数理解析研究所講究録
第 773 巻 1991 年 7-21

7Algebraic versus rigid cohomology with logarithmic coefficients:
the 1-dimensional example.
byFrancesco Baldassarri(Joint work with Bruno Chiarellotto)\S 1 Notation.
$(K, |-|)=$ a $com_{-1}p1ete$ , algebraically closed valued field extension of $(Q_{p}, |-|)$ , for someprime $p;|p|=p$$\mathcal{V}=the$ring of integers of $(K, |-|)$ .

$\mathcal{M}=the$maximal ideal of V.

$k=V/M$ , the residue field of $K$ .

For a V-scheme$T_{o}=T\cross v^{k}$$T_{K}=T\cross v^{K}$$T$

of finite presentation, we put:the special fiber of$T$, the generic fiber of$T$

ハ: the formal completion of $T$ along$T_{K}^{an}=the$.

$T_{O}$.

rigid analytic space associated to the K-scheme$\cdot$For a p-adic formal V-scheme$\mathcal{T}_{o}=T\cross v^{k}$the special fiber of$\mathcal{T}$$\mathcal{T}$$T_{K}$([BGR]).

of finite presentation, we put:.

the generic fiber of
it’s a rigid analytic space.
$\mathcal{T}_{K}=\mathcal{T}\cross v^{K}=$$\mathcal{T}$in the sense of Raynaud and Berthelot ([Ra], [Ber]):

For a separated V-scheme of finite presentation
an open immersion of rigid analytic spaces:$\hat{T}_{K}arrow T_{K}^{an}$$T$, using the previous notation, we get,which is an isomorphism when is proper over V ([Be]).
The following definition will play an important role in the sequel.
$T$

Definition 1.1 For$\gamma\in K$

the type$\rho(\gamma)$of$\gamma$is the radius of convergence of the series:$g_{\gamma}(x)= \sum_{\gamma\neq i=0}^{\infty}\frac{x^{i}}{\gamma-i}$

Notice that $\rho(\gamma)\in[0,1]$ and $\rho(\gamma)=\rho(\gamma+n),$
non-Liouville if $\rho(\gamma)=\rho(-\gamma)=1$ .

.

$\forall n\in$

Z. We say that

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

$\gamma$

is p-adically8\S 2 Main Result.
We consider:$Y=a$ proper smooth, connected V-scheme.
$Z=a$ divisor in $Y$ with normal crossing relative to V:$Z= \bigcup_{:=1}^{f}Z^{(i)}$where$Z^{(i)}$is a closed V-subscheme of$Y$, smooth, connected of codimension 1.

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

The previous hypotheses mean that there exists a finite covering of by affine open
subsets such that:via “coordinates”
i)is \’etale over.
ii) The ideal of $Z_{U}=Z$ xy $U=Z_{|U}$ inis generated by $x_{1}\ldots x_{\nu}=0$ where$\mathcal{U}$

$Y$

$U$

$U$

$(x_{1}, \ldots, x_{m})$$A_{\mathcal{V}}^{m}$$\mathcal{O}(U)$$\nu=\nu(U)$.

We also consider:a locally free finite Oy-module.
$\nabla=an$ integrable $Y_{K}/K$ connection on$\mathcal{E}_{\mathcal{V}}=$$Z_{K}$

So,$\mathcal{E}=\mathcal{E}_{\mathcal{V}}\otimes K$.

$\nabla$with logarithmic singularities along

is a morphism of abelian sheaves:$\nabla$:$\mathcal{E}arrow \mathcal{E}\otimes\Omega_{Y_{K}/K}^{1}<Z_{K}>$satisfying Leibnitz’s rule and the usual integrability condition. We recall that using theprevious notation we have, on
as before:$U\in \mathcal{U}$$\Omega^{1}Y_{K}K<Z_{K}>=\sum_{l=1}^{\nu}\mathcal{O}_{U_{K}}\frac{dx:}{x_{i}}+\sum_{i=\nu+1}^{m}\mathcal{O}_{U_{K}}dx_{j}$The hypercohomology of the de Rham complex of$D\mathcal{R}(X_{K}/K, (\mathcal{E}, \nabla))=$$(\mathcal{E}, \nabla)_{|X_{K}}$i.e. of the complex:$0arrow \mathcal{E}_{|X_{K}}arrow \mathcal{E}_{|X_{K}}\otimes\Omega_{X_{K}/K}^{1}arrow\ldots$

is, by definition, the algebraic cohomology of$X_{K}$

with coefficients in$H(X_{K}/K, (\mathcal{E}, \nabla))$$i^{From}$.

$(\mathcal{E}, \nabla)$, denoted by:.

the morphism of ringed sites:$\beta$one may deduce fromits de Rham complex$(\mathcal{E}, \nabla)$a: $X_{K}^{an}arrow X_{K}$ .

$X_{K}^{an}/K$-connection$(\mathcal{E}^{an}, \nabla^{an})$$D\mathcal{R}(X_{K}^{an}/K,$ $(\mathcal{E}^{an}, \nabla^{an})$. The hypercohomology of9is, by definition, the analytic cohomology (in the rigid analytic sense) of, denoted by:cients in$X_{K^{\iota}}^{a}$with coeffi-$(\mathcal{E}^{a}", V^{an})$$H^{\cdot}(X_{K}^{an}/K, (\mathcal{E}^{an},\nabla)^{an})$.

The morphism of ringed sites, , induces a natural morphism of complexe of sheaves :$s$$\beta$$\beta^{-1}D\mathcal{R}(X_{K}/K, (\mathcal{E}, \nabla))arrow \mathcal{D}\mathcal{R}(X_{K}^{an}/K, (\mathcal{E}^{an}, \nabla^{an}))$which gives a morphism in hypercohomology:$\overline{\beta}$:.

$H^{\cdot}(X_{K}/K, (\mathcal{E}, \nabla))arrow H^{\cdot}(X_{K}^{an}/K, (\mathcal{E}^{an}, \nabla)^{an})$Consider now the assumption:
The additive subgroup
generated by the exponents of monodromy ofaround the branches ofconsists of p-adically non-Liouville numbers.

$(NL)_{G}$$(\mathcal{E}, \nabla)$$\Lambda ofK$

$Z_{K}$In this setting we proved years ago ([Ba2]) a result of GAGA type:

Theorem 2.1. Under the assumption $(NL)_{G}$ the morphism$\overline{\beta}$

is an isomorphism.

Other results in the same direction may be found in [Bal],[Ct].
We now come to our main statement. Let$j_{0}$$j$:: $X_{o}arrow Y_{o}$ ,$X_{K}^{an}arrow Y_{K}^{an}$denote the corresponding open immersions. We notice thatis a strict neighborhood$tube$] [of[Ber]. Using the theory of Berthelot we may considerof thein$X_{K}^{an}$$X_{o}$$X_{o}$$\hat{Y}_{K}=Y_{K^{an}}$the -completion of the previous coefficients. We recall the definition. For $\lambda\in(0,1)$
sufficiently close to 1, we define$j\ddagger$$V_{\lambda}=\hat{Y}_{K}\backslash \bigcup_{1=1}^{f}[Z_{o}^{(i)}]_{\lambda}$

wheredenotes the closed tube of radius
We get open immersions:$[Z_{O}^{(i)}]_{\lambda}$$\lambda$of$Z_{o}^{(i)}$in$\hat{Y}_{K}$.

on$V_{\lambda}$$arrow^{j_{\lambda}}$$\hat{Y}_{K}$$V_{\lambda}$$\downarrow$$\nearrow$$X_{K}^{an}$the connection induced by
We denote byits de Rham complex (i.e. the restriction of$(\mathcal{E}_{\lambda}, \nabla_{\lambda})$$\mathcal{D}\mathcal{R}_{\lambda}$$(\mathcal{E}^{an}, \nabla^{a}")$, and by$\mathcal{D}\mathcal{R}(V_{\lambda}/K, (\mathcal{E}_{\lambda}, \nabla_{\lambda}))$$\mathcal{D}\mathcal{R}(X_{K}^{an}/K, (\mathcal{E}^{a}", \nabla^{an}))$to$V_{\lambda}$.

10

We then obtain a connection ($\mathcal{E}\dagger,$$\nabla\dagger_{)}$

on$\mathcal{E}\dagger=j_{0}\dagger \mathcal{E}=def\lim_{\lambdaarrow 1^{-}}j_{\lambda*}\mathcal{E}_{\lambda}arrow$

whose de Rham complex is:$\mathcal{D}\mathcal{R}(\hat{Y}_{K}/K, (\mathcal{E}, \nabla))=\lim_{\lambdaarrow 1^{-}}j_{\lambda*}\mathcal{D}\mathcal{R}_{\lambda}arrow$.

We introduce the hypothesis:$(SC)_{G}$The connection$(\mathcal{E}^{\uparrow,\dagger}\nabla)$

is overconvergent along

As in [Ber], we define the rigid cohomology of$X_{O}$$Z_{o}$.

with coefficients in ($H_{rig}(X_{o}/K, (\mathcal{E}^{\uparrow\dagger}\nabla))=H^{\cdot}(\hat{Y}_{K},$ $\mathcal{D}\mathcal{R}(\hat{Y}_{K}/K, (\mathcal{E}, \nabla))$$\mathcal{E}\dagger,$$\nabla\dagger_{)}$as:.

We also notice that, since
is quasi-compact and separated, the cohomology on
commutes with the direct limits ([Ber], [SGAIV]). Hence:$\hat{Y}_{K}$(2.2)$\hat{Y}_{K}$.

$H_{rig}(X_{o}/K, ( \mathcal{E}\dagger, \nabla^{\uparrow}))=H^{\cdot}(\hat{Y}_{K},\lim_{\lambda}j_{\lambda*}\mathcal{D}\mathcal{R}_{\lambda})arrow=\lim_{\lambda}arrow H^{\cdot}(\hat{Y}_{K},j_{\lambda*}\mathcal{D}\mathcal{R}_{\lambda})$We then have a natural morphism of complexes of sheaves on$j_{\lambda*}\mathcal{D}\mathcal{R}_{\lambda}$$\downarrow$$arrow$$\hat{Y}_{K}=Y_{K^{an}}$:$D\mathcal{R}(\hat{Y}_{K}/K, (\mathcal{E}^{\uparrow}, \nabla^{\uparrow}))$$\nearrow$$j_{*}\mathcal{D}\mathcal{R}(X_{K}^{an}/K, (\mathcal{E}^{an}, \nabla^{an}))$and the induced morphism in hypercohomology:$H^{\cdot}(V_{\lambda}/K, (\mathcal{E}_{\lambda}, \nabla_{\lambda}))$$\downarrow\overline{\alpha}_{\lambda}$$arrow$$H_{:},g(X_{o}/K, (\mathcal{E}^{\uparrow}, \nabla^{\uparrow}))$$\nearrow$$H^{\cdot}(X_{K}^{an}/K, (\mathcal{E}^{an}, \nabla^{an}))$(this makes sense because(resp.
) for any coherent sheaf$i*(resp.
j_{\lambda*})$(resp.
) and $q>0$ , since
is a quasi-Stein map [Bal],[K]).
Our main result is the following:$R^{q}j_{*}\mathcal{F}=0$$X_{K}^{an}$$V_{\lambda}$$R^{q}j_{\lambda*}\mathcal{F}=0$$\mathcal{F}$

on

11

is an isoTheorem 2.4. Under th assumptions $(NL)_{G}$ and $(SC)_{G}$ the morphism
morphism for any $\lambda\in(0,1)$ for which
is defin $ed$ (i.e. for any sufliciently close to1).
$e$$\overline{\alpha}_{\lambda}$$\lambda$$V_{\lambda}$

From (2.2) and the identification:$H^{\cdot}(V_{\lambda}/K, (\mathcal{E}_{\lambda}, \nabla_{\overline{\lambda}}))=H(\hat{Y}_{K},j_{\lambda*}\mathcal{D}\mathcal{R}_{\lambda}))$we obtain the corollaries:
Corollary 2.5. Under the assumptions of the theorem, the morphism$\overline{\alpha}$: $H^{\cdot}(X_{K}^{an}/K, (\mathcal{E}^{an}, \nabla^{an}))arrow H_{rig}(X_{o}/K,$$(\mathcal{E}\dagger,$$\nabla\dagger_{))}$

is an isomorphism.
Corollary 2.6. Under th assumptions of the theorem the morphism:$e$$ro\overline{\beta}$: $H^{\cdot}(X_{K}/K, (\mathcal{E}, \nabla))arrow H_{rig}(X_{o}/K,$$(\mathcal{E}\dagger,$$\nabla\dagger_{))}$

is an isomorphism (cf. 2.1).
Corollary 2.6 is our comparison theorem between algebraic and the rigid cohomology

with logarithmic coefficients.

12\S 3 The hypotheses $(NL)_{G}$ and $(SC)_{G}$We will illustrate by an example the role played by the two hyphoteses $(NL)_{G}$ and$(SC)_{G}$.

, perfectly analogous to the one of $Y=$ any proper
We consider the casesmooth V-scheme of relative dimension 1. We put: $D=D(O, 1^{-}),$ $D^{*}=D\backslash \{0\}$ . For$\lambda\in(0,1)$ we set $C_{\lambda}=\{x\in K|\lambda<|x|<1\}$ . So we now have:$Y=P_{\mathcal{V}}^{1}$$arrow^{jx}$$D$$C_{\lambda}$$\nearrow$$\downarrow$$D^{*}$

For$\gamma\in K$

we consider the complexes:a)$arrow^{\nabla_{\gamma}}$

$=$

$0$

$arrow$$\mathcal{O}(D)$$0$$arrow$$0$$arrow$$0$$arrow^{\nabla_{\gamma}}$

$=$

$0$

$arrow$$\frac{1}{z}\Omega^{1}(D^{*})$$\mathcal{O}(D^{*})$$\downarrow$$\downarrow$$\downarrow$c)$arrow$$\downarrow$$\downarrow$$\downarrow$b)$\frac{1}{z}\Omega^{1}(D)$$arrow^{\nabla_{\gamma}}$

$=$

$0$

$arrow$$\frac{1}{z}\Omega^{1}(C_{\lambda})$$\mathcal{O}(C_{\lambda})$where$\nabla_{\gamma}=d+\gamma\frac{dz}{z}$.

Theorem 3.1.
i) If
is not a positi $ve$ integer and $\rho(\gamma)=1$ , the inclusions:$\gamma\in K$$a)-b)\sim\rangle c)$

are homotopy $eq$ uivalences.

ii) If $\rho(-\gamma)=1$ then the complex a) has fini$te$Proof. i) Let us consider for example )$a$dimension$arrow ic$$al$cohomology.

). We construct a morphism of complexes$c)arrow a):R$$R( \sum_{n\in Z}a_{n}z^{n})=\sum_{n\geq 0}a_{n}z^{n}$$R( \sum_{n\in Z}a_{n}z^{n}\frac{dz}{z})=\sum_{n\geq 0}a_{n}z^{n}\frac{dz}{z}$13which is the left inverse of :$i$

$Roi=id_{a)}$while:$id_{c)}-ioR=\Delta_{\gamma}oH_{\gamma}+H_{\gamma}o\nabla_{\gamma}$

where the homotopy operator$H_{\gamma}$:$H_{\gamma}$

is

$0$

:

$c$

)

$arrow c$) $[-1]$in degrees 0,2, while:$H_{\gamma}( \sum_{n\in Z}a_{n}z^{n}\frac{dz}{z})=\sum_{n<0}\frac{a_{n}}{\gamma+n}z^{n}=(\sum_{n<0}a_{n}z^{n})*(g_{\gamma}(z^{-1})-\frac{1}{\gamma})$where $”*$ denotes the Hadamard product with respect to.
Now
is analytic for $|z|>\gamma$ , while, by the hypothesis on
isanalytic for $|z|>1$ . Thus
takes its values in. The same argument works for theother inclusions.
ii) We define: ) $arrow a$ ) $[-1]$$z^{-1}$$\sum_{n<0}a_{n}z^{n}$$\gamma,$$g_{\gamma}(z^{-1})$$\mathcal{O}(C_{\lambda})$$H_{\gamma}$$H^{\gamma}$$a$as$H^{\gamma}( \sum_{n\geq 0}a_{n}z^{n}\frac{dz}{z})=n\neq-\gamma\sum_{\hslash=0}^{\infty}\frac{a_{n}}{\gamma+n}z^{n}=-(\sum_{n\geq 0}a_{n}z^{n})*(g_{\gamma}(z))$

If $\rho(-\gamma)=1,$$H^{\gamma}$

takes its values in$\mathcal{O}(D)$.

. If is not a negative integer nor ,$0$$\gamma$$id_{a)}=\nabla_{\gamma}H^{\gamma}+H^{\gamma}\nabla_{\gamma}$

so that a) is acyclic. If $\gamma=-n_{O},$ $n_{O}\in\{0,1, \ldots\}$ then a) contains the subcomplex
d)$=$$0 arrow Kz^{n_{\Phi}}arrow Kz^{n_{o}}\frac{dz}{z}arrow 0$d)$\underline{j}$

$a$

).

The previous inclusion has an obvious retraction$r$$a)arrow fd)arrow ja)$which is the left inverse of i.e.:$j$$roj=id_{d)}$ ,while$id_{d)}-jor=\nabla_{\gamma}oH^{\gamma}+H^{\gamma}o\nabla_{\gamma}$

Q.E.D.

.

,

14

Corollary 3.2. Under the $ass$ umptions of the previous theorem the morphisms of complexes of abelian she aves on $D$ :a)$arrow^{\nabla_{\gamma}}$

$0$

$=$

$arrow$$\downarrow$b)$\downarrow$

$0$

$=$

$arrow$$arrow$$0$$arrow$$0$$arrow$$0$$\downarrow$$j_{*}\mathcal{O}_{D}$$\downarrow$c)$\Omega_{D/K}^{1}<0>$$\mathcal{O}_{D}$.

$arrow^{\nabla_{\gamma}}$$j_{*}\Omega_{D/K}^{1}$$\downarrow$$\downarrow$$arrow^{\nabla_{\gamma}}$

$0$

$=$

$arrow$$j_{\gamma*}\mathcal{O}_{C_{\lambda}}$$j_{\lambda*}\Omega_{C_{\lambda}/K}^{1}$

in $du$ ce isomorphisms in hypercohomology:$H^{\cdot}(D,\tilde{a}))arrow^{\sim}H^{\cdot}(D, \tilde{b}))arrow^{\sim}H(D,\tilde{c}))$

Proof. For any sheaf$\mathcal{F}$.

which appears in the above diagram:$H^{q}(D,\mathcal{F})=0$

for $q>0$ . (In fact $D,D^{*}$ are quasi-Stein while
is a quasi-Stein map). We are thenreduced to prove quasi-isomorphisms for the complexes of global sections, thus to thetheorem. Q.E.D.
We will show now discuss the role played by the hypothesis $(SC)_{G}$ .
Consider a system of linear differential equations of the form:$j_{\lambda}$$\frac{d}{dx}y=G(x)y$$S_{G}$

with $G(x)\in \mathcal{M}_{n}(K(x))$ . Let $Z_{K}=\{a_{1}, \ldots , a_{s}\}$ be the set of singular points of, sp } whereand let $Z_{o}=$ {sp$a_{1},$$\ldots$$S_{G}$

in$\hat{P}_{K}^{1}$$a_{f}$

sp:$\hat{P}_{K}^{1}arrow P_{k}^{1}$is the specialization map. Thendefines a connection on the sheaf
over. When is this connection overconvergent along ? If we intend to sick with thematrix , we are forced to assume that
contains $x=\infty$ ; then, Berthelot’s condition
involves:$S_{G}$$\mathcal{O}^{n}$$\hat{P}_{K}^{1}\backslash Z_{K}$$Z_{o}$$G$$Z_{o}$,$V_{\lambda}=\hat{P}_{K}^{1}\backslash \bigcup_{i=1}^{f}D(a_{i}, \lambda^{+})$and the matrices giving the action of$( \frac{d}{dx})^{m}$$\lambda\in(0,1)$

on solutions of$( \frac{d}{dx})^{m}y=G^{(m)}y$,$(G^{(\circ)}=I_{\tau\iota}, G^{(1)}=G, G^{(m+1)}= \frac{dG^{(m)}}{dx}+G^{(m)}G^{(1)})$$m\in N$

. It is:$S_{G}$:$X_{K}^{an}=$15$(SC)_{G}$(Berthelot) For each $\eta\in(0,1)\exists\lambda\in(0,1)$ , such that$\lim_{marrow\infty}\Vert\frac{G^{(m)}}{m!}\Vert_{V_{\lambda}}\eta^{m}=0$(where$\Vert-\Vert v_{\lambda}$

denotes the supnorm on$V_{\lambda}$, for matrices).

Recall the Gauss norm on $K(x)$ : it is defined on $K[x]$ as:$| \sum a_{i}x^{i}|_{Q}=Sup|a_{i}|$and extended to an absolute value $|-|g$ of $K(x)$ by multiplicativity. It depends essentiallyon the V-structure of: if a function $f\in K(x)$ has no poles in an open disk of radius
1, then$\hat{P}_{K}^{1}$$\mathcal{D}$$|f|_{Q}=\Vert f\Vert_{D}$.

So, condition $(SC)_{G}$ implies:$(SC)_{G}’,$$\forall\eta\in(0,1)$:$\lim_{marrow\infty}|\frac{G^{(m)}}{m!}|g\eta^{m}=0$.

Condition $(SC)_{G}’$ appears in the work of Dwork, Robba, Christol, Andr\’e ([Anl], [An2],[Ch], [Ch-Dw]), as that of convergence of the solutions ofin the generic disk of radius
1. This is motivated from the fact that the matrix function:$S_{G}$$\mathcal{U}_{G,t}(x)=\sum_{m=0}^{\infty}\frac{G^{(m)}(t)}{m!}(x-t)^{m}$is a fundamental solution matrix of$S_{G}$

at a generic unit$| \frac{G^{(m)}(t)}{m!}|=|\frac{G^{(m)}}{m!}|_{\mathcal{G}}$

$t$

and:

.

Let’s now check that $(SC)_{G}’\Rightarrow(SC)_{G}$ . Let us put, for each $\lambda\in(0,1)$ ,$(]X_{o}[=)W= \hat{P}_{K}^{1}\backslash \bigcup_{i=1}^{f}D(a_{i}, 1^{-})\subset V_{\lambda}$

Then $(SC)_{G}’$ certainly implies:$\lim_{marrow\infty}\Vert\frac{G^{(m)}}{m!}\Vert_{W}\eta^{m}=0$.

16

for each $\eta\in(0,1)$ . So we are left to consider separated annuli around the singular points.
We may assume:$a_{1},$$\ldots,$$a_{d}\in D=D(0,1^{-}),$$a_{d+1},$ $\ldots,$ $a_{f}\not\in D$.

Let$f(x)= \prod_{:=1}^{d}(x-a_{i})^{N}$

be such that$fG\in \mathcal{M}_{n}(\mathcal{O}(D))$$\lambda>\max:=1,\ldots,d|a_{i}|$we have:. Then also$f^{m}G^{(m)}\in \mathcal{M}_{n}(\mathcal{O}(D)$, for each $m\in$ N. For$\Vert\frac{G^{(m)}}{m!}[c_{\lambda}\leq\{|f^{m}\frac{G^{(m)}}{m!}\Vert c_{\lambda}|f^{-m}\Vert_{C_{\lambda}}\leq|f^{m}\frac{G^{(m)}}{m!}|_{Q}\lambda^{-mNd}=|\frac{G^{(m)}}{m!}|c^{\lambda^{-mNd}}$since $|f|_{Q}=1$ . So, if$\lambda$

is also$>\eta\pi^{1}\pi$, one has$\eta/\lambda^{Nd}\in(0,1)$

and$\lim_{marrow\infty}|\frac{G^{(m)}}{m!}|_{\mathcal{G}}(\frac{\eta}{\lambda^{Nd}})^{m}=0$$(SC)_{G}’$implies$\lim_{marrow\infty}\Vert\frac{G^{(m)}}{m!}|c_{\lambda}\eta^{m}=0$$(SC)_{G}$.

So:
Proposition 3.3. The system
is overconvergent along its polar divisor iff its solu tions
converge in th generic disk of radius one.
$S_{G}$$e$What we said should justify the following weaker, local, condition on a system on$D=D(0,1^{-})$ :$x \frac{d}{dx}y=Gy$$\mathcal{L}_{G}$with$G\in \mathcal{M}_{n}(\mathcal{O}(D))$. We put, as before,$x^{m}( \frac{d}{dx})^{m}y=G_{m}y$for $m\in N$ . We consider thc condition:$(SC)r\forall\eta\in(0,1)$, for each affinoid $V\subset D$$\lim_{marrow\infty}|\frac{G_{m}}{m!}\Vert_{V}\eta^{m}=0$

We also define the type of the system$\mathcal{L}_{G}$at$\rho=\prod\rho(\gamma)^{e_{\gamma}}$

if $\det(x-G(O))=\prod(x-\gamma)^{e_{\gamma}}$ .

Our main result in this framework is:

$0$

as:
$(\in[0,1])$

17, $G\in \mathcal{M}_{n}(O(D))$ satisfies condi tion $(SC)_{L}$ . Let
Theorem 3.4. Assume the system
at ,at . Then any formdly meromorphic column solution of
be the typ ofis p-adically meromorpAi for $|x|<\rho$ .
$\mathcal{L}_{G}$$e$$\rho$$\mathcal{L}_{G}$$0$$\mathcal{L}_{G}$

$y$

$0$

$c$

Corollary 3.5. Under th assumptions of the theorem, assume also that the eigenvalues$ofG(O)$ are p-adi cally non-Liouville. Then, the formally meromorphi solutions
at
of
$D$
are meromorpAic in .
$e$

$c$

$y$

$L_{G}$$0$Consider the condition:
The additive subgroup of $K$ generated by the eigenvalues of $G(O)$ , consistsof p-adically non-Liouville numbers.
$(NL)_{L}$$\Lambda$

Corollary 3.6. $Assume$ condi tions $(NL)_{L}$ and $(SC)_{L}$ hold forconsider the diagram of abelian sheaves on $D$ :a)$arrow$$\mathcal{O}_{D}^{n}[\frac{1}{x}]$$\Omega_{D/K}^{1}\otimes \mathcal{O}_{D}^{n}[\frac{1}{x}]$$\nabla_{G}=d+G\frac{d}{dx}$

,

$0$

$=$

$arrow$$j_{*}\mathcal{O}_{D}^{n}$$arrow$$0$$arrow$$0$$arrow$$0$$\downarrow$$\downarrow$.

$arrow^{\nabla_{G}}$$j_{*}(\Omega_{D/K}^{1}\otimes \mathcal{O}_{D^{*}}^{n})$$\downarrow$$\downarrow$$\downarrow$

c)

. For

$arrow^{\nabla_{G}}$

$0$

$=$

$\downarrow$b)$\mathcal{L}_{G}$$arrow^{\nabla_{\gamma}}$

$0$

$=$

The morphisms )$a$$arrow b$$arrow$$j_{\lambda*}\mathcal{O}_{C_{\lambda}}^{n}$$j_{\lambda*}(\Omega_{C_{\lambda}/K}^{1}\otimes \mathcal{O}_{D}^{n}.)$) $-c$ ) indu ce isomorphisms of hypercohomology groups $01^{r}erD$ .

Proof. It consists of the following steps:
1) Use the formal theory of logarithmic systems to formally reduce to upper-triangularsystems.
2) Use corollary 3.5 to show that the formal equivalence referred to in step 1), is infact convergent on $D$ .
So, we may assume that 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,
is an isomorphism). Q.E.D..
if needed (multiplication by$G$$x^{N}$

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 $\lambda\in(0,1)$ and that we had:
$V_{\lambda}=\hat{P}_{K}^{1}\backslash \bigcup_{i=1}^{f}D(a:, \lambda^{+})arrow\hat{P}_{K}^{1}j_{\lambda}$

We deal with a connection
lie in distinct residues classes).

$(\mathcal{E}, \nabla)$.

with logarithmic singularities at$a_{1},$$\ldots$,$a_{f}$(assumed to18

We are supposed to examine:$\mathcal{F}^{\cdot}=j_{*}D\mathcal{R}(X_{K}^{an}/K,$

We choose$\eta\in(\lambda, 1)$$(\mathcal{E}_{\lambda}, \nabla_{\lambda}):$$(\mathcal{E}^{an}, \nabla^{an})-\mathcal{F}_{\lambda}^{\cdot}=j_{\lambda*}\mathcal{D}\mathcal{R}(V_{\lambda}/K,$

and consider the admissible covering$\mathcal{W}$of$\mathcal{W}=\{V_{\eta}, D(a_{1},1^{-}), \ldots , D(a_{f}, 1^{-})\}$$\hat{P}_{K}^{1}$

.

:

.

We have that the restriction:$i_{|V_{\eta}}$:$\mathcal{F}_{1\dot{V}_{\eta}}\simeq \mathcal{F}_{\lambda|V_{\pi}}$is the identity, while on each disk $D(a:, 1^{-})$ we are in the situation of corollary 3.6.
Our result follows from the spectral sequence of hypercohomology associated to the covering :$\mathcal{W}$$E_{2}^{p,q}(\mathcal{F}^{\cdot})=H^{p}(\mathcal{W}, h^{q}(\mathcal{F}^{\cdot}))\Rightarrow H^{\cdot}(\hat{P}_{K}^{1}, \mathcal{F}^{\cdot})$where$h^{q}(\mathcal{F}^{\cdot})$

denotes the presheaf:$U H^{q}(U,\mathcal{F}^{\cdot})$).
(and similarly for
The morphism of spectral sequences:$\mathcal{F}_{\lambda}^{\cdot}$$E:(\mathcal{F}^{\cdot})arrow E:(\mathcal{F}_{\lambda})$is in fact an isomorphism at the$E_{2}$

level. (In fact, since$H^{q}(U,\mathcal{G})=0$

for any open setof the nerve ofwith coefficients inbicomplexes of). Q.E.D.
of$U$$\mathcal{W}$$\hat{P}_{K}^{1}$$\mathcal{W}$$\mathcal{F}^{\cdot}$$\forall q>0$

and any sheaf under consideration, the \v{C}ech, actually calculate the hypercohomology
and$\mathcal{G}$$\mathcal{F}_{\dot{\lambda}}$19\S 4 Hints for the general case.
We point ou some useful facts about the general case.
$t$I) Existence of tubular neighborhoods of radius 1 of

We may refine the covering of$\hat{Z}_{K}$in$\hat{Y}_{K}$([Ba-Ct3]).

$\hat{Y}_{K}$a$=\{\hat{U}_{K}\}_{U\in \mathcal{U}}$$\lambda\in(0,1)$ ,obtained from the original . This will be done in connection with a given
and assume
as in theorem 2. So, let
be as in section 2, with coordinates$\ldotsx_{\nu}=0$ .
$x_{1}=0,$
of equation, resp.,arethe branches ofmeeting, we put:. For
Let$\mathcal{U}$$V_{\lambda},$$(x_{1}, \ldots, x_{m})$$U\in \mathcal{U}$$Z_{K}$$\mathcal{T}_{U}=\{1, \ldots, \nu\},$$U_{S,\eta}=$$Z_{K}^{(1)},$$U_{K}${$\eta\in(\lambda, 1)$$S\subset \mathcal{T}_{U}$$p\in\hat{U}_{K}$$Z_{K}^{(\nu)}$$\ldots,$: $|x_{i}(p)|<1$ if $i\in S$ and $|x_{i}(p)|\geq\eta$ , if

$i\in \mathcal{T}_{U}\backslash S$}

the main point is:

Proposition 4.1.
is a trivial bundle in open unit polydisks of rela tive dimension$s=S$ over a smooth affinoid space $V_{S,\eta}=SpmA_{S,\eta}$ .
$U_{S,\eta}$

Proof. It is an immediate consequence of the following:
Lemma 4.2. Let $S,$ $Z,$$P$

be formal V-schemes of iini
$Z$

$Carrow$$\downarrow$$\nearrow$$te$presentation and

$P$

$S$

be a dosed immersion of S-objects where $Zarrow S$ (resp. $Parrow S$) is smooth of rela tive
dimension (resp. $d+s$ ). Assume (always true locally on $P$ ) that $S=SpfA,$ $P=SpfB$ ,$Z=SpfC$ are afllne and th at $C=B/J$ where $J=(f_{1}, \ldots, f_{s})$ is genera $ted$ by elements.
] [ denotes the closed immersion, there exists a retraction
Then, if :]and an isomorphism:
$d$

$s$

$i$

$Z_{K}rightarrow$$Z_{o}$$\sigma:$$p$$Z_{o}[parrow$$Z_{K}$$]Z_{O}[Parrow^{\sim}Z_{K}\cross D^{\delta}$

such that the diagram:$(:d_{\underline{Z_{K_{\rangle}}}},0)$$Z_{K}$$\downarrow i$$]Z_{o}[p$$Z_{K}\cross D^{s}$$\sim\nearrow$$arrow^{\sigma}$$I^{pr}1$$Z_{K}$

commutes.
Q.E.D.
II) The formal and convergent theory of systems with logarithmic singularities on
standard spaces ([Ba-Ct2]).

20

Here

$A$

is regular Tate K-algebra with no zero-divisors, and we define$D_{A}^{\iota}=SpmA\cross D^{\delta}$.

We consider systems of P.D.E. of the form:$s$$\partial y=G_{\partial}y$$\partial\in Der^{c}(A/K)$$\mathcal{L}_{G}$$x^{\alpha}( \frac{\partial}{\partial x})^{\alpha}y=G_{\alpha}y$$\alpha\in N^{\delta}$

satisfy the usual integrability conditions.
where. We then develop a refined formal
We consider conditions $(NL)_{L}$ and $(SC)_{L}$ on, ) and convergent (i.e. on) theory of such systems, analogous(i.e. on $A[x_{1},$to the one for ordinary systems.
$G_{\partial},$$G_{\alpha}\in M_{n}(\mathcal{O}(D_{A}^{\delta}))$$\mathcal{L}_{G}$$x_{\epsilon}I$$\ldots$$\mathcal{O}(D_{A}^{l})$References.
[A1] Andr\’e Y. : “ Sp\’ecialisation dans les disques singuliers r\’eguliers’’, in Groupe d’Etude
d’Analyse Ultram\’etrique 1985/86; Y.Amice, G.Christol, P.Robba, Eds. ; Publ. Math.
Universit\’e Paris VII.

[A2] Andr\’e Y. : “G-functions and Geometry”, Vieweg-Verlag, Bonn 1989.
[Bal] Baldassarri F. : “ Comparaison entre la cohomologie alg\’ebrique et la cohomologiepadique rigide \‘a coefficients dans un module diff\’erentiel I” Inv. Math. 87(1987),

83-99.
[Ba2] Baldassarri F. : “ Comparaison entre la cohomologie alg\’ebrique et la cohomologiepadique rigide \‘a coefficients dans un module diff\’erentiel II”. Math. Ann.280 (1988),

417-439.

[Ba-CtO] Baldassarri F.,Chiarellotto B. : “ On Christol’s theorem. A generalization to systemsof PDE’s with logarithmic singularities depending upon parameters”, to appear inProceedings of the Special Session on Algebraic Geometry and Number Theo ry, AMS
Publications, Series of Contemporary Mathematics.

[Ba-Ctl] Baldassarri F.,Chiarellotto B. :“ On Andr\’e’s transfert theorem.”, to appear in PTOceedings of the Special Session on Algebraic Geometry and Number Theory, AMS

Publications, Series of Contemporary Mathematics.

[Ba-Ct2] Baldassarri F.,Chiarellotto B. : “ Formal and p-adic theory of differential systems withlogarithmic singularities depending upon parameters”, article in preparation.
[Ba-Ct3] Baldassarri F.,Chiarellotto B. : “ Algebraic versus rigid cohomology with logarithmic
coefficients.”, to appear in Proceedings of the Barsotti Memorial Symposium. Abano
Terme (Padova), June 24-P7, J991, Perspectives in Mathematics,
Academic Press.

[Ber] Berthelot P. : “Cohomologie rigide et cohomologie rigide \’a support propre”. A paraitre
dans Ast\’erisque.

21[BGR] Bosch S.; Gu ntzer U.; Remmert R. : “Non archimedean Analysis”. Grundlehren derMath. Wissenschaften 261, Springer-Verlag, Berlin 1984.
[Ch] Christol G.: “Un th\’eor\‘eme de transfert pour les disques singuliers r\’eguliers’’, Ast\’erisque119-120 $(1984),151- 168$ .

[Ch-Dw] Christol G., Dwork B. : “ Effective p-adic bounds at regular singular points”, Duke
Math. J. Vol. 62 (1991), 689-720.

[Ct] Chiarellotto B. : “ A comparison theorem in p-adic cohomology”. Ann. di Mat. Purae Appl. (IV) 153 (1988), 115-132.
[K] Kiehl R. :“ Die de Rham kohomologie algebraischer Mannigfaltigkeiten uber einem
bewerteten Korper”. Publ. Math. I.H.E.S. 33 (1967) 5-20.
[Ra] Raynaud M (G\’eom\’etrie analitique rigide”. Bull. Soc. Math. France, M\’emoire 39-40(1974), 319-327.
[SGAIV] “Seminaire de Geometrie Algebrique. Theorie des Topos et Cohomologie etale desSch\’emas’’. M. Artin; A. Grothendieck L.N. in Math., 269-270 Springer-Verlag BerlinNew York 1972.

Author’s address:
Dipartimento di Matematica
Universit\‘a di Padova
Via Belzoni 7
I-35131 PADOVA-ITALY
E-mail: BALDASSARRI@PDMATI.UNIPD.IT
Fax: $(39- 49-)8758596$