(Ĝ(M ^{ˆ (v_{i})}), Ĝ(M ^{ˆ (v_{j})})), whence

\begin{matrix}NP_{Q′}( & π_{∗}(E, ∇)_{f^{′−1}(P)}) = NP (_{G}M)ˆ = \\ NP Ĝ(M ^{ˆ (v_{1})}) + LQ_{(δ−1)μ} = NP_{Q_{1}}((E, ∇)_{f^{−1}(P)}) + LQ_{(δ−1)μ} \end{matrix}

according to (II.4.3), and

\begin{matrix}NP_{Q′}( & \underline{End}(π_{∗}(E, ∇))_{f^{′−1}(f′(Q′))}) = NP (\underline{End}(_{G}M))ˆ = \\ NP_{Q_{1}}(\underline{End}(E, ∇)_{f^{−1}(P)}) + (δ − 1)μNP_{Q_{1}}((E, ∇)_{f^{−1}(P)}) + LQ_{(δ−1)^{2}μ^{2}} . \end{matrix}

By a similar computation,

NP_{Z′_{i}}(π_{∗}(E, ∇)_{|U′}) = NP_{Z_{i}}(E, ∇) + LQ_{(δ−1)μ} ,

NP_{Z′_{i}}(\underline{End}(π_{∗}(E, ∇))_{|U′}) = NP_{Z_{i}}(\underline{End}(E, ∇))+(δ−1)μNP_{Z_{i}}((E, ∇))+LQ_{(δ−1)^{2}μ^{2}} , and one obtains (6.6.3.2) (from (6.6.3.1)) by comparison.

6.6.4 A Kummer covering. Let S_{1}→^{τ}S be an etale covering such that U_{1} := U′ ×_{S} S_{1} ≅ P_{S_{1}}^{1} \ ∐_{0,1,...,r} σ_{i}(S_{1}), where σ_{i} are sections of \overline{f}′ given by x = θ_{i}, θ_{i} ∈ O(S_{1}), θ_{i} − θ_{j} ∈ (O(S_{1}))^{×} for 0 ≠ i ≠ j ≠ 0, σ_{0}(S_{1}) = {∞} × S_{1}. We shall also denote by τ (abusively) the induced étale covering U_{1} ⟶ U′. In order to be able to apply our result of (II.7.2) on cyclic vectors, which requires a differential module with Turrittin exponents equal to 1, we shall pull back π_{∗}(E, ∇)_{|U′} to a suitable Kummer étale covering U_{2} of U_{1}.

For j = 1, …, r, we denote by L_{j} = P_{S_{1}}^{1} →^{λ_{j}}P_{S_{1}}^{1} the ramified covering of degree μ! given by x_{j} ↦ x = x_{j}^{μ!} + θ_{j} (totally ramified at σ_{j}(S_{1}) and at σ_{0}(S_{1}), unramified elsewhere). We blow up Π_{S_{1}}L_{j} = (P^{1})_{S_{1}}^{r} along the closed subvariety (∞, …, ∞)_{S_{1}}, and consider the strict transform \overline{U}_{2} of (L_{1}×_{P_{S_{1}}^{1}} L_{2} …×_{P_{S_{1}}^{1}} L_{r}) ⊂

(P^{1})_{S_{1}}^{r}. We denote by \overline{ϵ} : \overline{U}_{2} ⟶ P_{S_{1}}^{1} the canonical finite morphism. We claim that:

(6.6.4.1) \overline{U}_{2} is smooth over S_{1},

(6.6.4.2) for each j = 0, 1, …, r, \overline{ϵ}^{−1}(σ_{j}(S_{1})) is a union of smooth divisors Z_{jk}, flat over S_{1}, with multiplicity μ!,

(6.6.4.3) for each Z_{jk}, \overline{ϵ} induces a morphism of smooth models (\overline{U}_{2}, Z_{ik}) ⟶ (P_{S_{1}}^{1}, σ_{j}(S_{1})),

(6.6.4.4) the natural morphism f_{2} : U_{2} := \overline{U}_{2} \ ∐ Z_{ik} ⟶ S_{1} is an elementary fibration,

(6.6.4.5) the restriction of \overline{ϵ} : U_{2}→^{ϵ}U_{1} is a Galois étale covering with Galois group

G = Gal( κ(S_{1})(x, (x − θ_{1})^{\frac{1}{μ!}} , …, (x − θ_{r})^{\frac{1}{μ!}} )/κ(S_{1})(x) ) ≅ (Z/μ!Z)^{r}.

Indeed, since the branch loci of the λ_{j} do not intersect each other at finite distance, (6.6.4.2) and (6.6.4.3) are clear except for j = 0, and \overline{U}_{2} is smooth over S_{1} outside \overline{ϵ}^{−1}(σ_{0}(S_{1})) = \overline{ϵ}^{−1}(\{∞\} × S_{1}). At ∞, we use the coordinates y_{j} = 1/x_{j} on L_{j} and y = 1/x on P_{S_{1}}^{1}, so that y_{j}^{μ!} = \frac{y}{1−θ_{j}y}. There is an affine cover (V_{i})_{i=1,...,r} of a neighborhood of \overline{ϵ}^{−1}(y = 0) in \overline{U}_{2} such that \{y_{i}, t_{ij} := y_{j}/y_{i} (j = 1, …, r, j ≠ i) \} are local coordinates on V_{i}. We have y_{i}^{μ!} = \frac{y}{1−θ_{i}y} , t_{ij}^{μ!} = \frac{1−θ_{i}y}{1−θ_{j}y}; in particular, the t_{ij} are unramified at y = 0, and properties (6.6.4.1, 2, 3, 4) are now clear. Finally, (6.6.4.1) implies that \overline{U}_{2} is normal. It is just the normalization of P_{S_{1}}^{1} in the Kummer extension κ(S_{1})(x, (x − θ_{1})^{\frac{1}{μ!}} , …, (x − θ_{r})^{\frac{1}{μ!}} )/κ(S_{1})(x), which implies (6.6.4.5).

We need the following slight generalization of (III.1.5):

Lemma 6.6.5. Given an elementary fibration

(6.6.5.1)  \begin{aligned}X & ↪ & \overline{X} & ↩ & Z \\ & ↘^{f} & ↓ \overline{f} & ↙ \\ & & S \end{aligned}

and a finite set of closed points {ξ} ⊂ X, there is an open neighborhood U of {ξ} (resp. T of f({ξ})), such that f induces a coordinatized elementary fibration U ⟶ T.

Proof. If {ξ} has only one element, this is (III.1.5). If f({ξ}) has only one element, this is obtained by a trivial modification of the proof of loc.cit.. In general, let us choose a closed S-embedding \overline{X} ⟶ P_{S}^{M}. For each ζ ∈ f(\{ξ\}), we denote by \{ξ_{1}^{ζ}, …, ξ_{r}^{ζ}\} the finite set f^{−1}\{ζ\} ∪ Z_{ζ} of closed points of \overline{X}_{ζ}. There exists an open dense subset U^{ζ} of the Grassmannian of lines in the dual projective space P̌^{M} such that for any D_{ζ} ∈ U^{ζ}: (a)_{ζ} D_{ζ} is a Lefschetz pencil; in particular, the axis of D_{ζ} (which has codimension 2 in P^{M}) does not cut \overline{X}_{ζ}, so that D_{ζ} gives rise to a finite morphism \overline{π}_{ζ} : \overline{X}_{ζ} ⟶ P^{1} ≃ D_{ζ}; (b)_{ζ} if i ≠ j, \overline{π}_{ζ}(ξ_{i}^{ζ}) ≠ \overline{π}_{ζ}(ξ_{j}^{ζ}); (c)_{ζ} for each i, \overline{π}_{ζ} is étale at each point of \overline{π}_{ζ}^{−1}\overline{π}_{ζ}(ξ_{i}^{ζ}).

There exists a relative line D/S in the dual projective space P̌_{S}^{M} such that D_{ζ} ∈ U^{ζ} for each ζ. Up to changing the coordinate in P^{1} ≃ D_{ζ}, we may assume moreover that (d)_{ζ} the hyperplane H_{∞} corresponding to ∞ ∈ D_{ζ} cuts \overline{X}_{ζ} transversally, and does not meet the points ξ_{i}^{ζ}. We then finish the proof as in loc.cit.

6.6.6 Up and down. We now return to the situation of (6.6.4). Since we may shrink S around ζ = f(ξ) (i.e. S_{1} around τ^{−1}({ζ})), we can apply the previous lemma to the elementary fibration f_{2} : U_{2} ⟶ S_{1} of (6.6.4.4). We find a divisor F_{2} in \overline{U}_{2}, which is an étale covering of S_{1} and does not contain ϵ^{−1}τ^{−1}\{π(ξ)\}, such that the restriction of f_{2} to V_{2} := U_{2}\F_{2}⟶^{f_{2}}S_{1} is a coordinatized elementary fibration:

(6.6.6.1)  \begin{aligned}V_{2} & ↪^{j_{2}} & \overline{V}_{2} = \overline{U}_{2} & ↩ & (\overline{U}_{2} \ U_{2}) ∐ F_{3} \\ ↓ π_{2} & & ↓ \overline{π}_{2} & & ↓ π′_{2} \\ V′_{2} & ↪^{j′_{2}} & P_{S_{1}}^{1} & ↩ & Z′_{2} \\ & ↘^{f′_{2}} & ↓^{pr} & ↙^{g′_{2}} \\ & & S_{1} \end{aligned}

where, moreover, (6.6.6.2) \overline{π}_{2} is étale above π′_{2}(\overline{U}_{2} \ U_{2}).

We may also replace F_{2} by the union of its conjugates under the Galois group G, hence assume that

(6.6.6.3) V_{2} is a G-covering of V_{1} := U_{1} \ ϵ(F_{2}). Since we may replace S_{1} by any finite étale covering, we may assume that

(6.6.6.4) f′_{2} is a rational elementary fibration. The conditions that π(ξ) ∉ τ\overline{ϵ}(F_{2}) and that F_{2} is disjoint from (\overline{U}_{2} \ U_{2}) imply that F̃ := \overline{π}^{−1}τ\overline{ϵ}F_{2} ⊂ \overline{U} is a divisor in U which does not contain ξ, and is étale finite over S; in particular, V := U \ F̃⟶^{f}S is an elementary fibration. The commutative diagram

(6.6.6.5)  \begin{matrix}U & →^{π} & U′ & ←^{τ} & U_{1} = U′_{S_{1}} & ←^{ϵ} & U_{2} \\ ↓ f & & ↓ f′ & & ↓ f_{1} & & ↓ f_{2} \\ S & = & S & ←^{τ} & S_{1} & = & S_{1} \end{matrix}

where the horizontal maps are étale coverings and the vertical maps are elementary fibrations, induces by restriction a commutative diagram

V = U \ F̃ →^{π} V′ ←^{τ} V_{1} = V′_{S_{1}}  ←^{ϵ} V_{2} = (U_{2} \ F_{2})
↓ f  ↓ f′  ↓ f_{1}  ↓ f_{2}
S  = S ←^{τ}  S_{1}  =  S_{1}

with the same properties, which extends to a commutative diagram

(6.6.6.6)  \begin{matrix}V & →^{π} & V′ & ←^{τ} & V_{1} = V′_{S_{1}} & ←^{ϵ} & V_{2} & →^{π_{2}} & V′_{2} \\ ↓ f & & ↓ f′ & & ↓ f_{1} & & ↓ f_{2} & & ↓ f′_{2} \\ S & = & S & ←^{τ} & S_{1} & = & S_{1} & = & S_{1} \end{matrix}

(since there is more danger of inflation than confusion with notation at this point, we use abusively the same letter for various restrictions of the same morphism).

Let us set (E_{2}, ∇_{2}) := ϵ^{∗}τ^{∗}π_{∗}((E, ∇)_{|V}). We have:

(6.6.6.7) τ^{∗}R_{DR}^{1}f_{∗}((E, ∇)_{|V}) ≅ R_{DR}^{1}f_{1∗}(τ^{∗}π_{∗}(E, ∇)_{|V}) ≅ R_{DR}^{1}f_{1∗}(ϵ_{∗}(E_{2}, ∇_{2}))^{G}

(6.6.6.8) R_{DR}^{1}f_{1∗}(ϵ_{∗}(E_{2}, ∇_{2})) ≅ R_{DR}^{1}f′_{2∗}(π_{2∗}(E_{2}, ∇_{2})).

6.6.7 Finiteness. We claim that for this choice V of a neighborhood of ξ, R_{DR}^{1}f_{∗}((E, ∇)_{|V}) is a coherent O_{S}-module. In view of (6.6.6.7), (6.6.6.8), it suffices to show that R_{DR}^{1}f′_{2∗}(π_{2∗}(E_{2}, ∇_{2})) is a coherent O_{S_{1}}-module. Let us list a few properties of the latter.

(6.6.7.1) π_{2∗}E_{2} extends to a locally free O_{P_{S_{1}}^{1}} -module. This follows from the fact that E_{|U} extends to a locally free O_{˜\overline{X}|S}-module (6.6.3) and from the fact that all maps in the upper row of (6.6.6.6) extend to finite flat morphisms of relative projective completions.

(6.6.7.2) The Turrittin ramification indices of π_{2∗}∇_{2} are all 1. Indeed, by the ramication property (6.6.4.2), the Turrittin indices of ϵ^{∗}τ^{∗}π_{∗}((E, ∇)_{|U}) at every branch of \overline{U}_{2} \ U_{2} are 1, and this connection has no singularity at the branches of F_{2}; (6.6.7.2) then follows from (6.6.6.2).

Let us write P_{S_{1}}^{1} \V′_{2} = ∐ W_{i}, where the W_{i} are images of sections σ_{i}′ of f′_{2} = pr.

(6.6.7.3) For any component W_{i} and any point Q′_{2} ∈ W_{i}, there are equalities of Newton polygons:

NP_{Q′_{2}}(π_{2∗}(E_{2}, ∇_{2})_{f_{2}^{′−1}(f′_{2}(Q′_{2}))}) = NP_{W_{i}}(π_{2∗}(E_{2}, ∇_{2})),
NP_{Q′}(\underline{End}(π_{2∗}(E_{2}, ∇_{2}))_{f_{2}^{′−1}(f′_{2}(Q′_{2}))}) = NP_{W_{i}}(\underline{End}(π_{2∗}(E_{2}, ∇_{2}))).

This follows from a calculation completely analogous to the one in (6.6.4). More precisely, we may assume that Q′_{2} ∈ \overline{π}_{2}(\overline{U}_{2}\U_{2}) (other points are regular singular and easier to handle). Then there is a unique point Q_{2} ∈ \overline{U}_{2} \ U_{2} such that \overline{π}_{2}(Q_{2}) = Q′_{2}, and a unique point Q ∈ \overline{U} \ U such that \overline{π}(Q) = τϵ(Q_{2}). If we set δ_{2} = deg\overline{π}_{2}, and using (II.4.3), we find

\begin{matrix}NP_{Q′_{2}}(π_{2∗}(E_{2} & , ∇_{2})_{f_{2}^{′−1}(f′_{2}(Q′_{2}))}) = \\ φ_{1,μ!}NP_{Q}(E, ∇)_{f^{−1}f(P)} + LQ_{(δδ_{2}−1)μ} & = NP_{W_{i}}(π_{2∗}(E_{2}, ∇_{2})) , \end{matrix}

NP_{Q′}(\underline{End}(π_{2∗}(E_{2}, ∇_{2}))_{f_{2}^{′−1}(f′_{2}(Q′_{2}))}) = φ_{1,μ!}NP_{Q}(\underline{End}(E, ∇)_{f^{−1}f(Q)})
+ (δδ_{2} − 1)μNP_{Q}((E, ∇)_{f^{−1}f(Q)}) + LQ_{(δδ_{2}−1)^{2}μ^{2}}
= NP_{W_{i}}(\underline{End}(π_{2∗}(E_{2}, ∇_{2}))) .

These three properties now allow us to apply our result of (II.7.2.9) on the existence of cyclic vectors in neighborhoods of singularities: there exists a smooth K-variety S′_{1} and a flat morphism h : S′_{1} → S_{1} such that

(6.6.7.4) S′_{1} is finite over h(S′_{1}),

(6.6.7.5) S_{1} \ h(S′_{1}) has codimension ≥ 2 in S_{1},

(6.6.7.6) for any ζ′ ∈ S′_{1}, there is an affine neighborhood U′ of the finite subset {σ_{i}′(ζ′)} of P_{S′_{1}}^{1} such that the inverse image of π_{2∗}E_{2} on W ∩ (V′_{2} ×_{S_{1}} S′_{1}) is cyclic with respect to (π_{2∗}∇_{2})(d/dx). Let us fix such a cyclic vector e, and denote by Λ the associated differential polynomial.

Property (6.6.7.3) also allows us to apply (II.7.1.2), and obtain that, up to shrinking U′, the leading coefficient of the indicial polynomial of Λ at W_{i} ×_{S_{1}} S′_{1} is a unit in O(W_{i} ×_{S_{1}} S′_{1}) ∩ U′.

We can now conclude, by the method of (III.4) (see especially the remark III.4.7.iii) that h^{∗}R_{DR}^{1}f′_{2∗}(π_{2∗}(E_{2}, ∇_{2})) is coherent over O_{S′_{1}}. Taking into account the flatness of h and (6.6.7.5), the purity lemma (III.4.9) now shows that R_{DR}^{1}f′_{2∗}(π_{2∗}(E_{2}, ∇_{2})) is coherent over O_{S_{1}}, and we conclude finally that R_{DR}^{1}f_{∗}((E, ∇)_{|V}) is a coherent O_{S}-module. Q.E.D.

6.7 Proof of 6.4. We claim that, with the same V →^{f}S as in (6.6.6), the canonical morphism

(R_{DR}^{j}f_{∗}(E, ∇)_{|V})^{an} ⟶ R_{DR}^{j}f_{∗}^{an}((E^{an}, ∇^{an})_{|V^{an}})

is an isomorphism in MIC(S^{an}). This is trivial for j ≠ 0, 1, and the case j = 0 follows from the argument in (5.2.6). Let us now consider the case j = 1. In view of (6.6.6.7), (6.6.6.8) and by finite flat descent, it suffices to show that the map

(R_{DR}^{1}f′_{2∗}(π_{2∗}(E_{2}, ∇_{2})))^{an} ⟶ R_{DR}^{1}f_{2∗}^{′an}(π_{2∗}^{an}(E_{2}^{an}, ∇_{2}^{an}))

is an isomorphism in MIC(S_{1}^{an}). The argument in (5.2.6) also shows that this map is injective. In order to prove the surjectivity, let us consider again the flat morphism h of (6.6.7), and perform the cyclic reduction as above.

We can now conclude, by the method of (5.1), (5.2) (see especially the remark 5.2.7.ii) that

h^{an∗}(R_{DR}^{1}f′_{2∗}(π_{2∗}(E_{2}, ∇_{2})))^{an} ⟶ h^{an∗}R_{DR}^{1}f_{2∗}^{′an}(π_{2∗}^{an}(E_{2}^{an}, ∇_{2}^{an}))

is surjective - hence in fact an isomorphism of coherent objects in MIC(S_{1}^{′an}). Taking into account the flatness of h and (6.6.7.5), the purity lemma (5.3.1) now shows that

(R_{DR}^{1}f′_{2∗}(π_{2∗}(E_{2}, ∇_{2})))^{an} ⟶ R_{DR}^{1}f_{2∗}^{′an}(π_{2∗}^{an}(E_{2}^{an}, ∇_{2}^{an}))

is an isomorphism, and we conclude finally that

(R_{DR}^{1}f_{∗}(E, ∇)_{|V})^{an} ⟶ R_{DR}^{1}f_{∗}^{an}((E^{an}, ∇^{an})_{|V^{an}})

is an isomorphism in MIC(S^{an}).

This achieves the proof of (6.4) and of the main theorem (6.1). Q.E.D.

6.8 Properties of the GAGA functor.

In cohomological degree 0, the statement of (6.1) reduces to the following:

Corollary 6.8.1. Let f : X ⟶ S be a smooth morphism of smooth algebraic K_{0}-varieties, with S = A_{0}(f). Then for any coherent O_{X}-module with integrable connection (E, ∇), the natural morphism

(f_{∗}E^{∇_{|\underline{Der}(X/S)}}) ⊗_{O_{S}} O_{S^{an}} ⟶ f_{∗}^{an}(E^{an})^{∇_{|\underline{Der}_{cont}(X^{an}/S^{an})}^{an}}

is an isomorphism in MIC(S^{an}).

Simplified proof. Indeed, in this special case, one can provide a much shorter argument. With the help of lemma (III.4.2.1) and remark (III.3.2.4.ii), we may reduce by dévissage to the case when f is a coordinatized elementary fibration, or even a rational elementary fibration (arguing as in (3.3) or (4.3)). The result then follows by application of (2.1) exactly as in (5.2.6). Q.E.D.

Corollary 6.8.2. Let X be a smooth algebraic variety over an algebraically closed subfield K_{0} ⊂ K which does not contain any Liouville number. The functor

( \begin{matrix}Coherent O_{X}-modules \\ with integrable connection \end{matrix} ) ⟶ ( \begin{matrix}Coherent O_{X_{K}^{an}}-modules \\ with integrable connection \end{matrix} )
(E, ∇) ⟼ (E_{K}^{an}, ∇_{K}^{an})

is fully faithful.

Proof. Apply (6.8.1) to internal \underline{Hom} as in (3.6). Q.E.D.

Remark 6.8.3. This functor is however not essentially surjective, even for X = A^{1}. Indeed, let h be any non-polynomial entire function. Then there is no entire function g such that \frac{g′}{g} + h is a polynomial. This means that the rank one connection defined by ∇(\frac{d}{dx})e = he is not algebraizable.

This is in contrast with the complex-analytic situation, where the analogous functor is in fact essentially surjective, but not fully faithful.

Corollary 6.8.4. Let (E, ∇),(E′, ∇′) be coherent O_{X}-modules with integrable connections over X. Then the natural map

Ext_{MIC(X_{K})}^{1}((E_{K}, ∇_{K}), (E′_{K}, ∇′_{K})) ⟶ Ext_{MIC(X_{K}^{an})}^{1}((E_{K}^{an}, ∇_{K}^{an}), (E′_{K}^{an}, ∇′_{K}^{an}))

is a bijection.

Proof. Apply (6.1) with j = 1 to internal \underline{Hom} as in (3.7). Q.E.D.

§7 The relative non-archimedean Turrittin theorem.

Let X ⟶ S be a rational elementary fibration over an affine base S as in (5.1.0) and let (M, ∇) be a coherent object of MIC(X).

Proposition 7.1. We make the following technical assumption:

(∗) The horizontal side of the Newton polygon of End(M_{P}) does not depend on the point P ∈ Z, except possibly on a subset of codimension 2 in Z.

There exists a finite surjective morphism S′ ⟶ S and a finite set of elements a ∈ O(S′) such that, for any i = 0,…,r, the O(X)[\frac{d}{dx}]-module Γ(X, M) is solvable in

O(S′)[[x − θ_{i}]][log(x − θ_{i}), (x − θ_{i})^{α}, exp(a(x − θ_{i})^{− \frac{k}{μ!}} )]_{α∈Δ, k∈Z_{>0}} .

Proof. This follows from the formal decomposition of M along the divisor Z_{i} established in (II.6) (for i = 0, we tacitly replace θ_{i} by ∞).

7.2. Let S ⊂ S^{an} be an affinoid subspace, X ⟶ S be the morphism obtained from f^{an} by base change. The morphism S^{′an} ⟶ S^{an} is finite, hence induces a finite morphism of affinoid spaces S′ ⟶ S for any irreducible component S′ of the inverse image of S in S^{′an} [BGR, 7.1.2, 9.4.4].

We have a Mittag-Leffler decomposition (induced by (III.4.5.2))

(7.2.1)  O(X) = O(A_{S}^{1 an}) ⊕ ⨁_{i=1}^{r} \frac{1}{x − θ_{i}} O(P_{S}^{1 an} \ σ_{i}(S)) ,

where

O(A_{S}^{1 an}) = {∑_{n≥0} a_{n,∞}x^{n} ∈ O(S)[[x]] , lim_{n→∞} ||a_{n,∞}||^{1/n} = 0 }

O(P_{S}^{1 an} \ σ_{i}(S)) = \{∑_{n≥0} \frac{a_{n,i}}{(x − θ_{i})^{n}} ∈ O(S)[[\frac{1}{x − θ_{i}}]] , lim_{n→∞} ||a_{n,i}||^{1/n} = 0 \}

for some (any) Banach norm || || on the Tate K-algebra O(S), and similarly for S′ instead of S. For any ϵ > 0, we introduce the affinoid tube in S^{′an} × (A^{1})^{an}

T_{S′,i,ϵ} = {(s, x) , s ∈ S′ , |x − θ_{i}(s)| ≤ ϵ } ,

and

T_{S′,i,ϵ}^{∗} = {(s, x) , s ∈ S′ , 0 < |x − θ_{i}(s)| ≤ ϵ } .

We define

C′_{S′,i} = ⋃_{ϵ>0} O(T_{S′,i,ϵ}^{∗}),

A_{S′,i} = ⋃_{ϵ>0} O(T_{S′,i,ϵ})[log(x − θ_{i}), (x − θ_{i})^{α}, exp(a(x − θ_{i})^{− \frac{k}{μ!}} )]_{α∈Δ, k∈Z_{>0}}

As in (5.2.6), we consider the Banach field K′, completion of the fraction field of O(S′) with respect to the spectral norm, and let M_{θ_{i}}^{(K′)} be the field of meromorphic functions at θ_{i} on the analytic curve over K′, obtained by extension of X to K′. Then, as in (5.2.4), we show that

A′_{S′,i} := C′_{S′,i} ∩ A_{S′,i} ⊂ M_{θ_{i}}^{(K′)} .

This in turn implies

\begin{aligned}A′_{S′,i} & = ⋃_{ϵ>0} O(T_{S′,i,ϵ})[\frac{1}{x − θ_{i}}] = \\ & \{ ∑_{n>>−∞} a_{n,i}(x − θ_{i})^{n} ∈ O(S′)((x − θ_{i})), \underline{lim}_{n→∞}||a_{n,i}||^{−1/n} > 0\} . \end{aligned}

A_{S′} = ∏_{i=0}^{r} A_{S′,i} , C′_{S′} = ∏_{i=0}^{r} C′_{S′,i} , ∂ = \frac{d}{dx} .

We remark that

• A_{S′,i} is free over A′_{S′,i}, and the (x − θ_{i})-adic completion of A′_{S′,i} is O(S′)((x − θ_{i})); it follows that A_{S′} is faithfully flat over A_{S′}^{∂} = C_{S′}^{∂} = O(S′)^{r+1};

• O(X) embeds diagonally in an obvious way into C′_{S′}, and A_{S′} ∩ O(X) = O(S)[x, \frac{1}{∏(x−θ_{i})}] ≅ O(X)⊗_{O(S)} O(S), as follows from the Mittag-Leffler decom-

position (5.1.6.1).

The following result is a generalization of the p-adic Turrittin theorem [B1] in the relative case:

Theorem 7.3. Under assumption (∗), the O(X)[\frac{d}{dx}]-module Γ(X, M) is solvable in A_{S′}.

Proof. We have to show that Γ(X, M) is solvable in A_{S′,i}, for every i = 0, …, r. From (7.1), the problem becomes to show that all formal series ∈ O(S′)[[x − θ_{i}]] involved in the solutions converge, in the sense that they define elements of O(T_{S′,i,ϵ}) for ϵ small enough, i.e. are of the form

∑_{n≥0} a_{n,i}(x − θ_{i})^{n}, \underline{lim}_{n→∞}||a_{n,i}||^{−1/n} > 0.

As in [B1] (or (II.6.ii)), we remark that these series are formal solutions of Γ(X, \underline{End} M), and that the Turrittin exponents of \underline{End} M belong to Δ ⊂ K_{0}. Then the same argument of indicial polynomials as in loc. cit. (due to Clark [Cl]) allows us to conclude from the fact that K_{0} does not contain Liouville numbers. Q.E.D.

Remark 7.4. It is tempting to try to apply (2.4) with C = A_{S′}C′_{S′} instead of the differential ring C being considered in (5.2.6) in order to obtain the surjectivity of φ_{1} without using (5.1.1). However, a problem arises with the stability of the differential ring A_{S′,i} under integration: we have to consider, as in (5.2.4.1), formal expressions of the type y = u^{−1}z^{−α} e^{−a z^{−k}} ∫ u z^{α} e^{a z^{−k}}dz, with z = (x − θ_{i})^{\frac{1}{μ!}} , u ∈ A′_{S′,i} or C′_{S′,i}, α ∈ Δ, and remark that y satisfies a homogenous differential equation of second order, whose indicial polynomial ϕ_{z=0}(T) equals ak(T − k − 1) if ka ≠ 0. This would lead us to assume that any non-zero element of the additive monoid generated by the elements a ∈ O(S′) occuring in (7.1) are units.

Appendix E: Riemann’s existence problem: an elementary approach.

Theorem E.1. Let X be a smooth complex algebraic variety. Then the natural functor

{algebraic étale coverings of X}⟶^{(∗)}{topological unramified finite coverings of X(C)}

is an equivalence of categories.

The problem of essential surjectivity of this functor, sometimes called Riemann’s existence problem, was settled by Grauert and Remmert [GR] and subsequently by Grothendieck using Hironaka’s resolution of singularities.

In this appendix, we present an “elementary proof” by reduction to the one-dimensional case.

E.2. Recall that a topological unramified covering of X(C) admits a canonical structure of an analytic variety Y, endowed with a holomorphic étale morphism π : Y ⟶ X^{an}. In the finite case, we have equivalences of categories

{finite étale coverings π : Y ⟶ X^{an}} ⟶
{coherent O_{X^{an}}-modules with integrable connection (E, ∇), endowed with a

horizontal (commutative) O_{X^{an}}-algebra structure × : E^{⊗2} ⟶ E }

⟶ {étale O_{X^{an}}-algebras (E, ×), coherent as O_{X^{an}}-modules}

given by

π ⟼ (R_{DR}^{0}π_{∗}(O_{Y}, d), ×) = (π_{∗}O_{Y}, ∇, ×) ⟼ (π_{∗}O_{Y}, ×) ,

where × is deduced from the π^{−1}O_{X^{an}}-algebra structure of O_{Y}. These equivalences also take place in the algebraic categories:

{finite étale coverings π : Y ⟶ X} ⟶

{coherent O_{X}-modules with integrable regular connection (E, ∇), endowed with a

horizontal (commutative) O_{X}-algebra structure × : E^{⊗2} ⟶ E }

⟶ {finite étale O_{X}-algebras E}

They permit to deduce that the functor (∗) is fully faithful, from the comparison theorem 3.6. (But this fact is of course much more elementary, cf. [SGA 4, Exp. XI, Prop. 4.3. ii]).

E.3. We will need an “extension lemma” for étale coverings, which, thanks to the dictionary E.2, will be obtained as a special case of a general extension lemma for regular connections.

Extension Lemma E.4. Let X be a smooth complex algebraic variety, and let L be a local system of finite dimensional complex vector spaces on X(C). Assume that there is a dense open subset U ⊂ X and a regular connection (E_{U}, ∇_{U}) on U such that (E_{U}^{an})^{∇_{U}^{an}} ≅ L_{|U(C)}. Then there is a regular connection (E, ∇) on X such that (E^{an})^{∇^{an}} ≅ L and (E, ∇)_{|U} ≅ (E_{U}, ∇_{U}).

Proof. Let Z be the smooth, purely 1-codimensional part of X \ U. We denote by j : U′ := U ⨿ Z ↪ X the canonical immersion, and by (Ẽ_{U}, ˜∇_{U}) the τextension of (E_{U}, ∇_{U}) to U′ (I.4.7). Because (E_{U}^{an})^{∇_{U}^{an}} extends to a local system on U′(C) (even on X(C)), the local monodromy around Z is trivial. With the help of [De, II, 3.11], and taking into account the fact that τ(0) = 0, we deduce that Res_{Z} ˜∇_{U} = 0, hence ˜∇_{U} is a genuine connection on U′ (without pole at Z). We claim that (E, ∇) := j_{∗}(Ẽ_{U}, ˜∇_{U}) satisfies the requirements. Indeed, since X \ U′ has codimension ≤ 2, E is coherent, according to [EGA IV, 5.11.4], and the natural map j^{∗}(E, ∇) ⟶ (Ẽ_{U}, ˜∇_{U}) is an isomorphism (which restricts to an isomorphism (E, ∇)_{|U} ⟶ (E_{U}, ∇_{U}) over U). On the other hand, (E^{an})_{|U′(C)}^{∇^{an}} ≅

(Ẽ_{U}^{an}) ˜∇_{U}^{an} ≅ L_{|U′(C)}. Since (X \ U′)(C) has topological codimension ≤ 4 in X(C), π_{1}((X \ U′)(C), x) ⟶ π_{1}(X(C), x) is an isomorphism, and we conclude that (E^{an})^{∇^{an}} ≅ L. Q.E.D.

Extension lemma E.5. Assume that there is an étale dominant morphism X′⟶^{φ}X finite over its image, and an étale covering Y′ ⟶ X′, such that Y^{′an} and Y ×_{X^{an}} X^{′an} are isomorphic coverings of X^{′an}. Then there exists an étale covering Y ⟶ X such that Y^{an} and Y are isomorphic coverings of X^{an}.

Proof. Let us first assume that φ is surjective, hence an étale covering. Let Y′′/Y′ be an étale covering such that the induced covering Y′′/X is Galois with group G. Then by the usual Galois theory of topological coverings, there is a subgroup H ⊂ G such that Y ≅ Y^{′′an}/H. It then suffices to take Y := Y^{′′an}/H.

This first step allows us to assume that φ is an open dominant immersion X′ ↪ X. Let (R_{DR}^{0}π_{∗}(O_{Y}, d), ×) be the triple associated to π as in D.2. We know from the assumption that the restriction to X′ of this triple is algebraizable; in particular, R_{DR}^{0}π_{∗}(O_{Y}, d) comes from a regular connection on X′. In virtue of the extension lemma E.4, R_{DR}^{0}π_{∗}(O_{Y}, d) itself comes from a regular connection (E, ∇) on X. By the comparison theorem 3.6, the horizontal multiplication × : E^{an⊗2} ⟶ E^{an} comes from a horizontal O_{X}-linear map × : E^{⊗2} ⟶ E, which endows E with the structure of a finite O_{X}-algebra. It then suffices to take Y := SpecE. Q.E.D.

E.6. Let us prove the essential surjectivity of (∗) by induction on dimX, assuming the (elementary) case dimX = 1. According to E.5, we may replace X by any open dense subset. Hence we may assume that X is the total space of an elementary fibration

X ↪^{j} \overline{X} ↩ Z
↘^{f} ↓ \overline{f} ↙
S

Let K be an algebraically closed subfield of C of finite transcendence degree over Q, such that this elementary fibration comes from an elementary fibration

X_{K} ↪^{j_{K}}  \overline{X}_{K} ↩ Z_{K}
↘^{f_{K}} ↓ \overline{f}_{K} ↙
S_{K}

defined over K. Let s ∈ S(C) induce a geometric generic point of S_{K}, and let us denote by K′ the algebraic closure of the image of κ(S_{K}) in C (via the embedding given by s). We notice that the smooth projective curve \overline{X}_{s} comes from \overline{X}_{K} ×_{S_{K}} SpecK′ by extension of scalars K′ ↪ C.

E.7. In virtue of Riemann’s existence theorem in dimension 1, we know that the finite étale covering of X_{s}^{an} induced by Y is algebraizable, and in fact comes from a finite covering \overline{T} ⟶ \overline{X}_{s} unramified outside Z_{s} (inducing a finite étale covering T ⟶ X_{s}). By a standard argument of (Weil) descent, such a covering is already defined over K′, i.e. comes from a finite covering \overline{T}_{K′} ⟶ \overline{X}_{K} ×_{S_{K}} SpecK′, unramified outside Z_{K} ×_{S_{K}} SpecK′ (thus inducing an étale covering T_{K′} ⟶ X_{K} ×_{S_{K}} SpecK′).

Moreover, by “spreading-out”, one can find an étale morphism S′_{K} ⟶ S_{K}, finite over its image, an embedding κ(S′_{K}) ↪ K′, a finite étale morphism T′_{K} ⟶ X_{K}×_{S_{K}} S′_{K} and an isomorphism of coverings T_{K′} ⟶^{∼}T′_{K}×_{S′_{K}} SpecK′. Let s′ ∈ S′_{K}(C) be the point above s defined by the embedding κ(S′_{K}) ↪ K′ ↪ C. Then the fiber at s′ of the morphism T′_{K} ⟶ S′_{K} satisfies

T_{Ks′}^{′an} ≅ T^{an} ≅ Y_{s} ≅ (Y ×_{S^{an}} S^{′an})_{s′} .

E.8. According to E.5, we can again replace X by an étale dominant neighborhood finite over its image in X, for instance (X_{K} ×_{S_{K}} S′_{K}) ⊗_{K} C or a Galois covering of this space lying above T′_{K} ⊗_{K} C. This allows to assume that Y_{s} is a trivial covering of X_{s}. Because any elementary fibration is locally topologically trivial, we have a homotopy long exact sequence, a fragment of which being

π_{1}(X_{s}^{an}, x) ⟶ π_{1}(X^{an}, x) ⟶ π_{1}(S^{an}, s) ⟶ 1

for any x ∈ X(C) lying above s. We see that Y comes from an analytic étale covering of S^{an}, which is algebraizable by the induction hypothesis. Therefore Y is algebraizable. Q.E.D.

It would be interesting to give a similar proof of the p-adic avatar, namely Lütkebohmert’s theorem [Lü2].

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