FORMAL AND p-ADIC THEORY OF DIFFERENTIAL SYSTEMS WITH LOGARITHMIC SINGULARITIES DEPENDING UPON PARAMETERS

by

Francesco Baldassarri and Bruno Chiarellotto

Dedicated to Professors

R. Gérard and A.H.M. Levelt.

INTRODUCTION

The present article is divided naturally into two parts. The first, and longer, part extends and systematizes the beautiful work of Gérard and Levelt on differential modules with regular singularities over a ring of formal power series with coefficients in a field of characteristic zero. The second part is of a global (over a relative polydisk of radius 1) p-adic nature and was our motivation for considering the problems dealt with in the first part. While the second part is addressed to the limited public of “p-adic geometers”, we believe that the first part will be of interest to a more general audience including algebraic geometers, differential algebraists, and, possibly, complex analysts. We have tried to make the first part completely self-contained, limiting our prerequisites to Bourbaki’s Commutative Algebra. We hope to have succeeded in clarifying even the “constant” case considered by Gérard-Levelt [G-L] and Yoshida-Takano [Y-T]. The main need for reconsidering their work in a more systematic context lies in the fact that one is forced to study families of differential modules parametrized by smooth algebraic varieties or analytic spaces. This necessity depends on the local structure of general coefficients on any geometric non-singular space having logarithmic singularities along a divisor with normal crossings.

The results in the first part have essentially been known to a few specialists for a number of years, but we felt obliged to give a systematic presentation for lack of a readily available reference. In particular we were originally led to the present organization of this material by a communication received from Prof. N. Katz several years ago. In fact, already in section 2 of [Ba] we gave a concise and partial exposition of such results, which it has seemed opportune to extend and enlarge here.

This paper is in final form and no version of it will be submitted for publication elsewhere.

AMS Subject Classification:12H05,12H25.

The second part of this article provides essential mathematical tools for solving the problem of comparing algebraic and rigid cohomologies with coefficients in an overconvergent crystal with logarithmic singularities along a divisor with normal crossings. A rigorous setting for the problem and an explanation of the role played by the present work, may be found in the article [Ba-Ct3]. Let K denote a complete non-archimedean valued field with ring of integers V and residue field k. Let Y be a proper smooth V-scheme, Z be a divisor in Y with normal crossings relative to V, and X be the complementary open V-sub-scheme of Y. The previous V-objects have generic fibers Y_{K} = Y ×_{V} K, Z_{K}, X_{K} defined over K and special fibers \overline{Y} = Y ×_{V} k, \overline{Z}, \overline{X}, which are k-schemes. One can then construct the proper smooth rigid analytic space Y_{K}^{an}, its normal crossing divisor Z_{K}^{an}, and the complementary admissible open set X_{K}^{an}. We consider coefficients in the category of isocrystals on \overline{X}/K, overconvergent along \overline{Z} (cf. [Be, 2.3.2]), which are realized algebraically by locally free O_{X_{K}} -modules of finite type equipped with an integrable X_{K}/K-connection having logarithmic singularities along Z_{K}. We have long been interested in comparing the rigid cohomology of such a crystal with the algebraic cohomology of its algebraic realization. We refer to the introduction of [Ba-Ct3] for more details. The link between [Ba-Ct3] and the present paper depends on the topological considerations of section 4 of [Ba-Ct3]. On the rigid space Y_{K}^{an} there exists an admissible covering by open sets of the type

SpA × D_{K}^{s} = D_{A}^{s}

where SpA is a smooth affinoid space (A is a regular affinoid K-algebra) and D_{K}^{s} is the standard open polydisk of radius 1. In terms of the natural “vertical coordinates” x_{1}, …, x_{s}, the divisor Z_{K}^{an} traces on this set the analytic closed set x_{1} … x_{s} = 0. This explains why in order to understand cohomology with general coefficients of arithmetic varieties, it is of primary importance to first discuss the structure of differential modules on the space D_{A}^{s}. The aim of this article is precisely to examine those differential modules on D_{A}^{s} which are induced by overconvergent crystals in the sense of Berthelot [Be] and have logarithmic singularities along the divisor x_{1} … x_{s} = 0 (cf. §6). In the present article such differential modules will be studied as modules over the ring A{{x}} of global analytic functions on D_{A}^{s}, rather than as sheaves on D_{A}^{s}. In particular, the notion of overconvergence of [Be] is replaced by the condition of definition 5.1, which turns out to follow from that of Berthelot for the type of coefficients that are of interest to us here ([Ba-Ct3, sect.6]). This should render the present article, with the possible exception of section 4, comprehensible from an elementary algebraic point of view. We point out that, despite the similar appearance, the p-adic results of the present paper represent a considerable improvement upon those of section 3 of [Ba]. In that paper in fact we were only concerned with the local structure of the differential modules under consideration, in a small tubular neighborhood of SpA × {0} in D_{A}^{s}, while here our classification holds globally over D_{A}^{s}. The prototype of the results of the present paper is the beautiful “transfer theorem” of Christol [Ch2, Th.1].

The plan of the paper is as follows. (Notice that the first part parallels the exposition given in [G-L].) In section 1 we consider a field K of characteristic 0, a commutative unitary K-algebra A, and a K-Lie-sub-algebra L (and A-sub-module) of the K-Lie-algebra (and A-module) Der(A/K) of K-linear derivations of A. What we have in mind in that section is the case of a noetherian regular K-algebra A, and of L = Der(A/K); so, an (A, L)-module is a generalization of a projective A-module of finite type, equipped with an A/K-connection. This explains the title given to section 1. For commuting horizontal endomorphisms of an (A, L)-module we develop the analogue of Jordan theory for commuting endomorphisms of a finite-dimensional vector-space over K. In particular the notions of semisimple and nilpotent horizontal endomorphisms of an (A, L)-module play a central role here. Section 2 deals with the category of projective limits of (A, L)-modules in the sense of section 1. The theory of Jordan is here developed for commuting endomorphisms in that category, in terms of topologically semisimple and topologically nilpotent endomorphisms. In section 3 we deal with the ring A[[x]] of formal power series in the variables x_{1}, …, x_{s} with coefficients in an algebra A of the type considered in the previous sections. In particular, we have in mind a K-Lie-algebra and A-module L of derivations of A. We introduce the K-Lie-algebra and A[[x]]-module L_{x}, generated in Der(A[[x]]/K) by the elements of L (acting trivially on the x_{i}’s) and by the A-linear derivations x_{i}∂/∂x_{i}, for i = 1, …, s. We then apply the results of the previous sections to the classification of the (A[[x]], L_{x})-modules which are projective modules of finite type over the ring A[[x]]. The main result is corollary 3.6.7, which shows that the modules under consideration are iterated extensions of certain elementary objects, which we call simple. This result is theorem 2.9 of [Ba]. As we said before, we felt that this result deserved more attention than it had been given before, and that its proof needed to be cast in a clear elementary and general setting. We hope to have now succeded in that intention. Section 4 deals with our notion of differential modules on a p-adic open polydisk of radius 1, D_{A}^{s}, as before, over the smooth affinoid space SpA, with logarithmic singularities along the coordinate divisor x_{1} … x_{s} = 0. We are forced to restrict our considerations to O_{D_{A}^{s}}-modules which are pull-backs to D_{A}^{s} of locally free O_{A}-modules of finite type, to avoid the difficulties of rigid K-theory. We are indebted to S. Bosch for warning us of those problems. Sheaves of the latter type are however equivalent to their modules of global sections, and this simplifies our work considerably. This also explains the need of introducing such apparently clumsy categories as Mod_{A}^{f}(A{{x}}, L_{A,x}) or Mod_{A,Σ}^{f}(A{{x}}, L_{A,x}) where the index A is supposed to remind one that the A{{x}}-modules under consideration are extensions to A{{x}} of (projective) A-modules (of finite type). The index Σ refers to the fact that the multiexponents of our differential modules are required, throughout the second part of this paper, to have entries in a given additive subgroup Σ of the field K consisting of p-adically non-Liouville numbers (cf. [Ba-Ct0, 1.2]) and containing the integers. In section 5 we examine the notion of overconvergence. We have already justified our definition 5.1. In this section we prove (Th. 5.6) that the full subcategory SCMod_{A}^{f}(A{{x}}, L_{A,x}) of Mod_{A}^{f}(A{{x}}, L_{A,x}) consisting of the overconvergent objects is stable under the bifunctors Hom_{A{{x}}}(−, −) and − ⊗_{A{{x}}} −. We also prove (Prop. 5.3) the local nature of overconvergence w.r.t. the strong Grothendieck topology of SpA. The main result of this section is however theorem 5.9, which we regard as a satisfactory generalization of previous results on p-adic convergence of formal power-series solutions of regular systems of P.D.E.’s (cf. [Ba-Ct0]). Finally, in section 6 we examine the structure of the objects of SCMod_{A,Σ}^{f}(A{{x}}, L_{A,x}). The p-adic results of this section are perfectly analogous to the formal results of section 3: every object of SCMod_{A,Σ}^{f}(A{{x}}, L_{A,x}) is an iterated extension of simple ones (corollary 6.5.2). We emphasize the improvement upon similar results in [Ba]: once again our present statements are global over D_{A}^{s}, while those of [Ba] were valid in a small tubular neighborhood of SpA × {0} in D_{A}^{s}. Our results are cast in a form suitable for direct application to [Ba-Ct3].

Aknowledgements. We are indebted to G. Gerotto for his careful reading of the manuscript and his helpful criticism. We wish to thank F. Sullivan for his assistance in clarifying the presentation of our results.

PART I: FORMAL THEORY.

§1 Differential modules without singularities.

1.1 Notation.

We introduce some notation: By |−|_{R} we will denote the archimedean absolute value in R, and also the norm: |α|_{R} = ∑_{i=1}^{s} |α_{i}|_{R}, for α ∈ R^{s}, any s. For any ring R, left R-modules will always be assumed to be unitary. K= a field of characteristic zero; A = a commutative K-algebra. For an A-module M, we will regard End_{K}(M) as an A-module via the position (af)(m) = a(f(m)), for a ∈ A, f ∈ End_{K}(M), m ∈ M. Der(A/K)= the set of K-linear derivations of A, regarded as an A-sub-module and a Lie sub-algebra over K (bracket [ , ]) of End_{K}A. If l, l′ ∈ Der(A/K) and a ∈ A we have

(1.1.1)  [al, l′] = a[l, l′] − l′(a)l .

L = a Lie sub-algebra over K and A-sub-module of Der(A/K). We put: C = A^{L} = {a ∈ A | l(a) = 0 ∀l ∈ L} , a (commutative) K-sub-algebra of A.

Definition 1.1.2. An (A, L)-module is an A-module M with an action L × M ⟶ M, (l, m) ⟼ l(m) satisfying:

(i)  [l, l′](m) = l(l′(m)) − l′(l(m))

(ii)  (al)(m) = a(l(m))

(iii)  l(am) = a(l(m)) + l(a)m

for l, l′ ∈ L, m ∈ M, and a ∈ A. Warning 1.1.2.1. Sometimes in the present paper L will denote a Lie subalgebra over K of Der(A/K) which may not be an A-sub-module of Der(A/K). When that is the case, we mean to replace L by AL, the A-sub-module of Der(A/K) generated by L, which is still a Lie sub-algebra over K of Der(A/K). In that situation therefore “(A, L)-module” will be synonymous of “(A, AL)- module”.

Definition 1.1.3. We regard the A-module A ⊕ L as a Lie algebra over K via the bracket:

[a_{1} ⊕ l_{1}, a_{2} ⊕ l_{2}] = (l_{1}(a_{2}) − l_{2}(a_{1})) ⊕ [l_{1}, l_{2}]

for any a_{1}, a_{2} ∈ A, l_{1}, l_{2} ∈ L .

Remark 1.1.4. Notice that

\begin{matrix}α_{0} : A & ⟶ A ⊕ L \\ a & ⟼ a ⊕ 0 , \end{matrix}

\begin{matrix}β_{0} : L ⟶ A ⊕ L \\ l ⟼ 0 ⊕ l , \end{matrix}

\begin{matrix}γ_{0} : A ⊕ L ⟶ L \\ a ⊕ l ⟼ l , \end{matrix}

are morphisms of Lie algebras over K (the Lie algebra A is commutative). We will regard the first two maps as imbeddings, and consequently A as a bilateral ideal of A⊕L. We also have a natural imbedding of Lie algebras over K and A-modules A ⊕ L ↪ End_{K}(A), where a ∈ A is identified with the map multiplication by a in A.

Lemma 1.1.5. The category of (A, L)-modules is equivalent to the one of Amodules M equipped with a morphism φ_{M} = φ of Lie algebras over K and of A-modules

A ⊕ L⟶^{φ}End_{K}M

such that φ_{M}(1_{A}) = id_{M}, the identity map of M.

Proof. Obvious. Q.E.D.

For a Lie algebra A over K, we will indicate by U_{K}(A) = T_{K}(A)/I its universal enveloping algebra and by {U_{K}^{(n)}(A)}_{n∈N} the natural filtration on U_{K}(A) ([Se, Chap.III]). We recall that T_{K}(A) denotes the tensor algebra of the K-vector space A, while I = two sided ideal of T_{K}(A) generated by [m, m′] − m ⊗ m′+ m′ ⊗ m, for any m, m′ ∈ A. If A is commutative, U_{K}(A) = S_{K}(A), the symmetric algebra of the K-module A. There is a canonical morphism of Lie algebras over K, ϵ = ϵ_{A} : A → U_{K}(A). In the special case of A = A ⊕ L, from the maps α_{0}, β_{0}, γ_{0} of remark 1.1.4, we get canonical morphisms of associative unitary K-algebras:

U_{K}(α_{0}) : U_{K}(A) = S_{K}(A) ⟶ U_{K}(A ⊕ L) ,
U_{K}(β_{0}) : U_{K}(L) ⟶ U_{K}(A ⊕ L) ,
U_{K}(γ_{0}) : U_{K}(A ⊕ L) ⟶ U_{K}(L) .

We will regard U_{K}(A ⊕ L) as an S_{K}(A)-module by identifying s ∈ S_{K}(A) with the map left multiplication by U_{K}(α_{0})(s) in U_{K}(A ⊕ L). We have a natural morphism of K-algebras

\begin{matrix}μ : S_{K}(A) & ⟶ A \\ a_{1} ⊗ ⋯ ⊗ a_{n} & ⟼ a_{1} ⋯ a_{n} . \end{matrix}

We denote by \overline{⊗} the product in U_{K}(A ⊕ L) and consider the bilateral ideal P of U_{K}(A ⊕ L) generated by the elements of the form

a\overline{⊗}l − al ,

for a ∈ A and l ∈ L. In the following definition we denote by \overline{U}_{K}^{(n)}(L) the S_{K}(A)-submodule of U_{K}(A ⊕ L) generated by the image via U_{K}(β_{0}) of U_{K}^{(n)}(L) in U_{K}(A ⊕ L).

Definition 1.1.6. We define a ring and A-module D(A, L) by

D(A, L) = A ⊗_{S_{K}(A)} (U_{K}(A ⊕ L)/P) .

There are natural unitary morphisms of rings

α : A ⟶ D(A, L) , β : U_{K}(L) ⟶ D(A, L) ,

induced by U_{K}(α_{0}) and U_{K}(β_{0}), respectively. The ring D(A, L) is the composite of the two subrings α(A) and β(U_{K}(L)). We have

(1.1.6.1)  β(al) = (αa)(βl) , (βl)(αa) = αl(a) + β(al) ,

for any a ∈ A and l ∈ L. The inclusion of A⊕L in End_{K}(A), produces a natural morphism of associative unitary K-algebras and A-modules

ι = ι_{(A,L)} : D(A, L) → End_{K}(A) .

We define a filtration by A-modules on D(A, L) by posing

D^{(n)}(A, L) = Im((\overline{U}_{K}^{(n)}(L) + P) → D(A, L)) .

For any m, n ∈ N, D^{(n)}(A, L) is an A-sub-module of D(A, L), D^{(0)}(A, L) = A and

D^{(m)}(A, L)D^{(n)}(A, L) ⊂ D^{(m+n)}(A, L) .

The associated graded ring gr^{·}(D(A, L)) of D(A, L) is easily seen to be commutative. The natural map

βϵ_{L} : L ⟶ D(A, L)

induces an A-linear map L ⟶ gr^{·}(D(A, L)) and therefore a map

(1.1.6.2)  S_{A}(L) ⟶ gr^{·}(D(A, L)) .

The map (1.1.6.2) is easily seen to be surjective. The canonical morphism ι will be understood in the sequel: for L ∈ D(A, L) and a ∈ A, the expression L(a) is used to denote ι_{(A,L)}(L)(a) ∈ A. By La we will instead denote the product of L and a in D(A, L). The ring D(A, L) is then equipped with a natural augmentation

(1.1.6.3)  \begin{matrix}κ = κ_{(A,L)} : D(A, L) & ⟶ A \\ L & ⟼ L(1_{A}) , \end{matrix}

which is a morphism of A-modules. The kernel D^{+}(A, L) of the augmentation map κ_{(A,L)} is the left ideal of D(A, L) spanned by L. We have:

(1.1.6.4)  D(A, L) = A ⊕ D^{+}(A, L) .

Remark 1.1.6.4. Aside from the well-known case of a regular K-algebra of finite type A, and of L = Der(A/K), we are unaware of general conditions that guarantee that (1.1.6.2) is an isomorphism (cf. [Se, Th.4.3]). We would likewise be interested in knowing under which conditions the canonical morphism ι_{(A,L)} is injective.

Proposition 1.1.7. If A is a noetherian ring and L is an A-module of finite type, D(A, L) is a left and right noetherian ring.

Proof. The commutative A-algebra gr^{·}(D(A, L)) is a homomorphic image of the symmetric algebra of the A-module L, hence it is noetherian. The result then follows from [Bj, Chap. 2, Prop. 6.1] Q.E.D.

Let Mod(A, L) indicate the category whose objects are (A, L)-modules and whose morphisms are morphisms of A-modules ϕ : M ⟶ N , such that ϕ(l(m)) = l(ϕ(m)) , for l ∈ L and m ∈ M. An (A, L)-module will be said to be of finite type if it is so as an A-module. We will denote by Mod^{f}(A, L) the full sub-category of Mod(A, L) consisting of objects of finite type. To a triple (A, L, K) as before we associate the left A-algebra D = D(A, L); for any extension field L of K we put: A_{L} = L ⊗_{K} A, L_{L} = L ⊗_{K} L ⊂ Der(A_{L}/L), D_{L} = D(A_{L}, L_{L}).

1.2. Let (A, L, K, C, D) be as in 1.1.

Lemma 1.2.1. Let M be a K-module. Then the following structures on M are equivalent:

a) a structure of left D-module;

b) a structure of (A, L)-module.

Proof. Obvious. Q.E.D.

Corollary 1.2.2. The category Mod(A, L) is isomorphic to the category of left D-modules. In particular, Mod(A, L) is an abelian category. If A is a noetherian ring, the category Mod^{f}(A, L) is also abelian.

For an (A, L)-module M, m ∈ M and L ∈ D, we will write indifferently Lm or L(m), meaning the action of L on m, via the left D-module structure of M. This applies in particular to the case of L ∈ L. When M = A, however, this may cause confusion, as we pointed out in the lines following definition 1.1.6: in this special case we will follow the convention of loc. cit.. It is well known how to define in Mod(A, L) (resp. Mod^{f}(A, L)) the internal bi-functors − ⊗_{A} − and Hom_{A}(−, −), as well as the functors: X ⟼ X^{∨} = Hom_{A}(X, A) and X ⟼ ∧_{A}^{p}X for every p ∈ N and X = an (A, L)-module. If M, N are (A, L)-modules, we will denote by Hom_{(A,L)}(M, N) the abelian group of morphisms between M and N in Mod(A, L). The ring A is canonically an object of Mod^{f}(A, L), via the natural action ((l, a) ⟼ l(a), for l ∈ L and a ∈ A) of L on A. We will put :

M^{∇} = {m ∈ M | L(m) = (κ(L))m ∀L ∈ D}

(cf. 1.1.6.2). Since the kernel of κ is the left ideal generated by L, it follows that

M^{∇} = {m ∈ M | l(m) = 0, ∀l ∈ L}.

In particular for A = M we have: A^{∇} = A^{L} = C.

1.3. The following hypotheses on (A, L, K) will play an important role in the sequel:

(1.3.i) A^{L} = K.

(1.3.ii) Every (A, L)-module of finite type P is a projective A-module.

1.4. We first recall the following (easy generalization of a) result of N.Katz:

Proposition 1.4.1. Let K be a field of characteristic 0 and let A be a noetherian commutative K-algebra. We put B = A[[x_{1}, …, x_{n}]], L = ∑_{i=1}^{n} B \frac{∂}{∂x_{i}} and D = D(B, L) = B[\frac{∂}{∂x_{1}} , …, \frac{∂}{∂x_{n}}] ⊂ End_{A}B. Let M be a left D-module of finite type as a B-module. Assume M/(x_{1}, …, x_{n})M is a projective A-module. Then M^{∇} is an A-sub-module of M, the natural map B ⊗_{A} M^{∇} ⟶ M is an isomorphism of B-modules and

(1.4.2)  L(b ⊗ m) = L(b) ⊗ m

if b ∈ B, m ∈ M^{∇} and L ∈ D. In particular, M is a projective B-module.

Proof. There exist elements f_{1}, … f_{r} ∈ A such that (f_{1}, … f_{r}) = A and (M/(x_{1}, …, x_{n})M) ⊗_{A} A_{f_{i}} is a free A_{f_{i}}-module, for i = 1, …, r. Let B̂_{f_{i}} (resp. M̂_{f_{i}}) denote the (x_{1}, …, x_{n})-adic completion of the fraction ring B_{f_{i}} (resp. M ⊗_{B} B̂_{f_{i}}); let D_{i} denote the ring D(B̂_{f_{i}}, L). Then M̂_{f_{i}} is a left D_{i}-module of finite type as a B̂_{f_{i}}-module. To prove that B ⊗_{A} M^{∇} ⟶ M is an isomorphism of B-modules, it is sufficient to prove that for each i = 1, …, r the natural map B̂_{f_{i}} ⊗_{A_{f_{i}}} M̂_{f_{i}}^{∇} ⟶ M̂_{f_{i}} is an isomorphism of B̂_{f_{i}}-modules (cf. section 3.2 below). In other words the assumption “M/(x_{1}, …, x_{n})M is a projective A-module (of finite type)” may be replaced by “M/(x_{1}, …, x_{n})M is a free A-module (of finite type)”. We can then follow step by step the proof of proposition 8.9 of [Ka]. Q.E.D.

We now give an example of a class of triples (A, L, K) as in (1.1) which satisfy hypotheses (1.3).

Definition 1.4.3. Let K and A be as in (1.1). We will say that A is regular if for each maximal ideal M ∈ Max(A), the local ring A_{M} is regular.

Proposition 1.4.4. Let (A, L, K) be as in (1.1). Let us assume that A is regular and that the triple (A, L, K) satisfies the following hypothesis: for each maximal ideal M of A, L generates over the M-completion Â_{M} of A_{M}, the left Â_{M}-module Der^{c}(Â_{M}/K) of M-adically continuous K-derivations of Â_{M}. Then the triple (A, L, K) satisfies conditions (1.3).

Proof. Condition (1.3.i) follows from the fact that if M ∈ MaxA then Â_{M} ≅ K[[t_{1}, …, t_{n}]], where (t_{1}, …, t_{n}) is a regular system of parameters of A_{M} and (Â_{M})^{L} = K. Thus A^{L} = K. This proves (1.3.i).

To prove (1.3.ii), let us consider a left D-module M of finite type as an A-module. We need to show that M is a projective A-module, i.e. that for each maximal ideal M of A, M_{M} is a free A_{M}-module (of finite type). But A_{M} is a Zariski ring for the M-adic topology [AC, Chap.III, 3.3.6] and therefore [loc. cit., 3.5.9] Â_{M} is faithfully flat over A_{M}. We are then reduced to the situation of (1.4.1).Q.E.D.

Proposition 1.4.5. Let (A, L, K) be a triple which satisfies the hypotheses (1.3), with A noetherian. Let B = A[[x_{1}, …, x_{n}]], and let us extend the elements of L to (x_{1}, …, x_{n})-adically continuous K[[x_{1}, …, x_{n}]]-linear derivations of B. Let L_{x} denote the B-sub-module of End_{K}B generated by L and the A-linear derivations \frac{∂}{∂x_{i}}, for i = 1, …, n. Then the triple (B, L_{x}, K) satisfies the assumptions of (1.3). If, in particular, (A, L, K) satisfies the hypotheses of proposition 1.4.4, so does (B, L_{x}, K).

Proof. The first assertion is a corollary of proposition 1.4.1; the second is obvious. Q.E.D.

1.4.6. Examples. We give a list of K-algebras A and Lie algebras L satisfying the assumptions of proposition 1.4.4.

1.4.6.1. A = the affine ring of a smooth affine algebraic variety defined over K, L = Der(A/K).

1.4.6.2. Let K = C be the field of complex numbers and (S, O_{S}) be a Stein manifold. Then we may take: A = O_{S}(S), L = the O_{S}(S)-module of global holomorphic vector fields on S.

1.4.6.3. We now regard K as a discrete topological ring. Let S = SpfA be a smooth affine formal scheme over K. Then A is a topological K-algebra and we may take L = Der^{c}(A/K), the A-module of continuous derivations of A/K.

1.4.6.4. Let K be a field of characteristic zero complete in a non-archimedean absolute value. Let A be a regular affinoid K-algebra ([BBKN]), endowed with the action of its continuous derivations. We will deal with this case later.

1.5. In what follows we keep (K, A, L, D) fixed and assume hypotheses (1.1) and (1.3).

Lemma 1.5.1. Let M be an object in Mod^{f}(A, L) and N be a sub-object of M in Mod(A, L). There exists an A-sub-module N′ of M such that M = N ⊕ N′ as A-modules. In particular, N is an A-module of finite type, hence it is a sub-object of M in Mod^{f}(A, L).

Proof. The quotient M/N in Mod(A, L) is an A-module of finite type, hence by (1.3.ii) it is projective. The canonical projection M → M/N in Mod(A, L) therefore admits an A-linear section. Q.E.D.

Corollary 1.5.2. Mod^{f}(A, L) is a full and thick abelian subcategory of Mod(A, L).

Lemma 1.5.3. Let M be an object of Mod^{f}(A, L) and φ an endomorphism in End_{(A,L)}M. Then φ satisfies a non-zero polynomial with coefficients in K.

Proof. The A-module M is projective of finite type hence locally free of finite type. Let f_{1}, …, f_{s} be generators of the trivial ideal A, such that M_{f_{i}} is free of finite type over A_{f_{i}} for i = 1, …, s. Then SpecA = ∪_{i=1}^{s}SpecA_{f_{i}} and it is enough to show that, for i = 1, …, s, there exists P_{i}(x) ∈ K[x] \ {0}, such that φ ⊗ 1_{A_{f_{i}}} ∈ End_{A_{f_{i}}} M_{f_{i}} satisfies P_{i}(φ ⊗ 1_{A_{f_{i}}}) = 0. In fact, if we then put P (x) = ∏_{i=1}^{s} P_{i}(x), we will have: P (φ ⊗ 1_{A_{f_{i}}}) = P (φ) ⊗ 1_{A_{f_{i}}} = 0 for i = 1, …, s; hence P (φ) = 0. So, let us suppose M ≅ A^{n} as an A-module. Let t indicate an indeterminate over K, and let us define K_{t} = K(t), A_{t} = A_{K(t)}, L_{t} = L_{K(t)} ⊂ Der(A_{t}/K_{t}) and D_{t} = D_{K_{t}} . The triple (A_{t}, L_{t}, K_{t}) still satisfies the hypotheses of (1.1) and (1.3). Consider M_{t} = A_{t} ⊗_{A} M, which is an object of Mod(A_{t}, L_{t}), and φ_{t} = 1_{A_{t}} ⊗_{A} φ ∈ End_{(A_{t},L_{t})}M_{t}. The map t·id_{M_{t}} : x ⟼ tx is an element of End_{(A_{t},L_{t})}M_{t}. We then define:

ψ_{t} = ∧_{A_{t}}^{n}(φ_{t} − t · id_{M_{t}}) : ∧_{A_{t}}^{n}M_{t} = N_{t} ⟶ N_{t},

an element of End_{(A_{t},L_{t})}N_{t} . For n ∈ N_{t}, ψ_{t}(n) = det(φ − t · id_{M})n = P (t)n, where P (t) is the characteristic polynomial of φ which is, a priori, an element of A[t]. But under the hypothesis (1.1) for (A, L), N_{t} is an object of Mod(A_{t}, L_{t}) free of rank one as an A_{t}-module and ψ_{t} ∈ End_{(A_{t},L_{t})}N_{t}. We get: ∂ψ_{t}n = ψ_{t}∂n for any n ∈ N_{t}, ∂ ∈ L. Thus there exists an element a_{ψ_{t}} of A_{t} such that for any n ∈ N_{t}:

ψ_{t}n = a_{ψ_{t}}n

and

∂ψ_{t}n = ∂(a_{ψ_{t}})n + a_{ψ_{t}}∂n = ψ_{t}(∂n) = a_{ψ_{t}}∂n.

As a consequence, we obtain that a_{ψ_{t}} ∈ A_{t}^{L_{t}} and P (t) ∈ A_{t}^{L_{t}} = K(t). It follows that P (t) ∈ A[t] ∩ K(t) = K[t], P (φ) = 0. Q.E.D.

Definition 1.5.4. In the situation of Lemma 1.5.3, we define the minimal polynomial of φ as the monic generator of the ideal of K[t] consisting of the polynomials P (t) ∈ K[t] such that P (φ) = 0.

1.6.

Definition 1.6.1. Let M be an object of Mod^{f}(A, L) and φ ∈ End_{(A,L)}M. We will say that φ is semisimple if the minimal polynomial of φ (which is an element of K[t]) is a product of distinct irreducible factors in K[t].

Clearly, φ is semisimple if and only if for one (i.e. for any) finite extension L of K, φ_{L} ∈ End_{(A_{L},L_{L})}M_{L} is semisimple. (In the previous formula M_{L} indicates L ⊗_{K} M).

Lemma 1.6.2. Let M be an object of Mod^{f}(A, L) and let φ_{1}, …, φ_{s} ∈ End_{(A,L)}M be a set of commuting endomorphisms of M. Let P_{i}(x) be the minimal polynomial of φ_{i} in K[x] for i = 1, …, s, and

(1.6.2.1)  P_{i}(x) = ∏_{j} P_{ij}^{e_{j}}(x)

be the decomposition of P_{i}(x) into a product of irreducible factors in K[x]. Then, for any choice j : i ⟼ j(i) of an irreducible factor P_{i, j(i)} of each P_{i}, the module:

(1.6.2.2)  M(j) = ⋂_{i=1}^{s} Ker_{M}P_{i, j(i)}^{e_{j(i)}}(φ_{i})

is a sub-object of M in Mod^{f}(A, L) which is φ_{i}-stable for each i = 1, …, s and

(1.6.2.3)  M = ⨁_{j} M(j)

in Mod^{f}(A, L).

Proof. The proof is the same as in the case of vector spaces over K (i.e. of A = K, L = 0). Q.E.D.

Proposition 1.6.3. Let M be an object of Mod^{f}(A, L) and let φ ∈ End_{(A,L)}M. Then the following are equivalent:

i) φ is semisimple. ii) For any sub-object N of M such that φN ⊂ N, there exists a sub-A-module N′ of M such that φN′ ⊂ N′ and such that M = N ⊕ N′ as A-modules.

Proof. ii) ⇒ i). Let P (x) = ∏ P_{j}^{e_{j}}(x) the decomposition into irreducible factors of the minimal polynomial in K[x]. By (1.6.2), we have (for M(j) = Ker(P_{j}(φ)^{e_{j}})):

M = ⨁ M(j)

and the minimal polynomial of φ_{|M(j)} is P_{j}^{e_{j}}(x). Condition ii) holds for M = M(j) and φ = φ_{|M(j)}. We can then assume that P (x) = Q^{e}(x), where Q(x) is irreducible in K[x]. Consider W = Ker(Q(φ)) : W is a non zero φ-stable sub-object of M in Mod^{f}(A, L). Let W′ be a φ-stable A-submodule of M such that M = W ⊕W′ as A- modules. But then Q(φ)^{e−1}W′ ⊂ W i.e. Q(φ)^{e−1}W′ ⊂ W ∩ W′ = {0}. It follows that Q(φ)^{sup(e−1,1)} = 0, hence e = 1. We conclude that φ is semisimple.

i) ⇒ ii) We will prove a stronger result:

Proposition 1.6.4. Let M be an object of Mod^{f}(A, L) and φ_{1}, …, φ_{s} ∈ End_{(A,L)}M be a set of commuting semisimple endomorphisms of M. Let N be a sub-object of M in Mod(A, L) stabilized by every φ_{j}, j = 1, …, s. Then, there exists a sub-A-module N′ of M of finite type stabilized by the φ_{j}’s such that M = N ⊕ N′ as A-modules.

Proof. Let P_{j}(x) be the minimal polynomial of φ_{j} in K[x]. Suppose:

P_{j}(x) = ∏_{λ∈S_{j}⊂\overline{K}}(x − λ) .

Let L denote the splitting field of P_{1}(x), …, P_{s}(x) over K. We will indicate by a sub-index L the L-linear extension of objects defined over K. Consider for λ = (λ_{1}, …, λ_{s}) ∈ S = S_{1} × … × S_{s}:

M_{L}(λ) = ⋂_{j=1}^{s} Ker_{M_{L}}(φ_{jL} − λ_{j}) .

So we have (cf. (1.6.2))

M_{L} = ⨁_{λ∈S} M_{L}(λ)

in Mod^{f}(A, L). If N is an (A, L)-submodule of M which is stabilized by all φ_{j}’s, then the minimal polynomial of φ_{j|N} divides P_{j}(x) in K[x] for j = 1, …, s. As a consequence we obtain that, if

N_{L}(λ) = ⋂_{j=1}^{s} Ker_{N_{L}}(φ_{jL|N_{L}} − λ_{j}) ,

then

N_{L}(λ) = N_{L} ∩ M_{L}(λ)

and

N_{L} = ⨁_{λ∈S} N_{L}(λ) .

If \overline{φ}_{j} denotes the element of End_{(A,L)}(M/N) induced by φ_{j}, we can analogously define

(M/N)_{L}(λ) = ⋂_{j=1}^{s} Ker_{(M/N)_{L}}(\overline{φ}_{jL} − λ_{j}) = M_{L}(λ)/N_{L}(λ) ,

and we also get

(M/N)_{L} = ⨁_{λ∈S}(M/N)_{L}(λ) .

Every N_{L}(λ) is a sub-object of M_{L}(λ) in Mod^{f}(A_{L}, L_{L}). Consider the short exact sequence:

0 → N_{L}(λ) ⟶ M_{L}(λ)⟶^{π_{λ}}M_{L}(λ)/N_{L}(λ) = (M/N)_{L}(λ) → 0.

Now M_{L}(λ)/N_{L}(λ) is a projective A_{L}-module: in fact M/N is an object of Mod^{f}(A, L), therefore A-projective. Therefore (M/N)_{L} is A_{L}-projective as well as its direct summand M_{L}(λ)/N_{L}(λ). Therefore there exists an A_{L}-linear section s_{λ} of the canonical projection π_{λ}

s_{λ} : (M/N)_{L}(λ) ⟶ M_{L}(λ) .

We put G = Gal(L/K). It is then clear that the sections s_{λ}, λ ∈ S, may be chosen in such a way for every σ ∈ G and λ ∈ S we have:

s_{λ^{σ}}σ = σs_{λ}

where σ : M_{L}(λ) → M_{L}(λ^{σ}) is the canonical A-linear morphism induced by σ ∈ G on M_{L} = L ⊗_{K} M. We then define N′_{L}(λ) = Im(s_{λ}) for λ ∈ S and

N′_{L} = ∑_{λ∈S} N′_{L}(λ) = ⨁_{λ∈S} N′_{L}(λ) .

The A_{L}-module N′_{L} is stabilized by φ_{jL}, j = 1, …, s (these endomorphisms act on each N′_{L}(λ) as multiplications by an element in L) and M_{L} = N_{L} ⊕ N′_{L}. On the other hand G acts on N′_{L} and the action commutes with that of the φ_{jL}’s. It follows that N′_{L} = L ⊗_{K} N′ for a sub-A-module N′ of M with the properties of the statement. Q.E.D.

As an immediate corollary we state:

Corollary 1.6.5. In the notation of the previous proposition, let

\overline{φ}_{j} : M/N ⟶ M/N j = 1, …, s

be the endomorphisms induced by φ_{j} : M → M. There exists an A-linear section

s : M/N ⟶ M

of the canonical projection M → M/N such that:

φ_{j}s = s\overline{φ}_{j}

j = 1, …, s.

1.7. We can refine the considerations of 1.6. Let Φ_{1}, …, Φ_{s} represent commutative indeterminates over A and let us put A[Φ] = A[Φ_{1}, …, Φ_{s}]. Each l ∈ L acts as a K-derivation of A[Φ] extending the K-derivation l of A, by imposing l(Φ_{j}) = 0, for j = 1, …, s. So, L ⊂ Der(A[Φ]/K) and we define D[Φ] = D(A[Φ] , L). Let Mod(A, Φ) (resp. Mod(A, L, Φ)) indicate the category of left A[Φ]-modules (resp. D[Φ]-modules). Let Mod^{f}(A, Φ) (resp. Mod^{f}(A, L, Φ)) denote the full subcategory of Mod(A, Φ) (resp. Mod(A, L, Φ)), consisting of the objects which are of finite type as A-modules. We will say that an object M of Mod^{f}(A, L, Φ) is semisimple if for every j = 1, …, s the endomorphism φ_{j} : M → M defined as the multiplication by Φ_{j}

(1.7.1)  m ⟼ φ_{j}(m) = Φ_{j}m ∈ M ,

which is an element of End_{(A,L)}M, is semisimple.

The full sub-category of Mod^{f}(A, L, Φ) consisting of semisimple objects will be indicated by SS(A, L, Φ). The categories Mod(A, Φ), Mod(A, L, Φ), Mod^{f}(A, L, Φ) and SS(A, L, Φ) (and Mod^{f}(A, Φ) if A is noetherian) are all abelian (use corollary 1.5.2).

It follows that an object in Mod^{f}(A, L) endowed with endomorphisms (resp. semisimple endomorphisms) φ_{1}, …, φ_{s} which commute with each other, may be regarded as an object of Mod^{f}(A, L, Φ) (resp. SS(A, L, Φ)) via the construction (1.7.1) above. In particular, if N is any sub-object of M in Mod^{f}(A, L) stabilized by the semisimple endomorphisms φ_{1}, …, φ_{s}, then N (resp. M/N) may be viewed as a sub-object (resp. quotient) of M in SS(A, L, Φ).

In this new setting Proposition 1.6.4 becomes:

Corollary 1.7.2. Let π : M → N be an epimorphism in SS(A, L, Φ). Then there exists a A[Φ]-linear section s : N → M of π, π ∘ s = id_{N}.

The next proposition will be of use in the proof of the main result of section 2, namely proposition 2.1.4.

Proposition 1.7.3. Let

D. = \begin{matrix}0 & & 0 \\ ↓ & & ↓ \\ M_{1} & ⟶ & N_{1} & ⟶ & 0 \\ ↓ & & ↓ \\ M & ⟶^{u} & N & ⟶ & 0 \end{matrix}

be an exact commutative diagram in SS(A, L, Φ). Let π_{M} : M → M/M_{1} = M_{2} and π_{N} : N → N/N_{1} = N_{2} be the canonical projections in SS(A, L, Φ) and let \overline{u} : M/M_{1} = M_{2} → N/N_{1} = N_{2} be the morphism induced by the morphism u in D. . Then for any section s_{N} : N/N_{1} = N_{2} → N of π_{N} in Mod(A, Φ) there exists a section s_{M} : M/M_{1} = M_{2} → M of π_{M} in Mod(A, Φ) such that:

u ∘ s_{M} = s_{N} ∘ \overline{u}.

Proof. The digram D. may be completed into the following exact diagram in SS(A, L, Φ):

\overline{D}. = \begin{aligned}& & 0 & & 0 & & 0 \\ & & ↓ & & ↓ & & ↓ \\ 0 & ⟶ & P_{1} & ⟶ & M_{1} & ⟶ & N_{1} & ⟶ & 0 \\ & & ↓ & & ↓ & & ↓ \\ 0 & ⟶ & P & ⟶^{j} & M & ⟶^{u} & N & ⟶ & 0 \\ & & ↓ & & ↓^{π_{M}} & & ↓^{π_{N}} \\ 0 & ⟶ & P_{2} & ⟶ & M_{2} & ⟶^{\overline{u}} & N_{2} & ⟶ & 0 \\ & & ↓ & & ↓ & & ↓ \\ & & 0 & & 0 & & 0 \end{aligned} .

By assumption we are given an A[Φ]-linear section s_{N} : N_{2} ⟶ N of π_{N} : N → N_{2} = N/N_{1}. Corollary 1.7.2 guarantees the existence of A[Φ]-linear sections of P → P_{2} and u : M → N, say s_{P} and t, respectively. These sections permit to define a A[Φ]-linear map s_{M} : M_{2} → M by

s_{M}(m_{2}) = ts_{N} \overline{u}(m_{2}) + js_{P} (m_{2} − π_{M}ts_{N} \overline{u}(m_{2}))

for m_{2} ∈ M_{2}. In fact:

π_{M}s_{M}(m_{2}) = π_{M}ts_{N} \overline{u}(m_{2}) + m_{2} − π_{M}ts_{N} \overline{u}(m_{2}) = m_{2}

and

us_{M}(m_{2}) = uts_{N} \overline{u}(m_{2}) = s_{N} \overline{u}(m_{2}),

for each m_{2} ∈ M_{2}. Q.E.D.

1.8.

Proposition 1.8.1. Let M be an object of Mod^{f}(A, L) and φ ∈ End_{(A,L)}M be an endomorphism. Then there exists a unique decomposition

φ = ^{s}φ +^{n}φ

with ^{s}φ, ^{n}φ in End_{(A,L)}M, such that:

a) ^{s}φ is semisimple, ^{n}φ is nilpotent;

b) [^{n}φ,^{s}φ] = 0.

Proof. By Galois descent we may assume that the minimal polynomial of φ factors completely in K. We will denote it by

P (x) = ∏_{λ∈S⊂K}(x − λ)^{s_{λ}}.

Now M = ⊕_{λ∈S}Ker(φ − λ)^{s_{λ}} in Mod(A, L), so we are reduced to the case of P (x) = (x − λ)^{s_{λ}}. In this situation ^{s}φ = λ and ^{n}φ = φ − λ is the desired decomposition. Q.E.D.

Corollary 1.8.2. Let φ_{1}, …, φ_{s} be elements in End_{(A,L)}M with [φ_{j}, φ_{h}] = 0 ∀j, h ∈ {1, …, s}. Then:

[^{s}φ_{j},^{s}φ_{h}] = [^{n}φ_{j},^{s}φ_{h}] = [^{n}φ_{j},^{n}φ_{h}] = 0

for each j, h ∈ {1, …, s} .

Proof. We may assume that the minimal polynomials of the φ_{j}’s are completely reducible in K. Let ∏_{λ∈S_{i}⊂K}(x − λ)^{s_{i,λ}} be the minimal polynomial of φ_{i} for i = 1, …, s. In Mod^{f}(A, L) we have

M = ⨁_{λ∈K^{s}} M(λ)

where

M(λ) = ⋂_{j=1}^{s} Ker(φ_{j} − λ_{j})^{s_{j,λ_{j}}} .

For each x ∈ M(λ) we then have :

sφ_{i}(x) = λ_{i}x, ^{n}φ_{i}(x) = φ_{i}(x) − λ_{i}x

for i = 1, …, s. Q.E.D.

Corollary 1.8.3. Let M be an object of Mod^{f}(A, L), and let φ_{1}, …, φ_{s} ∈ End_{(A,L)}M be commuting endomorphisms of M. Let ^{s}φ_{1}, …,^{s} φ_{s} be as in 1.8.2. For λ = (λ_{1}, …, λ_{s}) ∈ K^{s} we put

M(λ) = ⋂_{i=1}^{s} Ker(^{s}φ_{i} − λ_{i}) .

For N_{i} ∈ N greater or equal to the exponent of nilpotence of^{n}φ_{i}, for i = 1, …, s, we have:

M(λ) = ⋂_{i=1}^{s} Ker(φ_{i} − λ_{i})^{N_{i}} .

Also, M(λ) is a sub-object of M in Mod^{f}(A, L) stable under φ_{i}, ^{s}φ_{i}, ^{n}φ_{i} for i = 1, …, s. If the minimal polynomial P_{i}(x) ∈ K[x] of φ_{i} has all of its zeros in K, for all i = 1, …, s. we have:

M = ⨁_{λ∈K^{s}} M(λ)

in Mod^{f}(A, L).

Corollary 1.8.4. For any λ ∈ K^{s} the assignment M ⟼ M(λ) introduced in the previous corollary defines an exact functor

Mod^{f}(A, L, Φ) ⟶ Mod^{f}(A, L, Φ) .

Definition 1.8.5. In the situation of the previous corollary, we will say that λ = (λ_{1}, …, λ_{s}) ∈ K^{s} is a multieigenvalue of (φ_{1}, …, φ_{s}), if M(λ) ≠ 0.

The expression: “the multieigenvalues of (φ_{1}, …, φ_{s}) all are in K^{s}”, is meant to indicate that the minimal polynomial P_{i}(x) ∈ K[x] of φ_{i} has all of its zeros in K, for all i = 1, …, s.

Proposition 1.8.6. Let M be a non zero object of Mod^{f}(A, L) and let φ_{1}, …, φ_{s} ∈ End_{(A,L)}M be nilpotent commuting endomorphisms of M. Then

N = ⋂_{i=1}^{s} Ker φ_{i}

is a non-zero sub-object of M in Mod^{f}(A, L).

Proof. We may restrict ourselves to the case s = 1, φ = φ_{1}. Let p ∈ N be such that φ^{p} = 0 ≠ φ^{p−1}. Then {0} ≠ Im φ^{p−1} ⊂ Ker φ = N. Q.E.D.

§2 Projective limits of differential modules without singularities.

2.1. We retain the notation of the previous section. In particular, (A, L, K, D) are as in 1.1, and are subject to the hypotheses (1.3).

Definition 2.1.1. Let MF(A, L) denote the category whose objects are pairs (V, {F^{i}V }_{i∈N}), where:

i) V is a left D-module; ii) {F^{i}V }_{i∈N} is a decreasing filtration of V, by left D-sub-modules F^{i}V, i ∈ N, and F^{0}V = V. iii) V is separated and complete in the D-linear topology for which a fundamental system of open neighborhoods of 0 is given by {F^{i}V }_{i∈N}. iv) For any i ∈ N , V/F^{i}V is an A-module of finite type . A morphism between two objects (V, {F^{i}V }_{i∈N}) and (W, {F^{i}W }_{i∈N}) in MF(A, L) is a morphism of left D-modules

φ : V ⟶ W

such that φF^{i}V ⊂ F^{i}W for any i ∈ N.

Whenever no confusion will possibly arise, an object (V, {F^{i}V }_{i∈N}) of MF(A, L) will be simply denoted by V. In the additive category MF(A, L) every morphism has a kernel and a cokernel. Given a morphism W ⟶ V in MF(A, L) we will say that W is a sub-object of V when W ⟶ V is a set-theoretical inclusion and is a kernel of its cokernel (i.e. the canonical map of W = Coim(W ⟶ V) to Im(W ⟶ V) is an isomorphism). This is the case precisely when W is a closed left D-sub-module of V and F^{i}W = W ∩ F^{i}V for each i ∈ N. We will denote by Hom_{MF(A,L)}(V, W) the abelian group of morphisms from V to W in MF(A, L). Let φ : V → W be a morphism in MF(A, L); for any i ∈ N we denote by φ(i) : V/F^{i}V → W/F^{i}W the morphism induced by φ . We will also call:

π_{i} = π_{i,V} : V ⟶ V/F^{i}V

and

θ_{i} = θ_{i,V} : V/F^{i}V ⟶ V/F^{i−1}V

for i ∈ N, the canonical projections.

Definition 2.1.2. Let V be an object in MF(A, L) and φ ∈ End_{MF(A,L)}V be an endomorphism. We will say that φ is topologically semisimple (resp. topologically nilpotent) if, for each i ∈ N, φ(i) is semisimple (resp. nilpotent) as a morphism in Mod^{f}(A, L).

Proposition 2.1.3. Let V be an object in MF(A, L) and φ ∈ End_{MF(A,L)}V be an endomorphism. Then the following facts are equivalent:

i) φ is topologically semisimple; ii) If H is a sub-object of V in MF(A, L) stabilized by φ, then there exists a closed A-sub-module L of V such that φ(L) ⊂ L and V = L ⊕ H as A-modules.

Proof. ii) ⇒ i). Fix a p ∈ N. Consider H_{p} a D-sub-module of V/F^{p}V stabilized by φ(p). The D-sub-module H = π_{p}^{−1}H_{p} ⊃ F^{p}V of V is closed and stable under φ. Let L be as in ii). Then L_{p} = π_{p}(L) is a φ(p)-stable A-sub-module of V/F^{p}V and

V/F^{p}V = H_{p} ⊕ L_{p}.

i) ⇒ ii). We will prove a stronger result:

Proposition 2.1.4. Let φ_{1}, …, φ_{s} ∈ End_{MF(A,F)}V be topologically semisimple commuting endomorphisms of V. Let H be a sub-object of V in MF(A, F) such that φ_{j}H ⊂ H for j = 1, …, s. Then there exists a closed A-sub-module L of V such that φ_{j}L ⊂ L for each j = 1, …, s and V = H ⊕ L as A-modules.

Proof. For any p ∈ N we put H_{p} = π_{p}(H). We have H_{0} = {0} and we put L_{0} = {0}. By proposition 1.6.3 it is possible to find an A-sub-module, L_{1}, of V/F^{1}V stabilized by φ_{j}(1), j = 1, …, s such that V/F^{1}V = H_{1} ⊕ L_{1} as A-modules. We may assume to have constructed L_{0}, L_{1}, …, L_{n−1} such that for each j = 1, …, n − 1 the following conditions hold: I_{j}) L_{j} is a A-sub-module of V/F^{j}V and

V/F^{j}V = H_{j} ⊕ L_{j}

II_{j}) φ_{i}(j)(L_{j}) ⊂ L_{j}, for i = 1, … s . III_{j}) θ_{j}(L_{j}) ⊂ L_{j−1} (cf. (2.1.1)).

For p ∈ N, V/F^{p}V and H_{p} may be viewed as objects of SS(A, L, Φ) and the Φ-action is given by Φ_{i}π_{p}v = π_{p}φ_{i}v = φ_{i}(p)π_{p}v for i = 1, …, s. Under this point of view θ_{p} may be interpreted as a morphism of D[Φ]-modules. We consider the exact commutative diagram in SS(A, L, Φ):

D. = \begin{matrix}0 & & 0 \\ ↓ & & ↓ \\ H_{n} & ⟶ & H_{n−1} & ⟶ & 0 \\ ↓ & & ↓ \\ V/F^{n}V & ⟶^{u} & V/F^{n−1}V & ⟶ & 0 \end{matrix}

and the A[Φ]-linear section s_{n−1} of the projection

V/F^{n−1}V ⟶ (V/F^{n−1}V)/H_{n−1}

corresponding, via I_{n−1}), to L_{n−1} ⟶ (V/F^{n−1}V), l ⟼ 0⊕l. Proposition 1.7.3 (applied to the previous diagram and the section s_{n−1}) provides a section s_{n} of the canonical projection:

V/F^{n}V ⟶ (V/F^{n}V)/H_{n}

whose image, L_{n}, satisfies I_{n}), II_{n}) and III_{n}). We can then put

L = ⋂_{n∈N} π_{n}^{−1}(L_{n}).

Q.E.D.

2.2.

Proposition 2.2.1. Let V be an object of MF(A, L) and φ ∈ End_{MF(A,L)}V be an endomorphism of V. Then there exist two endomorphisms ^{s}φ and ^{n}φ in End_{MF(A,L)}V such that:

i) ^{s}φ (resp. ^{n}φ) is topologically semisimple (resp. topologically nilpotent); ii) φ = ^{s}φ +^{n}φ iii) [^{s}φ,^{n}φ] = 0. The morphisms ^{n}φ and ^{s}φ are uniquely determined by i) and ii).

Proof. We use (1.8.1) on setting ^{s}φ(i) = ^{s}(φ(i)) and ^{n}φ(i) = ^{n}(φ(i)) for each i ∈ N. This uniquely defines ^{s}φ and ^{n}φ. Q.E.D.

Corollary 2.2.2. Let φ_{1}, …, φ_{s} be endomorphisms in End_{MF(A,L)}V, such that [φ_{i}, φ_{j}] = 0 for i, j = 1, …, s. Then

[^{s}φ_{i},^{s}φ_{j}] = [^{s}φ_{i},^{n}φ_{j}] = [^{n}φ_{i},^{n}φ_{j}] = 0

for i, j = 1, …, s.

Proof. It is a consequence of (1.8.2). Q.E.D.

Corollary 2.2.3. We consider α ∈ K and x ∈ V. Then

sφ(x) = αx ⇔ lim_{N→∞}(φ − α)^{N}(x) = 0.

Corollary 2.2.4. Let φ_{1}, …, φ_{s} be as in (2.2.2). Suppose that for each i ∈ N the multieigenvalues of (φ_{1}(i), …, φ_{s}(i)) all are in K^{s}. For λ = (λ_{1}, …, λ_{s}) ∈ K^{s}, we set :

V (λ) = ⋂_{i=1}^{s} Ker(^{s}φ_{i} − λ_{i}).

Then V (λ) is a sub-object of V in MF(A, L) stable under ^{s}φ_{j}, ^{n}φ_{j}, for j = 1, …, s, and

W = ⨁_{λ∈K^{s}} V (λ)

is dense in V.

Proof. It is a consequence of (1.8.3). Q.E.D.

§3. Differential modules with logarithmic singularities depending upon parameters.

3.1. Let R be a commutative ring with 1, let M be a R-module, δ a derivation in R, ∇ an endomorphism of M as an abelian group such that ∇(rm) = δ(r)m+ r∇(m) for each r ∈ R and m ∈ M. For the convenience of the reader, we recall explicitly the following result of Gérard and Levelt [G-L, Prop. 2.1].

Lemma 3.1.1. In the previous notation we consider N ∈ N. Let α and β be elements of R such that: δ(α) = δ(β) = 0. If f ∈ R and m ∈ M we have:

(∇ − α − β)^{N}(fm) = ∑_{i=0}^{N} \binom{N}{i}\{(δ − α)^{i}f\}\{(∇ − β)^{N−i}m\}.

3.2. We will assume from now on that the K-algebra A is noetherian. We consider B = A[[x_{1}, …, x_{s}]], I = ∑_{i=1}^{s} Bx_{i}. By [AC, III, 2.13, Lemme 3, (ii)], I is contained in the radical of B. It follows that B equipped with the Iadic topology is a Zariski ring ([AC, III, 3.3, Prop.6]). If P ∈ MaxA we put P[[x]] = PB + I ∈ MaxB. It is then clear that the map:

(3.2.1)  \begin{matrix}MaxA & ⟶ & MaxB \\ P & ⟼ & P[[x]] \end{matrix}

is a bijection. For P ∈ MaxA (resp. f ∈ A), let B̂_{P} (resp. B̂_{f}) be the separated completion of B_{P[[x]]} (resp. B_{f} = S^{−1}B, for S = {1, f, f^{2}, …}) in the IB_{P[[x]]}-adic (resp. IB_{f}-adic) topology. We have B̂_{P} = A_{P}[[x_{1}, …, x_{s}]] (resp. B̂_{f} = A_{f}[[x_{1}, …, x_{s}]]) while B̂_{P} (resp. B̂_{f}) is faithfully flat over B_{P[[x]]} (resp. B_{f}) (cf. [AC, III, 3.4, Th.3, (iii)] for flatness, and [AC, I, 3.5, Prop.9, e)] together with (3.2.1) above, for faithful flatness). If M is a B-module of finite type and P ∈ MaxA (resp. f ∈ A), we will denote by M̂_{P} (resp. M̂_{f}) the separated completion of M_{P[[x]]} (resp. M_{f}) w.r.t. the IB_{P[[x]]}-adic (resp. IB_{f}- adic) topology. Then:

M̂_{P} = B̂_{P} ⊗_{B} M ( resp. M̂_{f} = B̂_{f} ⊗_{B} M ).

Lemma 3.2.2. Let M be a B-module of finite type. Then:

a) (Nakayama’s lemma) If the classes of e_{1}, …, e_{p} ∈ M in M/IM generate the A-module M/IM, then e_{1}, …, e_{p} generate M as B-module.

b) The following facts are equivalent:

i) M is a projective B-module; ii) for each P ∈ MaxA, M̂_{P} is a free B̂_{P}-module; iii) there exist a finite number of elements f_{1}, …, f_{r} in A such that

SpecA = ⋃_{i=1}^{r} SpecA_{f_{i}}

and

M̂_{f_{i}} = A_{f_{i}}[[x_{1}, …, x_{s}]] ⊗_{B} M

is a free A_{f_{i}}[[x_{1}, …, x_{s}]]-module for every i = 1, …, r; iv) there exists a finite family {F_{1}, …, F_{r}} of elements of B such that

SpecB = ∪_{i=1}^{r}SpecB_{F_{i}}

and

M_{F_{i}} = B_{F_{i}} ⊗_{B} M

is a free B_{F_{i}}-module for every i = 1, …, r;

v) M is a flat B-module.

Proof. Assertion a) is the well-known Nakayama’s Lemma, which applies here since I is contained in the radical of B. To prove b) one can use theorem 1 of [AC, II, 5.2] and the following facts: I) For the equivalence i) ⇔ ii), we already pointed out that MaxB = {P[[x]], P ∈ MaxA} while B̂_{P} is faithfully flat over B_{P[[x]]} (cf. [AC, I, 3.6, Prop.12]). II) To show now that iii) ⇔ iv), let us put f_{i} = F_{i}(0, …, 0) ∈ A, for i = 1, …, r. Then (F_{1}, …, F_{r}) = B ⇔ (f_{1}, …, f_{r}) = A. (Proof of the last statement: implication from left to right is obvious. For the other direction, assume the ideal (f_{1}, …, f_{r}) is equal to A and (F_{1}, … F_{r}) ≠ B. Then there exists P[[x]] ∈ MaxB, such that (F_{1}, …, F_{r}) ⊂ P[[x]]. This implies that f_{i} ∈ P ∈ MaxA for i = 1, …, r, absurd.) We recall that the completion of B_{F_{i}} in the IB_{F_{i}}-topology is B̂_{f_{i}} = A_{f_{i}}[[x_{1}, …, x_{s}]], which then is faithfully flat over B_{F_{i}}. To prove iv) ⇒ iii) we take the completion of the free B_{F_{i}}-module B_{F_{i}} ⊗_{B} M with respect to the IB_{F_{i}}-adic topology, for any i = 1, …, r. To prove that iii) ⇒ iv) we use the fact that B̂_{f_{i}} is faithfully flat over B_{F_{i}} for i = 1, …, r. III) The equivalence i) ⇔ v) follows from [AC, II, 5.2, Cor.2]. Q.E.D.

We explicitly state the obvious:

Corollary 3.2.3. Let P ∈ SpecA. If M is a projective B-module of finite rank n in P[[x]], then M/IM is A-projective of rank n in P.

We will also need the rings A[x] = A[x_{1}, ⋯, x_{s}], B′ = B_{x_{1}···x_{s}} = B[(x_{1}x_{2} … x_{s})^{−1}] and A[x]′ = A[x][(x_{1}x_{2} ⋯ x_{s})^{−1}]. The ideal (x_{1}, x_{2}, …, x_{s}) in A[x] will still be denoted by I. For d ∈ D and y = ∑_{α∈Z^{s}} a_{α}x^{α} ∈ B′, we define:

dy = ∑_{α∈Z^{s}}(da_{α})x^{α}.

Let Der^{c}(B/K) denote the B-module and Lie algebra over K of the K-derivations of B which are I-adically continuous. Let L_{x_{1}...x_{s}} = L_{x} denote the Lie subalgebra over K of Der^{c}(B/K) (and of Der(A[x]/K)) generated by L and by the A-derivations x_{1} \frac{∂}{∂x_{1}} , …, x_{s} \frac{∂}{∂x_{s}}, defined by:

x_{i} \frac{∂}{∂x_{i}} (x_{j}) = δ_{ij}x_{j}

for i, j = 1, …, s. We recall the warning 1.1.2.1.

As in (1.1.1), on replacing (A, L, K) by (B, L_{x}, K) (resp. (A[x], L_{x}, K)), we get the abelian category Mod(B, L_{x}) (resp. Mod(A[x], L_{x})). Since B (resp. A[x]) is noetherian Mod^{f}(B, L_{x}) (resp. Mod^{f}(A[x], L_{x})) is also abelian. Every element of L_{x} can be uniquely extended to a K-derivation of B′; so we regard L_{x} as contained in Der(B′/K). We may then consider the abelian categories Mod(B′, L_{x}) and Mod^{f}(B′, L_{x}). Any (B, L_{x})-module (resp. (B′, L_{x})-module) may canonically be regarded as an (A, L)-module (resp. (B, L_{x})-module ).

Proposition 3.2.4. Assume the triple (A, L, K) satisfies the assumptions of proposition 1.4.4. Then so does the triple (B′, B′L_{x}, K).

Proof. It is an immediate consequence of proposition 1.4.5, since ⊕_{i=1}^{s}B′x_{i} \frac{∂}{∂x_{i}} = ⊕_{i=1}^{s}B′ \frac{∂}{∂x_{i}}, Q.E.D.

3.2.5 Examples. We show by the first two examples that the triple (B, L_{x}, K) will not in general satisfy the assumptions of (1.3). 3.2.5.1. For any triple (A, L, K) satisfying the assumptions of (1.3) consider the object T of Mod^{f}(B, L_{x}) by the conditions: T = A as an (A, L)-module, x_{i}∂/∂x_{i}(t) = 0 and g(x)t = g(0)t, for any g ∈ B, t ∈ T, i = 1, …, s. Then T is a torsion B-module. 3.2.5.2. Assume s > 1 and for any triple (A, L, K) satisfying the assumptions of (1.3), consider the ideal I = (x_{1}, …, x_{s}) of B. Then I is a sub-object of B in Mod^{f}(B, L_{x}), which is not B-projective. Notice that I contains the sub-object S = x_{1} ⋯ x_{s}B, which is B-projective, and such that I/S is a torsion B-module.

3.2.5.3. Let D = D(A[x], L_{x}). For λ ∈ K^{s} and α ∈ N^{s}, we define a (A[x], L_{x})- module N_{λ}^{(α)} by

N_{λ}^{(α)} = D/ ∑_{i=1}^{s} D(x_{i} \frac{∂}{∂x_{i}} − λ_{i})^{α_{i}} .

Then N_{λ}^{(α)} is an object of Mod^{f}(A[x], L_{x}) which is a projective A[x]-module. By 1_{N_{λ}^{(α)}}, we denote the canonical image of 1_{D} in N_{λ}^{(α)}.

3.2.6. We are now led to introduce the full (additive) sub-category Mod^{fp}(B, L_{x}) of Mod^{f}(B, L_{x}) consisting of the objects which are projective B-modules of finite type. An A[x]-module M of finite type will be said to be locally free on A, if for each prime ideal P of SpecA, A_{P}[x] ⊗_{A[x]} M is a free A_{P}[x]-module. It is known (Quillen’s patching theorem, [L]) that, A being noetherian, every A[x]-module M of finite type locally free on A, is of the form M = M_{0} ⊗_{A} A[x], for a finite projective A-module M_{0}, unique up to isomorphism. Viceversa, if an object M of Mod^{f}(A[x], L_{x}) is of the form M = M_{0} ⊗_{A} A[x] for an A-module M_{0}, then the A-module M_{0} is necessarily projective and of finite type. This is because M_{0} is the underlying A-module of the (A, L)-module M/IM. The full (additive) sub-category of Mod^{f}(A[x], L_{x}) consisting of the objects which are locally free on A, will be denoted by Mod_{A}^{f}(A[x], L_{x}). There is of course a natural functor (I-adic completion):

(3.2.6.1)  Mod^{f}(A[x], L_{x}) ⟶ Mod^{f}(B, L_{x}) ,

which we denote M ⟼ M̂ = B⊗_{A[x]}M, sending Mod_{A}^{f}(A[x], L_{x}) to Mod^{fp}(B, L_{x}).

For an object M of Mod(B, L_{x}) (resp. Mod(A[x], L_{x})) and for i = 1, …, s, we define the A-endomorphisms:

\begin{aligned}∇_{M,i} = ∇_{i} : & M & ⟶ & M \\ & m & ⟼ & x_{i} \frac{∂}{∂x_{i}} m . \end{aligned}

In the following definition C will denote any of the A-algebras B, A[x], A[[x]][\frac{1}{x}] = B′, A[x][\frac{1}{x}] = A[x]′.

Definition 3.2.6.2. Let M be an object of Mod(C, L_{x}). Then an element m ∈ M is called horizontal relative to A if:

∇_{i}(m) = 0

for i = 1, …, s. The A-submodule of M consisting of the elements horizontal relative to A, will be denoted by M^{∇_{C/A}}.

Remark 3.2.7. For any object M of Mod(A[x], L_{x}) and any λ ∈ K^{s}, α ∈ N^{s}, we have a canonical isomorphism of (A, L)-modules

Hom_{A[x]}(N_{λ}^{(α)}, M)^{∇_{A[x]/A}} ⟶^{∼} ⋂_{i=1}^{s} Ker_{M}(∇_{i} − λ_{i})^{α_{i}} ,

sending φ to φ(1_{N_{λ}^{(α)}}).

Let M be an object of Mod^{f}(B, L_{x}); we put F^{i}M = I^{i}M for each i ∈ N. By [AC, III, 3.4, Th. 3 (ii)], M is separated and complete in the I-adic topology, hence (M, {F^{i}M}_{i∈N}) may be regarded as an object of MF(A, L). It is endowed with the commuting endomorphisms ∇_{1} = ∇_{M,1}, …, ∇_{s} = ∇_{M,s} ∈ End_{MF(A,L)}M. This holds in particular for B = (B, {I^{i}}_{i∈N}), where ∇_{B,i} = x_{i}∂/∂x_{i}, for i = 1, …, s. If φ : M → N is a morphism in Mod^{f}(B, L_{x}), we have φ(F^{i}M) ⊂ F^{i}N for i ∈ N hence φ defines a morphism of M into N as objects of MF(A, L). Thus Mod^{f}(B, L_{x}) may be viewed as a sub-category (not full, in general) of MF(A, L). A morphism W ⟶ V in Mod^{f}(B, L_{x}) is strict ([AC, III, 3.2, Cor. of Th.2]) and has closed image ([AC, III, 3.3, Prop.6, c)]). Notice however that the previous imbedding of Mod^{f}(B, L_{x}) in MF(A, L) is not compatible with the operation of taking kernels. We are given a natural functor from Mod(B, L_{x}) to Mod(B′, L_{x}), which, for the underlying B-modules, is simply the extension of scalars from B to B′: M ⟼ M′ = B′ ⊗_{B} M. For a (B, L_{x})-module M, there is a canonical morphism of (B, L_{x})-modules

(3.2.8)  M ⟶ M′ = B′ ⊗_{B} M .

The kernel of the previous morphism in Mod(B, L_{x}) will be denoted by T_{x}(M). If M is a B-module of finite type, T_{x}(M) is a sub-object of M in Mod(B, L_{x}) which is an A-module of finite type. We have

T_{x}(M) = {m ∈ M | (x_{1} ⋯ x_{s})^{r}m = 0 , for some r ∈ N} .

Similar definitions will be understood for the natural functor

Mod(A[x], L_{x}) ⟶ Mod(A[x]′, L_{x}) .

Lemma 3.2.9. Let M be an object of Mod^{f}(B, L_{x}). Then there exists a subobject N of M in Mod^{f}(B, L_{x}) such that T_{x}(N) = 0 and T_{x}(M/N) = M/N.

Proof. There exists r ∈ N such that (x_{1} ⋯ x_{s})^{r}T_{x}(M) = 0. We may then take N = (x_{1} ⋯ x_{s})^{r}M. Q.E.D.

3.3. For an object M in Mod^{f}(B, L_{x}) let

(3.3.1)  ∇_{i} = ∇_{M,i} = ^{s}∇_{M,i} +^{n}∇_{M,i} = ^{s}∇_{i} +^{n}∇_{i}

for i = 1, …, s be the decomposition of proposition 2.2.1 applied to the endomorphism φ_{i} = ∇_{M,i} in MF(A, L).

The following theorem is essentially propositon 3.2 of [G-L]. We reproduce it here with its proof for the convenience of the reader.

Theorem 3.3.2. In the previous notation, we have:

s∇_{i}(fm) = x_{i} \frac{∂f}{∂x_{i}} m + f^{s}∇_{i}(m)

for i = 1, …, s, f ∈ B, m ∈ M.

Proof. Let p be an element of N. We fix an index i ∈ {1, …, s} and write ∇ instead of ∇_{i} and ∇(p) : M/I^{p}M → M/I^{p}M for the morphism of (A, L)-modules induced by ∇ = ∇_{i}.

We need to check:

(3.3.2.1)  ^{s}∇(p)(fm) = x_{i} \frac{∂f}{∂x_{i}} m + f^{s}∇(p)(m)

for m ∈ M/I^{p}M and f ∈ B. By A-linearity it is enough to check formula (3.3.2.1) for f = x^{r} = x_{1}^{r_{1}} … x_{s}^{r_{s}} and r ∈ N^{s}. For λ ∈ K, we denote:

(M/I^{p}M)(λ) = Ker_{M/I^{p}M}(^{s}∇(p) − λ),

hence

M/I^{p}M = ⨁_{λ∈K}(M/I^{p}M)(λ)

and we may assume m ∈ (M/I^{p}M)(λ). We then have:

(3.3.2.2)  x_{i} \frac{∂f}{∂x_{i}} m + f^{s}∇(p)(m) = (r_{i} + λ)fm.

To check that ^{s}∇(p)(fm) = (r_{i} + λ)fm we only have to show, according to (2.2.3), that

(3.3.2.3)  (∇(p) − (λ + r_{i}))^{N}(x^{r}m) = 0

if N >> 0. This follows from the formula in Lemma 3.1.1. Q.E.D.

Corollary 3.3.3. For any object M of Mod^{f}(B, L_{x}), the maps ^{n}∇_{i}, for i = 1, …, s, are B-linear topologically nilpotent endomorphisms of M.

Proof. ^{n}∇_{i} = ∇_{i} −^{s}∇_{i}, for i = 1, …, s. Q.E.D.

N.B. 3.3.4. We will see later (cf. corollary 3.4.13) that the ^{n}∇_{i}’s are actually nilpotent.

3.4. Let M be an object of Mod^{f}(B, L_{x}); we keep the notation of section 3.3. We may apply corollary 1.8.3 to the (A, L)-module M/IM endowed with the endomorphisms φ_{1} = ∇_{1}(1), …, φ_{s} = ∇_{s}(1). If the multieigenvalues of (φ_{1} = ∇_{1}(1), …, φ_{s} = ∇_{s}(1)) all are in K^{s}, we obtain a decomposition of M/IM as an (A, L)-module:

(3.4.1)  M/IM = ⨁_{λ∈K^{s}}(M/IM)(λ) ,

where for λ = (λ_{1}, …, λ_{s}) ∈ K^{s}

(3.4.2)  (M/IM)(λ) = ⋂_{i=1}^{s} Ker_{M/IM}(^{s}∇_{i}(1) − λ_{i}) .

Definition 3.4.3. A multieigenvalue of (φ_{1} = ∇_{1}(1), …, φ_{s} = ∇_{s}(1)) in K^{s}, is called a multiexponent of M in K^{s}. If P ∈ SpecA, the multiplicity e(λ, M, P) = e(λ, P) in P of the multiexponent λ of M will be the rank in P of the projective A-module (M/IM)(λ).

We will indicate by E(M) the set of multiexponents of M in K^{s}. The expression “the multiexponents of M all are in K^{s}” means that (3.4.1) holds.

N.B. 3.4.3.1. When the rank in P of the projective A-module (M/IM)(λ) is independent of P ∈ SpecA, it will be called the multiplicity of λ and it will be denoted by e(λ, M) or, briefly, by e(λ). In the second chapter we could in fact assume that SpecA is connected (and more, with no restriction in view of our future applications, that A is an integral domain).

The notions just introduced for objects of Mod^{f}(B, L_{x}) will also be applied to objects of Mod^{f}(A[x], L_{x}), via the I-adic completion functor (3.2.6.1).

Definition 3.4.4. For any additive subgroup Σ of K, we will denote by Mod_{Σ}^{f}(B, L_{x}) (resp. Mod_{Σ}^{fp}(B, L_{x}), Mod_{Σ}^{f}(A[x], L_{x}), Mod_{A,Σ}^{f}(A[x], L_{x}) ) the full additive subcategory of Mod^{f}(B, L_{x}) (resp. Mod^{fp}(B, L_{x}), Mod^{f}(A[x], L_{x}), Mod_{A}^{f}(A[x], L_{x})) consisting of the objects M such that the multiexponents of M are all in Σ^{s}, i.e. (3.4.1) holds and (M/IM)(λ) = 0 if λ ∉ Σ^{s}.

We will fix in the sequel an additive subgroup Σ of K containing Z. Notice that unless Σ = A = K, our notation for the categories Mod_{Σ}^{f}(A[x], L_{x}), Mod_{A}^{f}(A[x], L_{x}) and Mod_{A,Σ}^{f}(A[x], L_{x}) should never lead to confusion. In case Σ = A = K, we will try to make ourselves understood by using the three distinct symbols Σ, A, K, even if they denote the same object. We point out that Mod_{Σ}^{f}(B, L_{x}) (resp. Mod_{Σ}^{f}(A[x], L_{x})) is an abelian category, and that Mod_{Σ}^{f}(B, L_{x}) (resp. Mod_{Σ}^{f}(A[x], L_{x})) and Mod_{Σ}^{fp}(B, L_{x}) (resp. Mod_{A,Σ}^{f}(A[x], L_{x})) are stable under the bifunctors − ⊗_{B} − and Hom_{B}(−, −) (resp. − ⊗_{A[x]} − and Hom_{A[x]}(−, −)).

3.4.5.

Definition 3.4.5.1. For λ, μ ∈ K^{s} we put:

λ ≤ μ ⇔ μ − λ ∈ N^{s}

and

λ ≡ μ ⇔ μ − λ ∈ Z^{s}.

If λ, μ ∈ K^{s} and λ ≡ μ, we will say that λ and μ are congruent; an equivalence class of ≡ in any subset E of K^{s} will be called a congruence class of E.

Let now M be an object of Mod_{Σ}^{f}(B, L_{x}) or of Mod_{Σ}^{f}(A[x], L_{x}). We will put:

C(M) = { congruence classes of E(M) } and we will denote by:

\begin{aligned}c & : & E(M) & ⟶ & C(M) \\ & & λ & ⟼ & c(λ) \end{aligned}

the canonical projection. From this viewpoint, c ⊂ E(M), so that each c ∈ C(M) is a finite set, partially ordered by the relation ≤. On the other hand, we may, and will, regard C(M) as a subset of K^{s}/ ≡, in a natural way. We will also define S(M) as the set of maps:

\begin{aligned}μ & : & C(M) & ⟶ & K^{s} \\ & & c & ⟼ & μ(c) \end{aligned}

such that μ(c(λ)) ≥ λ for each λ ∈ E(M). We introduce a partial ordering in S(M) via the position:

(3.4.5.2)  μ ≤ μ′ ⟺ μ(c) ≤ μ′(c) ∀ c ∈ C(M) ,

for any μ, μ′ ∈ S(M).

For P ∈ SpecA the multiplicity in P of a class c ∈ K^{s}/ ≡, viewed as a class of multiexponents of M, will then be the positive integer:

e(c, M, P) = e(c, P) = ∑_{λ∈c} e(λ, M, P).

When e(c, M, P) is independent of P ∈ SpecA, we will also denote it by e(c, M) = e(c).

Proposition 3.4.6. Let M be an object of Mod^{f}(B, L_{x}) and r = 0, 1, …, be a natural number. Then there exists an A-sub-module M_{r} of M closed w.r.t. the I-adic topology, such that ^{s}∇_{i}(M_{r}) ⊂ M_{r} for i = 1, …, s and such that:

M = M_{r} ⊕ I^{r+1}M

as A-modules.

Proof. One applies proposition 2.1.4 to the case V = M, φ_{i} = ^{s}∇_{i} and H = I^{r+1}M, endowed with the filtration F^{i}H = H ∩ I^{i}M, for i ∈ N. Q.E.D.

Corollary 3.4.7. In the notation of the proposition, for any r ∈ N, we define a structure of A[Φ]-module on M_{r} (resp. M/I^{r+1}M) as follows. Let m be an element of M_{r} (resp. M/I^{r+1}M); we put Φ_{i}m = ^{s}∇_{i}(m) (resp. = ^{s}∇_{i}(r+1)(m)) for i = 1, …, s. Then the canonical projection M → M/I^{r+1}M induces an isomorphism of A[Φ]-modules :

M_{r} ⟶ M/I^{r+1}M.

In particular, for each λ ∈ K^{s} we have (cf. (3.4.2))

(3.4.7.1)  M_{r}(λ) = ⋂_{i=1}^{s} Ker_{M_{r}}(^{s}∇_{i} − λ_{i}) ≅ (M/I^{r+1}M)(λ) ,

as A-modules. For any λ ∈ K^{s}, (M/I^{r+1}M)(λ) is an (A, L)-sub-module of M/I^{r+1}M.

Let us assume that the multiexponents of M all are in K^{s}. Then the multieigenvalues of (^{s}∇_{1}(r+1), …,^{s}∇_{s}(r+1)) are of the form λ+α, where λ ∈ E(M) and α ∈ N^{s}, |α|_{R} ≤ r. From (3.4.1) we then deduce

(3.4.7.2)  M_{r} = ⨁_{λ∈K^{s}} M_{r}(λ) .

Proof. The only point that deserves some care is the assertion that the multieigenvalues of (^{s}∇_{1}(r + 1), …,^{s}∇_{s}(r + 1)) are integral translates of the multiexponents of M. Let us denote by \overline{M}_{0} (resp. \overline{M}_{0}(λ)) the canonical image of M_{0} (resp. M_{0}(λ)) in M/I^{r+1}M. Then clearly

M/I^{r+1}M = ∑_{λ∈E(M)} ∑_{α} x^{α}\overline{M}_{0}(λ) ,

where the sum runs over α ∈ N^{s}, |α|_{R} ≤ r. But for any α, λ as before x^{α}\overline{M}_{0}(λ) ⊂ (M/I^{r+1}M)(α + λ). In particular, for any λ′ ∈ K^{s}

(3.4.7.3)  (M/I^{r+1}M)(λ′) = ∑_{λ∈E(M)} ∑_{α∈N^{s},α+λ=λ′} x^{α}\overline{M}_{0}(λ) ,

Q.E.D.

Corollary 3.4.8. In proposition 3.4.6 assume the multiexponents of M are in K^{s}. We may then take for any r ∈ N

(3.4.8.1)  M_{r} ⊂ ∑_{α,λ} x^{α}M_{0}(λ) ,

where α ∈ N^{s}, |α|_{R} ≤ r and λ ∈ E(M). For λ ∈ K^{s} we then have

(3.4.8.2)  M_{r}(λ) ⊂ ∑_{λ′} x^{λ−λ′}M_{0}(λ′) ,

where the sum runs over the set of λ′ ∈ E(M), such that λ ≥ λ′ and |λ−λ′|_{R} ≤ r.

Proof. The key point is that the A-sub-module ∑_{α,λ} x^{α}M_{0}(λ) of M that appears in (3.4.8.1) is stable under the action of ^{s}∇_{1}, …,^{s}∇_{s} and generates M modulo I^{r+1}M. The canonical projection M ⟶ M/I^{r+1}M then induces a surjective A-linear map of ∑_{α,λ} x^{α}M_{0}(λ) onto the projective A-module M/I^{r+1}M. This is in fact an epimorphism in SS(A, L, Φ), where φ_{i} =^{s} ∇_{i} (resp. φ_{i} =^{s} ∇_{i}(r+1)). We may take M_{r} as the image of any A[Φ]-linear section of that epimorphism (cf. (1.7.2)). The other assertion follows immediately. Q.E.D.

Corollary 3.4.9. Let M, M_{0} be as in proposition 3.4.6 and, for r ∈ N, let M_{r} be as in the previous corollary. For λ ∈ K^{s}, let

(3.4.9.1)  r(λ) = sup{|λ − λ′|_{R} | λ′ ∈ E(M), λ′ ≤ λ} .

Then if r ≥ r(λ)

(3.4.9.2)  M_{r}(λ) = ∑_{λ′} x^{λ−λ′}M_{0}(λ′) ,

where the sum runs over the set of λ′ ∈ E(M), such that λ ≥ λ′. We conclude that if r ≥ r(λ)

(3.4.9.3)  M_{r}(λ) = ⋂_{i=1}^{s} Ker_{M}(^{s}∇_{i} − λ_{i})

independently of the choice of M_{0} as in the proposition.

Proof. The key point is that

(3.4.9.4)  (⋂_{i=1}^{s} Ker_{M}(^{s}∇_{i} − λ_{i})) ∩ I^{r+1}M = {0}

if r ≥ r(λ). This follows from formula (3.4.7.3), which implies that (I^{r}M/I^{r+1}M)(λ) = 0 if r > r(λ), hence that (I^{r+1}M/I^{r+h}M)(λ) = 0 if r ≥ r(λ) and h ∈ Z_{>0}. So from

M_{r}(λ) ⊂ ⋂_{i=1}^{s} Ker_{M}(^{s}∇_{i} − λ_{i})

we deduce equality if r ≥ r(λ). Q.E.D.

Definition 3.4.9.5. Let M be an object of Mod^{f}(B, L_{x}) and let us choose M_{0} as in (3.4.6). For λ ∈ K^{s} we put

M(λ) = ⋂_{i=1}^{s} Ker_{M}(^{s}∇_{i} − λ_{i}) = ∑_{λ′∈E(M),λ′≤λ} x^{λ−λ′}M_{0}(λ′) .

We also define:

M[λ] = A[x]M(λ)

(resp.

M⟨λ⟩ = BM(λ) ) ,

the A[x]-sub-module (resp. B-sub-module) of M generated by M(λ).

Remark 3.4.9.6. Let M be as in definition 3.4.9.5. From the formula in that definition we see that

M(λ) = ∑_{λ′∈E(M),λ′≤λ} x^{λ−λ′}M_{0}(λ′) .

In particular, if λ ∈ E(M) is minimal for ≤, M(λ) = M_{0}(λ).

Corollary 3.4.10. Let M, M_{0} be as in proposition 3.4.6 and let us consider d ∈ L_{x} such that [d,^{s}∇_{i}] = 0 for i = 1, …, s and λ ∈ K^{s}. Then we have: i)

dM(λ) ⊂ M(λ) .

ii) The A[x]-sub-module M^{0} of M generated by M_{0} is a sub-object of M in Mod^{f}(A[x], L_{x}). If the multiexponents of M all are in K^{s}, we have

M^{0} = ∑_{λ∈K^{s}} M(λ) ,

so that M^{0} does not depend upon the choice of M_{0} as in proposition 3.4.6.

Proof. The first assertion is clear. The second follows from the fact that among the d’s with the property mentioned in part i), are the elements of L and s∇_{i}, ^{n}∇_{i}, ∇_{i}, for i = 1, …, s. Q.E.D.

Definition 3.4.10.1. Let M be an object of Mod_{Σ}^{f}(B, L_{x}). The (A[x], L_{x})-sub-module M^{0} of M introduced in part ii) of corollary 3.4.10, will be called the core of M and denoted by Co(M).

Example 3.4.10.2. For any object M of Mod^{f}(B, L_{x}) and λ ∈ K^{s} the core of M⟨λ⟩ is M[λ].

Corollary 3.4.11. Let M be an object of Mod_{Σ}^{f}(B, L_{x}) and λ ∈ K^{s}. Then M[λ] (resp. M⟨λ⟩) is a sub-object of M in Mod_{Σ}^{f}(A[x], L_{x}) (resp. Mod_{Σ}^{f}(B, L_{x})). The only multiexponent of M⟨λ⟩ is λ.

Proof. The first assertion is a direct consequence of the previous corollary and of Leibnitz’s rule. The second assertion is obvious. Q.E.D.

Corollary 3.4.12. The assignment M ⟼ Co(M) defines an exact fully-faithful functor

Co : Mod_{Σ}^{f}(B, L_{x}) ⟶ Mod_{Σ}^{f}(A[x], L_{x}) .

The I-adic completion functor is an exact left quasi-inverse functor of Co. The functor Co restricts to a functor from Mod_{Σ}^{fp}(B, L_{x}) to Mod_{A,Σ}^{f}(A[x], L_{x}).

Proof. The I-adic completion functor is exact because B is flat over A[x]. On the other hand, exactness of Co follows from corollary 1.8.4. Obviously, for any object M of Mod_{Σ}^{f}(B, L_{x}), M ≅ Ĉo(M). Q.E.D.

Corollary 3.4.13. For any object M of Mod^{f}(B, L_{x}), the maps ^{n}∇_{i}, for i = 1, …, s, are B-linear nilpotent endomorphisms of M.

Proof. By corollary 3.3.3, ^{n}∇_{i} is topologically nilpotent in the I-adic topology of M, for i = 1, …, s. We have M = ∑_{λ∈E(M)} BM(λ), and, according to corollary 3.4.10, M(λ) is stable under the action of the ^{n}∇_{i}’s. On the other hand, M(λ) is discrete in M, so that the restriction of ^{n}∇_{i} to M(λ) is nilpotent. The conclusion follows. Q.E.D.

Corollary 3.4.14. Let M be an object of Mod^{f}(B, L_{x}) For i = 1, …, s, let α_{i} ∈ N be bigger than the exponent of nilpotence of ^{n}∇_{i} and α = (α_{1}, …, α_{s}). Then we have

M(λ) = ⋂_{i=1}^{s} Ker_{M}(∇_{i} − λ_{i})^{α_{i}} ,

for any λ ∈ K^{s}. Therefore (cf. (3.2.5.3) and (3.2.7)) for any λ ∈ K^{s}, the map φ ⟼ φ(1_{N_{λ}^{(α)}}) induces an isomorphism

Hom_{A[x]}(N_{λ}^{(α)}, M)^{∇_{B/A}} ⟶^{∼}M(λ) .

If M is an object of Mod_{Σ}^{f}(B, L_{x})

Co(M) = ⋃_{λ∈E(M)+N^{s}}(⋂_{i=1}^{s} Ker_{M}(∇_{i} − λ_{i})^{α_{i}}) .

Proof. We only have to prove the first assertion. Both sides in that equality are I-adically closed. All we have to prove is then equality of their images in M/I^{r+h}M for all h ∈ Z_{>0}. By corollary 1.8.3,

⋂_{i=1}^{s} Ker_{M}(^{s}∇_{i}(r + h) − λ_{i}) = ⋂_{i=1}^{s} Ker_{M}(∇_{i}(r + h) − λ_{i})^{α_{i}} .

This proves our claim. Q.E.D.

3.5. We now give conditions on an object M of Mod_{Σ}^{f}(B, L_{x}) for being a projective B-module. We first recall how the assumption that M be a projective B-module simplifies the situation.

Proposition 3.5.1. Let M be an object of Mod_{Σ}^{fp}(B, L_{x}) and let M_{0} be chosen as in proposition 3.4.6. Then the following statements hold:

i) For any P ∈ SpecA, the rank in P[[x]] of the projective B-module M equals the rank in P of the projective A-module M_{0}. ii) The canonical map B ⊗_{A} M_{0} ⟶ M is an isomorphism of B-modules. iii) For each λ ∈ E(M), let BM_{0}(λ) denote the B-sub-module of M spanned by M_{0}(λ). Then BM_{0}(λ) is projective and

M = ⨁_{λ∈E(M)} BM_{0}(λ) ≅ ⨁_{λ∈E(M)} B ⊗_{A} M_{0}(λ) ,

as B-modules.

Proof. See corollary 3.2.3. Q.E.D.

Lemma 3.5.2. Let M be an object of Mod_{Σ}^{f}(B, L_{x}). Assume that:

i) For λ, λ′ ∈ E(M) λ ≠ λ′ implies λ ≢ λ′; ii) T_{x}(M) = 0. Then M is a projective B-module and there is a unique A-sub-module M_{0} of M as in (3.4.6).

Proof. Certainly M = ∑_{λ∈E(M)} BM(λ). We may assume that A is a local ring. For any λ′ ∈ K^{s}, either M(λ′) = 0 or λ′ may be written as λ + α, with uniquely determined α ∈ N^{s} and λ ∈ E(M). Consider a relation

∑_{λ∈E(M)} ∑_{α∈N^{s}} x^{α}m_{λ,α} = 0 ,

with m_{λ,α} ∈ M(λ). We then deduce x^{α}m_{λ,α} = 0, and therefore m_{λ,α} = 0, for each α ∈ N^{s} and λ ∈ E(M). This proves that an A-basis of ∑_{λ∈E(M)} M(λ) = ⊕_{λ∈E(M)}M(λ) is B-free, hence is a B-basis of M. Q.E.D.

Corollary 3.5.3. Let M be an object of Mod_{Σ}^{f}(B, L_{x}) and λ ∈ K^{s}. Assume T_{x}(M) = 0. Then M[λ] (resp. M⟨λ⟩) is an A[x]-module locally free on A (resp. a projective B-module).

Definition 3.5.4. Let M be an object of Mod_{Σ}^{f}(B, L_{x}) and μ ∈ S(M). We define

M[μ] = ∑_{c∈C(M)} M[μ(c)]

(resp.

M⟨μ⟩ = ∑_{c∈C(M)} M⟨μ(c)⟩ )

(cf. definition 3.4.9.5).

Corollary 3.5.5. Let M, μ be as in definition 3.5.4. Then M[μ] (resp. M⟨μ⟩) is a sub-object of M in Mod_{Σ}^{f}(A[x], L_{x}) (resp. Mod_{Σ}^{f}(B, L_{x})) and a projective A[x]-module (resp. B-module). The quotient M/M[μ] (resp. M/M⟨μ⟩) is an A[x]-module (resp. a B-module) annihilated by a monomial in x_{1}, …, x_{s}.

3.6.

Corollary 3.6.1. Let M be an object of Mod_{Σ}^{fp}(B, L_{x}) and M^{0} = Co(M) be as in ii) of (3.4.10). Let λ̃ ∈ E(M) be minimal and μ ∈ S(M). We put:

M^{n}(λ̃) = {m ∈ M(λ̃) | ^{n}∇_{i}(m) = 0 , for i = 1, …, s} =
= {m ∈ M | ^{s}∇_{i}(m) = λ̃_{i}m and ^{n}∇_{i}(m) = 0 , for i = 1, …, s} =
= {m ∈ M | ∇_{i}(m) = λ̃_{i}m and ^{n}∇_{i}(m) = 0 , for i = 1, …, s} ,
M^{n}⟨λ̃⟩ = BM^{n}(λ̃) ,
M^{n}[λ̃] = A[x]M^{n}(λ̃) .

Then we have:

i) M⟨λ̃⟩ is a non-zero sub-object of M in Mod_{Σ}^{f}(B, L_{x}); both M⟨λ̃⟩ and the quotient M/M⟨λ̃⟩ in Mod_{Σ}^{f}(B, L_{x}) are projective B-modules. i)′ M[λ̃] is a non-zero sub-object of M^{0} in Mod_{Σ}^{f}(A[x], L_{x}); both M[λ̃] and the quotient M^{0}/M[λ̃] in Mod_{Σ}^{f}(A[x], L_{x}) are A[x]-modules locally free on A. ii) M^{n}⟨λ̃⟩ is a non-zero sub-object of M⟨λ̃⟩ in Mod_{Σ}^{f}(B, L_{x}); both M^{n}⟨λ̃⟩ and the quotient M⟨λ̃⟩/M^{n}⟨λ̃⟩ in Mod_{Σ}^{f}(B, L_{x}) are projective B-modules. ii)′ M^{n}[λ̃] is a non-zero sub-object of M[λ̃] in Mod_{Σ}^{f}(A[x], L_{x}); both M^{n}[λ̃] and the quotient M[λ̃]/M^{n}[λ̃] in Mod_{Σ}^{f}(A[x], L_{x}) are A[x]-modules locally free on A. iii) M⟨μ⟩ is a sub-object of M in Mod_{Σ}^{f}(B, L_{x}); it is a projective B-module and T_{x}(M/M⟨μ⟩) = M/M⟨μ⟩. iii)′ M[μ] is a sub-object of M^{0} in Mod_{Σ}^{f}(A[x], L_{x}); it is an A[x]-module locally free on A and T_{x}(M^{0}/M[μ]) = M^{0}/M[μ].

Proof. The crucial point for proving the first four assertions is the fact that, for any choice of M_{0} as in (3.4.6), M(λ̃) = M_{0}(λ̃) (cf. definition 3.4.9.5 and corollary 1.8.4) and therefore

(3.6.1.1)  M⟨λ̃⟩ = BM_{0}(λ̃) , M[λ̃] = A[x]M_{0}(λ̃) .

This also shows that

M^{n}(λ̃) = {m ∈ M_{0}(λ̃) | ^{n}∇_{i}(m) = 0 , for i = 1, …, s} .

In particular, considering (3.5.1), we obtain the B-module decompositions:

(3.6.1.2)  M/M⟨λ̃⟩ ≅ ⨁_{λ∈E(M)\{λ̃}} B ⊗_{A} M_{0}(λ) ,

(3.6.1.3)  M^{0}/M[λ̃] ≅ ⨁_{λ∈E(M)\{λ̃}} A[x] ⊗_{A} M_{0}(λ) ,

which show that the modules considered in i) and i)′ are projective. For the assertions ii) and ii)′, we use the B-module decompositions

M⟨λ̃⟩/M^{n}⟨λ̃⟩ ≅ B ⊗_{A} (M(λ̃)/M^{n}(λ̃)) ,
M[λ̃]/M^{n}[λ̃] ≅ A[x] ⊗_{A} (M(λ̃)/M^{n}(λ̃)) ,

and the fact that the A-module M_{0}(λ̃)/M_{0}^{n}(λ̃) is projective since it is an object of Mod^{f}(A, L). The remaining assertions were already in (3.5.5). Q.E.D.

Remark 3.6.1.4. According to corollary 3.5.3, M[λ] (resp. M⟨λ⟩) is a subobject of M^{0} (resp. M) in Mod_{A,Σ}^{f}(A[x], L_{x}) (resp. Mod_{Σ}^{fp}(B, L_{x})), with no assumption on λ ∈ K^{s}. The additional hypothesis that λ̃ ∈ E(M) is minimal is here needed to guarantee, in view of (3.6.1.1,2,3), that also the quotient M^{0}/M[λ] (resp. M/M⟨λ⟩) is an object of Mod_{A,Σ}^{f}(A[x], L_{x}) (resp. Mod_{Σ}^{fp}(B, L_{x})).

Corollary 3.6.2. Assume that the object M of Mod_{Σ}^{fp}(B, L_{x}) satisfies the assumptions of lemma 3.5.2. Then M_{0} is uniquely determined by (3.4.6) and is stable under the action of the elements of L and of ^{n}∇_{i}, ^{s}∇_{i}, ∇_{i} for i = 1, …, s. Let us denote by \overline{M} the (A, L)-module M/IM. The canonical projection π_{1} induces an isomorphism of (A, L)-modules:

π_{1} : M_{0} ⟶ \overline{M}

such that:

π_{1}^{s}∇_{i} = ^{s}∇_{i}(1)π_{1}, π_{1}^{n}∇_{i} = ^{n}∇_{i}(1)π_{1} π_{1}∇_{i} = ∇_{i}(1)π_{1}

for i = 1, …, s.

Proof. Follows from corollary 3.4.10. Q.E.D.

Definition 3.6.3. Let N be an object of Mod^{f}(A, L, Φ). We associate to N an object N_{B}^{∗} (resp. N_{A[x]}^{∗}) of Mod^{fp}(B, L_{x}) (resp. Mod_{A}^{f}(A[x], L_{x})) on setting N_{B}^{∗} = B ⊗_{A} N (as a B-module), (resp. N_{A[x]}^{∗} = A[x] ⊗_{A} N (as an A[x]-module))

∂(b ⊗ n) = ∂b ⊗ n + b ⊗ ∂n

and

∇_{i}(b ⊗ n) = x_{i} \frac{∂b}{∂x_{i}} ⊗ n + b ⊗ Φ_{i}n

for ∂ ∈ L, i = 1, …, s, b ∈ B (resp. b ∈ A[x]) and n ∈ N. If there exists λ ∈ K^{s} such that

Φ_{i}n = λ_{i}n

for each n ∈ N, i = 1, …, s, the associated object N_{B}^{∗} (resp. N_{A[x]}^{∗}) of Mod^{fp}(B, L_{x})

(resp. Mod_{A}^{f}(A[x], L_{x})) will be called a λ-simple object.

Corollary 3.6.4. Let M be an object of Mod_{Σ}^{f}(B, L_{x}). We regard \overline{M} = M/IM as an A[Φ]-module (so an object of Mod^{f}(A, L, Φ), cf. (1.7)) by way of:

Φ_{i}\overline{m} = ∇_{i}(1)\overline{m} ,

for i = 1, …, s and \overline{m} ∈ \overline{M}.

i) In the situation of (3.5.2), M is a projective B-module and is isomorphic to (\overline{M})_{B}^{∗}. ii) For any object M of Mod_{Σ}^{f}(B, L_{x}) and any μ ∈ S(M) sufficiently big (cf. (3.4.5.2)), the assumptions of (3.5.2) are satisfied by the object M⟨μ⟩ defined in (3.5.4). iii) For any object M of Mod_{Σ}^{fp}(B, L_{x}) and any μ ∈ S(M), the assumptions of (3.5.2) are satisfied by the object M⟨μ⟩ defined in (3.5.4).

Proof. Part i) follows from corollary 3.6.2 and the previous definition. In part ii), we first choose any μ′ ∈ S(M) and a monomial x^{α}, α ∈ N^{s} such that x^{α}T_{x}(M) = 0. We then take μ = μ′ + α ∈ S(M) and apply remark 3.4.9.6. Part iii) is obvious. Q.E.D.

Corollary 3.6.5. Let M be an object of Mod_{Σ}^{fp}(B, L_{x}) and let λ̃ ∈ E(M) be minimal with respect to ≤. Let also M^{0} = Co(M) be as in ii) of (3.4.10), M^{n}(λ̃) be as in (3.6.1) and \overline{M} = M/IM be as in (3.6.4). We have:

i) The sub-object M^{n}⟨λ̃⟩ (resp. M^{n}[λ̃]) of M (resp. M^{0}), defined in (3.6.1) is λ̃-simple. ii) The quotient M/M^{n}⟨λ̃⟩ (resp. M^{0}/M^{n}[λ̃]) in Mod_{Σ}^{f}(B, L_{x}) (resp. Mod_{Σ}^{f}(A[x], L_{x})) is an object of Mod_{Σ}^{fp}(B, L_{x}) (resp. Mod_{A,Σ}^{f}(A[x], L_{x})). iii) Let

\overline{M}^{n}(λ̃) = ⋂_{i=1,...,s} Ker_{\overline{M}(λ̃)}(^{n}∇_{i}(1)) =

= ( ⋂_{i=1,...,s} Ker\overline{M}(^{s}∇_{i}(1) − λ̃_{i})) ∩ ( ⋂_{i=1,...,s} Ker\overline{M}(^{n}∇_{i}(1))) ,

a sub-object of \overline{M} in Mod^{f}(A, L, Φ). The canonical projection π_{1} induces an isomorphism in Mod^{f}(A, L):

π_{1} : M^{n}(λ̃) ⟶ \overline{M}^{n}(λ̃)

such that:

π_{1}∇_{i} = ∇_{i}(1)π_{1}

(both actions coincide with multiplication by λ̃_{i}) for i = 1, …, s. iv) There exists a unique morphism in Mod^{f}(B, L_{x}) (resp. Mod^{f}(A[x], L_{x})):

iλ̃ : \overline{M}^{n}(λ̃)_{B}^{∗} ⟶ M

(resp.

iλ̃ : \overline{M}^{n}(λ̃)_{A[x]}^{∗} ⟶ M^{0} )

which is compatible with the canonical projections to \overline{M}^{n}(λ̃) ↪ \overline{M}. The morphism iλ̃ is a monomorphism, its image is M^{n}⟨λ̃⟩, (resp. M^{n}[λ̃]) and its cokernel is an object of Mod_{Σ}^{fp}(B, L_{x}) (resp. Mod_{A,Σ}^{f}(A[x], L_{x})).

Definition 3.6.6. We define the length of an object M of Mod_{Σ}^{fp}(B, L_{x}) as a natural number τ(M) by the following conditions:

i) τ(0) = 0; ii) In the situation of part iv) of corollary 3.6.5, τ(M) = τ(Coker(iλ̃)) + 1.

It is easy to check that definition 3.6.6 is good.

Corollary 3.6.7. Let M be an object of Mod_{Σ}^{fp}(B, L_{x}) and M^{0} = Co(M) be as in ii) of (3.4.10). Then:

i) M^{0} is an iterated extension in Mod^{f}(A[x], L_{x}) of λ-simple objects, for λ ∈ E(M); ii) Let R ⊂ K^{s} denote any set of representatives of C(M) in K^{s}. In Mod_{A,Σ}^{f}(A[x]′, L_{x}) we have

(M^{0})′ ≅ ⊕_{λ∈R}(N_{λ})′ ,

where N_{λ} denotes a λ-simple object of Mod_{A,Σ}^{f}(A[x], L_{x}).

Proof. For the first assertion we use induction on the length, τ(M) (3.6.6), applying repeatedly part iv) of corollary 3.6.5. The second part follows from part iii)′ of corollary 3.6.1, since it is obvious in the case M = M[μ], for some μ ∈ S(M). Q.E.D.

3.7. We know from (an easy generalization to several variables of) [Ch2, Prop.1.1], that there exist uniquely determined integers c_{α,β} and d_{α,β}, for α ≥ β in N^{s} such that, as operators on any (K[x_{1}, …, x_{s}], ⊕_{i=1}^{s}K[x_{1}, …, x_{s}]\frac{∂}{∂x_{i}})-module:

(3.7.1)  (x∂/∂x)^{α} = ∏_{i=1}^{s}(x_{i}∂/∂x_{i})^{α_{i}} = ∑_{β≤α} c_{α,β}x^{β}(\frac{∂}{∂x})^{β} ,

(3.7.2)  x^{α}(∂/∂x)^{α} = ∏_{i=1}^{s} x_{i}^{α_{i}}(∂/∂x_{i})^{α_{i}} = ∑_{β≤α} d_{α,β}(x\frac{∂}{∂x})^{β} .

Let now M be a (K[x_{1}, …, x_{s}], ⊕_{i=1}^{s}K[x_{1}, …, x_{s}]x_{i} \frac{∂}{∂x_{i}})-module. We will define x^{α}(∂/∂x)^{α}, as a K-linear operator on M, via (3.7.2). Then (3.7.1) holds true in M and we have

(3.7.3) \frac{1}{α!}x^{α}(\frac{∂}{∂x})^{α}(fm) = ∑_{β≤α} \frac{1}{β!}x^{β}(\frac{∂}{∂x})^{β}(f)\frac{1}{(α − β)!}x^{α−β}(\frac{∂}{∂x})^{α−β}(m) ,

for any f ∈ K[x_{1}, …, x_{s}] and m ∈ M. As operators on M, we also have

(3.7.4)  x^{α}(\frac{∂}{∂x})^{α}x_{i}(∂/∂x_{i}) = α_{i}x^{α}(\frac{∂}{∂x})^{α} + x^{α+δ_{i}}(\frac{∂}{∂x})^{α+δ_{i}} ,

where δ_{i} ∈ N^{s} denotes the standard unit vector.

If M in particular is an object of Mod(B, L_{x}) (resp. of Mod(A[x], L_{x})) as defined in this section, then formula (3.7.3) holds for any f ∈ B (resp. f ∈ A[x]).

PART II: p-ADIC THEORY.

§4 Differential modules on a relative open polydisk with logarithmic singularities along the coordinate divisor.

4.1 Notation.

In this section we tie together the formal results of part I with the p-adic theory of systems of PDE’s with logarithmic singularities depending upon parameters of [Ba-Ct0]. Our aim is to describe the structure of certain objects in that category of systems, globally over a relative open polydisk of radius 1.

We first introduce some notation (cf. [Ba-Ct0]) and specialize some definitions given in part I.

Let p be a prime number and K be a field of characteristic zero complete under a non-archimedean absolute value

| | : K ⟶ R_{≥0}

extending the p-adic absolute value of Q_{p} and normalized by |p| = p^{−1}. We put, as in [Be], Γ_{0}^{∗} = |K^{×}| = a multiplicative subgroup of R_{>0}, and Γ^{∗} = Γ_{0}^{∗} ⊗_{Z} Q.

We fix an additive subgroup Σ of the additive group of K consisting of p-adically non-Liouville numbers and containing Z [Ba-Ct0, 1.2].

Let A be a regular affinoid K-algebra ([BGR, 7.3.2], [BBKN]); A is in particular a regular noetherian ring. We denote by S = SpA the smooth affinoid space associated to A [BGR,7.3.2]. As in the case of Zariski spaces, S is a finite disjoint union of open connected affinoid sub-domains: S = ∪_{j=1}^{b}S_{j}, where for j = 1, …, b, S_{j} = SpA_{j}, is the maximal spectrum of an integral regular affinoid K-algebra A_{j} (cf. [BGR, 7.1.2, Cor.8], [Fr, 3.6, ex.26]). So, A ≅ ⊕_{j=1}^{b}A_{j}, a direct sum of rings. We denote by | − |_{A} its supremum semi-norm: since A is reduced, it is a Banach K-algebra norm on A [BGR, 6.2.1, Prop.4]. All K-algebra Banach norms on A are equivalent [BGR, 6.1.3, Prop.2]. We denote by L_{A} the K-Lie algebra of continuous K-derivations on A. As we observed in (1.4.6.4), the triple (A, L_{A}, K) satisfies hypotheses 1.3.

We recall that for any affinoid K-algebra C, an element f ∈ C is invertible in C iff f does not vanish at any point of X = SpC. This is in fact a consequence of Hilbert’s Nullstellensatz for affinoid algebras [BGR, 7.1.3, Prop.1].

Let φ : D_{A}^{s} ⟶ S be the trivial bundle in open polydisks of radius 1 and relative dimension s over the K-affinoid base space S = SpA [Be, 0.1.5.2]. We have

D_{A}^{s} = SpA × D(0, 1^{−})^{s} ,

where φ = pr_{1} is the fibration map. We then use natural “relative” coordinates x_{1}, …, x_{s} on D_{A}^{s} and denote by i : S ⟶ D_{A}^{s} , p ⟼ (p, 0), for p ∈ S, the natural closed immersion (“zero section”). In D_{A}^{s} we consider the “coordinate divisor” Z of equation x_{1} … x_{s} = 0. We sometimes shortly denote by O the structural sheaf O_{D_{A}^{s}} of D_{A}^{s}.

The ring O(D_{A}^{s}) of global analytic functions on D_{A}^{s} may be identified with a sub-ring

A{{x}} = A{{x_{1}, …, x_{s}}}

(A{{x}}_{1} in the notation of [Ba-Ct0]) of the ring of formal power series A[[x]] = A[[x_{1}, …, x_{s}]] given by:

(4.1.1) A{{x}} = { ∑_{α∈N^{s}} a_{α}x^{α}, a_{α} ∈ A | lim_{|α|_{R}→+∞} |a_{α}|_{A}η^{|α|_{R}} = 0 , ∀η ∈ [0, 1)} .

We pointed out in [Ba-Ct3, 5.10.1] that A{{x}} is a flat A-module (cf. lemma 4.1.6 below, for a short proof of this assertion).

For ϵ ∈ Γ^{∗}, let D_{A}^{s}(ϵ) denote the trivial bundle in closed polydisks of radius ϵ over S. We have, for a ∈ K with |a| = ϵ^{m}, m ∈ N:

(4.1.2)  \begin{matrix}O(D_{A}^{s}(ϵ) & ) = A\{x_{1}, …, x_{s}, y_{1}, …, y_{s}\}/(x_{i}^{m} − ay_{i}, i = 1, …, s) = \\ = \{ ∑_{α∈N^{s}} a_{α}x^{α}, a_{α} ∈ A | lim_{|α|_{R}→+∞} |a_{α}|_{A}ϵ^{|α|_{R}} = 0 \} . \end{matrix}

The latter is a K-affinoid algebra complete under the K-algebra norm:

(4.1.2.1)  | ∑ a_{α}x^{α}|_{ϵ} = sup_{α} |a_{α}|_{A}ϵ^{|α|_{R}} .

From (4.1.2), we see that an element f ∈ O(D_{A}^{s}(ϵ)) is divisible by a monomial x^{α}, α ∈ N^{s} in O(D_{A}^{s}(ϵ)) iff it is so in A[[x_{1}, …, x_{s}]]. It’s known from [R, Chap. II, 3, Th.2] that O(D_{A}^{s}(ϵ)) is flat over A (cf. [Ba-Ct3, 5.9.2]). For ϵ ∈ Γ^{∗} ∩ (0, 1), there are natural restriction maps (imbeddings of rings) A{{x}} ↪ O(D_{A}^{s}(ϵ)), corresponding to the open immersions D_{A}^{s}(ϵ) ⊂ D_{A}^{s}. If we choose a sequence ϵ_{n} ∈ Γ^{∗} ∩ (0, 1), n ∈ N, ϵ_{n} ↑ 1, then we have

(4.1.3)  A{{x}} = lim_{⟵ n} O(D_{A}^{s}(ϵ_{n})) = ⋂_{ϵ∈Γ^{∗}∩(0,1)} O(D_{A}^{s}(ϵ)) .

In fact {D_{A}^{s}(ϵ_{n})}_{n} is an admissible affinoid covering of D_{A}^{s} [Ta]. We deduce from the previous formula that an element f ∈ A{{x}} is invertible (resp. divisible by a monomial x^{α}, α ∈ N^{s}) in A{{x}} iff f does not vanish at any point x ∈ D_{A}^{s} (resp. iff it is so in A[[x_{1}, …, x_{s}]]).

We emphasize the natural ring imbeddings:

(4.1.3.1)  A{{x}} ↪ O(D_{A}^{s}(ϵ)) ↪ A[[x_{1}, …, x_{s}]] ,

that present A[[x_{1}, …, x_{s}]] as the (x_{1}, …, x_{s})-adic completion of the ring A{{x}}. We recall however that the ring A{{x}} is not noetherian; we ignore, in particular, whether or not A[[x_{1}, …, x_{s}]] is flat over A{{x}}.

The ring A{{x}} is equipped with the family of norms |−|_{ϵ}, for ϵ ∈ (0, 1), introduced in (4.1.2.1). These norms define a complete uniform structure which induces the projective limit topology of A{{x}} (cf. (4.1.3)), where each O(D_{A}^{s}(ϵ)) is equipped with its topology as an affinoid K-algebra. The imbeddings (4.1.3.1) may be extended to functions with meromorphic singularities along Z; we then get an imbedding of rings

(4.1.3.2)  A\{\{x\}\}[\frac{1}{x}] = A\{\{x\}\}′ ↪ A[[x_{1}, …, x_{s}]][\frac{1}{x}] = B′ .

On the other hand from the closed immersion i : S ⟶ D_{A}^{s} we obtain the natural surjection

ρ_{0} : A{{x}} ⟶ A
f(x) ⟼ f(0) .

4.1.4. Let M be an A{{x}}-module (resp. M_{0} be an A-module). We define i^{∗}M = M ⊗_{A{{x}}} A, where the tensor product is taken via ρ_{0} : A{{x}} ⟶ A (resp. φ^{∗}M_{0} = M_{0} ⊗_{A} A{{x}}). For any A-module M_{0} there is a canonical isomorphism i^{∗}(φ^{∗}M_{0}) ≅ M_{0}. Let Mod(A{{x}}) be the abelian category of A{{x}}- modules and Mod_{A}^{f}(A{{x}}) be the full (additive) sub-category of Mod(A{{x}}) whose objects are of the type M = φ^{∗}M_{0}, for an A-module M_{0} of finite type. Since A{{x}} is flat over the noetherian ring A (cf. [Ba-Ct3, 5.10.1] or (4.1.6) below), by [AC, I.2.10, Prop.11]

(4.1.4.1)  Hom_{A{{x}}}(φ^{∗}M_{0}, φ^{∗}N_{0}) ≅ φ^{∗}(Hom_{A}(M_{0}, N_{0})) ,

for A-modules M_{0}, N_{0} of finite type. Under the same assumptions

Hom_{A[[x]]}((φ^{∗}M_{0})⊗_{A{{x}}}A[[x]], (φ^{∗}N_{0})⊗_{A{{x}}}A[[x]]) ≅ Hom_{A}(M_{0}, N_{0})⊗_{A}A[[x]] ≅

(4.1.4.2)

≅ Hom_{A{{x}}}(φ^{∗}M_{0}, φ^{∗}N_{0}) ⊗_{A{{x}}} A[[x]] ,

as may again be seen by using flatness of A[[x]] over A. We point out that formula 4.1.4.2 would be a-priori clear, if we knew flatness of A[[x]] over A{{x}}.

Let M_{0} be an A-module of finite type. Then on M_{0} all complete A-module norms (also called A-Banach norms) are equivalent [BGR, 3.7.3, Prop.3] and define equivalent structures of K-Banach space on M_{0}. We denote by M_{0}˜ the coherent (cf. [BGR, 9.4]) O_{SpA}-module associated to M_{0}.

For a morphism of K-rigid analytic spaces, f : X ⟶ Y, we freely use the functor f^{∗} defined for O_{Y} -modules. In the particular case of f = i and f = φ, the previous use of the notation i^{∗} and φ^{∗} should create no confusion, in that it was there used for the modules of global sections of certain sheaves, rather than for the sheaves themselves. We need however to investigate in some detail the relation between the two meanings of the functors i^{∗} and φ^{∗}.

An object M = φ^{∗}M_{0} of Mod_{A}^{f}(A{{x}}) defines a coherent O_{D_{A}^{s}}-module M = φ^{∗}(M_{0}˜ ).

Lemma 4.1.5. Let M_{0} be an A-module of finite type and | − |_{M_{0}} denote any complete A-module norm on M_{0}. Let

φ_{ϵ} : D_{A}^{s}(ϵ) ⟶ S ,

denote the usual fibration in closed polydisks of radius ϵ ∈ Γ^{∗} ∩ (0, 1). Then i)

φ_{ϵ}^{∗}(M_{0}˜ )(D_{A}^{s}(ϵ)) = M_{0} ⊗_{A} O(D_{A}^{s}(ϵ)) .

We define, for simplicity:

φ_{ϵ}^{∗}(M_{0}) = φ_{ϵ}^{∗}(M_{0}˜ )(D_{A}^{s}(ϵ)) .

ii) We have a canonical imbedding of A-modules

φ_{ϵ}^{∗}(M_{0}) ↪ ∏_{α∈N^{s}} M_{0} ⊗ x^{α} .

iii) The elements of φ_{ϵ}^{∗}(M_{0}) are precisely the elements

(m_{α} ⊗ x^{α})_{α} ∈ ∏_{α∈N^{s}} M_{0} ⊗ x^{α} ,

such that

(4.1.5.1)  lim_{|α|_{R}→+∞} |m_{α}|_{M_{0}}ϵ^{|α|_{R}} = 0 .

iv) For m ∈ φ_{ϵ}^{∗}(M_{0}) we put

(4.1.5.2)  ||m||_{ϵ} = sup_{α∈N^{s}} |m_{α}|_{M_{0}}ϵ^{|α|_{R}} ,

if m = (m_{α} ⊗ x^{α})_{α}. Then || − ||_{ϵ} is a O(D_{A}^{s}(ϵ))-Banach norm on φ_{ϵ}^{∗}(M_{0}).

v) Every element m ∈ φ_{ϵ}^{∗}(M_{0}) can be uniquely written as a convergent sum

m = ∑_{α∈N^{s}} m_{α}x^{α} ,

with m_{α} ∈ M_{0}.

Proof. This is essentially Lemma 2.2.12 of [Be]. Q.E.D.

Lemma 4.1.6. Let M_{0} be an A-module of finite type and M_{0}˜ be the associated coherent O_{S}-module. Let T = SpC be an affinoid sub-domain of S = SpA. We recall the fibration

φ : D_{A}^{s} ⟶ S

and consider the coherent sheaf M = φ^{∗}(M_{0}˜ ) on D_{A}^{s} . We have:

M(φ^{−1}T) = M_{0} ⊗_{A} C{{x}} .

In particular (cf. [Ba-Ct3, Cor. 5.10, i)])

(4.1.6.1)  φ^{∗}(M_{0}) = M(D_{A}^{s}) = lim_{⟵ n} M(D_{A}^{s}(ϵ_{n})) .

for any sequence ϵ_{n} as in (4.1.3).

Proof. We first recall the proof of (4.1.6.1), as it appears in [Ba-Ct3, Remark 5.10.1]. The second equality of that statement follows from the fact that M is a sheaf on D_{A}^{s} and that {D_{A}^{s}(ϵ_{n})}_{n} is an admissible affinoid covering of D_{A}^{s}. We now get to the first equality. Let U = SpR be any affinoid open set of D_{A}^{s}. Then, by the maximum modulus principle, U ⊂ D_{A}^{s}(ϵ) for some ϵ ∈ Γ^{∗} ∩ (0, 1). Therefore ([BGR,7.3.2, Cor.6], [R, Chap. II, 3, Th.2]) the morphisms A ⟶ O(D_{A}^{s}(ϵ)) ⟶ R are flat. So, the functor (φ_{|U})^{∗} is an exact functor from the category of coherent O_{S}-modules to that of coherent O_{U}-modules. The functor φ^{∗} is therefore an exact functor from the category of coherent O_{S}-modules to that of coherent O_{D_{A}^{s}}-modules. One then recalls from [K, Satz 2.4], that the functor Γ(D_{A}^{s}, −) is also an exact functor from the category of coherent O_{D_{A}^{s}}- modules to that of O_{D_{A}^{s}}(D_{A}^{s})-modules. In the end M_{0} ⟼ M(D_{A}^{s}) represents an exact functor Ψ(M_{0}) in the A-module of finite type M_{0}. One then considers a presentation of M_{0}

(4.1.6.2)  F′ ⟶ F ⟶ M_{0} ⟶ 0 ,

with F′ and F finite and free over A. By applying to (4.1.6.2) on the one hand the exact functor Ψ(−) and on the other the a priori right-exact functor (−) ⊗_{A} O_{D_{A}^{s}}(D_{A}^{s}), and noticing that the two coincide on free A-modules of finite type, one deduces the first equality of (4.1.6.1). We wish to point out that this reasoning also proves flatness of O_{D_{A}^{s}}(D_{A}^{s}) over A.

We now come to the remaining assertions of the lemma. Let us denote by

φ_{C} : D_{C}^{s} ⟶ T ,

the fibration induced by φ on T, and let us put M_{0}_{C} = M_{0} ⊗_{A} C. Then, clearly,

φ^{∗}(M_{0}˜ )_{|D_{C}^{s}} = φ_{C}^{∗}(M_{0}_{C}˜ ) .

From (4.1.6.1) applied to C, we get

φ_{C}^{∗}(M_{0}_{C}˜ )(D_{C}^{s}) = M_{0}_{C} ⊗_{C} C{{x}} .

Q.E.D.

Corollary 4.1.7. Let M_{0} be an A-module of finite type and | − |_{M_{0}} denote any complete A-module norm on M_{0}. Let

φ : D_{A}^{s} ⟶ S ,

denote the standard fibration in open polydisks of radius 1. Then i)

φ^{∗}(M_{0}˜ )(D_{A}^{s}) = M_{0} ⊗_{A} O(D_{A}^{s}) = φ^{∗}(M_{0}) .

ii) We have a canonical imbedding of A-modules

φ^{∗}(M_{0}) ↪ ∏_{α∈N^{s}} M_{0} ⊗ x^{α} .

iii) The elements of φ^{∗}(M_{0}) are precisely the elements

(m_{α} ⊗ x^{α})_{α} ∈ ∏_{α∈N^{s}} M_{0} ⊗ x^{α} ,

such that

(4.1.7.1)  lim_{|α|_{R}→+∞} |m_{α}|_{M_{0}}ϵ^{|α|_{R}} = 0 , ∀ϵ ∈ (0, 1) .

iv) For m ∈ φ^{∗}(M_{0}) and ϵ ∈ (0, 1) we put

(4.1.7.2)  ||m||_{ϵ} = sup_{α∈N^{s}} |m_{α}|_{M_{0}}ϵ^{|α|_{R}} ,

if m = (m_{α} ⊗ x^{α})_{α}. Then (φ^{∗}(M_{0}), || − ||_{ϵ}) is a normed A{{x}}-module [BGR, 2.1.1, Def.1] if A{{x}} is viewed as a normed ring via the norm | − |_{ϵ}. The family of norms {|| − ||_{ϵ}}_{ϵ∈(0,1)} defines a complete uniform structure which induces the projective limit topology of φ^{∗}(M_{0}) (cf. (4.1.6.1)).

v) Every element m ∈ φ^{∗}(M_{0}) can be uniquely written as a convergent sum in the projective limit topology

m = ∑_{α∈N^{s}} m_{α}x^{α} ,

with m_{α} ∈ M_{0}.

Corollary 4.1.8. Let ψ : M ⟶ N be an injective morphism in Mod_{A}^{f}(A{{x}}). For any ϵ ∈ (0, 1) ∩ Γ^{∗} the morphism ψ_{ϵ} : M_{ϵ} ⟶ N_{ϵ} obtained from ψ by the base-change A{{x}} ⊂ O(D_{A}^{s}(ϵ)) is injective.

Proof. (Bosch) Let Ψ : M ⟶ N be the morphism of coherent O_{D_{A}^{s}}- modules corresponding to ψ via the construction of lemma 4.1.6. Let then K denote the kernel of Ψ. Then

0 ⟶ K ⟶ M ⟶ N

is an exact sequence of coherent O_{D_{A}^{s}}-modules and, since D_{A}^{s} is a rigid Stein space,

0 ⟶ Γ(D_{A}^{s}, K) ⟶ M ⟶ N

is an exact sequence of A{{x}}-modules [K, Satz 2.4]. On the other hand application of the functor Γ(D_{A}^{s}, −) to the previous sequence and (4.1.6.1) show that Γ(D_{A}^{s}, K) = 0, since ψ : M ⟶ N is injective. But then [K, Satz 2.4] implies that K = 0. We conclude that Ψ : M ⟶ N is a monomorphism of abelian sheaves on D_{A}^{s}, and therefore the morphism ψ_{ϵ}, obtained from Ψ by applying the left-exact functor Γ(D_{A}^{s}(ϵ), −) is also injective. Q.E.D.

Corollary 4.1.9. Let ψ : M ⟶ N be a morphism in Mod_{A}^{f}(A{{x}}). Let M_{0} (resp. N_{0}) be the A-module of finite type such that M = ψ^{∗}(M_{0}) (resp. N = ψ^{∗}(N_{0})) and | − |_{M_{0}} (resp. | − |_{N_{0}}) denote any complete A-module norm on M_{0} (resp. N_{0}). For ϵ ∈ (0, 1) ∩ Γ^{∗}, let || − ||_{M,ϵ} (resp. || − ||_{N,ϵ}) be the norm on M_{ϵ} = M ⊗_{A{{x}}} O(D_{A}^{s}(ϵ)) (resp. N_{ϵ} = N ⊗_{A{{x}}} O(D_{A}^{s}(ϵ))) introduced in (4.1.5.2), by use of the norm | − |_{M_{0}} (resp. | − |_{N_{0}}) on M_{0} (resp. N_{0}). Let ψ_{ϵ} : M_{ϵ} ⟶ N_{ϵ} be the morphism obtained from ψ by the base-change A{{x}} ⊂ O(D_{A}^{s}(ϵ)). Then:

i) ψ is continuous if M and N are equipped with the inverse limit topology of projective systems of normed modules over affinoid K-algebras. ii) For any ϵ ∈ (0, 1) ∩ Γ^{∗}, there exists a positive constant C_{ϵ} such that

||ψ(m)||_{N,ϵ} ≤ C_{ϵ}||m||_{M,ϵ} ,

for each m ∈ M. iii) For any ϵ ∈ (0, 1) ∩ Γ^{∗}, ψ_{ϵ} is injective. There exists a positive constant C′_{ϵ} such that

||m||_{M,ϵ} ≤ C′_{ϵ}||ψ(m)||_{N,ϵ} ,

for each m ∈ M.

Proof. We consider the commutative diagram

(4.1.9.1)  \begin{matrix}M & ⟶^{ψ} & N \\ ↓ & & ↓ \\ M_{ϵ} & ⟶^{ψ_{ϵ}} & N_{ϵ} . \end{matrix}

where the vertical maps are injective. The norm || − ||_{M,ϵ} is induced on M by its inclusion in M_{ϵ}. Assertion ii) then follows from [BGR, 3.7.3, Prop. 2], applied to the map ψ_{ϵ}. Assertion i) is a consequence of ii). The first assertion in iii) is precisely lemma 4.1.8. As for the second, we recall [loc. cit., Prop.3 and Cor.5] that the injective morphism M_{ϵ} ⟶ ψ_{ϵ}(M_{ϵ}) is an isomorphism, so that the norm || − ||_{M,ϵ} of M_{ϵ} is equivalent to the norm induced on M_{ϵ} by the norm || − ||_{N,ϵ} via the inclusion of M_{ϵ} in N_{ϵ}. This concludes the proof. Q.E.D.

Remark 4.1.10. The norms || − ||_{ϵ} introduced in corollary 4.1.7 on an object M of Mod_{A}^{f}(A{{x}}), depend upon the original choice of the complete A-module norm |−|_{M_{0}} on M_{0}. However, by corollary 4.1.8, two different choices of complete A-module norm on M_{0}, produce two norms, say ||−||_{ϵ} and ||−||′_{ϵ} which are related by:

|| − ||′_{ϵ} ≤ C_{ϵ}|| − ||_{ϵ} ≤ C′_{ϵ}|| − ||′_{ϵ} ,

for two positive constants C_{ϵ} and C′_{ϵ}. So, we will assume in the sequel that a choice of a complete A-module norm |−|_{M_{0}} has been made for any M_{0} as before. Our notation will not keep track of that choice.

4.2. We will still denote by L_{A} the K-Lie algebra of continuous derivations of A/K when extended to A{{x}} by the rule

d( ∑_{α∈N^{s}} a_{α}x^{α}) = ∑_{α∈N^{s}}(da_{α})x^{α}

for d ∈ L_{A} and ∑ a_{α}x^{α} ∈ A{{x}}. We also consider K-Lie-sub-algebra L_{A,x} of Der(A{{x}}/K) (and of Der(A[x]/K), Der(A[[x]]/K)) generated by L_{A} and the A-derivations x_{1} \frac{∂}{∂x_{1}} , …, x_{s} \frac{∂}{∂x_{s}}.

Let Mod_{A}^{f}(A{{x}}, L_{A,x}) denote the full (additive) sub-category of the abelian category Mod(A{{x}}, L_{A,x}) consisting of the objects M whose underlying A{{x}}- module is an object of Mod_{A}^{f}(A{{x}}). As in (3.2.6), one can argue that for every object M = M_{0} ⊗ A{{x}} in Mod_{A}^{f}(A{{x}}, L_{A,x}) the finite A-module M_{0} (welldefined up to isomorphism by M) is projective.

For an object M of Mod_{A}^{f}(A{{x}}, L_{A,x}) and for i = 1, …, s, we consider the K-endomorphisms

(4.2.1)  \begin{aligned}∇_{M,i} = ∇_{i} : & M & ⟶ & M \\ & m & ⟼ & x_{i} \frac{∂}{∂x_{i}} m . \end{aligned}

We then obtain a natural functor (cf. 1.7)

i_{diff}^{∗} : Mod_{A}^{f}(A{{x}}, L_{A,x}) ⟶ Mod^{f}(A, L_{A}, Φ) ,

which coincides with the i^{∗} defined in (4.1.4) at the level of underlying A{{x}}- modules, while the action of Φ_{i} on i_{diff}^{∗}M is induced by the action of ∇_{M,i}, for i = 1, …, s. We also define a functor

φ_{diff}^{∗} : Mod^{f}(A, L_{A}, Φ) ⟶ Mod_{A}^{f}(A{{x}}, L_{A,x}) ,

which again coincides with the φ^{∗} defined in (4.1.4) at the level of underlying A-modules, in the following way. Let N be an object of Mod^{f}(A, L_{A}, Φ) and b ⊗ n ∈ φ^{∗}N, b ∈ A{{x}}, n ∈ N; we put

∂(b ⊗ n) = ∂b ⊗ n + b ⊗ ∂n

and

∇_{i}(b ⊗ n) = x_{i} \frac{∂b}{∂x_{i}} ⊗ n + b ⊗ Φ_{i}n

for ∂ ∈ L_{A} and i = 1, …, s. The previous definition is clearly analogous to definition 3.6.7. If there exists λ ∈ K^{s} such that

Φ_{i}n = λ_{i}n

for each n ∈ N , i = 1, …, s, the associated object φ_{diff}^{∗}(N) of Mod_{A}^{f}(A{{x}}, L_{A,x}) will be called a simple object (or a λ-simple object, if specification of λ is necessary). 4.2.2. We define the multiexponents of an object M of Mod_{A}^{f}(A{{x}}, L_{A,x}) as the multieigenvalues of i_{diff}^{∗}M. We define the multiplicity of a multiexponent (at a point a of SpA) as in the case of objects of Mod_{A}^{f}(A[[x]], L_{A,x}), and we use the same notation. We will say that M has all of its multiexponents in Σ^{s}, when this assumption holds for the multiexponents of i_{diff}^{∗}(M). We will indicate by E(M) the set of the multiexponents of M. Let Mod_{A,Σ}^{f}(A{{x}}, L_{A,x}) denote the full (additive) sub-category of Mod_{A}^{f}(A{{x}}, L_{A,x}) consisting of the objects whose multi-exponents are all in Σ^{s}. Definition 3.4.5.1 introduces a partial ordering in K^{s}, and in particular in E(M), for any object M of Mod_{A,Σ}^{f}(A{{x}}, L_{A,x}).

4.2.3. The inclusion A{{x}} ↪ A[[x]] = B of 4.1 provides a natural functor

(4.2.3.1)  \begin{matrix}Mod^{f}(A\{\{x\}\}, L_{A,x}) & ⟶ Mod^{f}(B, L_{A,x}) \\ M & ⟼ M ⊗_{A\{\{x\}\}} B . \end{matrix}

When M = φ^{∗}(M_{0}), with M_{0} of finite type and locally free over A, the previous functor coincides with completion w.r.t. the (x_{1} …, x_{s})-adic topology, which we will denote by M ⟼ M̂. We restrict that functor to

(4.2.3.2)  \begin{matrix}Mod_{A}^{f}(A\{\{x\}\}, L_{A,x}) & ⟶ Mod^{fp}(B, L_{A,x}) \\ M & ⟼ M̂ = M ⊗_{A\{\{x\}\}} B . \end{matrix}

For an object M of Mod_{A}^{f}(A{{x}}, L_{A,x}) we obviously have E(M) = E(M̂). We also define C(M) = C(M̂) and S(M) = S(M̂) (cf. (3.4.5)). We can restrict the functor (4.2.3.2) also to

(4.2.3.3)  \begin{matrix}Mod_{A,Σ}^{f}(A\{\{x\}\}, L_{A,x}) & ⟶ Mod_{Σ}^{fp}(B, L_{A,x}) \\ M & ⟼ M̂ . \end{matrix}

(cf. (3.4.4) of part I). For an affinoid subdomain SpC of SpA, represented by the morphism of affinoid K-algebras A ⟶ C, we obtain the following commutative diagram

(4.2.3.4)  \begin{matrix}Mod_{A}^{f}(A\{\{x\}\}, L_{A,x}) & ⟶^{} & ˆ  Mod^{fp}(A[[x]], L_{A,x}) \\ ↓ & & ↓ \\ Mod_{C}^{f}(C\{\{x\}\}, L_{C,x}) & ⟶^{} & ˆ  Mod^{fp}(C[[x]], L_{C,x}) , \end{matrix}

where the vertical maps

M_{0} ⊗_{A} A{{x}} ⟼ M_{0} ⊗_{A} C{{x}}

and

M_{0} ⊗_{A} A[[x]] ⟼ M_{0} ⊗_{A} C[[x]]

are deduced from A ⟶ C (cf. (4.1.6)). None of the functors in diagram (4.2.3.4) affects the multiexponents: therefore diagram (4.2.3.4) induces a similar diagram for differential modules with multiexponents in Σ.

4.2.4. The natural inclusion maps

(4.2.4.1)  A[x] ↪ A{{x}} ↪ B

give origin to functors

(4.2.4.2) Mod_{A}^{f}(A[x], L_{A,x}) ⟶ Mod_{A}^{f}(A{{x}}, L_{A,x})⟶^{}ˆ Mod^{fp}(B, L_{A,x}) .

All the categories in (4.2.4.2) are additive and are endowed with well-known internal bifunctors Hom(−, −) and −⊗−. The functors in (4.2.4.2) are compatible with the previous internal bifunctors. We may restrict (4.2.4.2) to

Mod_{A,Σ}^{f}(A[x], L_{A,x}) ⟶ Mod_{A,Σ}^{f}(A{{x}}, L_{A,x})⟶^{}ˆ Mod_{Σ}^{fp}(B, L_{x}) .

4.2.5. We will also consider the natural extension functor (cf. (4.1.3.2))

(4.2.5.1)  \begin{matrix}Mod_{A}^{f}(A\{\{x\}\}, L_{A,x}) & ⟶ Mod(A\{\{x\}\}′, L_{A,x}) \\ M = M_{0} ⊗_{A} A\{\{x\}\} & ⟼ M′ = M_{0} ⊗_{A} A\{\{x\}\}′ . \end{matrix}

Then Mod_{A}^{f}(A{{x}}′, L_{A,x}) will denote the essential image of the previous functor. In a similar way we will understand the sub-category Mod_{A,Σ}^{f}(A{{x}}′, L_{A,x}) of Mod_{A}^{f}(A{{x}}′, L_{A,x}). The inclusions (4.2.4.1) induce inclusions

(4.2.5.2)  A[x]′ ↪ A\{\{x\}\}′ ↪ A[[x]][\frac{1}{x}] = B′ .

From (4.2.5.2) we obtain the following commutative diagram

(4.2.5.3)  \begin{matrix}Mod_{A}^{f}(A\{\{x\}\}, L_{A,x}) & ⟶^{} & ˆ  Mod^{fp}(B, L_{A,x}) \\ ↓ & & ↓ \\ Mod_{A}^{f}(A\{\{x\}\}′, L_{A,x}) & ⟶ & Mod(B′, L_{A,x}) . \end{matrix}

We clearly have a similar diagram for differential modules with multiexponents in Σ.

§5 Overconvergence.

We recall that, for any α ∈ N^{s}, the action of x^{α}(∂/∂x)^{α} is well defined, by (3.7), on any object M of Mod(A{{x}}, L_{A,x}). This action satisfies formula (3.7.3), for any f ∈ A{{x}}. We also recall the notation of corollary 4.1.7, and, in particular, the convention of remark 4.1.10.

Definition 5.1. An object M of Mod_{A}^{f}(A{{x}}, L_{A,x}) will be said to be overconvergent if the following condition is satisfied (SC)_{L} ∀m ∈ M and ∀ϵ , η ∈ (0, 1):

lim_{|α|_{R}→∞} ||x^{α}(∂/∂x)^{α}(m)/α!||_{ϵ}η^{|α|_{R}} = 0 .

From corollary 4.1.9, ii), we easily deduce that that if a morphism φ : M ⟶ N in Mod_{A}^{f}(A{{x}}, L_{A,x}) is a surjective map and condition (SC)_{L} holds for M, then it also holds for N. On the other hand from 4.1.9, iii) we see that, if φ : M ⟶ N is an injective map, and condition (SC)_{L} holds for N, then it also holds for M.

For an object M of Mod_{A}^{f}(A{{x}}), and ϵ ∈ (0, 1) the norm || − ||_{ϵ} induces a norm (sup-norm on the unit ball) on the space of continuous || − ||_{ϵ}-bounded Klinear operators on M. This norm will still be denoted by || − ||_{ϵ}. As a particular case, on the space M_{n}(A{{x}}) of n × n matrices with entries in A{{x}}, we will in general use the norms ||(a_{ij})||_{ϵ} = sup_{i,j} |a_{ij}|_{ϵ}.

Proposition 5.2. The following assertions are equivalent for an object M of Mod_{A}^{f}(A{{x}}, L_{A,x}).

i) M is overconvergent; ii) condition (SC)_{L} holds for m ∈ M running over a system of generators of M; iii) ∀ϵ , η ∈ (0, 1)

lim_{|α|_{R}→∞} ||x^{α}(∂/∂x)^{α}/α!||_{ϵ}η^{|α|_{R}} = 0 .

Proof. All statements are immediate consequences of (3.7.3) and the fact that for any f ∈ A{{x}}, β ∈ N^{s} and ϵ ∈ Γ^{∗} ∩ (0, 1)

|\frac{1}{β!}x^{β}(\frac{∂}{∂x})^{β}(f)|_{ϵ} ≤ |f|_{ϵ}

(cf. (4.1.2.1)). Q.E.D.

We denote by SCMod_{A}^{f}(A{{x}}, L_{A,x}) (resp. SCMod_{A,Σ}^{f}(A{{x}}, L_{A,x})) the full (additive) subcategory of Mod_{A}^{f}(A{{x}}, L_{A,x}) (resp. Mod_{A,Σ}^{f}(A{{x}}, L_{A,x})) consisting of the overconvergent objects.

Lemma 5.2.1. Let M_{0} be a projective A-module of finite type, M = M_{0} ⊗_{A} A{{x}} be the corresponding object of Mod_{A}^{f}(A{{x}}), and M̂ = M ⊗_{A{{x}}} B be the (x_{1}, …, x_{s})-adic completion of M. Consider a finite affinoid covering {U_{j} = SpA_{j}}_{j=1,...,b} of S = SpA. For j = 1, …, b, we put M_{0j} = M_{0} ⊗_{A} A_{j}, an object of Mod^{fp}(A_{j}), M_{j} = M_{0j} ⊗_{A_{j}} A_{j}{{x}}, the corresponding object of Mod_{A_{j}}^{f}(A_{j}{{x}}), and M̂_{j} = M_{j} ⊗_{A_{j}{{x}}} A_{j}[[x]], the (x_{1}, …, x_{s})-adic completion of M_{j}. For j = 1, …, b, we denote by ρ_{j} : M ⟶ M_{j} (resp. ρ̂_{j} : M̂ ⟶ M̂_{j}) the natural maps. We obtain the commutative diagrams:

(5.2.1.1)  \begin{matrix}M & ↪ & M̂ \\ ↓^{ρ_{j}} & & ↓ρ̂_{j} \\ M_{j} & ↪ & M̂_{j} . \end{matrix}

Then an element of m̂ ∈ M̂ is in M iff ρ̂_{j}(m̂) is in M_{j} for all j = 1, …, b.

Proof. Let | − |_{M_{0}} (resp. | − |_{M_{0j}}, for j = 1, …, b) be any A-Banach (resp. A_{j}-Banach) norm on M_{0} (resp. M_{0j}). We consider the continuous morphism

\begin{matrix}∏ ρ_{j} : M_{0} & ⟶ ∏ M_{0j} \\ m & ⟼ ∏ ρ_{j}(m) , \end{matrix}

where the right hand term is endowed with the product norm of the norms | − |_{M_{0j}} for j = 1, …, b. We recall [BGR, 7.3.2, Cor.5] that the morphism ∏ ρ_{j} is injective. Then, by Tate’s Acyclicity Theorem, ([BGR, 8.2], [Ta]), the set-theoretic image of ∏ ρ_{j} is a closed subspace of ∏ M_{0j}. Therefore |m|′_{M_{0}} = sup_{j=1,...,b} |ρ_{j}(m)|_{M_{0j}}, for any m ∈ M_{0}, defines an A-Banach norm | − |′_{M_{0}} on M_{0}. Then |−|_{M_{0}} and |−|′_{M_{0}} are equivalent. The statement then follows from the fact that for any ϵ ∈ Γ^{∗} ∩ (0, 1) the two norms obtained on φ_{ϵ}^{∗}(M_{0}) via (4.1.5.2) from | − |_{M_{0}} and | − |′_{M_{0}} are also equivalent. Q.E.D.

We may also state a local version of definition 5.1.

Proposition 5.3. Let M = M_{0} ⊗_{A} A{{x}} be an object of Mod_{A}^{f}(A{{x}}, L_{A,x}). Consider a finite affinoid covering {U_{j} = SpA_{j}}_{j=1,...,b} of S = SpA. For j = 1, …, b, let M_{0j} and M_{j} be as in lemma 5.2.1. Then M satisfies (SC)_{L} if and only if M_{j} satisfies (SC)_{L}, for all j = 1, …, b.

Proof. It follows from the same arguments used in the proof of lemma 5.2.1. Q.E.D.

Remark 5.4. Let M be an object of Mod_{A}^{f}(A{{x}}, L_{A,x}) and suppose that the underlying A{{x}}-module is free with basis e = (e_{1}, …, e_{n}); (so e is a row of elements of M). We write

(5.4.1)  x^{α}(\frac{∂}{∂x})^{α}(e) = eG_{α}(x) ,

for matrices G_{α}(x) ∈ M_{n}(A{{x}}). In particular, G_{0} = I_{n}. We wish to point out that as a consequence of (3.7.4) we have

(5.4.2)  α_{i}G_{α}(x) + G_{α+δ_{i}}(x) = ∑_{β+γ=α} \frac{α!}{β!γ!}G_{β}(x)x^{γ}(\frac{∂}{∂x})^{γ}(G_{δ_{i}}) ,

for i = 1, …, s. On the other hand clearly

(5.4.3)  α_{i}G_{α}(x) + G_{α+δ_{i}}(x) = G_{δ_{i}}(x)G_{α}(x) + x_{i} \frac{∂}{∂x_{i}} G_{α}(x) .

Then M is overconvergent iff

(5.4.4)  lim_{|α|_{R}→∞} ||\frac{1}{α!}G_{α}||_{ϵ}η^{|α|_{R}} = 0 ,

for all ϵ ∈ Γ^{∗} ∩ (0, 1), η ∈ (0, 1). This is clearly equivalent to the assertion that the formal power series

(5.4.5)  S_{M,e}(x, t) = ∑_{α∈N^{s}} G_{α}(x)\frac{t^{α}}{α!} ∈ M_{n}(A[[x, t]]) ,

where M_{n}(A[[x, t]]) = M_{n}(A[[x_{1}, …, x_{s}, t_{1}, …, t_{s}]]) is actually in M_{n}(A{{x, t}}) = M_{n}(O(D_{A}^{2s})). From formula 5.4.2 one easily deduces that the series S_{M,e}(x, t) ∈ M_{n}(A[[x, t]]) satisfies the system of P.D.E.’s

(5.4.6)  (1 + t_{i})\frac{∂}{∂t_{i}} S_{M,e}(x, t) = S_{M,e}(x, t)G_{δ_{i}}(xt + x) ,

for i = 1, …, s, where δ_{i} denotes the standard unit s-vector. From formulas 5.4.3 and 5.4.6 we then obtain

(5.4.7)  x_{i} \frac{∂}{∂x_{i}} S_{M,e}(x, t) = S_{M,e}(x, t)G_{δ_{i}}(xt + x) − G_{δ_{i}}(x)S_{M,e}(x, t) .

for i = 1, …, s. Let s_{M,e}(x, t) = det S_{M,e}(x, t) and g_{δ_{i}}(z) = Tr G_{δ_{i}}(z). Then s_{M,e}(x, t) ∈ A[[x, t]] satisfies the system of P.D.E.’s

(5.4.8)  \begin{aligned}(1 + t_{i})\frac{∂}{∂t_{i}} s_{M,e}(x, t) & = g_{δ_{i}}(xt + x)s_{M,e}(x, t) \\ \frac{∂}{∂x_{i}} s_{M,e}(x, t) & = x_{i}^{−1}(g_{δ_{i}}(xt + x) − g_{δ_{i}}(x))s_{M,e}(x, t) \end{aligned}

for i = 1, …, s. Notice that the integrability conditions imply that x_{i}|\frac{∂}{∂x_{j}} g_{δ_{i}}(x), for i ≠ j. Therefore (5.4.8) is a system of P.D.E.’s with holomorphic coefficients and no singularities on D_{A}^{2s}. Assume now that M is overconvergent, so that the matrix S_{M,e}(x, t) is holomorphic in D_{A}^{2s}. The solution s_{M,e}(x, t) of (5.4.8), holomorphic in D_{A}^{2s}, cannot possibly vanish on D_{A}^{2s}. This is well-known when A = K is a field. The general case can immediately be reduced to the previous one. Therefore (cf. the discussion in 4.1), s_{M,e}(x, t) is a unit of A{{x, t}} and we conclude that if S_{M,e}(x, t) is in M_{n}(A{{x, t}}), then it is automatically in GL_{n}(A{{x, t}}).

Lemma 5.5.1. Let M be an object of Mod_{A}^{f}(A{{x}}, L_{A,x}) and suppose that the underlying A{{x}}-module is free with basis e, as in remark 5.4. Let M^{∨} = Hom_{A{{x}}}(M, A{{x}}) be the dual object of Mod_{A}^{f}(A{{x}}, L_{A,x}), and let e^{∨} denote the basis of M^{∨} dual to e. Then

S_{M,e}(x, t)^{t}S_{M^{∨},e^{∨}}(x, t) = I_{n} .

As a consequence, M is overconvergent iff so is M^{∨}.

Proof. The first statement is purely formal, while the second follows from the previous remark. Q.E.D.

Corollary 5.5.2. The category SCMod_{A}^{f}(A{{x}}, L_{A,x}) is stable under the functor M ⟼ M^{∨}. Ditto for the subcategory SCMod_{A,Σ}^{f}(A{{x}}, L_{A,x})).

Proof. This follows from the lemma with the help of proposition 5.3. Q.E.D.

Theorem 5.6. The category SCMod_{A}^{f}(A{{x}}, L_{A,x}) is stable under the bifunctors Hom_{A{{x}}}(−, −) and − ⊗_{A{{x}}} − . The same assertion holds for the subcategory SCMod_{A,Σ}^{f}(A{{x}}, L_{A,x})).

Proof. By the previous corollary, we are reduced to proving the assertion for − ⊗_{A{{x}}} −. Let then M, N be objects of Mod_{A}^{f}(A{{x}}, L_{A,x}) (resp. Mod_{A,Σ}^{f}(A{{x}}, L_{A,x})). Suppose they both satisfy (SC)_{L}. Then for m ⊗ n ∈ M ⊗_{A{{x}}} N and α ∈ N^{s}

\frac{1}{α!}x^{α}(\frac{∂}{∂x})^{α}(m ⊗ n) = ∑_{β≤α} \frac{1}{β!}x^{β}(\frac{∂}{∂x})^{β}(m) ⊗ \frac{1}{(α − β)!}x^{β}(\frac{∂}{∂x})^{α−β}(n) .

To check condition (SC)_{L} for M ⊗ N, we write M = φ^{∗}(M_{0}), N = φ^{∗}(N_{0}), so that M ⊗ N = φ^{∗}(M_{0} ⊗_{A} N_{0}). Given A-Banach norms | − |_{M_{0}} on M_{0} and | − |_{N_{0}} on N_{0}, we use on M_{0} ⊗_{A} N_{0} the tensor A-Banach norm | − |_{M_{0}⊗_{A}N_{0}} [BGR, 2.1.7 and 3.7.3, Prop.6]. This has the property that, for any m ∈ M_{0} and n ∈ N_{0},

|m ⊗ n|_{M_{0}⊗_{A}N_{0}} ≤ |m|_{M_{0}}|n|_{N_{0}} .

Given ϵ ∈ Γ^{∗} ∩ (0, 1), we then derive from (4.1.5.2) norms || − ||_{M,ϵ}, || − ||_{N,ϵ}, || − ||_{M⊗N,ϵ} on M, N, M ⊗ N, respectively. For these norms, and any m ∈ M, n ∈ N, we clearly have

||m ⊗ n||_{M⊗N,ϵ} ≤ ||m||_{M,ϵ}||n||_{N,ϵ} .

The assertion now follows immediately. Q.E.D.

From now on definition 3.2.6.2 is meant to be extended to the case of C = A{{x}} and C = A{{x}}′.

Remark 5.7. Let M be an object of Mod_{A}^{f}(A{{x}}, L_{A,x}) and suppose that the underlying A{{x}}-module is free with basis e, as in remark 5.4. An element m = ey of M, for y = y(x) ∈ A{{x}}^{n}, is horizontal relative to A iff, in the notation of [Ba-Ct0, 4.4], y(x) satisfies the standard system of PDE’s on A{{x}} relative to A with logarithmic singularities at Z

(5.7.1)  x_{i} \frac{∂}{∂x_{i}} y = −^{t}G_{δ_{i}}(x)y ,

for i = 1, …, s. Using (5.5.1), one immediately checks that M is overconvergent iff, in the notation of [Ba-Ct0, 4.5], the system (5.7.1) is in _{A,s}S_{A_{1}}. We will refer to (5.7.1) as to “the standard system associated to the sections of M horizontal relative to A, w.r.t. the basis e”. We deduce from [Ba-Ct0, 5.2] that the multiexponents of an object of SCMod_{A}^{f}(A{{x}}, L_{A,x}) are automatically in Z_{p}^{s}. Therefore SCMod_{A,Σ}^{f}(A{{x}}, L_{A,x}) = SCMod_{A,Σ∩Z_{p}}^{f}(A{{x}}, L_{A,x}).

Remark 5.8. A λ-simple object (4.2) M of Mod_{A}^{f}(A{{x}}, L_{A,x}) is overconvergent if and only if λ ∈ Z_{p}^{s}. According to the previous remark, we only have to show the “if” part. By proposition 5.3, we may assume M to be free with basis e = (e_{1}, …, e_{n}). Then

(5.8.1)  S_{M,e}(x, t) = ∑_{α∈N^{s}} \binom{λ}{α}I_{n}t^{α} = ∑_{α∈N^{s}} ∏_{i=1}^{s} \binom{λ_{i}}{α_{i}}I_{n}t^{α} ,

which is an element of M_{n}(Z_{p}[[t]]) ⊂ M_{n}(A{{x, t}}).

Theorem 5.9. Let M be an object of SCMod_{A,Σ}^{f}(A{{x}}, L_{A,x}) and let M̂ be its (x_{1}, …, x_{s})-adic completion, an object of Mod_{Σ}^{fp}(B, L_{x}). The natural imbedding

M ↪ M̂

induces an A-linear isomorphism

M^{∇_{A{{x}}/A}} ⟶^{∼}M̂^{∇_{B/A}} .

Proof. Write M = M_{0} ⊗_{A} A{{x}} and let m̂ ∈ M̂ = M_{0} ⊗_{A} A[[x]] be horizontal relative to A. We need to show that m̂ ∈ M. By lemma 5.2.1, this may be checked locally on A. We may then assume that M_{0} is free of rank, say, n. At this point the problem amounts to showing that a formal solution in A[[x]]^{n} of a system of the type _{A,s}S_{A_{1}} with multiexponents non Liouville numbers is automatically an element of A{{x}}^{n}: this is theorem 4.9.1 of [Ba-Ct0]. Q.E.D.

§6 Structure of overconvergent differential modules.

In this section we show how the formal results on the structure of differential modules with logarithmic singularities depending upon parameters given in the first part of this paper extend to the category SCMod_{A}^{f}(A{{x}}, L_{A,x}), under the condition of p-adic non-Liouvilleness of all relevant multiexponents. We first give a sufficient condition for overconvergence.

Proposition 6.1. Let M be an object of Mod_{A}^{f}(A{{x}}, L_{A,x}). Suppose there exists an A-sub-module of finite type \overline{M} ⊂ M such that the following conditions hold:

\overline{M} ⊗_{A} A\{\{x\}\} = M ;

x_{i} \frac{∂}{∂x_{i}} \overline{M} ⊆ \overline{M} ,

for i = 1, …, s;

∂\overline{M} ⊆ \overline{M} ,

for all ∂ ∈ L_{A}. Suppose furthermore that the multiexponents of M are in Z_{p}^{s}. Then M is an object of SCMod_{A}^{f}(A{{x}}, L_{A,x}).

Proof. By proposition 5.3, the statement is local w.r.t. the strong Grothendieck topology of SpA, which is finer than the Zariski topology. On the other hand \overline{M} is a projective A-module of finite type because it is isomorphic as an A-module to the (A, L_{A})-module M/(x_{1}, …, x_{s})M. We may then suppose \overline{M} A-free and we will pick an A-basis e of \overline{M} which will then also be an A\{\{x\}\}-basis of M. It then follows that the actions of the various x_{i} \frac{∂}{∂x_{i}}, for i = 1, …, s, on M will be represented in terms of the basis e by matrices with entries in A and eigenvalues in Z_{p}, namely

x_{i} \frac{∂}{∂x_{i}} e = eG_{δ_{i}}

for i = 1, …, s, with G_{δ_{i}} ∈ M_{n}(A).

At this point, by remark 5.7, it is enough to check that the standard system associated to the sections of M horizontal relative to A w.r.t. the basis e

(6.1.1)  x_{i} \frac{∂}{∂x_{i}} y = −^{t}G_{δ_{i}}y

for i = 1, …, s, is in _{A,s}S_{A_{1}} [Ba-Ct0, 4.5]. After possible further localization w.r.t. the Grothendieck topology of SpA, we may also assume that A is an integral domain. The norm | − |_{A} is then the sup of a finite number l of nonequivalent absolute values | − |_{j}, for j = 1, …, l [Ba-Ct0, 4.1]. For each j = 1, …, l we denote by K_{j} the completion of the quotient field Q(A) of A w.r.t. the absolute value |−|_{j}. As we noticed in [Ba-Ct0, 5.1], we may reduce to proving the claim when A is replaced by any of the fields K_{j} for j = 1, …, l. We are therefore reduced to the case A = K. We now apply the following obvious generalization of [Ch1, Prop. 6.3.11] to several variables. For α = (α_{1}, …, α_{s}) ∈ N^{s} and y a solution of (6.1.1), we write

(α!)^{−1}x^{α} \frac{∂^{α}}{∂x^{α}} y = G_{α}^{∗}y ,

where G_{α}^{∗} ∈ M_{n}(A) and from (6.1.1) we get

G_{α}^{∗} = ∏_{i=1,...,s} \binom{−^{t}G_{δ_{i}}}{α_{i}} .

The conclusion follows from [Ch1, loc. cit.] Q.E.D.

We reformulate the previous proposition in the following way.

Corollary 6.1.2. Let N be an object of Mod^{f}(A, L_{A}, Φ) whose multiegenvalues all are in Z_{p}^{s}. Then the object φ_{diff}^{∗}(N) of Mod_{A}^{f}(A{{x}}, L_{A,x}) is overconvergent. In particular, if λ ∈ Z_{p}^{s}, a λ-simple object of Mod_{A}^{f}(A{{x}}, L_{A,x}) is automatically an object of SCMod_{A}^{f}(A{{x}}, L_{A,x}).

Corollary 6.1.3. If λ ∈ Z_{p}^{s} and α ∈ N^{s}, the (A{{x}}, L_{A,x})-module

N_{λ}^{c,(α)} = N_{λ}^{(α)} ⊗_{A[x]} A{{x}}

(cf. example 3.2.5.3) is an object of SCMod_{A,Z_{p}}^{f}(A{{x}}, L_{A,x}).

Proof. We take in the proposition M = N_{λ}^{c,(α)} and \overline{M} = ∑_{β∈N^{s},β≤α} A(x\frac{∂}{∂x})^{β}(1_{N_{λ}^{(α)}}). Q.E.D.

6.2. Let now M be an object of SCMod_{A,Σ}^{f}(A{{x}}, L_{A,x}) and let M̂ denote as before the (x_{1}, …, x_{s})-adic completion of M, viewed as an object of Mod_{Σ}^{fp}(A[[x]], L_{A,x}). We will regard M as canonically imbedded in M̂. We recall the (A[x], L_{A,x})-submodule Co(M̂) of M̂ introduced in definition 3.4.10.1. The action of x_{i} \frac{∂}{∂x_{i}}, for i = 1, …, s, on M̂, was denoted by ∇_{i} all over the first part of this paper. By (3.4.6) there exists a formal decomposition

(6.2.1)  M̂ = M_{0} ⊕ IM̂ ,

where M_{0} is a projective A-module of finite type stable under the “semisimple part”^{s}∇_{i} of the various ∇_{i}, for i = 1, …, s. We recall that, M̂ being a projective B-module, by proposition 3.5.1, M̂ = BM_{0} = M_{0} ⊗_{A} B, as a B-module. The multiexponents of M are in Σ^{s}, hence in K^{s}, so that (3.4.7.2) holds. Overconvergence of M implies that in fact those multiexponents are in (Σ ∩ Z_{p})^{s}. We will show that, for any possible decomposition as in (6.2.1), M_{0}, which is a priori only known to be an A-sub-module of M̂, is in fact contained in M (⊂ M̂). We will now exhibit two proofs of this fact. The first is contained in the following proposition.

Proposition 6.2.2. Let M be an object of SCMod_{A,Σ}^{f}(A{{x}}, L_{A,x}) and let M̂ denote its (x_{1}, …, x_{s})-adic completion. Then Co(M) ⊂ M.

Proof. It’s a consequence of corollary 3.4.14 and theorem 5.9. Q.E.D. We now come to a second proof of the fact that M_{0} ⊂ M. Let us pick a

μ ∈ S(M) as in corollary 3.6.1, part iii). Since Z ⊂ Σ, μ(c) ∈ (Σ ∩ Z_{p})^{s}, for all c ∈ C(M). We recall that the sub-object M̂⟨μ⟩ of M̂ was defined in (3.5.4). We also define, as in (3.6.4), an object N of Mod^{f}(A, L_{A}, Φ) by N = M̂⟨μ⟩/IM̂⟨μ⟩. Then, according to part ii) of corollary 3.6.4, we have

(6.2.3)  M̂⟨μ⟩ ≅ N_{B}^{∗} ,

where the functor N ⟼ N_{B}^{∗} was defined in (3.6.3). The previous isomorphism is uniquely determined by the condition that it induces the identity N = M̂⟨μ⟩/IM̂⟨μ⟩, via the base-change B = A[[x]] ⟶ A, f(x) ⟼ f(0). In particular, the decomposition (6.2.1) applied to M̂⟨μ⟩, gives

(6.2.4)  M̂⟨μ⟩ = \overline{N} ⊕ IM̂⟨μ⟩ ≅ N ⊗_{A} B ,

where now \overline{N} denotes the A-sub-module of M̂⟨μ⟩, which is the isomorphic image in M̂⟨μ⟩ of N, via the isomorphism (6.2.3). The A-sub-module \overline{N} of M̂⟨μ⟩ is stable under the action of the elements of the elements of L_{A} and of x_{i}∂/∂x_{i}, for i = 1, …, s. The projection map M̂⟨μ⟩ ⟶ M̂⟨μ⟩/IM̂⟨μ⟩ = N, may then viewed as inducing the given isomorphism \overline{N} ≅ N, compatible with the action of the elements of L_{A} and of x_{i}∂/∂x_{i}, for i = 1, …, s. Notice that we have imbeddings of (A[x], L_{A,x})-modules, identical on \overline{N} modulo the identification \overline{N} ≅ N:

(6.2.5) M̂[μ] = N_{A[x]}^{∗} ↪ φ_{diff}^{∗}(N) = A{{x}} ⊗_{A} N ↪ M̂⟨μ⟩ = N_{B}^{∗} ↪ M̂ .

Theorem 6.3. Let M be an object of SCMod_{A,Σ}^{f}(A{{x}}, L_{A,x}), μ ∈ S(M). Then, for any possible choice of the isomorphism (6.2.3), the image \overline{N} of N = M̂⟨μ⟩/IM̂⟨μ⟩ in M̂ is contained in M.

Proof. We know that M and φ_{diff}^{∗}(N) are objects of SCMod_{A,Σ}^{f}(A{{x}}, L_{A,x}). We consider

Hom_{A{{x}}}(φ_{diff}^{∗}(N), M)

which is also an object of SCMod_{A,Σ}^{f}(A{{x}}, L_{A,x}) (cf. Theorem (5.6)). By (4.1.4.2) we have

\begin{matrix}(Hom_{A\{\{x\}\}}(φ_{diff}^{∗}(N) & , M)) ⊗_{A\{\{x\}\}} A[[x]] = Hom_{A[[x]]}((φ_{diff}^{∗}(N)) ⊗_{A\{\{x\}\}} A[[x]], M̂) = \\ = Hom_{A[[x]]}(M̂⟨μ⟩, M̂) . \end{matrix}

According to our previous discussion, and in particular to (6.2.5), there exists an element κ of Hom_{A[[x]]}(M̂⟨μ⟩, M̂) which represents a sub-object M̂⟨μ⟩ ↪ M̂. The element κ ∈ Hom_{A[[x]]}(M̂⟨μ⟩, M̂) is in particular horizontal and, a fortiori, horizontal relative to A. Since Hom_{A{{x}}}(φ_{diff}^{∗}(N), M) is an object of SCMod_{A,Σ}^{f}(A{{x}}, L_{A,x}), we may apply theorem 5.9 to deduce that κ is actually in Hom_{A{{x}}}(φ_{diff}^{∗}(N), M). We conclude that κ(N) ⊂ M and therefore that \overline{N} is contained in M. Q.E.D.

Corollary 6.3.1. Let M be an object of SCMod_{A,Σ}^{f}(A{{x}}, L_{A,x}). Then for any choice of M_{0} as in (6.2.1), we have M_{0} ⊂ M and

M = A{{x}}M_{0} = M_{0} ⊕ (x_{1}, …, x_{s})M ≅ M_{0} ⊗_{A} A{{x}} .

Proof. We keep the notation of the theorem and recall from corollary 3.5.5 that there exists a monomial x^{α} in x_{1}, …, x_{s} such that, for any m ∈ M_{0}, x^{α}m ∈ A[x]\overline{N}. Since \overline{N} ⊂ M and M = M_{0} ⊗_{A} A\{\{x\}\} is identified in M̂ = M_{0} ⊗_{A} A[[x]] as described in part iii) of corollary 4.1.7, we deduce the first assertion of the statement. The remaining part of the statement is clear. Q.E.D.

Proposition 6.4. Let M be an object of SCMod_{A,Σ}^{f}(A{{x}}, L_{A,x}). Then Co(M̂) is an (A[x], L_{A,x})-submodule of M and an object of Mod_{A,Σ}^{f}(A[x], L_{A,x}).

We have: i)

M = A{{x}}Co(M̂) ≃ A{{x}} ⊗_{A[x]} Co(M̂) ;

ii) Co(M̂) is an iterated extension in Mod_{A,Σ}^{f}(A[x], L_{A,x}) of λ-simple objects, for λ ∈ E(M); iii) Let R ⊂ K^{s} denote any set of representatives of C(M) in K^{s}. In Mod_{A,Σ}^{f}(A[x]′, L_{x}) we have

Co(M̂)′ ≅ ⊕_{λ∈R}(N_{λ})′ ,

where N_{λ} denotes a λ-simple object of Mod_{A,Σ}^{f}(A[x], L_{x}).

Proof. According to proposition 6.2.2, Co(M̂) ⊂ M. Also, since M̂ is a projective B-module, M̂ = M_{0} ⊗_{A} B, for any M_{0} as in 6.2.1 (cf. (3.5.1)). So, Co(M̂) = M_{0}⊗_{A}A[x], and M = M_{0}⊗_{A}A{{x}}. All the remaining results already appear in corollary 3.6.7. Q.E.D.

Theorem 6.5. Let M be an object of SCMod_{A,Σ}^{f}(A{{x}}, L_{x}) and let M_{0} = i_{diff}^{∗}M be its image in Mod_{Σ}^{f}(A, L, Φ). Let λ̃ ∈ E(M) be minimal for ≤. Let us put

M_{0}^{n}(λ̃) = (⋂_{i=1}^{s} Ker_{M_{0}}(^{s}φ_{i} − λ̃_{i})) ∩ (⋂_{i=1}^{s} Ker_{M_{0}}(^{n}φ_{i})) =

= (⋂_{i=1}^{s} Ker_{M_{0}}(φ_{i} − λ̃_{i})) ∩ (⋂_{i=1}^{s} Ker_{M_{0}}(^{n}φ_{i})) .

Then M_{0}^{n}(λ̃) is a non-zero sub-object of M_{0} in Mod^{f}(A, L, Φ). For any point a ∈ SpA, let r_{a}(λ̃) denote the rank in a of the projective A-module M_{0}^{n}(λ̃). There is a unique morphism

χλ̃ : φ_{diff}^{∗}(M_{0}^{n}(λ̃)) ⟶ M

in Mod(A{{x}}, L_{x}), such that

i_{diff}^{∗}(χλ̃) : M_{0}^{n}(λ̃) ⟶ M_{0} ,

is the natural inclusion M_{0}^{n}(λ̃) ↪ M_{0}. The morphism χλ̃ is injective and its cokernel Coker(χλ̃) in Mod(A{{x}}, L_{x}) is an object of SCMod_{A,Σ}^{f}(A{{x}}, L_{x}). For any point a ∈ SpA and μ ∈ Σ^{s}, we have:

e_{a,μ,Coker(χλ̃)} = e_{a,μ,M} , if μ ≠ λ̃ ;
e_{a,λ̃,Coker(χλ̃)} = e_{a,λ̃,M} − r_{a}(λ̃) .

Proof. All statements follow directly from part i)′ of corollary 3.6.1 together with the fact that M_{0} ⊂ M. Q.E.D.

Definition 6.5.1. We define the length of an object M of SCMod_{A,Σ}^{f}(A{{x}}, L_{x}) as a natural number τ(M) by the following conditions:

i) τ(0) = 0; ii) In the situation of theorem 6.5, τ(M) = τ(Coker(χλ̃)) + 1.

It is easy to check that definition 6.5.1 is good and that, for an object M of SCMod_{A,Σ}^{f}(A{{x}}, L_{x}), τ(M) = τ(M̂) (cf. (3.6.6)).

Corollary 6.5.2. Let M be an object of SCMod_{A,Σ}^{f}(A{{x}}, L_{A,x}). Then M is an iterated extension of λ-simple objects in SCMod_{A,Σ}^{f}(A{{x}}, L_{A,x}), for λ ∈ E(M).

Proof. We use induction on the length τ(M) of M, applying repeatedly theorem 6.5. Q.E.D.

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Authors’ address: Dipartimento di Matematica Università di Padova Via Belzoni 7 I-35131 PADOVA - ITALY E-mail:

“BALDASSARRI@PDMAT1.UNIPD.IT” “CHIARELLOTTO@PDMAT1.UNIPD.IT”

Fax: (39-49-)8758596