Algebraic versus rigid cohomology with logarithmic coefficients.

by

Francesco Baldassarri and Bruno Chiarellotto

Dedicated to Iacopo Barsotti, in memoriam.

§1 Notation.

(K, | − |) = a complete valued field extension of (Q_{p}, | − |), for some prime p; |p| = p^{−1}. The image of K \ {0} in R is a multiplicative subgroup Γ of the multiplicative group of positive real numbers (see the beginning of section 2 for more assumptions on K).

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}, and any s.

V= the ring of integers of (K, | − |).

m = the maximal ideal of V.

k = V/m, the residue field of K.

For a V-scheme T locally of finite type, we put:

T_{k} = T ×_{V} k, the special fiber of T

T_{K} = T ×_{V} K , the generic fiber of T.

T̂ = the p-adic completion of T

(T̂, OT̂) = (T_{k}, lim_{⟵} O_{T} /p^{n}O_{T}) ,

which we sometimes call “the formal completion of T along T_{k}”.

T_{K}^{an} = the rigid analytic space associated to the K-scheme T_{K} ([BGR]).

We follow notation and conventions of [Be]. In particular, unless otherwise specified, for all formal V-schemes (X , O_{X}) appearing in this article and n =

This paper is in final form and no version of it will be submitted for publication elsewhere. A preliminary version has appeared in the Proceedings of the Symposium on Special Differential Equations, held at RIMS, Kyoto, in Sept. 1991.

1991 Mathematics Subject Classification: 14F30, 12H25.

1, 2, …, (X , O_{X} /p^{n}O_{X}) will be an ordinary V-scheme and O_{X} = lim_{⟵} O_{X} /p^{n}O_{X}. If necessary, we will qualify such formal V-schemes as “p-adic”. For a p-adic formal V-scheme T locally of finite type, we put:

T_{k} = T ×_{V} k, the special fiber of T .

T_{K} = the generic fiber of T in the sense of Raynaud and Berthelot ([Ra], [Be]): it’s a rigid analytic space.

For a separated V-scheme of finite type T, using the previous notation, we get an open immersion of rigid analytic spaces

T̂_{K} ⟶ T_{K}^{an} ,

which is an isomorphism when T is proper over V ([Be, 0.3.5]).

The following definition will play an important role in the sequel.

Definition 1.1 For γ ∈ K the type ρ(γ) of γ is the radius of convergence of the series

g_{γ}(x) = ∑_{γ≠i=0}^{∞} \frac{x^{i}}{γ − i} .

We point out that ρ(γ) ∈ [0, 1] and ρ(γ) = ρ(γ + n), ∀n ∈ Z. We say that γ is p-adically non-Liouville if ρ(γ) = ρ(−γ) = 1.

Remark. Algebraic numbers are p-adically non-Liouville.

§2 Main Result.

From now on we assume that the field K is algebraically closed, so that Γ is a dense subgroup of the multiplicative group R_{>0}. We fix a discretely valued complete sub-field K′ of K, with valuation ring V′, uniformizing parameter π, and residue field k′. The field K′ (resp. ring V′) will play the role of “field (resp. ring) of definition” for all the algebraic objects we introduce. A base-change from K′ to K (resp. from V′ to V ) will sometimes be denoted by an appended index K (resp. V ), but more frequently will be understood. In other words, throughout the present article, the algebraic objects under consideration are defined over K′ or V′, but all constructions we perform on them, unless otherwise specified, are in fact performed over (i.e. after base-change to ) K or V, respectively. This point of view is not strictly necessary, but it does save us some work of extending results which are at present only available over a noetherian base.

We consider: Y = a proper smooth V′-scheme of finite type, with geometrically connected fibers.

Z = a divisor in Y with normal crossings relative to V′

Z = ⋃_{i=1}^{r} Z^{(i)}

where each Z^{(i)} is a smooth closed V′-subscheme of Y, of codimension 1, with geometrically connected fibers.

X = the open V′-subscheme of Y, complementary to Z in Y.

The previous hypotheses mean that there exists a finite covering U of Y by affine open subsets U such that:

i) U is étale over A_{V′}^{m} via “coordinates” (x_{1}, …, x_{m}).

ii) The ideal of Z_{U} = Z ×_{Y} U = Z_{|U} in O(U) is generated by x_{1} … x_{ν} where ν = ν(U).

We also consider: E_{V′} = a locally free O_{Y} -module of finite type. ∇ = an integrable Y_{K′}/K′-connection on E = E′_{V} ⊗ K′ with logarithmic singularities along Z_{K′}. This means that ∇ is a morphism of abelian sheaves

∇ : E ⟶ E ⊗ Ω_{Y_{K′} /K′}^{1} <Z_{K′} >

satisfying Leibnitz’s rule and the usual integrability condition. We recall that, using the previous notation, we have on U ∈ U as before

Ω_{Y_{K′} /K′}^{1} <Z_{K′} >_{|U_{K′}} = ∑_{i=1}^{ν} O_{U_{K′}} \frac{dx_{i}}{x_{i}} + ∑_{j=ν+1}^{m} O_{U_{K′}} dx_{j}.

The hypercohomology of the de Rham complex of (E_{K}, ∇_{K})_{|X_{K}} = (E, ∇)_{|X_{K}} i.e. of the complex

DR(X_{K}/K, (E, ∇)) = 0 → E_{|X_{K}} ⟶ E_{|X_{K}} ⊗ Ω_{X_{K}/K}^{1} ⟶ …

is, by definition, the algebraic cohomology of X_{K} with coefficients in (E, ∇), denoted by

H^{•}(X_{K}/K, (E, ∇)).

From the morphism of ringed sites

β : X_{K}^{an} ⟶ X_{K}

one may deduce from (E, ∇) a X_{K}^{an}/K-connection (E^{an}, ∇^{an}). The hypercohomology of its de Rham complex

DR(X_{K}^{an}/K, (E^{an}, ∇^{an}))

is, by definition, the analytic cohomology (in the rigid analytic sense) of X_{K}^{an} with coefficients in (E^{an}, ∇^{an}), denoted by

H^{•}(X_{K}^{an}/K, (E^{an}, ∇^{an})).

The morphism of ringed sites β induces a natural morphism of complexes of sheaves

β^{−1}DR(X_{K}/K, (E, ∇)) ⟶ DR(X_{K}^{an}/K, (E^{an}, ∇^{an}))

which gives a morphism in hypercohomology

\overline{β} : H^{•}(X_{K}/K, (E, ∇)) ⟶ H^{•}(X_{K}^{an}/K, (E^{an}, ∇^{an})).

Consider now the assumption:

(NL)_{G} The additive subgroup Λ of K generated by the exponents of monodromy of (E, ∇) around the branches of Z_{K} consists of p-adically non-Liouville numbers.

In this setting we proved years ago ([Ba2]) a result of GAGA type:

Theorem 2.1. Under the assumption (NL)_{G} the morphism \overline{β} is an isomorphism.

Other results in the same direction may be found in [Ba1],[Ct].

We now come to our main statement. Let

j_{k} : X_{k} ⟶ Y_{k},
j : X_{K}^{an} ⟶ Y_{K}^{an}

denote the corresponding open immersions. We point out that X_{K}^{an} is a strict neighborhood of the tube ]X_{k}[ of X_{k} in Ŷ_{K} = Y_{K}^{an} [Be]. Using the theory of

Berthelot we may consider the j†_{k}-completion of the previous coefficients. We recall the definition. For λ ∈ (0, 1) ∩ Γ sufficiently close to 1, we define

(2.1.1)  V_{λ} = Ŷ_{K} \ ⋃_{i=1}^{r}[Z_{k}^{(i)}]_{λ}

where [Z_{k}^{(i)}]_{λ} denotes the closed tube of radius λ of Z_{k}^{(i)} in Ŷ_{K}. (The definition of [Z_{k}^{(i)}]_{λ} is independent of the chosen coordinate covering U = {(U, (x_{1}, …, x_{m}))} for λ sufficiently close to 1; cf. [Be,1.1]). We get open immersions:

(2.1.2)  \begin{matrix}V_{λ} & ⟶^{j_{λ}} & Ŷ_{K} \\ ↓ & ↗ \\ X_{K}^{an} \end{matrix}

We denote by (E_{λ}, ∇_{λ}) the connection induced by (E^{an}, ∇^{an}) on V_{λ}, and by DR(V_{λ}/K, (E_{λ}, ∇_{λ})) = DR_{λ} its de Rham complex (i.e. the restriction of DR(X_{K}^{an}/K, (E^{an}, ∇^{an})) to V_{λ}). We then obtain a connection (E†, ∇†) on

E† = j†_{k}E =_{def} lim_{\overrightarrow{λ→1^{−}}} j_{λ∗}E_{λ}

whose de Rham complex is

DR(Ŷ_{K}/K, (E†, ∇†)) = lim_{\overrightarrow{λ→1^{−}}} j_{λ∗}DR_{λ}.

We introduce the hypothesis ([Be]):

(SC)_{G} The connection (E†, ∇†) is overconvergent along Z_{k}.

This means ([Be, 2.2.13]) that for each coordinate open set (U, x_{1}, …, x_{m}) ∈ U, the following condition holds.

(SC)_{U} ∀η ∈ (0, 1) there exists λ ∈ (0, 1) ∩ Γ such that for each affinoid open subset V ⊂ Û_{K} ∩ V_{λ} and any section e ∈ Γ(V, E^{an})

lim_{|α|_{R}⟶+∞}||(α!)^{−1}(∇^{an}(∂/∂x))^{α}(e)||_{V} η^{|α|_{R}} = 0 ,

where || − ||_{V} denotes any Banach norm on Γ(V, E^{an}), and (∇^{an}(∂/∂x))^{α} = ∏_{i=1}^{m}(∇^{an}(∂/∂x_{i}))^{α_{i}}.

As in [Be], we define the rigid cohomology of X_{k} with coefficients in (E†, ∇†) as

H_{rig}^{•}(X_{k}/K, (E†, ∇†)) = H^{•}(Ŷ_{K}, DR(Ŷ_{K}/K, (E†, ∇†))).

We recall that, since Ŷ_{K} is quasi-compact and separated, the cohomology on Ŷ_{K} commutes with the direct limits ([Be], [SGAIV]). Hence:

(2.2) H_{rig}^{•}(X_{k}/K, (E†, ∇†)) = H^{•}(Ŷ_{K}, lim_{\overrightarrow{λ}} j_{λ∗}DR_{λ}) = lim_{\overrightarrow{λ}} H^{•}(Ŷ_{K}, j_{λ∗}DR_{λ}).

We then have natural morphisms of complexes of sheaves on Ŷ_{K} = Y_{K}^{an}:

j_{λ∗}DR_{λ}  ⟶ DR(Ŷ_{K}/K, (E†, ∇†))
↑  ↗
j_{∗}DR(X_{K}^{an}/K, (E^{an}, ∇^{an}))

and the induced morphisms in hypercohomology:

(2.3)  \begin{matrix}H^{•}(V_{λ}/K, (E_{λ}, ∇_{λ})) & ⟶ & H_{rig}^{•}(X_{k}/K, (E†, ∇†)) \\ ↑ \overline{α}_{λ} & ↗ \\ H^{•}(X_{K}^{an}/K, (E^{an}, ∇^{an})) \end{matrix}

(this makes sense because R^{q}j_{∗}F = 0 (resp. R^{q}j_{λ∗}F = 0) for any coherent sheaf F on X_{K}^{an} (resp. V_{λ}) and q > 0. This holds since j (resp. j_{λ}) is a quasi-Stein map, i.e. the inverse image via j (resp. j_{λ}) of any affinoid subdomain of Ŷ_{K}/K is quasi-Stein [Ba1],[K3]).

Our main result is the following.

Theorem 2.4. Under the assumptions (NL)_{G} and (SC)_{G} the morphism \overline{α}_{λ} is an isomorphism for any λ ∈ (0, 1) ∩ Γ for which V_{λ} is defined (i.e. for any λ sufficiently close to 1).

From (2.2) and the identification

H^{•}(V_{λ}/K, (E_{λ}, ∇_{λ})) = H^{•}(Ŷ_{K}, j_{λ∗}DR_{λ}))

we obtain the following corollaries.

Corollary 2.5. Under the assumptions of the theorem, the morphism

\overline{α} : H^{•}(X_{K}^{an}/K, (E^{an}, ∇^{an})) ⟶ H_{rig}^{•}(X_{k}/K, (E†, ∇†))

is an isomorphism.

Corollary 2.6. Under the assumptions of the theorem the morphism

\overline{α} ∘ \overline{β} : H^{•}(X_{K}/K, (E, ∇)) ⟶ H_{rig}^{•}(X_{k}/K, (E†, ∇†))

is an isomorphism (cf. 2.1).

Corollary 2.6 is our comparison theorem between algebraic and the rigid cohomology with logarithmic coefficients.

§3 The 1-dimensional example: role of hypotheses (NL)_{G} and (SC)_{G}.

We will illustrate by an example the role played by the two hypotheses (NL)_{G} and (SC)_{G}. This section is not strictly necessary to the logical development of the proof of theorem 2.4, given in the subsequent sections. In dimension 1, however, the techniques are considerably simpler while the main ideas involved in the proof of the comparison theorem appear more clearly. So, we prefer to start by presenting an independent treatment of the 1-dimensional case. In this section the assumption that our picture is defined over the noetherian ring V′ is irrelevant and may be dropped. We consider the case Y = P_{V}^{1}, perfectly analogous to the one of Y = any proper smooth V-scheme of relative dimension 1. We put:

D = D(0, 1^{−}) = {x ∈ K | |x| < 1} , D^{∗} = D \ {0} .

For ϵ , η ∈ [0, 1) ∩ Γ, ϵ ≤ η we set :

C_{ϵ} = {x ∈ K | ϵ < |x| < 1} ,
C(ϵ, η) = {x ∈ K | ϵ ≤ |x| ≤ η} ,
D(η) = D(0, η^{+}) = {x ∈ K | |x| ≤ η} .

In particular, D^{∗} = C_{0}, D(η) = C(0, η).

We have a diagram of open immersions, for any λ ∈ [0, 1) ∩ Γ,

\begin{matrix}C_{λ} & ⟶^{j_{λ}} & D \\ ↓ & ↗ \\ D^{∗} \end{matrix}

For γ ∈ K we consider the complexes

a) =  0 ⟶  O(D)  ⟶^{∇_{γ}}  \frac{1}{z}Ω^{1}(D) ⟶ 0
↓  ↓  ↓
b) =  0 ⟶ O(D)[z^{−1}] ⟶^{∇_{γ}} Ω^{1}(D)[z^{−1}] ⟶ 0
↓  ↓  ↓
c) =  0 ⟶ O(D^{∗}) ⟶^{∇_{γ}} Ω^{1}(D^{∗}) ⟶ 0
↓  ↓  ↓
d) =  0 ⟶ O(C_{λ}) ⟶^{∇_{γ}} Ω^{1}(C_{λ}) ⟶ 0

where

∇_{γ} = d + γ \frac{dz}{z} .

Theorem 3.1.

i) If γ ∈ K is not a positive integer and ρ(γ) = 1, the inclusions

a) ↪ b) , a) ↪ c) and a) ↪ d)

are homotopy equivalences. ii) If ρ(−γ) = 1 then the complex a) has finite dimensional cohomology. More precisely, under the present assumption,

H^{i}(a)) ≅ \{ \begin{matrix}K, & if γ ∈ Z_{≤0} \\ 0, & if γ ∉ Z_{≤0} \end{matrix} ,

for i = 0, 1. iii) If ρ(γ) = 1 the inclusions

b) ↪ c) ↪ d)

are homotopy equivalences. iv) If ρ(γ) = ρ(−γ) = 1, we have

H^{i}(b)) ≅ H^{i}(c)) ≅ H^{i}(d)) ≅ \{ \begin{matrix}K, & if γ ∈ Z \\ 0, & if γ ∉ Z \end{matrix} ,

for i = 0, 1.

Proof. i) Let us consider for example a)^{\underrightarrow{i}} d). We construct a morphism of complexes d)^{\underrightarrow{R}} a)

R(∑_{n∈Z} a_{n}z^{n}) = ∑_{n≥0} a_{n}z^{n}

R(∑_{n∈Z} a_{n}z^{n} \frac{dz}{z}) = ∑_{n≥0} a_{n}z^{n} \frac{dz}{z}

which is a left inverse of i:

Ri = id_{a)}

while

id_{d)} − iR = ∇_{γ}H_{γ} + H_{γ}∇_{γ}

where the homotopy operator H_{γ}

H_{γ} : d) → d)[−1]

is 0 in degrees 0,2, while

H_{γ}(∑_{n∈Z} a_{n}z^{n} \frac{dz}{z}) = ∑_{n<0} \frac{a_{n}}{γ + n}z^{n} = (∑_{n<0} a_{n}z^{n}) ∗ (g_{γ}(z^{−1}) − \frac{1}{γ})

where “ ∗ ” denotes the Hadamard product with respect to z^{−1}. (In the previous formula \frac{1}{γ} must be replaced by 0, if γ = 0.) Now ∑_{n<0} a_{n}z^{n} is analytic for |z| > λ, while, by the hypothesis on γ, g_{γ}(z^{−1}) is analytic for |z| > 1. Thus H_{γ} takes its values in O(C_{λ}). The same argument works for the other inclusions.

ii) We define

H^{γ} : a) ⟶ a)[−1]

as

H^{γ}(∑_{n≥0} a_{n}z^{n} \frac{dz}{z}) = ∑_{{n=0 \atop n≠−γ}}^{∞} \frac{a_{n}}{γ + n}z^{n} = −(∑_{n≥0} a_{n}z^{n}) ∗ (g_{−γ}(z)).

If ρ(−γ) = 1, H^{γ} takes its values in O(D). If γ is not a negative integer nor 0,

id_{a)} = ∇_{γ}H^{γ} + H^{γ}∇_{γ}

so that a) is acyclic. If γ = −n_{∘}, n_{∘} ∈ {0, 1, …} then a) contains the sub-complex

e)  =  0 → Kz^{n_{∘}} ⟶ Kz^{n_{∘}} \frac{dz}{z} → 0 ,

The inclusion e)^{\underrightarrow{j}} a) has an obvious retraction a)^{\underrightarrow{r}} e), namely

r(∑_{n≥0} a_{n}z^{n}) = a_{n_{∘}}z^{n_{∘}} , r(∑_{n≥0} a_{n}z^{n} \frac{dz}{z}) = a_{n_{∘}}z^{n_{∘}} \frac{dz}{z} .

Then r is a left inverse of j, i.e.

rj = id_{e)} ,

while

id_{d)} − jr = ∇_{γ}H^{γ} + H^{γ}∇_{γ}.

iii) and iv) We may reduce to i) and ii), via multiplication by some integral power of z, which is an isomorphism of each of b) , c) and d). Q.E.D.

Corollary 3.2. Consider the following morphisms of complexes of abelian sheaves on D

ã) =  0 ⟶  O_{D}  ⟶^{∇_{γ}} Ω_{D/K}^{1} < 0 > ⟶ 0
↓  ↓  ↓
b̃) =  0 ⟶ O_{D}[z^{−1}] ⟶^{∇_{γ}} Ω_{D/K}^{1}[z^{−1}] ⟶ 0
↓  ↓  ↓
c̃) =  0 ⟶ j_{∗}O_{D^{∗}}  ⟶^{∇_{γ}} j_{∗}Ω_{D^{∗}/K}^{1} ⟶ 0
↓  ↓  ↓
d̃) =  0 ⟶ j_{λ∗}O_{C_{λ}}  ⟶^{∇_{γ}} j_{λ∗}Ω_{C_{λ}/K}^{1} ⟶ 0

and the corresponding morphisms in hypercohomology

(3.2.1)  H^{•}(D, ã)) ⟶ H^{•}(D, b̃)) ⟶ H^{•}(D, c̃)) ⟶ H^{•}(D, d̃)) .

All the previous hypercohomology groups vanish in degrees i ≠ 0, 1. Under the assumptions of cases i), ii), iii), iv) of the theorem, respectively, we have:

i) All the morphisms in (3.2.1) are isomorphisms. ii) H^{•}(D, ã)) is a finite dimensional K-vector space. More precisely, under the present assumption

H^{i}(D, ã)) ≅ \{ \begin{matrix}K, & if γ ∈ Z_{≤0} \\ 0, & if γ ∉ Z_{≤0} \end{matrix} ,

for i = 0, 1. iii) H^{•}(D, b̃)) ⟶^{∼} H^{•}(D, c̃)) ⟶^{∼} H^{•}(D, d̃)) are isomorphisms. iv)

H^{i}(D, b̃)) ≅ H^{i}(D, c̃)) ≅ H^{i}(D, d̃)) ≅ \{ \begin{matrix}K, & if γ ∈ Z \\ 0, & if γ ∉ Z \end{matrix} ,

for i = 0, 1.

Proof. For any sheaf F which appears in the above diagram

H^{q}(D, F) = 0

for q > 0. (In fact D,D^{∗} are quasi-Stein while j_{λ} is a quasi-Stein map). We are then reduced to proving quasi-isomorphisms for the complexes of global sections, thus to the theorem. Q.E.D.

We will now discuss the role played by the hypothesis (SC)_{G}.

Consider a system of linear differential equations of the form

S_{G}  \frac{d}{dx}y = G(x)y

with G(x) ∈ M_{n}(K(x)). We assume, with no restriction, that the poles of G are in K. Let Z_{K} = {a_{1}, …, a_{r}} be the set of singular points of S_{G} in P̂_{K}^{1} and let Z_{k} = {sp a_{1}, …, sp a_{r}} where

sp : P̂_{K}^{1} ⟶ P_{k}^{1}

is the specialization map. Then S_{G} defines a connection on the sheaf O^{n} over X_{K}^{an} = P̂_{K}^{1} \Z_{K}. When is this connection overconvergent along Z_{k}? If we intend to stick with the matrix G, we are forced to assume that Z_{k} contains x = ∞; then, Berthelot’s condition ([Be, 2.2.13]) involves

V_{λ} = P̂_{K}^{1} \ ⋃_{i=1}^{r} D(a_{i}, λ^{+}), λ ∈ (0, 1) ∩ Γ ,

and the matrices giving the action of (\frac{d}{dx})^{m} on solutions of S_{G}

(\frac{d}{dx})^{m}y = G^{(m)}y, m ∈ N

(G^{(∘)} = I_{n}, G^{(1)} = G, G^{(m+1)} = \frac{dG^{(m)}}{dx} + G^{(m)}G^{(1)}). It is :

(SC)_{G} (Berthelot) For each η ∈ (0, 1), there exists λ ∈ (0, 1) ∩ Γ such that

lim_{m→∞} ||\frac{G^{(m)}}{m!} ||_{V_{λ}}η^{m} = 0

(where || − ||_{V_{λ}} denotes the sup-norm on V_{λ}, for matrices).

We recall the Gauss norm on K(x). It is defined on K[x] as

| ∑ b_{i}x^{i}|_{G} = Sup_{i}|b_{i}|

and is extended to an absolute value |−|_{G} of K(x) by multiplicativity. It depends essentially on the V-structure of P̂_{K}^{1}: if a function f ∈ K(x) has no poles in an open disk D of radius 1, then

|f|_{G} = ||f||_{D}.

So, condition (SC)_{G} implies

(SC)′_{G} ∀η ∈ (0, 1)

lim_{m→∞} |\frac{G^{(m)}}{m!} |_{G}η^{m} = 0 .

Condition (SC)′_{G} appears in the work of Dwork, Robba, Christol, André ([An1], [An2], [Ch1], [Ch-Dw]), as that of convergence of the solutions of S_{G} in the generic disk of radius 1. This is motivated by the fact that the matrix function

U_{G,t}(x) = ∑_{m=0}^{∞} \frac{G^{(m)}(t)}{m!}(x − t)^{m}

is a fundamental solution matrix of S_{G} at a generic unit t and

|\frac{G^{(m)}(t)}{m!}| = |\frac{G^{(m)}}{m!} |_{G}.

Let’s now check that (SC)′_{G} ⇒ (SC)_{G}. Let us put, for each λ ∈ (0, 1) ∩ Γ,

(]X_{k}[=)W = P̂_{K}^{1} \ ⋃_{i=1}^{r} D(a_{i}, 1^{−}) ⊂ V_{λ} .

Then (SC)′_{G} certainly implies

lim_{m→∞} ||\frac{G^{(m)}}{m!} ||_{W} η^{m} = 0

for each η ∈ (0, 1). So we are then reduced to considering separately disjoint annuli around the singular points. We may assume

a_{1}, …, a_{d} ∈ D = D(0, 1^{−}) , a_{d+1}, …, a_{r} ∉ D.

Let

f(x) = ∏_{i=1}^{d}(x − a_{i})^{N}

be such that fG ∈ M_{n}(O(D)). Then also f^{m}G^{(m)} ∈ M_{n}(O(D)), for each m ∈ N. For λ > max_{i=1,...,d} |a_{i}| we have:

||\frac{G^{(m)}}{m!} ||_{C_{λ}} ≤ ||f^{m} \frac{G^{(m)}}{m!} ||_{C_{λ}}||f^{−m}||_{C_{λ}} ≤ |f^{m} \frac{G^{(m)}}{m!} |_{G}λ^{−mNd} = |\frac{G^{(m)}}{m!} |_{G}λ^{−mNd},

since |f|_{G} = 1. So, if λ is also > η^{\frac{1}{Nd}}, one has η/λ^{Nd} ∈ (0, 1) and

(SC)′_{G}  lim_{m→∞} |\frac{G^{(m)}}{m!} |_{G}(\frac{η}{λ^{Nd}})^{m} = 0

implies

(SC)_{G}  lim_{m→∞} ||\frac{G^{(m)}}{m!} ||_{C_{λ}}η^{m} = 0.

So:

Proposition 3.3. The system S_{G} is overconvergent along its polar divisor iff its solutions converge in the generic disk of radius one.

What we said should justify the following local condition on a system on D = D(0, 1^{−}):

L_{G}  x\frac{d}{dx}y = Gy

with G ∈ M_{n}(O(D)). We put, as before,

x^{m}(\frac{d}{dx})^{m}y = G_{m}y

for m ∈ N. We consider the condition:

(SC)_{L} ∀η ∈ (0, 1) and ∀ϵ ∈ (0, 1) ∩ Γ

lim_{m→∞} ||\frac{G_{m}}{m!} ||_{D(ϵ)}η^{m} = 0 .

We also define the type ρ ∈ [0, 1] of the system L_{G} at 0 as:

ρ = ∏ ρ(γ)^{e_{γ}} ,

if det(x − G(0)) = ∏(x − γ)^{e_{γ}}.

Our main result in this framework is the following.

Theorem 3.4. ([Ba-Ct1, Th.3.4]) Assume the system L_{G} , G ∈ M_{n}(O(D)) satisfies condition (SC)_{L}. Let ρ be the type of L_{G} at 0. Then any formally meromorphic column solution y of L_{G} at 0, is p-adically meromorphic for |x| < ρ.

Corollary 3.5. Under the assumptions of the theorem, assume also that the eigenvalues of G(0) are p-adically non-Liouville. Then, the formally meromorphic solutions y of L_{G} at 0 are meromorphic in D.

Consider the condition:

(NL)_{L} The additive subgroup Λ of K generated by the eigenvalues of G(0), consists of p-adically non-Liouville numbers.

Corollary 3.6. Assume conditions (NL)_{L} and (SC)_{L} hold for L_{G}. For ∇_{G} = d + G\frac{d}{dx}, consider the diagram of abelian sheaves on D:

a) =  0 ⟶ O_{D}^{n}[\frac{1}{x}] ⟶^{∇_{G}} Ω_{D/K}^{1} ⊗ O_{D}^{n}[\frac{1}{x}] ⟶ 0
↓  ↓  ↓
b) =  0 ⟶ j_{∗}O_{D^{∗}}^{n} ⟶^{∇_{G}} j_{∗}(Ω_{D^{∗}/K}^{1} ⊗ O_{D^{∗}}^{n}) ⟶ 0
↓  ↓  ↓
c) =  0 ⟶ j_{λ∗}O_{C_{λ}}^{n} ⟶^{∇_{G}} j_{λ∗}(Ω_{C_{λ}/K}^{1} ⊗ O_{C_{λ}}^{n}) ⟶ 0

The morphisms a) ↪ b) ↪ c) induce isomorphisms of hypercohomology groups over D. All of the previous hypercohomology groups are finite dimensional Kvector spaces.

Proof. It consists of the following steps:

1) Use the formal theory of logarithmic systems to formally reduce to uppertriangular systems.

2) Use corollary 3.5 (or [Ba-Ct1, Cor. 3.5]) to show that the formal equivalence referred to in step 1), is in fact convergent on D.

So, we may assume that G is upper-triangular.

3) Reduce to systems of rank 1, via the spectral sequence of filtered complexes.

4) Apply the corollary to theorem 3.1, after translating the exponents by an integer, if needed (multiplication by x^{N} is an isomorphism). Q.E.D..

We now come to our main result:

Corollary 3.7. Theorem 2.4 and its corollaries hold for Y = P^{1}.

Proof. Recall that in theorem 2.4 we fixed a λ ∈ (0, 1) ∩ Γ and that we had:

V_{λ} = P̂_{K}^{1} \ ⋃_{i=1}^{r} D(a_{i}, λ^{+})^{\underrightarrow{j_{λ}}} P̂_{K}^{1}.

We deal with a connection (E, ∇) with logarithmic singularities at a_{1}, …, a_{r} (assumed to lie in distinct residues classes). Our task is to examine

F^{•} = j_{∗}DR(X_{K}^{an}/K, (E^{an}, ∇^{an}))^{\underrightarrow{i}} F_{λ}^{•} = j_{λ∗}DR(V_{λ}/K, (E_{λ}, ∇_{λ})) .

We choose η ∈ (λ, 1) and consider the admissible covering W of P̂_{K}^{1}:

W = {V_{η}, D(a_{1}, 1^{−}), …, D(a_{r}, 1^{−})}.

We have that the restriction

i_{|V_{η}} : F_{|V_{η}}^{•} ≃ F_{λ}^{•}_{|V_{η}}

is the identity, while on each disk D(a_{i}, 1^{−}) we are in the situation of corollary 3.6. Our result follows from the spectral sequence of hypercohomology associated to the covering W:

E_{2}^{p,q}(F^{•}) = H^{p}(W, h^{q}(F^{•})) ⇒ H^{•}(P̂_{K}^{1}, F^{•})

where h^{q}(F^{•}) denotes the presheaf

U ⟼ H^{q}(U, F^{•})

(and similarly for F_{λ}^{•}). The morphism of spectral sequences:

E_{•}^{•,•}(F^{•}) ⟶ E_{•}^{•,•}(F_{λ}^{•})

is an isomorphism at the E_{2} level. (In fact, since

H^{q}(U, G) = 0 ∀q > 0

for any open set U of the nerve of W and any sheaf G under consideration, the Čech bicomplexes of W with coefficients in F^{•} and F_{λ}^{•}, actually calculate the hypercohomology of P̂_{K}^{1}). Q.E.D.

§4 Existence of tubular neighborhoods of radius 1 of Ẑ_{K} in Ŷ_{K}.

We come back to the hypotheses of section 2. In particular, we recall that the ring V′ is noetherian. We need here to refine the affinoid covering of Ŷ_{K′}

Û = {Û_{K′}}_{U∈U}

obtained from the originally chosen coordinate covering U of Y. This will be done in connection with a given V_{λ}, λ ∈ (0, 1) ∩ Γ, as in theorem 2.4, which we fix throughout this section. We start by refining the Zariski covering U of Y. We may then assume that for each U ∈ U, we have a presentation of R = O(U):

(4.0)  ψ : V′[X_{1}, …, X_{N}]/(F_{m+1}(X), …, F_{N}(X))⟶^{∼}R ,

with ψ(det(∂F_{m+i}/∂X_{m+j})_{1≤i,j≤N−m}) invertible in R and ψ(X_{i}) = x_{i}, for i = 1, …, m. We recall that the ideal of Z_{U} in R is supposed to be generated by x_{1} … x_{ν} with ν = ν(U). We fix one such U, and R, ψ as in (4.0), correspondingly. We denote by Û , R̂ , ψ̂ their p-adic completions, respectively.

The following definition (up to and including formula (4.1.2)_{n}) applies under the general hypotheses of section 1 (i.e. K is any complete extension of Q_{p}). For K-rigid analytic spaces we use the notion of smoothness of a morphism given in [Be, 0.1.11]; we fix a notion suitable for p-adic formal V-schemes as follows.

Definition 4.1. A morphism ϕ : P ⟶ S of formal p-adic V-schemes locally of finite type is said to be smooth of relative dimension d if there exist two open coverings {P_{α}}_{α} of P and {S_{α}}_{α} of S consisting of formal affine V-schemes P_{α} = SpfB_{α} , S_{α} = SpfA_{α} with the following properties: (i) ϕ(P_{α}) ⊂ S_{α} , (ii) The induced morphism ϕ_{α} : P_{α} ⟶ S_{α} corresponds to a morphism of topological V- algebras ϕ_{α}^{∗} : A_{α} ⟶ B_{α} that can be factored through a commutative diagram (where we skipped the index α)

(4.1.1)  \begin{aligned}& & A\{X_{1}, …, X_{N}\} \\ & ↗ & & ↘ ψ^{∗} \\ A & & ⟶^{ϕ^{∗}} & & B \end{aligned}

where ψ^{∗} is surjective, Ker ψ^{∗} = (F_{d+1}(X), …, F_{N}(X)), and

(4.1.2)  ψ^{∗}(det(∂F_{d+i}/∂X_{d+j})_{1≤i,j≤N−d})

is invertible in B.

It is clear that if ϕ : P ⟶ S is a smooth morphism of V-schemes locally of finite type, the p-adic completion ϕ̂ : P̂ ⟶ Ŝ of ϕ is a smooth morphism of p-adic formal V-schemes locally of finite type.

For n = 0, 1, …, we put V_{n} = V/π^{n+1}V and let X ⟼ X_{n} denote the functor X ⟼ X ×_{V} V_{n}, from p-adic formal V-schemes (resp. locally of finite type) to ordinary V_{n}-schemes (resp. locally of finite type). Then, if ϕ : P ⟶ S is a smooth morphism of p-adic formal V-schemes locally of finite type, the morphism ϕ_{n} : P_{n} ⟶ S_{n} is smooth, for any n = 0, 1, . . .. In fact, on tensoring the diagrams (4.1.1) and (4.1.2) by V_{n}, we obtain diagrams

(4.1.1)_{n}  \begin{aligned}& & A_{n}[X_{1}, …, X_{N}] \\ & ↗ & & ↘ ψ_{n}^{∗} \\ A_{n} & & ⟶^{ϕ_{n}^{∗}} & & B_{n} \end{aligned}

where A_{n} = A/π^{n+1}A, B_{n} = B/π^{n+1}B, ψ_{n}^{∗} is surjective, Ker ψ_{n}^{∗} = (F_{d+1}(X), …, F_{N}(X)), and

(4.1.2)_{n}  ψ_{n}^{∗}(det(∂F_{d+i}/∂X_{d+j})_{1≤i,j≤N−d})

is invertible in B_{n}. So, B_{n} is a smooth A_{n}-algebra, i.e. a A_{n}-algebra of finite presentation, which is a formally smooth topological A_{n}-algebra, when both A_{n} and B_{n} are given the discrete topology.

We are now back to the assumption that K is algebraically closed. For the fixed U ∈ U, we let T_{U} = {1, …, ν}, S ⊂ T_{U}. For η ∈ (λ, 1) ∩ Γ, we put:

(4.1.3) U_{S,η} = {p ∈ Û_{K} | |x_{i}(p)| < 1 if i ∈ S and |x_{i}(p)| ≥ η, if i ∈ T_{U} \ S} .

We have:

(4.1.4)  U_{S,η} ∩ V_{λ} = {p ∈ U_{S,η} | |x_{i}(p)| > λ ∀ i ∈ S } .

The main point is the following:

Proposition 4.2. U_{S,η} is a trivial bundle in open unit polydisks of relative dimension s = card S over the smooth affinoid space

V_{S,η} = {p ∈ Û_{K} : |x_{i}(p)| = 0, if i ∈ S and |x_{i}(p)| ≥ η, if i ∈ T_{U} \ S} .

Remark 4.2.1. V_{S,η} is the generic fiber (over K) of a formal V′-scheme of finite presentation Z_{S,η} = SpfC_{S,η}. In fact, if |π|^{a} = η^{b}, for a, b ∈ N , (a, b) = 1, then

C_{S,η} = R̂{T_{i} , i ∈ T_{U} \ S}/(x_{j} , j ∈ S; ((x_{j}T_{j})^{b} − π^{a}) , j ∈ T_{U} \ S) .

We wish to point out that, while the K-affinoid space V_{S,η} is smooth, the V′- formal scheme Z_{S,η} is in general not smooth since x_{j} is not invertible in C_{S,η}, for j ∈ T_{U} \ S.

Proof. We put C = C_{S,η} and define topological V′-algebras topologically of finite presentation A, B and F, by

A = V′{X_{j}, T_{j} ; j ∈ T_{U} \ S}/((X_{j}T_{j})^{b} − π^{a}; j ∈ T_{U} \ S) ,
F = V′{X_{1}, …, X_{N}}{T_{j} ; j ∈ T_{U} \ S}/((X_{j}T_{j})^{b} − π^{a}; j ∈ T_{U} \ S) =
= A{X_{i}; i ∈ S}{X_{ν+1}, X_{ν+2}, …, X_{N}} ,
B = R̂{T_{j} , j ∈ T_{U} \ S}/(((x_{j}T_{j})^{b} − π^{a}) , j ∈ T_{U} \ S) =
= F/(F_{m+1}(X), …, F_{N}(X)) .

We observe that

C = F/((F_{m+1}(X), …, F_{N}(X)) + (X_{j}, j ∈ S))

We then obtain the following diagram of topological V′-algebras:

A
↓
F
↙  ↘
B  ⟶  C

We put S = SpfA, P = SpfB, Z = SpfC, Q = SpfF. It is then obvious that Q is a formal affine space over S, while the closed immersions P ⟶ Q, Z ⟶ Q show that P and Z are relatively smooth over S. Also, the morphism Z ⟶ P is a closed immersion.

We are then reduced to the following statement.

Lemma 4.3. Let S = SpfA, P = SpfB, Z = SpfC be affine formal V′-schemes (p-adic and) of finite type and let

\begin{matrix}Z & ↪ & P \\ ↓ & ↙ \\ S \end{matrix}

be a closed immersion of S-objects where Z ⟶ S (resp. P ⟶ S) is smooth of relative dimension d (resp. d + s). Assume (always true locally on P) that C = B/J where J = (f_{1}, …, f_{s}) is generated by s elements. Let ]Z_{k}[_{P} denote the open tube of radius 1 of Z_{k} in the generic fiber P_{K} of P over K. Then, if i : Z_{K} ↪]Z_{k}[_{P} denotes the canonical closed immersion of K-rigid analytic spaces, there exists a retraction σ :]Z_{k}[_{P} → Z_{K} and an isomorphism

]Z_{k}[_{P} ⟶^{∼}Z_{K} × D^{s}

such that the diagram

\begin{matrix}Z_{K} & ⟶^{(id_{Z_{K}} ,0)} & Z_{K} × D^{s} \\ ↓^{i} & ^{∼} ↗ & ↓^{pr_{1}} \\ ]Z_{k}[_{P} & ⟶^{σ} & Z_{K} \end{matrix}

commutes.

Proof. We rely upon the construction of [Be, 0.2.6, 0.2.7], which applies to locally noetherian formal V′-schemes, extending Raynaud’s definition of the generic fiber to formal V′-schemes not necessarily p-adic and of finite type. This is one of the main motivations for assuming that all algebraic objects are defined over the noetherian ring V′. We point out that in this argument, and nowhere else in the present paper, we use formal V′-schemes which are not p-adic formal V-schemes of finite type. As in [Be, loc. cit.], we denote by P̂ the formal completion of P along Z_{k′}. The ideal of B defining Z_{k′} is J′ = (π, f_{1}, …, f_{s}), so that P̂ = SpfB̂_{J′}, where B̂_{J′} = lim_{⟵}_{n}B/J′^{n}. Berthelot shows that it is possible to

associate functorially to such a formal V′-scheme a K′-rigid analytic space P̂_{K′} (not quasi-compact in the present case). The canonical morphism

P̂_{K′} → P_{K′}

is an open immersion of K′-rigid analytic spaces and, by [Be, Prop.0.2.7], it induces an isomorphism

P̂_{K′} →^{∼}]Z_{k′}[_{P} .

We now apply [EGA, Chap.0, 19.5.4] to A_{n} = A/π^{n+1}A, B_{n} = B/π^{n+1}B, J_{n} = ideal generated in B_{n} by the images of f_{1}, …, f_{s}, C_{n} = C/π^{n+1}C = B_{n}/J_{n}. Clearly, B_{n} and C_{n} are smooth A_{n}-algebras, so that B_{n} endowed with the J_{n}- adic topology is formally smooth over the discrete ring A_{n}. We conclude that lim_{⟵}_{m}B_{n}/J_{n}^{m} ≅ C_{n}[[T_{1}, …, T_{s}]], endowed with the (T_{1}, …, T_{s})-adic topology.

Now, the sequence (π, f_{1}, …, f_{s}) in B is regular, so that

(π, f_{1}, …, f_{s})^{2n} ⊂ (π)^{n} + (f_{1}, …, f_{s})^{n} ,

and the two fundamental systems of open ideals {(π)^{n} + (f_{1}, …, f_{s})^{m}}_{m,n} and {(π, f_{1}, …, f_{s})^{n}}_{n} determine the same linear topology on B. Then, B̂_{J′} =

lim_{⟵}_{n}(lim_{⟵}_{m}B_{n}/J_{n}^{m}) ≅ lim_{⟵}_{n}C_{n}[[T_{1}, …, T_{s}]] ≅ C[[T_{1}, …, T_{s}]]. The topology on

C[[T_{1}, …, T_{s}]] is the (π, T_{1}, …, T_{s})-adic. The retraction σ :]Z_{k}[_{P} → Z_{K} then corresponds, via the construction of Berthelot and base-field extension to K, to the inclusion C ↪ C[[T_{1}, …, T_{s}]], while

(SpfC[[T_{1}, …, T_{s}]])_{K} = Z_{K} × D^{s} .

Q.E.D.

§5 Systems with logarithmic singularities on relative open polydisks of radius 1. ([Ba-Ct2]).

The purpose of this section is to state the main p-adic result of the paper [Ba-Ct2], which should appear very soon. That paper is dedicated to the classification of systems with logarithmic singularities, depending upon parameters. It is divided into two parts. The first contains the Gérard-Levelt formal theory of such systems ([G-L], [Ba2]), over an algebra with derivations (i.e. the parameter space is a smooth “space” equipped with a connection, typically a non-singular algebraic variety or a smooth affinoid space). In the second part of that paper, we tie together the formal theory of the first part, with the results “à la Christol” of [Ba-Ct0], which established conditions to ensure p-adic convergence “in the large” of formal power series solutions of systems of the present type. We obtain the classification of systems with logarithmic singularities along the coordinate divisor on an open polydisk of radius 1, relative to a smooth affinoid base-space. Here, the two notions of local overconvergence and of non-Liouvilleness of the exponents of monodromy, play a crucial role. We are indebted to S. Bosch for pointing out to us the difficulties arising in the study of coherent or even locally free sheaves of finite type over a non-affinoid rigid analytic space, typically over an open polydisk or polyannulus T relative to an affinoid space S, via φ : T ⟶ S. These difficulties forced us to restrict our considerations to coherent sheaves on T of the form φ^{∗}F, for a coherent O_{S}-module F. This avoids the problem of the possible existence of O_{T} -modules which are locally free of finite type over (an admissible covering of) T, but not necessarily over the inverse image on T of any admissible covering of S. Also, the correspondence between coherent sheaves and modules of finite type over O_{T} (T), works properly when restricted to sheaves of the previous type. This restriction represents on the other hand no loss of generality in view of our “global” purposes.

We consider a regular affinoid K-algebra A, denote by |−|_{A} its supremum norm and define

D_{A}^{s} = SpmA × D^{s} ,

the trivial fiber bundle in open polydisks of radius 1 with “vertical coordinates” (x_{1}, …, x_{s}) over the smooth affinoid space S = SpmA. We denote by φ = pr_{1} : D_{A}^{s} ⟶ S the fibration, and by i : S ⟶ D_{A}^{s}, the closed immersion (“zero section”) p ⟼ (p, 0), for p ∈ S. Let L be the divisor x_{1}x_{2} … x_{s} = 0. We sometimes abbreviate O_{D_{A}^{s}} into O; we write Ω_{D_{A}^{s}/K}^{1} < L > (resp. Ω_{D_{A}^{s}/S}^{1} <L>) to denote the O_{D_{A}^{s}}-module of absolute (resp. relative to S) differentials (in the sense of rigid analytic geometry) with logarithmic singularities along L. For ϵ , η ∈ [0, 1) ∩ Γ, ϵ ≤ η, we set (cf. section 3):

C_{A,ϵ}^{s} = S × C_{ϵ}^{s} ,
C_{A}^{s}(ϵ, η) = S × C(ϵ, η)^{s} ,
D_{A}^{s}(η) = S × D(η)^{s} = C_{A}^{s}(0, η) .

Then i(S) = D_{A}^{s}(0). Finally, we put

D^{∗}_{A}^{s} = S × D^{∗}^{s} = C_{A,0}^{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}}} of the ring of formal power series A[[x]] = A[[x_{1}, …, x_{s}]]. Namely, we have:

(5.1) 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 have analogously for η ∈ Γ ∩ [0, 1) and a ∈ K with |a| = η:

O(D_{A}^{s}(η)) = A{x_{1}, …, x_{s}, y_{1}, …, y_{s}}/(x_{i} − ay_{i}, i = 1, …, s) =

(5.2)  = A{x_{1}/a, …, x_{s}/a} = A{x/a} =

= { ∑_{α∈N^{s}} a_{α}x^{α}, a_{α} ∈ A | lim_{|α|_{R}→+∞} |a_{α}|_{A}η^{|α|_{R}} = 0 } .

We emphasize the canonical imbedding of rings:

(5.3)  T : O(D_{A}^{s}(η)) ↪ A[[x_{1}, …, x_{s}]] .

There is a natural restriction map ρ_{η} : A{{x}} ⟶ O(D_{A}^{s}(η)), which is simply ∑_{α∈N^{s}} a_{α}x^{α} ⟼ ∑_{α∈N^{s}} a^{|α|_{R}}a_{α}(x/a)^{α}, if we use the previous representation for the ring O(D_{A}^{s}(η)). In particular, the immersion i : S ⟶ D_{A}^{s} corresponds to the natural augmentation ρ_{0} : A{{x}} ⟶ A, f(x) ⟼ f(0).

If we choose a sequence η_{n} ∈ Γ∩(0, 1), n ∈ N, η_{n} ↑ 1, then we have naturally

(5.4)  A{{x}} = lim_{⟵ n} O(D_{A}^{s}(η_{n})) .

To describe O(C_{A}^{s}(ϵ, η)) explicitly, for ϵ , η ∈ Γ ∩ (0, 1), ϵ ≤ η, we write for α ∈ Z^{s}, α = α_{+} − α_{−}, with α_{+}, α_{−} ∈ N^{s} and |α_{+}|_{R} + |α_{−}|_{R} = |α|_{R}. Then, if a, b ∈ K with |a| = η , |b| = ϵ

O(C_{A}^{s}(ϵ, η)) = A{x_{1}, …, x_{s}, y_{1}, …, y_{s}, z_{1}, …, z_{s}}/(x_{i}−ay_{i}, x_{i}z_{i}−b, i = 1, …, s) =

(5.5)  = A{x_{1}/a, …, x_{s}/a, b/x_{1}, …, b/x_{s}} = A{x/a, b/x} =

= { ∑_{α∈Z^{s}} a_{α}x^{α}, a_{α} ∈ A | lim_{|α|_{R}→+∞} |a_{α}|_{A}η^{|α_{+}|_{R}}ϵ^{−|α_{−}|_{R}} = 0 } .

We point out the existence of a canonical monomorphism of A-modules

(5.6)  T : O(C_{A}^{s}(ϵ, η)) ↪ ∏_{α∈Z^{s}} Ax^{α} ,

determined by the conditions of sending ∑_{α∈Z^{s}} a_{α}x^{α} to (a_{α}x^{α})_{α}. Of course, this map coincides with that of (5.3) when ϵ = 0. There is a natural restriction map, for any δ ∈ [0, ϵ) ∩ Γ , ρ_{ϵ,η}^{δ} : O(C_{A,δ}^{s}) ⟶ O(C_{A}^{s}(ϵ, η)), which is simply ∑_{α∈Z^{s}} a_{α}x^{α} ⟼ ∑_{α∈Z^{s}} a^{|α_{+}|_{R}}b^{−|α_{−}|_{R}}a_{α}(x/a)^{α_{+}}(b/x)^{α_{−}}, if we use the previous representation for the rings O(C_{A,δ}^{s}) and O(C_{A}^{s}(ϵ, η)). If we choose sequences ϵ_{n}, η_{n} ∈ Γ ∩ (δ, 1), n ∈ N, ϵ_{n} < η_{n}, η_{n} ↑ 1, ϵ_{n} ↓ δ, then we have naturally

(5.7)  O(C_{A,δ}^{s}) = lim_{⟵ n} O(C_{A}^{s}(ϵ_{n}, η_{n})) .

The restriction mappings ρ_{ϵ,η}^{δ} are compatible with T , so that we obtain a canonical monomorphism of A-modules

(5.8)  T : O(C_{A,δ}^{s}) ↪ ∏_{α∈Z^{s}} Ax^{α} ,

whose image consists of the elements (a_{α}x^{α})_{α} such that

lim_{|α|_{R}→+∞} |a_{α}|_{A}η^{|α_{+}|_{R}}ϵ^{−|α_{−}|_{R}} = 0 ,

for all δ < ϵ < η < 1. For an A-module of finite type M, we denote by M˜ the corresponding coherent sheaf on S.

Lemma 5.9. Let φ : T ⟶ S denote one of the fibrations:

φ_{|C_{A}(ϵ,η)^{s}} : C_{A}(ϵ, η)^{s} ⟶ S ,

for ϵ , η ∈ Γ ∩ [0, 1), ϵ ≤ η. Let M be an A-module of finite type and let | − |_{M} denote any Banach norm on M. Then i)

(φ^{∗}(M˜ ))(T) = M ⊗_{A} O_{T} (T) .

ii) The map T of (5.6) induces a canonical monomorphism of A-modules

T_{M} : (φ^{∗}(M˜ ))(T) ⟶ ∏_{α∈Z^{s}} M ⊗ x^{α} .

iii) The image of T_{M} consists precisely of the elements

(m_{α} ⊗ x^{α})_{α} ∈ ∏_{α∈Z^{s}} M ⊗ x^{α} ,

such that

(5.9.1)  lim_{|α|_{R}→+∞} |m_{α}|_{M}η^{|α_{+}|_{R}}ϵ^{−|α_{−}|_{R}} = 0 .

iv) For m ∈ (φ^{∗}(M˜ ))(T) we put

||m||_{ϵ,η} = sup_{α∈Z^{s}} |m_{α}|_{M}η^{|α_{+}|_{R}}ϵ^{−|α_{−}|_{R}} ,

if T_{M}(m) = (m_{α} ⊗ x^{α})_{α}. Then || − ||_{ϵ,η} (written || − ||_{η}, when ϵ = 0) is a Banach norm in (φ^{∗}(M˜ ))(T).

v) Every element m of (φ^{∗}(M˜ ))(T) can be uniquely written as a convergent sum:

m = ∑_{α∈Z^{s}} m_{α}x^{α} ,

with m_{α} ∈ M.

Proof. The first assertion is simply the definition of φ^{∗}(M˜ ). The remaining assertions are very easy and their proof is omitted. This lemma should be compared with [Be, 2.2.12]. Q.E.D.

Remark 5.9.2. From part ii) of Lemma 5.9 one can deduce an elementary proof of flatness of O_{T} (T) over A.

Corollary 5.10. Let φ : T ⟶ S denote one of the fibrations:

φ : D_{A}^{s} ⟶ S ,

φ_{|C_{A,ϵ}^{s}} : C_{A,ϵ}^{s} ⟶ S ,

for any ϵ ∈ [0, 1) ∩ Γ. Let M be an A-module of finite type. Then: i)

(φ^{∗}(M˜ ))(T) = M ⊗_{A} O_{T} (T) .

ii) Let U be an affinoid open subset of T and let ρ_{U}^{T} : O(T) ⟶ O(U) denote the restriction map. Then

(φ^{∗}(M˜ ))(U) = (φ^{∗}(M˜ ))(T) ⊗_{O(T)} O(U) ,

where the tensor product is taken w.r.t. ρ_{U}^{T}.

Proof. We give the proof for fiber spaces of the second type, the case of T = D_{A}^{s} being perfectly analogous. We write M for φ^{∗}(M˜ ). We choose sequences ϵ_{n}, η_{n} ∈ Γ ∩ (ϵ, 1), n ∈ N, ϵ_{n} < η_{n}, η_{n} ↑ 1, η_{n} ↓ ϵ, to have as in (5.7):

O(C_{A,ϵ}^{s}) = lim_{⟵ n} O(C_{A}^{s}(ϵ_{n}, η_{n})) .

By the lemma, every element m(n) of M(C_{A}^{s}(ϵ_{n}, η_{n})) = M ⊗_{A} O(C_{A}^{s}(ϵ_{n}, η_{n})) can be uniquely written as

m(n) = ∑_{α∈Z^{s}} m_{α}(n) ⊗ x^{α} ,

with m_{α}(n) ∈ M, and

lim_{|α|_{R}→+∞} |m_{α}(n)|_{M}η_{n}^{|α_{+}|_{R}}ϵ_{n}^{−|α_{−}|_{R}} = 0 .

Then, {m(n)}_{n} represents an element m of

lim_{⟵ n}M ⊗_{A} O(C_{A}^{s}(ϵ_{n}, η_{n})) = M(T) ,

precisely when m_{α}(n) = m_{α} is independent of n, for n ∈ N. Now, let g_{1}, …, g_{N} be a system of A-generators of M. We may assume ([BGR, 3.7.3.3]) that |−|_{M} is the residue norm of the corresponding map A^{N} ⟶ M. We pick positive real numbers l_{α} for α ∈ Z^{s} in such a way that

lim_{|α|_{R}→+∞} l_{α}δ^{−|α_{−}|_{R}} = 0 ,

for all δ > 0. Then we may write

m_{α} = ∑_{i=1}^{N} a_{α,i}g_{i} ,

with

( sup_{i=1,...,N} |a_{α,i}|_{A}) − |m_{α}|_{M} < l_{α} ,

by definition of the residue norm. In the end a_{i} = ∑_{α∈Z^{s}} a_{α,i}x^{α} ∈ O(C_{A,ϵ}^{s}), for i = 1, …, N, so that

m = ∑_{i=1}^{N} g_{i} ⊗ a_{i} = ∑_{i=1}^{N} a_{i}g_{i} ∈ M ⊗_{A} O(C_{A,ϵ}^{s}) .

This proves part i). Part ii) is obvious. Q.E.D.

Remark 5.10.1. S. Bosch suggested to us the following alternative proof of part i) of Corollary 5.10. One first deduces from remark 5.9.2 that, in the notation of the corollary, the functor φ^{∗} is an exact functor from the category of coherent O_{S}-modules to that of coherent O_{T} -modules. One then recalls from [K3, Satz 2.4], that the functor Γ(T, −) is also an exact functor from the category of coherent O_{T} -modules to that of O_{T} (T)-modules. In the end the l.h.s. of i) of Corollary 5.10, represents an exact functor Ψ(M) in the A-module of finite type M. One then considers a presentation of M

(5.10.2)  F′ ⟶ F ⟶ M ⟶ 0 ,

with F′ and F finite and free over A. By applying to (5.10.2) on the one hand the exact functor Ψ(−) and on the other the a priori right-exact functor (−) ⊗_{A} O_{T} (T), and noticing that the two coincide on free A-modules of finite type, one deduces part i) of Corollary 5.10. We wish to point out that this reasoning also proves flatness of O_{T} (T) over A, in the notation of the corollary.

Remark 5.11. The exact sequences of locally free O_{D_{A}^{s}}-modules

(5.11.1)  0 ⟶ φ^{∗}Ω_{SpmA/K}^{1} ⟶ Ω_{D_{A}^{s}/K}^{1} ⟶ Ω_{D_{A}^{s}/SpmA}^{1} ⟶ 0 ,

and

(5.11.2)  0 ⟶ φ^{∗}Ω_{S/K}^{1} ⟶ Ω_{D_{A}^{s}/K}^{1} <L>⟶ Ω_{D_{A}^{s}/S}^{1} <L>⟶ 0 ,

are the image via the functor φ^{∗} of exact sequences of locally free O_{S}-modules of finite type.

We may now abandon sheaves and express our results in terms of A-modules, for the rest of this section. Let M be an A{{x}}-module (resp. 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 = M ⊗_{A} A{{x}}). Let Mod(A{{x}}) be the 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 φ^{∗}M_{0}, for an A-module M_{0} of finite type; the morphisms of Mod_{A}^{f}(A{{x}}) are simply morphisms of A{{x}}-modules. An object M of Mod_{A}^{f}(A{{x}}) defines a coherent O-module M. Under the present assumptions, we have M(D_{A}^{s}) = M.

For any object M = φ^{∗}M_{0} of Mod_{A}^{f}(A{{x}}), the choice of a Banach norm |−|_{M_{0}} on M_{0} determines, for any ϵ , η ∈ Γ ∩ [0, 1), a seminorm ||−||_{ϵ,η} = ||−||_{ϵ,η,M} on M via the construction of iv) of lemma 5.9. (Again we write ||−||_{η} = ||−||_{η,M} instead of ||−||_{0,η} = ||−||_{0,η,M}, in case ϵ = 0.) We remark that the equivalence class of the seminorm ||−||_{ϵ,η} is independent of the chosen Banach norm |−|_{M_{0}} on M_{0}.

We now come to the description of our results on logarithmic singularities. We fix an additive subgroup Σ of the additive group of K consisting of

p-adically non-Liouville numbers.

For any K-algebra C, let Der(C/K) denote the K-Lie-algebra of K-derivations of C. Let L denote the K-Lie-algebra of continuous derivations of A/K, extended to A{{x}} by the condition

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

for d ∈ L and ∑_{α∈N^{s}} a_{α}x^{α} ∈ A{{x}}. Let L_{x} be the K-Lie-sub-algebra and A{{x}}-sub-module of Der(A{{x}}/K) generated by L and by the A-derivations x_{1} \frac{∂}{∂x_{1}} , …, x_{s} \frac{∂}{∂x_{s}}.

Definition 5.12. Let C denote a commutative unitary K-algebra and G a K-Lie sub-algebra and C-sub-module of Der(C/K). A (C, G)-module is a left C-module M with an action G × M ⟶ M, (g, m) ⟼ g(m) satisfying

(i)  [g, g′](m) = g(g′(m)) − g′(g(m))
(ii)  (cg)(m) = c(g(m))
(iii)  g(cm) = cg(m) + g(c)m

for g, g′ ∈ G, m ∈ M, and c ∈ C. A morphism of (C, G)-modules is a Clinear map commuting with the action of G. The abelian group of morphisms of (C, G)-modules from M to N, will be denoted Hom_{(C,G)}(M, N). We denote by Mod(C, G) (resp. Mod^{f}(C, G)) the category of (C, G)-modules (resp. whose underlying C-module is of finite type).

It is easily seen ([Ba-Ct2, 1.2.2]) that, under the assumptions of the definition, Mod(C, G) (and Mod^{f}(C, G) if C is noetherian) is an abelian category. It is also well-known that every object of Mod^{f}(A, L) is a projective A-module ([Ba-Ct2, 1.4.4]). Let M be an object of Mod^{f}(A, L); an element φ ∈ End_{(A,L)}M then satisfies a polynomial with coefficients in K ([Ba-Ct2, 1.5.3]). We say that φ ∈ End_{(A,L)}M is semisimple if the minimal polynomial of φ (which is an element of K[x]) is a product of distinct irreducible factors. We recall Proposition 1.8.1 of [Ba-Ct2] and its corollaries:

Proposition 5.13. 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.

Corollary 5.14. 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} .

Corollary 5.15. 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 5.14. For λ = (λ_{1}, …, λ_{s}) ∈ K^{s} we put

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

For N ∈ N sufficiently big, we have

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

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 and we have

M = ⨁_{λ∈K^{s}} M(λ)

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

Definition 5.16. In the situation of the previous corollary, we will say that λ = (λ_{1}, …, λ_{s}) ∈ K^{s} is a multi-eigenvalue of (φ_{1}, …, φ_{s}), if M(λ) ≠ 0. For any point a ∈ SpmA and any λ ∈ K^{s}, we will define the multiplicity in a of λ as a multi-eigenvalue of M as the rank in a of the projective A-module M(λ). If SpmA (i.e. SpecA) is connected, we omit the reference to a particular point a of SpmA.

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. Let Mod^{f}(A, L, Φ) indicate the category of (A[Φ], L)-modules which are of finite type as A-modules.

It follows that an object in Mod^{f}(A, L) endowed with endomorphisms φ_{1}, …, φ_{s} which commute with each other is precisely an object of Mod^{f}(A, L, Φ), when φ_{i} is identified with the multiplication by Φ_{i}, for i = 1, …, s. The multi-exponents of an object M of Mod^{f}(A, L, Φ) are then meant to be the multi-eigenvalues of φ_{1}, …, φ_{s} on M, viewed as an object of Mod^{f}(A, L). We define multiplicities of multi-exponents of M as in (5.16). For any point a ∈ SpmA and any λ ∈ K^{s}, we write e_{a,λ} = e_{a,λ,M} for the multiplicity in a of λ as a multi-exponent of M; if SpmA is connected, we simply write e_{λ} = e_{λ,M}

We define Mod_{Σ}^{f}(A, L, Φ) to be the full subcategory of Mod^{f}(A, L, Φ) consisting of objects on which (φ_{1}, …, φ_{s}) have all their multi-eigenvalues in Σ^{s}. The categories Mod^{f}(A, L, Φ) and Mod_{Σ}^{f}(A, L, Φ) are easily seen to be abelian.

Let Mod_{A}^{f}(A{{x}}, L_{x}) denote the full (additive) sub-category of the abelian category Mod(A{{x}}, L_{x}), consisting of the objects M, whose underlying A{{x}}- module is an object of Mod_{A}^{f}(A{{x}}).

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

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

We then obtain a natural functor

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

which coincides with the previously defined i^{∗} 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, Φ) ⟶ Mod_{A}^{f}(A{{x}}, L_{x}) ,

which again coincides with the previously defined φ^{∗} at the level of underlying A-modules, in the following way. Let N be an object of Mod^{f}(A, L, 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 and i = 1, …, s. 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_{x}) will be called a simple object (or a λ-simple object, if specification of λ is necessary).

We define the multi-exponents of an object M of Mod_{A}^{f}(A{{x}}, L_{x}) as the multi-eigenvalues of i_{diff}^{∗}M. We will say that M has all of its multi-exponents in Σ^{s}, when this assumption holds for i_{diff}^{∗}(M). We will indicate with E(M) the set of the multiexponents of M. Let Mod_{A,Σ}^{f}(A{{x}}, L_{x}) denote the full (additive) sub-category of Mod_{A}^{f}(A{{x}}, L_{x}) consisting of the objects whose multi-exponents are all in Σ^{s}. For λ, μ ∈ K^{s} we put

λ ≤ μ ⇔ μ − λ ∈ N^{s} .

This defines a partial ordering in K^{s}, and in particular in E(M), for any object M of Mod_{A,Σ}^{f}(A{{x}}, L_{x}).

In the following definition we make use of the fact that for α ∈ N^{s} and for suitable integers d_{α,β}, β ≤ α,

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

as operators on any object M of Mod(A{{x}}, L_{x}) (cf. [Ch2, Prop.1.1], [Ba-Ct2, 3.7.2]).

Definition 5.18. An object M of Mod_{A}^{f}(A{{x}}, L_{x}) will be said to be overconvergent if the following condition is satisfied:

(SC)_{L}  \begin{aligned}∀m ∈ M & and ∀ϵ , η ∈ (0, 1), ϵ ∈ Γ \\ & lim_{|α|_{R}→∞} ||x^{α}(∂/∂x)^{α}(m)/α!||_{ϵ}η^{|α|_{R}} = 0 . \end{aligned}

It’s easy to show that condition (SC)_{L} only needs to be checked for m in a system of A{{x}}-generators of M. Also, if N ⟶ M (resp. (M ⟶ N) is a monomorphism (resp. an epimorphism) in Mod_{A}^{f}(A{{x}}, L_{x}), and if condition (SC)_{L} holds for M, then it also holds for N. We denote by SCMod_{A}^{f}(A{{x}}, L_{x}) (resp. SCMod_{A,Σ}^{f}(A{{x}}, L_{x})) the full (additive) subcategory of Mod_{A}^{f}(A{{x}}, L_{x}) (resp. Mod_{A,Σ}^{f}(A{{x}}, L_{x})) consisting of overconvergent objects. We proved in [Ba-Ct0, 5.2] that an object of SCMod_{A,K}^{f}(A{{x}}, L_{x}) automatically has multiexponents in Z_{p}^{s}. Therefore SCMod_{A,Σ}^{f}(A{{x}}, L_{x}) = SCMod_{A,Σ∩Z_{p}}^{f}(A{{x}}, L_{x}). On the other hand, one easily checks that a λ-simple object of Mod_{A}^{f}(A{{x}}, L_{x}) is overconvergent if and only if λ ∈ Z_{p}^{s}.

The main result of [Ba-Ct2] is essentially the following:

Theorem 5.19. ([Ba-Ct2, Thm.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

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

Then M_{0}^{n}(λ̃) is a non-zero sub-object of M_{0} in Mod_{Σ}^{f}(A, L, Φ). For any point a ∈ SpmA, 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 a monomorphism in Mod(A{{x}}, L_{x}) 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 ∈ SpmA 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) .

Definition 5.20. 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 5.19, τ(M) = τ(Coker(χλ̃)) + 1.

It is easy to check that definition 5.20 is good.

§6 Local comparison theorem.

The aim of this section is to finally prove theorem 2.4. We fix λ ∈ (0, 1) ∩ Γ sufficiently close to 1 and a coordinate open set (U, x_{1}, …, x_{m}) ∈ U with Z_{K} ∩ U_{K} = V (x_{1} … x_{ν(U)}), as in section 2. We then apply the construction of section 4, to obtain an admissible covering of Û_{K} consisting of open sets U_{S,η} (cf. Proposition 4.2) isomorphic to trivial fiber bundles in open polydisks of radius 1 over a smooth affinoid base, like the D_{A}^{s} of section 5. We remarked in (4.1.4) that, in this trivialization, V_{λ} ∩ D_{A}^{s} = C_{A,λ}^{s}, while the coordinates (x_{1}, …, x_{s}) on D_{A}^{s} are induced by some of the coordinates (x_{1}, …, x_{ν(U)}) on U and Ẑ_{K} ∩ D_{A}^{s} = L. Without contradicting the global notation of section 2, we will denote by j : D_{A}^{∗s} ↪ D_{A}^{s} and j_{λ} : C_{A,λ}^{s} ↪ D_{A}^{s} the canonical open immersions. The diagram (2.1.2) then traces on D_{A}^{s} the diagram

(6.0)  \begin{matrix}C_{A,λ}^{s} & ⟶^{j_{λ}} & D_{A}^{s} \\ ↓ & ↗ j \\ D_{A}^{∗s} \end{matrix}

We must check that the sheaf with connection (E^{an}, ∇^{an}) induces on D_{A}^{s} an object of SCMod_{A,Σ}^{f}(A{{x}}, L_{x}), namely E^{an}(D_{A}^{s}) equipped with the action of ∇^{an}(∂), for each continuous derivation ∂ of A, and of ∇^{an}(x_{i}∂/∂x_{i}), for i = 1, …, s. All verifications are trivial except perhaps the one of the local condition of overconvergence (SC)_{L} given in definition 5.8. To that end, we need the following lemma.

Lemma 6.1. Let P denote a formal affine V-scheme finite over a formal affine space Â_{V}^{m} and flat. Let T_{k} be a Zariski closed subset of the special fiber P_{k} of P, not containing any component of P_{k}. Let W = P_{K}\]T_{k}[_{P} denote the complement of the open tube of T_{k} in P. Then for any function f ∈ O(P_{K}), we have

||f||_{P_{K}} = ||f||_{W} .

Proof. We know from [BGR, 6.2.1.4] that, unless ||f||_{P_{K}} = 0 (in which case we have nothing to prove), there exist c ∈ K and m ∈ N such that ||cf^{m}||_{P_{K}} = 1. We may then assume ||f||_{P_{K}} = 1. Then ([BGR, 6.3.4.1]) f is integral over the ring of power-bounded elements of O((Â_{V}^{m})_{K}) which trivially coincides with O(Â_{V}^{m}). Hence f is integral over O(P). Then there exists a finite morphism of affine formal V-schemes σ : P′ ⟶ P, whose generic fiber is an isomorphism σ_{K} : P′_{K} ⟶^{∼}P_{K} such that P′ is flat and f ∈ O(P′). Since σ is finite, so is σ_{k}, and, by the dimension formula ([Ma, 14.C, Th.23] and [Ma, 14.H, Cor.3]) σ_{k}^{−1}(T_{k}) = T′_{k} also does not contain any component of P′_{k}. On the other hand, we have a commutative diagram

(6.1.1)  \begin{aligned}& |P′_{K}| = |P_{K}| \\ sp_{P′} ↙ & & ↘ sp_{P} \\ |P′_{k}| & ⟶^{σ_{k}} & |P_{k}| \end{aligned}

where |S| denotes the underlying topological space to any ringed space S, while sp_{P}, sp_{P′} denote the specialization maps. In the previous diagram all maps are surjective, when restricted to the subsets of closed points (cf. [Be, 1.1.5]). Then

W = P_{K}\]T_{k}[_{P} = P′_{K}\]T′_{k}[_{P′} .

This shows that we may assume f ∈ O(P) and ||f||_{P_{K}} = 1 to start with. The proof then reduces to showing that ||f||_{W} < 1 is impossible. In fact, let \overline{f} ∈ O(P_{k}) denote the restriction of f to P_{k} and let x ∈ |P_{K}|. Then, if \overline{x} = sp_{P} (x) is the corresponding closed point of P_{k},

|f(x)| < 1 ⇔ \overline{f}(\overline{x}) = 0 .

If ||f||_{W} < 1, from the surjectivity of

sp_{P} : |W | ⟶ { closed points of P_{k} \ T_{k} } ,

we deduce that \overline{f}(\overline{x}) = 0 for all closed points \overline{x} of P_{k} \ T_{k}. But then (since every k-scheme of finite type is of Jacobson [EGA, I.6.4.6]), \overline{f} = 0 on P_{k} \ T_{k}, hence \overline{f} = 0 on P_{k}. Therefore, we would have |f(x)| < 1 ∀x ∈ P_{K}, which contradicts the Maximum Modulus Principle on P_{K}. Q.E.D.

We immediately deduce the following statement.

Corollary 6.2. In the situation of the lemma, let M be a coherent sheaf of O_{P} -modules and M_{K} = sp_{P}^{∗} M denote the corresponding coherent sheaf on P_{K}. Then, for any choice of Banach norms || − ||_{P_{K}} on M_{K}(P_{K}) and || − ||_{W} on M_{K}(W), there exist constants C, C′ > 0, such that

||m||_{P_{K}} < C||ρ_{W}^{P_{K}} (m)||_{W} < C′||m||_{P_{K}} ,

for all m ∈ M_{K}(P̂_{K}), where ρ_{W}^{P_{K}} : M_{K}(P_{K}) ⟶ M_{K}(W) denotes the restriction map for M_{K}.

We now reconsider condition (SC)_{U} of section 2. We observe that for any affinoid open sub-domain V of Û_{K}, for any section e ∈ Γ(V, E^{an}) and α ∈ N^{m},

x^{α}(∇^{an}(∂/∂x))^{α}(e) ∈ Γ(V, E^{an}) .

Then, let W denote the affinoid open sub-domain of Û_{K}

W = Û_{K} \ ⋃_{i=1}^{r}]Z_{k}^{(i)}[_{P} .

Obviously, W ⊂ Û_{K} ∩ V_{λ}, ∀λ ∈ (0, 1) ∩ Γ. By Corollary 6.2, for any choice of Banach norms || − ||Û_{K} on E_{K}^{an}(Û_{K}) and || − ||_{W} on E_{K}^{an}(W), there exist constants C, C′ > 0, such that, for any e ∈ Γ(Û_{K}, E^{an}) and α ∈ N^{m},

||(α!)^{−1}x^{α}(∇^{an}(∂/∂x))^{α}(e)||Û_{K} < C||(α!)^{−1}x^{α}(∇^{an}(∂/∂x))^{α}(ρÛ_{K} ; W e)||_{W} <

< C′||(α!)^{−1}x^{α}(∇^{an}(∂/∂x))^{α}(e)||Û_{K} .

Condition (SC)_{U} then implies the following condition:

(SC)′_{U} ∀η ∈ Γ ∩ (0, 1) and for any section e ∈ Γ(Û_{K}, E^{an}),

lim_{|α|_{R}⟶+∞}||(α!)^{−1}x^{α}(∇^{an}(∂/∂x))^{α}(e)||Û_{K} η^{|α|_{R}} = 0 ,

for any choice of a Banach norm || − ||Û_{K} on E^{an}(Û_{K}).

It is now clear that condition (SC)_{L} holds for M = E^{an}(Û_{K}).

On the other hand it is clear that an object M of SCMod_{A,Σ}^{f}(A{{x}}, L_{x}) determines on D_{A}^{s} a locally free O_{D_{A}^{s}}-module of finite type M, equipped with an integrable D_{A}^{s}/K-connection ∇ = ∇_{M} with logarithmic singularities along Z_{K} ∩ D_{A}^{s} = L.

If M = E^{an}(D_{A}^{s}) is obtained from (E^{an}, ∇^{an}) as before, then (M, ∇_{M}) = (E^{an}, ∇^{an})_{|D_{A}^{s}} = (E_{|D_{A}^{s}}^{an}, ∇_{|D_{A}^{s}}^{an}).

Given an object M of SCMod_{A,Σ}^{f}(A{{x}}, L_{x}), we may now consider the complexes of abelian sheaves on D_{A}^{s}

DR_{mer}^{∗}(M) = 0 ⟶ M[(x_{1} … x_{s})^{−1}] ⟶ M ⊗ Ω_{D_{A}^{s}/K}^{1}[(x_{1} … x_{s})^{−1}] ⟶ …,

DR_{ess}^{∗}(M) = j_{∗}DR(D_{A}^{∗s}/K, (M, ∇_{M})_{|D_{A}^{∗s}}) =
= j_{∗}(0 ⟶ M_{|D_{A}^{∗s}} ⟶ M_{|D_{A}^{∗s}} ⊗ Ω_{D_{A}^{∗s}/K}^{1} ⟶ …) ,
DR_{λ}(M) = j_{λ∗}DR(C_{A,λ}^{s}/K, (M, ∇_{M})_{|C_{A,λ}^{s}}) =
= j_{λ∗}(0 ⟶ M_{|C_{A,λ}^{s}} ⟶ M_{|C_{A,λ}^{s}} ⊗ Ω_{C_{A,λ}^{s}/K}^{1} ⟶ …) .

The previous complexes are related by the natural monomorphisms

(6.3)  DR_{mer}^{∗}(M) ↪ DR_{ess}^{∗}(M) ↪ DR_{λ}(M) .

Obviously, Theorem 2.4 can be deduced, via the spectral sequence argument of Corollary 3.7, from the following result:

Theorem 6.4. For any object M of SCMod_{A,Σ}^{f}(A{{x}}, L_{x}), the morphisms (6.3) induce isomorphisms of hypercohomology groups

(6.4.1) H^{•}(D_{A}^{s}, DR_{mer}^{∗}(M))⟶^{∼}H^{•}(D_{A}^{s}, DR_{ess}^{∗}(M))⟶^{∼}H^{•}(D_{A}^{s}, DR_{λ}(M)) .

Proof. We apply theorem 5.9 to M and a minimal multi-exponent λ̃ of M. We obtain a sequence

0 ⟶ M(λ̃) ⟶ M ⟶ Q(λ̃) ⟶ 0

in SCMod_{A,Σ}^{f}(A{{x}}, L_{x}), which is exact in the abelian category Mod(A{{x}}, L_{x}). Here M(λ̃) denotes a λ̃-simple object of SCMod_{A,Σ}^{f}(A{{x}}, L_{x}), such that

τ(Q(λ̃)) = τ(M) − 1 .

If we denote by M (resp. M(λ̃), resp. Q(λ̃)) the sheaf on D_{A}^{s} associated to M (resp. M(λ̃), resp. Q(λ̃)), we deduce a diagram, with exact rows, of complexes of abelian sheaves on D_{A}^{s}

a) = 0 ⟶ DR_{mer}^{∗}(M(λ̃)) ⟶ DR_{mer}^{∗}(M) ⟶ DR_{mer}^{∗}(Q(λ̃)) ⟶ 0
↓  ↓  ↓  ↓
b) = 0 ⟶ DR_{ess}^{∗}(M(λ̃)) ⟶ DR_{ess}^{∗}(M) ⟶ DR_{ess}^{∗}(Q(λ̃)) ⟶ 0
↓  ↓  ↓  ↓
c) = 0 ⟶ DR_{λ}(M(λ̃)) ⟶ DR_{λ}(M) ⟶ DR_{λ}(Q(λ̃)) ⟶ 0

We are then reduced, by induction on the length τ(M) of M, to proving the statement for a simple object. So, let γ ∈ Σ^{s}, and let M be a γ-simple object of SCMod_{A,Σ}^{f}(A{{x}}, L_{x}). After possible localization over an admissible covering of S, we may assume that M = φ^{∗}(M_{0}), for a free A-module M_{0} = ⊕_{i=1}^{N}Ae_{i} of finite type. We may then apply the techniques of [Ka, 3.2] (cf. [Ba2, 4]), i.e. the spectral sequence of relative cohomology, to reduce to comparison of de Rham cohomology relative to A. In view of calculation of de Rham cohomology relative to A, one may replace the simple object M = ⊕_{i=1}^{N}A{{x}}e_{i} (direct sum in the category of A{{x}}-modules!) of SCMod_{A,Σ}^{f}(A{{x}}, L_{x}) with an object M′ having the same underlying A{{x}}-module and the same action of x_{i}∂/∂x_{i}, for i = 1, …, s as M, but trivial (i.e. zero) action of L on the chosen basis elements e_{1}, …, e_{N}. Then M′ is a direct sum in the category SCMod_{A,Σ}^{f}(A{{x}}, L_{x}) of simple objects whose underlying A{{x}}-module is free of rank 1. The case of such an object will be treated in proposition 6.6 below. Q.E.D.

For γ ∈ K^{s} we consider the de Rham complexes:

a) = 0 ⟶  O(D_{A}^{s})  ⟶^{∇_{γ}}  ⊕_{i}O(D_{A}^{s})\frac{dx_{i}}{x_{i}}  ⟶ …
↓  ↓  ↓
b) = 0 ⟶ O(D_{A}^{s})[(x_{1} … x_{s})^{−1}] ⟶^{∇_{γ}} Ω_{D_{A}^{s}/S}^{1}(D_{A}^{s})[(x_{1} … x_{s})^{−1}] ⟶ …
↓  ↓  ↓
c) = 0 ⟶  O(D_{A}^{∗s})  ⟶^{∇_{γ}}  Ω_{D_{A}^{s}/S}^{1}(D_{A}^{∗s})  ⟶ …
↓  ↓  ↓
d) = 0 ⟶  O(C_{A,λ}^{s})  ⟶^{∇_{γ}}  Ω_{D_{A}^{s}/S}^{1}(C_{A,λ}^{s})  ⟶ …

where

(6.5)  ∇_{γ} = d + ∑_{i=1}^{s} γ_{i} \frac{dx_{i}}{x_{i}} , d = d_{D_{A}^{s}/S}.

The following proposition is a p-adic version “in the large” of a result of Deligne’s [De, Lemme 6.9], which we already discussed in the past in a locally analytic p-adic setting (cf. [Ba2, 5]). For the convenience of the reader we repeat Deligne’s arguments step by step.

Proposition 6.6.

i) If γ_{i} ∈ K is not a positive integer and ρ(γ_{i}) = 1, for all i = 1, …, s, the inclusions

a) ↪ b) , a) ↪ c) and a) ↪ d)

are homotopy equivalences. ii) If there exists i ∈ {1, …, s} such that ρ(−γ_{i}) = 1 and γ_{i} ∉ Z_{≤0}, then the complex a) is homotopically trivial. If γ ∈ Z_{≤0}^{s}, we have:

H^{i}(a)) ≅ A^{\binom{s}{i}} , for i = 0, 1, …, s ,

and is zero otherwise. iii) If ρ(γ_{i}) = 1, for all i = 1, …, s, the inclusions

b) ↪ c) ↪ d)

are homotopy equivalences. iv) If ρ(γ_{i}) = 1, for all i = 1, …, s, and ρ(−γ_{j}) = 1, for some j = 1, …, s, the cohomology of the complexes b), c), d) vanishes unless γ ∈ Z^{s}. If γ ∈ Z^{s},

H^{i}(b)) ≅ H^{i}(c)) ≅ H^{i}(d)) ≅ A^{\binom{s}{i}} , for i = 0, 1, …, s ,

and are zero otherwise.

Proof. The proof is only technically more intricate than that of theorem 3.1.

i) We restrict to the case of a)^{\underrightarrow{i}} d). We write an element ω ∈ Ω_{D_{A}^{s}/S}^{q}(C_{A,λ}^{s}) as

ω = ∑_{Q} f_{Q} \frac{dx^{Q}}{x^{Q}} ,

with f_{Q} ∈ O(C_{A,λ}^{s}), where Q varies over the subsets of {1, …, s} of cardinality q; here

\frac{dx^{Q}}{x^{Q}} = \frac{dx_{i_{1}}}{x_{i_{1}}} ∧ … ∧ \frac{dx_{i_{q}}}{x_{i_{q}}} ,

if Q = {i_{1}, …, i_{q}} with i_{1} < … < i_{q}.

Then:

∇_{γ}(ω) = ∑_{i=1}^{s} d_{i}ω ,

d_{i}ω = ∑_{Q} \frac{x_{i}∂f_{Q}}{∂x_{i}} \frac{dx_{i}}{x_{i}} ∧ \frac{dx^{Q}}{x^{Q}} + γ_{i} \frac{dx_{i}}{x_{i}} ∧ ω .

The previous formulas exhibit a) = Ω_{D_{A}^{s}/S}^{•} < L > as a sub-complex of d) = Ω_{D_{A}^{s}/S}^{•}(C_{A,λ}^{s}). We write, as in (5.8):

f_{Q} = ∑_{α∈Z^{s}} a_{α,Q}x^{α} ,

with

lim_{|α|_{R}→+∞} |a_{α,Q}|_{A}η^{|α_{+}|_{R}}ϵ^{−|α_{−}|_{R}} = 0 ,

for all λ < ϵ < η < 1.

We then put, for i = 1, …, s :

R_{i}(f_{Q}) = ∑_{α_{i}≥0} a_{α,Q}x^{α} ,

R(f_{Q}) = ∑_{α∈N^{s}} a_{α,Q}x^{α} ,

R_{i}(ω) = ∑_{Q} R_{i}(f_{Q})\frac{dx^{Q}}{x^{Q}} ,

R(ω) = ∑_{Q} R(f_{Q})\frac{dx^{Q}}{x^{Q}} .

One then checks: I) R_{i} is an endomorphism of the complex d). II) R_{1}, …, R_{s} commute with each other and their product is R. III) R is a retraction of d) to a). We also put, for i = 1, …, s,

H_{γ,i}(f_{Q}) = ∑_{α_{i}<0} a_{α,Q}(γ_{i} + α_{i})^{−1}x^{α} .

We observe that the condition ρ(γ_{i}) = 1 implies that for any constant C > 1, there exists a constant κ_{C} such that |γ_{i} − n|^{−1} < κ_{C}C^{n} for all n ∈ Z_{>0}. Let us pick λ < ϵ < η < 1 and consider

|a_{α}|_{A}|γ_{i} + α_{i}|^{−1}η^{|α_{+}|_{R}}ϵ^{−|α_{−}|_{R}}

for α_{i} < 0. Let C be such that 1 < C < ϵ/λ, so that ϵ/C > λ; then:

|a_{α}|_{A}|γ_{i} + α_{i}|^{−1}η^{|α_{+}|_{R}}ϵ^{−|α_{−}|_{R}} < κ_{C}C^{−α_{i}}η^{|α_{+}|_{R}}ϵ^{−|α_{−}|_{R}} ≤

≤ κ_{C}C^{|α_{−}|_{R}}η^{|α_{+}|_{R}}ϵ^{−|α_{−}|_{R}} = κ_{C}η^{|α_{+}|_{R}}(ϵ/C)^{−|α_{−}|_{R}} .

We remark that λ < ϵ/C < η < 1. Therefore:

lim_{|α|_{R}→+∞} |a_{α}|_{A}|γ_{i} + α_{i}|^{−1}η^{|α_{+}|_{R}}ϵ^{−|α_{−}|_{R}} = 0 ,

for all λ < ϵ < η < 1, and H_{γ,i}(f_{Q}) ∈ O(C_{A,λ}^{s}). We therefore obtain a map

H_{γ,i} : Ω_{D_{A}^{s}/S}^{q}(C_{A,λ}^{s}) ⟶ Ω_{D_{A}^{s}/S}^{q−1}(C_{A,λ}^{s})

on setting:

H_{γ,i}(ω) = ∑_{i∈Q}(−1)^{k_{Q,i}}H_{γ,i}(f_{Q})\frac{dx^{Q\\{i\}}}{x^{Q\\{i\}}} ,

for k_{Q,i} = card(Q ∩ {1, …, i − 1}).

One then verifies that:

IV) 1 − R_{i} = d_{i}H_{γ,i} + H_{γ,i}d_{i} . V) d_{i}H_{γ,j} + H_{γ,j}d_{i} = 0, if i ≠ j . VI) 1 − R_{i} = ∇_{γ}H_{γ,i} + H_{γ,i}∇_{γ} . Since 1 is homotopically equivalent to R_{i}, 1 is also homotopically equivalent

to the product R of the R_{i}’s. Therefore R is a homotopy inverse of a)^{\underrightarrow{i}} d). This finishes the proof of i).

ii) We keep the previous notation but assume here that ω ∈ Ω_{D_{A}^{s}/S}^{q} < L > (D_{A}^{s}/S), i.e. a_{α,Q} = 0 for any Q, unless α ∈ N^{s}. We define, for i = 1, …, s :

H_{i}^{γ}(f_{Q}) = ∑_{α_{i}≠−γ_{i}} a_{α,Q}(γ_{i} + α_{i})^{−1}x^{α} .

We must check that if ρ(−γ_{i}) = 1 then H_{i}^{γ}(f_{Q}) ∈ O(D_{A}^{s}/S). So, let η ∈ (0, 1). Let C > 1 be such that Cη = η′ < 1. Then: |a_{α}|_{A}|α_{i}+γ_{i}|^{−1}η^{|α|_{R}} ≤ |a_{α}|_{A}κ_{C}C^{α_{i}}η^{|α|_{R}} ≤ κ_{C}|a_{α}|_{A}C^{|α|_{R}}η^{|α|_{R}} = κ_{C}|a_{α}|_{A}η′^{|α|_{R}} . We conclude that:

lim_{|α|_{R}→+∞} |a_{α}|_{A}η^{|α|_{R}} = 0 ∀η ∈ [0, 1) ,

so that H_{i}^{γ}(f_{Q}) ∈ A{{x}}.

We now define , for i = 1, …, s :

H_{i}^{γ}(ω) = ∑_{i∈Q}(−1)^{k_{Q,i}}H_{i}^{γ}(f_{Q})\frac{dx^{Q\\{i\}}}{x^{Q\\{i\}}} ,

r_{i}^{γ}(f_{Q}) = ∑_{α,Q,α_{i}=−γ_{i}} a_{α,Q}x^{α} ,

r^{γ}(f_{Q}) = ∑_{α,Q,α=−γ} a_{α,Q}x^{α} ,

r_{i}^{γ}(ω) = ∑_{Q} r_{i}^{γ}(f_{Q})\frac{dx^{Q}}{x^{Q}} ,

r^{γ}(ω) = ∑_{Q} r^{γ}(f_{Q})\frac{dx^{Q}}{x^{Q}} .

We observe that r_{i}^{γ} = 0, if γ_{i} ∉ Z_{≤0}. One then checks: VII) r_{i}^{γ} is an endomorphism of the complex a). VIII) r_{1}^{γ}, …, r_{s}^{γ} commute with each other and their product is r^{γ}. IX) H_{i}^{γ}d_{i} + d_{i}H_{i}^{γ} = 1 − r_{i}^{γ} . X) H_{i}^{γ}d_{j} + d_{j}H_{i}^{γ} = 0 , if i ≠ j . XI) H_{i}^{γ}∇_{γ} + ∇_{γ}H_{i}^{γ} = 1 − r_{i}^{γ} . We conclude that, if for one particular index i = 1, …, s, ρ(−γ_{i}) = 1 and

γ_{i} ∉ Z_{≤0}, the complex a) is homotopically equivalent to zero.

Now assume γ ∈ Z_{≤0}^{s}. Then a) contains the sub-complex:

e) = 0 ⟶ Ax^{−γ} ⟶^{0} ⊕_{i=1}^{s} Ax^{−γ} \frac{dx_{i}}{x_{i}} ⟶^{0} ⊕_{1≤i<j≤s} Ax^{−γ} \frac{dx_{i}}{x_{i}} ∧ \frac{dx_{j}}{x_{j}} ⟶^{0} ….

The inclusion e)^{\underrightarrow{j}} a) admits the retraction a)^{\underrightarrow{r^{γ}}} e). We have:

r^{γ}j = id_{e)} ,

while each r_{i}^{γ} is homotopically equivalent to id_{a)}, hence so is their product r^{γ}. This proves ii). The remaining assertions are obvious. 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 E-mail: CHIARELLOTTO@PDMAT1.UNIPD.IT

Fax: (39-49-)8758596