Global applications of relative
Applications globales de
(ϕ, Γ)-modules
(ϕ, Γ)-modules
I
relatifs I
Fabrizio ANDREATTA & Adrian IOVITA
ABSTRACT In this paper, given a smooth proper scheme X over a p-adic DVR and
a p-power torsion étale local system L on it, we study a family of sheaves associated
to the cohomology of local, relative (ϕ, Γ)-modules of L and their cohomology. As
applications we derive descriptions of the étale cohomology groups on the geometric
generic fiber of X with values in L, as well as of their classical (ϕ, Γ)-modules, in
terms of cohomology of the above mentioned sheaves.
ABSTRAIT Étant donné un schéma propre et lisse X défini sur un corps de valuation discrete et un système local L, étale, de torsion sur X on étudie une famille de
faisceaux associés à la cohomologie des (ϕ, Γ)-modules locaux, relatifs de L et leur
cohomologie. Comme application on déduit une description des groupes de cohomologie étales sur la fibre générique géometrique de X à valeurs dans L, et de leurs
(ϕ, Γ)–modules classiques en termes de la cohomologie des faisceaux mentionés plus
haut.
1 Introduction.
Let p be a prime integer, K a finite extension of Qp and V its ring of integers. In
[Fo], J.-M. Fontaine introduced the notion of (ϕ, Γ)–modules designed to classify p-adic
representations of the absolute Galois group GV of K in terms of semi-linear data. More
precisely, if T is a p-adic representation of GV , i.e. T is a finitely generated Zp -module
(respectively a Qp -vector space of finite dimension) with a continuous action of GV , one
associates to it a (ϕ, Γ)–module, denoted DV (T ). This is a finitely generated module
over a local ring of dimension two AV (respectively a finitely generated free module
over BV := AV ⊗Zp Qp ) endowed with a semi-linear Frobenius endomorphism ϕ and
a commuting, continuous, semi-linear action of the group ΓV := Gal(K(µp∞ )/K) such
that (DV (T ), ϕ) is étale. This construction makes the group whose representations we
wish to study simpler with the drawback of making the coefficients more complicated.
It could be seen as a weak arithmetic analogue of the Riemann–Hilbert correspondence
between representations of the fundamental group of a complex manifold and vector
bundles with integrable connections. The main point of this construction is that one
may recover T with its GV -action directly from DV (T ) and, therefore, all the invariants
which can be constructed from T can be described, more or less explicitly, in terms of
DV (T ). For example
2000 Mathematics Subject Classification 11G99,14F20,14F30.
Key words: (ϕ, Γ)-modules, étale cohomology, Fontaine sheaves, Grothendieck topologies, comparison isomorphisms.
Mots clé: (ϕ, Γ)-modules, cohomologie étale, faisceaux de Fontaine, topologies de Grothendieck,
isomorphismes de comparaison.
1
(∗) one can express in terms of DV (T ) the Galois cohomology groups Hi (K, T ) =
H (GV , T ) of T .
i
More precisely, let us choose a topological generator γ of ΓV and consider the complex
d
d
0
1
C • (T ) : DV (T ) −−→
DV (T ) ⊕ DV (T ) −−→
DV (T )
¡
¢
where d0 (x) = (1 − ϕ)(x), (1 − γ)(x) and d1 (a, b) = (1 − γ)(a) − (1 − ϕ)(b). It is proven
in [He] that for each i ≥ 0 there is a natural, functorial isomorphism
Hi (C • (T )) ∼
= Hi (GV , T ).
Moreover, for i = 1 this isomorphism was made explicit in [CC2]: let (x, y) be a 1-cocycle
for the complex C • (T ) and choose b ∈ A ⊗Zp T such that (ϕ − 1)(b) = x. Define the
map C(x,y) : GV −→ A ⊗Zp T by
C(x,y) (σ) = (σ 0 − 1)/(γ − 1)y − (σ − 1)b,
where σ 0 is the image of σ in ΓV . One can prove that the image of C(x,y) is in fact
contained in T , that C(x,y) is a 1-cocycle whose cohomology class [C(x,y) ] ∈ H1 (GV , T )
only depends on the cohomology class [(x, y)] ∈ H1 (C • (T )). Moreover, the isomorphism
H1 (C • (T )) ∼
= H1 (GV , T ) above is then defined by [(x, y)] 7→ [C(x,y) ].
As a consequence of (∗) we have explicit descriptions of the exponential map of PerrinRiou (or more precisely its “inverse” (see [Fo], [Ch], [CC2]) and an explicit relationship
with the “other world” of Fontaine’s modules: DdR (T ), Dst (T ), Dcris (T ) (see [CC2], [Be]).
Despite being a very useful tool, in fact the only one which allows the general classification of integral and torsion p-adic representations of GV , the (ϕ, Γ)–modules have
an unpleasant limitation. Namely, DV (T ) could not so far be directly related to geometry when T is the GV -representation on a p-adic étale cohomology group (over K) of
some smooth proper algebraic variety defined over K. Here is a relevant passage from
the Introduction to [Fo]: “Il est clair que ces constructions sont des cas particuliers de
constructions beaucoup plus générales. On doit pouvoir remplacer les corps que l’on
considère ici par des corps des fonctions de plusieurs variables ou certaines de leurs
complétions. En particulier (i) la loi de réciprocité explicite énoncée au no. 2.4 doit
se généraliser et éclairer d’un jour nouveau les travaux de Kato sur ce sujet; (ii) ces
constructions doivent se faisceautiser et peut être donner une approche nouvelle des
théorèmes de comparaison entre les cohomologies p-adiques.”
The first part of the program sketched above, i.e. the construction of relative (ϕ, Γ)–
modules was successfully carried out in [An]. The main purpose of the present article
is to continue Fontaine’s program. In particular various relative analogues, local and
global, of (∗) are proven.
Let us first point out that in the relative situation, over a “small”-V -algebra R (see
§2) there are several variants of (ϕ, Γ)–module functors, denoted DR (−) (arithmetic),
2
e R (−) (tilde–arithmetic), D
e R (−) (tilde–geometric) and their overDR (−) (geometric), D
†
†
†
e (−) and D
e † (−). For simplicity of exposition
convergent counterparts DR (−), DR (−), D
R
R
e R (−).
let us explain our results in terms of DR (−) and D
I) Local results. This is carried on in §3 together with the appendices §7 and §8. Let
R be a “small” V -algebra. Fix an algebraic closure Ω of R and let η be the associated
geometric generic point of Spec(R). Denote by R the union of all normal finite extensions
of R contained in Ω, which are étale R–algebras after inverting
p. Let M
¢ be a finitely
alg ¡
generated Zp -module with continuous action of GR := π1 Spm(RK , η) and let D :=
e R (M ). Then D is a finitely generated A
e -module endowed with commuting actions
D
R
of a semi-linear Frobenius ϕ and a linear action of the group ΓR (see §2.) As in the
classical case, ΓR is a much smaller group that GR . It is the semidirect product of ΓV
and of a group isomorphic to Zdp where d is the relative dimension of R over V .
Let C • (ΓR , D) be the standard complex of continuous cochains computing the continuous ΓR -cohomology of D and denote TR• (D) the mapping cone complex of the morphism (ϕ − 1): C • (ΓR , D) −→ C • (ΓR , D). Then, Theorem 3.3 states that we have natural
isomorphisms, functorial in R and M ,
¡
¢
Hicont (GR , M ) ∼
= Hi TR• (D) for all i ≥ 0.
The maps are defined in §3 in an explicit way, following Colmez’s description in the
classical case. The input of Fontaine’s construction of the classical (ϕ, Γ)–modules was
to replace modules over perfect, non–noetherain rings with modules over smaller rings:
“C’est d’ailleurs [...] que j’ai compris l’intérêt qu’il avait à ne pas remplacer k((π)) par sa
clôture radicielle” Indeed, “[...]ceci permet d’introduire des techniques différentielles”.
Motivated by the same needs, in view of applications to comparison isomorphisms, we
e R (M ) over the ring A
e , which is
show in appendix §7 that one can replace the module D
R
e R (M ) over the noetherian,
not noetherian, with the smaller (ϕ, ΓR )–module DR (M ) ⊂ D
regular domain AR of dimension d + 1. We show that the natural map
e R (M )) for all i ≥ 0
Hicont (ΓR , DR (M )) −−→ Hicont (ΓR , D
is an isomorphism. The proof follows and slightly generalizes the Tate-Sen method in
[AB]. In particular, one has a natural isomorphism
¢
¡
Hicont (GR , M ) ∼
= Hi TR• (DR (M )) for all i ≥ 0,
¡
¢
where TR• DR (M ) is the mapping cone complex of the map (ϕ − 1): C • (ΓR , DR (M )) →
C • (ΓR , DR (M )).
II) Global results. This is carried on in §4, §5, §6. The setting for §4 and §5 is the
following. Let X be a smooth, proper, geometrically irreducible scheme of finite type
over V and let L denote a locally constant étale sheaf of Z/ps Z-modules (for some
s ≥ 1) on the generic fiber XK of X. Let X denote the formal completion of X along its
3
rig
special fiber and let XK
be the rigid analytic space attached to XK . Fix a geometric
generic point η = Spm(CX ) and set L the fiber of L at η.
To each U → X étale such that U is affine, U = Spf(RU ), with RU a small V -algebra and
a choice of local parameters (T1 , T2 , . . . , Td ) of RU (as in §2) we attach the relative (ϕ, Γ)–
e U (L) := D
e R (L). However, the association U −→ D
e U (L) is not functorial
module D
U
e
because of the dependence of DU (L) on the choice of the local parameters. In other
words the relative (ϕ, Γ)–module construction does not sheafify.
¡
¢
e R (L)
Nevertheless due to I) above, for every i ≥ 0, the association U −→ Hi TR•U (D
U
is functorial and we denote by Hi (L) the sheaf on Xet associated to it. In §4 we prove
Theorem 4.1: there is a spectral sequence
E2p,q = Hq (Xet , Hp (L)) =⇒ Hp+q (XK,et , L).
We view this result as a global analogue of (∗): the étale cohomology of L is calculated
in terms of local relative (ϕ, Γ)–modules attached to L.
The proof of Theorem 4.1 follows a roundabout path which was forced on us by lack of
enough knowledge on étale cohomology of rigid analytic spaces. More precisely, for an
algebraic, possibly infinite, extension M of K contained in K, Faltings defines in [Fa3]
a Grothendieck topology XM on X (see also §4). The local system L may be thought of
as a sheaf on XM and it follows from [Fa3], see 4.6, that there is a natural isomorphism:
¡
¢
¡
¢
(∗∗) Hi XM , L ∼
= Hi XM,et , L ,
for all i ≥ 0. The main tool for proving (∗∗) is the result: every point x ∈ XK has
a neighborhood W which is K(π, 1). Such a result, although believed to be true, is
yet unproved in the rigid analytic setting. Therefore the proof of Theorem 4.1 goes as
rig
associated to L. We define
follows. Let Lrig be the locally constant étale sheaf on XK
b
the analogue Grothendieck
topology
XM on X , prove that there is a spectral sequence
¡
¢
b M , Lrig ), then compare Hi (XM , L) to
with E2p,q = Hq Xet , Hp (Lrig ) abutting to Hp+q (X
¡
¢
¡
¢
b M , Lrig ) and Hi XM,et , L to Hi X rig , Lrig and in the end use Faltings’ result
Hi (X
M,et
(∗∗).
In §5 we introduce a certain family
sheaves which we call Fontaine sheaves
¡ ¢of continuous
+
and which we denote by O , R O , Ainf (O ). There are algebraic and analytic variants
b M . We would like to remark
of these: the first are sheaves on XM and the second on X
that the local sections of the Fontaine sheaves are very complicated and they are not
bM
relative Fontaine rings. Continuous cohomology of continuous sheaves on XM and X
e V (Wi ),
respectively is developed
interpretation of D
¡
¢ in §5. As an application a geometric
i
s
where Wi = H XK et , L , for L an étale local system of Z/p Z-modules on XK as above
is given. More precisely, it is proven in §5 that there is a natural isomorphism of classical
(ϕ, Γ)–modules:
³
³ ¡
¡
¢´
¢´
∼
b K , Lrig ⊗ A+ O
e V Hi X
D
,
L
.
Hi X
=
∞
inf
K,et
b K∞
X
4
Finally, in §6 we relax our global assumptions. Now X denotes a formal scheme topologically of finite type over V , smooth and geometrically irreducible, not necessarily
rig
algebrizable, and XK
denotes its rigid analytic generic fiber.
In §6 we set up the basic theory for comparison isomorphisms between the different
p-adic cohomology theories in this analytic setting. Our main result is that, if Lrig is a
rig
p-power torsion local system on XK
and AF ont is one of the analytic Fontaine sheaves on
b M listed above, then the cohomology groups Hi (X
b M , Lrig ⊗ AF ont ) can be calculated
X
as follows. Let us first recall that we fixed a geometric generic point η = Spm(CX ),
where CX is a complete, algebraically closed field which can be chosen as in 4.5. For
each étale morphism U −→ X such that U is affine, U = Spf(RU ) with RU a small
V -algebra, let RU denote the union of all normal RU -algebras contained in CX which
are finite and étale over RU after inverting p. Write π1alg (UM , η) for the Galois group
of RU ⊗V M ⊂ RU ⊗V K. Let AF ont (RU ⊗ K) denote the Fontaine ring constructed
starting with the pair (RU , RU ) as in [Fo] and denote by L, as before, the fiber of Lrig
at η. One can show that the association U −→ Hi (π1alg (UM , η), L ⊗ AF ont (RU ⊗ K)) is
i
(Lrig ⊗ AF ont ) the sheaf on Xet associated to it.
functorial and denote HM
¡
¢
Notice that, due to the generalized Tate-Sen method of §7, if AFont = A+
inf O X
bK ,
the inflation defines an isomorphism:
¡
¢
¡
¢
¡ alg
¢
e R (L) ∼
Hi ΓRU , DRU (L) ∼
(R
⊗
K)
.
= Hi ΓRU , D
= Hi π1 (U, η), L ⊗ A+
U
U
inf
³
¡
¢´
i
Hence, the sheaf HM
Lrig ⊗ A+
O
is defined locally in terms of Γ-cohomology
inf
bK
X
of relative (ϕ, Γ)–modules.
It is proved (Theorem 6.1) that there exists a spectral sequence
p
b M , Lrig ⊗ AF ont ).
(Lrig ⊗ AF ont )) =⇒ Hp+q (X
(∗ ∗ ∗) E2p,q = Hq (Xet , HM
At this point we would like to remark that our results in §6 are distinct from those of
Faltings in [Fa1], [Fa2], [Fa3]. Namely let us consider the following diagram of categories
N
and functors:
b
vX
,M,∗
b M )N −−−
Sh(X
−−−→ Sh(Xet )N
α↓
β↓
H0 (Xet ,−)
Sh(Xet )
−−−−−−→
AbGr
where if C is a Grothedieck topology then we denote by Sh(C) the category of sheaves
of abelian groups on C and by Sh(C)N the category of continuous sheaves on C (see §5).
We also denote α = lim vbX ,M,∗ and β = lim H0 (Xet , −).
←
←
We analyze the spectral sequence attached to the composition of functors: H0 (Xet , −) ◦
lim(b
vX ,M,∗ )N while it appears, although very little detail is given, that Faltings considers
←
the composition of the other two functors in the above diagram (in the algebraic setting).
We believe that our point of view is appropriate for the applications to relative (ϕ, Γ)–
modules that we have in mind.
5
The analysis in §6 and the spectral sequence (∗ ∗ ∗) have already been used in order
to construct a p-adic, overconvergent, finite slope Eicher-Shimura isomorphism and to
give a new, cohomological construction of p-adic families of finite slope modular forms
in [IS].
In a sequel paper (“Global applications of relative (ϕ, Γ)-modules, II”) we plan to first
extend the constructions and results in §6 of the present paper to formal schemes over V
with semi-stable special fiber and use them in order to prove comparison isomorphisms
between the different p-adic cohomology theories involving Fontaine sheaves in such
analytic settings. We believe that we would be able to carry on this project for spaces
like: the p-adic symmetric domains, their étale covers (in the cases where good formal
models exist), the p-adic period domains of Rapoport-Zink, etc.
Acknowledgements: We thank A. Abbes, V. Berkovich and W. Niziol for interesting discussions pertaining to the subject of this paper. We thank O. Brinon and
J. Pottharst for several useful remarks. Part of the work on this article was done when
the first autor visited the Department of Mathematics and Statistics of Concordia University and the second author visited the IHES and il Dipartimento di matematica pura
ed applicata of the University of Padova. Both of us would like to express our gratitude
to these institutions for their hospitality.
Contents.
1.
2.
3.
4.
5.
6.
7.
8.
Introduction.
Preliminaries.
Galois cohomology and (ϕ, Γ)–modules.
Étale cohomology and relative (ϕ, Γ)–modules.
A geometric interpretation of classical (ϕ, Γ)–modules.
The cohomology of Fontaine sheaves.
Appendix I: Galois cohomology via the Tate–Sen method.
Appendix II: Artin–Schreier theory.
References.
I. Local Theory.
2 Preliminaries.
2.1 The basic rings. Let V be a complete discrete valuation ring, with perfect residue
field k of characteristic p and with fraction field K = Frac(V ) of characteristic 0. Let v
be the valuation on V normalized so that v(p) = 1. Let K ⊂ K be an algebraic closure
6
of K with Galois group Gal(K/K) =: GV and denote by V the normalization of V
in K. Define the tower
K0 := K ⊂ K1 = K(ζp ) ⊂ · · · ⊂ Kn = K(ζpn ) ⊂ · · ·
where ζpn is a primitive pn –th root of unity and ζppn+1 = ζpn for every n ∈ N. Let Vn
be the normalization of V in Kn and define K∞ := ∪n Kn . Write ΓV := Gal(K∞ /K)
and HV := Gal(K/K∞ ) so that ΓV = GV /HV .
We also fix a field extension K∞ ⊂ M ⊂ K so that K ⊂ M is Galois with
group Gal(M/K). We let W be the normalization of V in M . The two important
cases are M = K∞ with W = V∞ and M = K with W = V .
Let© R be a V –algebra
such that k ⊂ R ⊗V k is geometrically integral.
Let R0 =
ª
£
¤
V T1±1 , . . . , Td±1 be p–adic completion of the polynomial algebra V T1±1 , . . . , Td±1 .
Assume that R is obtained from R0 iterating finitely many times the following operations:
ét) the p–adic completion of an étale extension;
loc) the p–adic completion of the localization with respect to a multiplicative system;
comp) the completion with respect to an ideal containing p.
Define
·
1
pn
−1
pn
1
pn
−1
pn
Rn := R ⊗ Vn T1 , T1 , . . . , Td , Td
V
¸
,
R∞ := ∪n Rn .
Let R be the direct limit of a maximal chain of normal R∞ -algebras,
which are
h i
1
domains and, after inverting p, are finite and étale extensions of R∞ p .
Let m ∈ N and let S be a Rm –algebra such that S is finite as Rm –module and Rm£ ⊂ S¤
is étale after inverting p. Define Sn as the normalization of S ⊗Rm Rn in S ⊗Rm Rn p−1
for every n ≥ m. Let S∞ := ∪n≥m Sn .
0
Write Sn0 for the normalization of Sn in Sn ⊗Vn M and S∞
for the normalization
0
0
0
0
= R∞ ⊗V∞ W .
of S∞ in S∞ ⊗V∞ M . We put S := Sm . Note that R = R ⊗V W and R∞
2.2 Proposition. There exist constants 0 < ε < 1 and N = N (S) ∈ N, depending
p
on S, and there exists an element pε of VN of valuation ε such that Sn+1
+ pε Sn+1 ⊂
p
0
ε 0
ε 0
ε
0
Sn + p Sn+1 (as subrings of Sn+1 ) and Sn+1 + p Sn+1 ⊂ Sn + p Sn+1 (as subrings
0
) for every n ≥ N .
of Sn+1
Proof: The claim concerning Sn+1 follows from [An, Cor. 3.7]. It follows from [An,
Prop. 3.6] that there exists a decreasing sequence of rational
{δn (S)} such
¡ 0 numbers
¢
δn (S)
0
0
that p
annihilates the trace map Tr: Sn → HomR0 n Sn , R n . This implies that
δn (S) 0
p
Sn+1 ⊂ Sn0 ⊗R0 n R0 n+1 ; see loc. cit. This, and the fact that the proposition holds
for R0 by direct check, allows to conclude; see the proof of [An, Cor. 3.7] for details.
7
0
is an integral
2.3 Definition. For every R–subalgebra S ⊂ R as in 2.1 such that S∞
domain, viewed as a subring of R, define
µ · ¸
· ¸¶
µ
· ¸
· ¸¶
1
1
1
1
GS := Gal R
/S
, ΓS := Gal S∞
/S
p
p
p
p
and
HS := Ker (GS → ΓS ) .
Analogously, let
· ¸¶
µ · ¸
1
0 1
GS := Gal R
/S
,
p
p
Γ0S
and
· ¸¶
µ
· ¸
1
0
0 1
:= Gal S∞
/S
p
p
µ
HS := Ker (GS →
Γ0S )
= Gal
0
S∞
· ¸
· ¸¶
1
0 1
/S
.
p
p
0
is an integral domain, the map HS /HS → Gal(M/K∞ ) is an isomorphism.
Since S∞
Furthermore, ΓS is isomorphic to the semidirect product of ΓV and of Γ0S . The latter is
a finite index subgroup of Γ0R ∼
e1 , . . . , γ
ed be topological generators of Γ0R .
= Zdp . We let γ
2.4 RAE. Following Faltings [Fa1, Def. 2.1] we say that an extension R∞ ⊂ S∞ is
almost étale if it is finite and étale after inverting p and if, for every
n¤∈ N, the element
£ −1
1
n
p p e∞ is in the image of S∞ ⊗R∞ S∞ . Here e∞ ∈ S∞ ⊗£R∞ S¤∞ p £ is the
¤ canonical
−1
−1
idempotent splitting the multiplication map S∞ ⊗R∞ S∞ p
→ S∞ p .
We say that such extension satisfies (RAE), for refined almost étaleness, if the following holds. £For every
n£ ≥ m¤ let en be the diagonal idempotent associated to the étale
¤
−1
extension Rn p
⊂ Sn p−1 . There exists ` ∈ N, independent of m, such that there
`
`
exists an element p pn of Vn of valuation p`n and p pn en lies in the image of Sn ⊗Rn Sn .
We assume that (RAE) holds for every extension R∞ ⊂ S∞ arising as in 2.1. If this
holds, we say that R or equivalently Spf(R) is small.
2.5 Remark. It is proven in [An, Prop. 5.10 & Thm. 5.11] £that (RAE) holds
¤ if R is of
Krull dimension ≤ 2 or if the composite of the extensions V T1±1 , · · · , Td±1 → R0 → R
is flat and has geometrically regular fibers. For example, this holds if R is obtained by
taking the completion
with respect
£ ±1
¤ to an ideal containing p of the localization of an
±1
étale extension of V T1 , · · · , Td ; see [An, Prop. 5.12].
e S , ES , E
e S 0 and E0 . Let S be as in 2.1. Define
2.6 The rings E
∞
S
∞
¡
¢
¡
¢
e + := lim S∞ /pε S∞ ,
e +0 := lim S 0 /pε S 0
E
E
∞
∞
S∞
S∞
←
←
8
where the inverse limit is taken with respect to Frobenius. Using 2.2 define the generalized ring of norms,
e+
e +0
E+
E0+
S ⊂ E S∞ ,
S ⊂ E S∞
e + (resp. in E
e +0 ) such that
as the subring consisting of elements (a0 , . . . , an , . . .) in E
S∞
S∞
an is in Sn /pε Sn (resp. Sn0 /pε Sn0 ) for every n ≥ N (S).
e +0 and E0+ ) are endowed with a Frobenius hoe + , E+ (resp. E
By construction E
S
S∞
S
S∞
momorphism ϕ and a continuous action of ΓS (resp. Γ0S ). £Denote
¤ by ² the+ element
£ −1 ¤
+
+
−1
e S 0 := E
e 0 π
e S := E
e
(1, ζp , . . . , ζpn , . . .) ∈ EV and by π := ² − 1. Put E
,E
,
∞
S∞ π
S∞
∞
£
£
¤
¤
0+
+
E0S := ES π −1 and ES := ES π −1 .
e + , if it
By abuse of notation for α ∈ Q, we write π0 α for a = (a0 , a1 , . . . , an , . . .) ∈ E
V∞
exists, such that v(ai ) =
α
pi
p
for i À 0. For example, π = π0p−1 ; see [AB, Prop. 4.3(d)].
1
1
2
For every i = 1, . . . , d, let xi := (Ti , Tip , Tip , · · ·) ∈ E+
R0 . The following hold:
n
p ε +
ε
1) there exists N (S) ∈ N such that the map E+
S /π0 ES → Sn /p Sn and the map
n
e + /π p ε E
e + → S∞ /pε S∞ , sending (a0 , . . . , an , . . .) 7→ an , are isomorphisms for
E
0
S∞
S∞
every n ≥ N (S) (see [An, Thm. 5.1]);
+
+
2) E+
S is a normal ring, it is finite as ER –module and it is an étale extension of ER ,
after inverting π, of degree equal to the generic degree of Rm ⊂ S (see [An, Thm. 4.9
& Thm. 5.3]);
e + is normal and coincides with the π–adic completion of the perfect closure of E+
3) E
S∞
S
(see [An, Cor. 5.4]);
e + → E+ ⊗ + E
e+
e+
4) there exists ` ∈ N and maps π ` E
S∞
S
ER R∞ → ES∞ which are isomor£
£
¤
¤
e + π −1 = E
e + ⊗ + E+ π −1 (see [An,
phisms after inverting π. In particular, E
S∞
R∞ E R S
Lem. 4.15]);
5) consider the ring
d
lim S
∞ :=
∞←n
n¡
o
¢
¡ (m+1) ¢p
(m)
d
x(0) , x(1) , . . . , x(m) , . . . |x(m) ∈ S
,
x
=
x
,
∞
d
where S
∞ is the p–adic completion of S∞ , the transition maps are defined by raising
d
to the p–th power, the multiplicative structure is induced by the one on S
∞ and the
additive structure is defined by
¡
¢
n
(. . . , x(m) , . . .) + (. . . , y (m) , . . .) = . . . , lim (x(m+n) + y (m+n) )p , . . . .
n→∞
e+
d
The natural map lim∞←n S
∞ → ES∞ is a bijection (see [An, Lem. 4.10]).
It follows from (1), see [An, Cor. 4.7], that
E+
V
∼
= k∞ [[πK ]]
and
E+
R0
½
¾
1
1
+
∼
,
= EV x1 , . . . , xd , , . . . ,
x1
xd
where k∞ is the residue field of V∞ and πK = (. . . , τn , τn+1 , . . .), with τi ∈ Vi for i À 0,
9
p
±1
is a system of uniformizers satisfying τi+1
≡ τi mod pε . The convergence in x±1
1 , . . . , xd
+
+
is relative to the π-adic topology on E+
V . Eventually ER is obtained from ER0 iterating
the operations
ét) the π–adic completion of an étale extension;
loc) the π–adic completion of the localization with respect to a multiplicative system;
comp) the completion with respect to an ideal containing a power of π.
In particular, {πK , x1 , . . . , xd } is an absolute p–basis of E+
R.
2.7 Lemma. Let S be as in 2.1. The following hold:
n
n
e +0 /π p ε E
e +0 → S 0 /pε S 0 and E0+ /π p ε E0+ → S 0 /pε S 0 , given by
1. the maps E
∞
∞
n
n
0
0
S∞
S∞
S
S
+
e
(a0 , . . . , an , . . .) 7→ an , are isomorphisms for n ≥ N (S). In particular, ER∞
coincides
0
0+
e+ ⊗ + E
e+
with the π–adic completion of E
R∞ E
eV∞ W and ER coincides with the π–adic
e+
completion of E+
R ⊗E+ EW ;
V
e + , E+ → E
e +0 and E+ → E0+ are faithfully flat. For
2. the extensions
→ E
R∞
R
R∞
R
R
+
every finitely generated ER –module M , the base change of M via any of the above
extensions is π–adically complete and separated;
e +0 → E+ ⊗ + E
e +0 → E
e +0 and π ` E0+ → E+ ⊗ + E0+ → E0+ .
3. we have maps π ` E
E+
R
S∞
S
ER
R∞
They are isomorphisms after inverting π;
S∞
S
S
ER
R
S
Proof: Statements (1) and (3) follow from 2.4 arguing as in the proofs of [An, Thm. 5.1]
and [An, Lem. 4.15] respectively.
n
εpn + ∼
ε
ε
e + /π εp E
e+ ∼
(2) By 2.6 we have E+
/p
R
and
E
n
n
0
R /π0 n ER = R
W
W = W/p W .n One de¡
¢
εp
0+ ∼
ε
e +0 /π εp E
e +0 ∼
duces form (1) that E0+
0
R /π0 ER = Rn /p Rn ⊗Vn W and that ER∞
R∞ =
¡
¢
ε
R∞ /p R∞ ⊗V∞ W . By construction R∞ ⊗V∞ W is a free Rn ⊗Vn W –module with baα1
pm
αd
pm
e +0 ) is the
} for m ≥ n and 0 ≤ αi < pm−n . Hence, E0+
R0 (resp. ER∞
1
¡
¢ 1
e + (resp. of ∪n E+ ⊗ + E
e + [x pn , . . . , x pn ]). The E+ –
+ E
π–adic completion E+
⊗
1
R EV
W
R EV
W
V
d
+
e
algebra E is the π–adic completion of finite, normal and generically separable extensis {T1
· · · Td
W
+
sions of the DVR E+
V . Those are free as EV –module. We may then apply [An, Lem. 8.7]
to conclude.
h i
Given an R∞ –algebra S∞ , finite and étale over R∞
there exists a Rm –algebra S, finite and étale over Rm
in 2.1, is the normalization of S ⊗Rn R∞ .
1
p
, there exists m ∈ N and
1
p
such that S∞ , defined as
h i
2.8 Theorem. The functor S∞ → E+
S defines an equivalence of categories from the
category R∞ -AED of R∞ –algebras which are normal domains, finite and étale over R∞
+
after inverting p to the category E+
R -AED of ER –algebras, which are normal domains,
finite and étale after inverting π. In particular, this realizes HR as the Galois group
of ER .
10
Proof: See [An, Thm. 6.3].
Let E+
be ∪S∞ E+
where the union is taken over all R∞ –subalgebras S∞ ⊂ R
R
£ −1 ¤ S
£
¤
such that S∞ p
is finite étale over R∞ p−1 . Similarly, define E0+
to be the union
R
¡
¢
+
0+
e = lim R/pε R , where the inverse limit is taken with respect to Frobe∪S∞ ES . Let E
R
£
¤
e + . Denote E := E+ π −1 ,
nius. It coincides with the π–adic completion of ∪S∞ E
S∞
R
R
£
¤
£
¤
e + π −1 .
e := E
E0 := E0+ π −1 and E
R
R
R
R
2.9 Proposition. Let S be as in 2.3. Then,
b HS = S
b HS = S
0
d
d
a) R
∞ and R
∞ (here c denotes the p–adic completion);
b) we have
¡
E+
R
and
¢HS
¡
= E+
S,
E0R
¢HS
¡ + ¢HS
e
e+ ,
E
=E
S∞
R
S
EH
= ES ,
R
= E0S ,
¡
e+
E
R
¢HS
e +0 ,
=E
S∞
e HS = E
eS
E
∞
R
¡ ¢ HS
e
e S0 ;
E
=E
R
∞
c) the maps
e R ⊗ ES −−→ E
eS ,
E
∞
∞
ER
E0R ⊗ ES −−→ E0S ,
ER
e R0 ⊗ ES −−→ E
e S0
E
∞
∞
ER
e W → E0 and E
eS ⊗
e
are isomorphisms. In particular, the maps ES ⊗EV E
∞
S
eV∞ EW →
E
e S 0 are injective with dense image.
E
∞
b HS = S
d
Proof: (a) The fact that R
∞ is proven in [An, Lem. 6.13].
(b) The equalities in the first displayed formula hold due to [An, Prop. 6.14]. Those in
e +0 ) can be
e + (resp. E
the second displayed formula follow arguing as in loc. cit. In fact, E
S∞
R
b
0
d). To get the last two equalities
written as in 2.6(5) as the limit lim R (resp. lim S
∞
∞←n
HS
∞←n
b
0
d
in the second display one is left to prove that R
= S
∞ . This follows arguing as
¡
¢ HS
in [An, Lem. 6.13]. The fact that the inclusion E0S ⊂ E0R
is an equality can be
e +0 since the latter is faithfully flat by 2.7(2).
checked after base change via E0+
R → ER∞
¡ + ¢ HS
¡ 0+ ¢HS
e +0 ⊂ E
e
e +0 .
e +0 ∼
0
0+ E
But E0S ⊗E0+ E
⊗
=E
E
by
2.7(3)
and
E
=
S
R∞
S∞
R∞
ER
∞
R
R
R
(c) The first equality follows from 2.6(4). The others follow from 2.7(3). In the
case S = R the last statement follows from 2.7(1). The general case follows from this,
the equalities just proven and the fact that ER ⊂ ES is finite étale by 2.8.
¡ ¢
eS , A
e † , AS , A† , A
e † 0 , A0 and A0† . Define A
e S0 , A
e := W E
e .
2.10 The rings A
∞
S
S∞
S
S∞
S
R
R
∞
It is endowed with the following topology, called
e + }n as fundamental
e the topology having {π n E
The weak topology. Consider on E
R
R ¡
¢
e
system of neighborhoods of 0. On the truncated Witt vectors Wm E
R we consider
11
¡ ¢
∼
e
the product topology via the isomorphism Wm E
R =
components.
¡ ¢ Eventually, ¡the ¢weak topology is defined
e .
e = lim Wm E
ogy W E
R
R
¡ ¢m
e
E
given by the phantom
R
as the projective limit topol-
∞←m
e + :=
Alternatively, let π := [ε] − 1 where [ε] is the Teichmüller lift of ε. Put A
R
¡ ¢+
ne
h e+
e
W ER . For every n and h ∈ N define Un,h := p AR + π AR . The weak topology
e has {Un,h }n,h∈N as fundamental system of neighborhoods.
on A
R
e → Q ∪ {∞} by vE (z) = ∞ if z = 0 and vE (z) = p max{n ∈
Define vE : E
R
p−1
e + }. For z = P [zk ]pk ∈ A
e and N ∈ N we put
Q|π −n z ∈ E
k
R
R
≤N
vE
(z) := inf{vE (zk )|0 ≤ k ≤ N }.
For every N ∈ N we have
≤N
e +;
(i) vE
(x) = +∞ ⇔ x ∈ pN +1 A
R
≤N
≤N
≤N
(ii) vE
(xy) ≥ vE
(x) + vE
(y);
≤N
≤N
≤N
≤N
≤N
(iii) vE (x + y) ≥ min(vE (x), vE
(y)) with equality if vE
(x) 6= vE
(y);
≤N
≤N
≤N
≤N
p
(iv) vE (π) = p−1 and vE (πx) = vE (π) + vE (x);
¢
≤N ¡
≤N
(v) vE
ϕ(x) = pvE
(x);
¡
¢
≤N
≤N
(vi) vE γ(x) = vE (x) for every γ ∈ GR .
The second claim in (iii) and property (v) follow since
e + and each E
e + is normal by
adic completion of ∪S∞ E
S∞
S∞
e /pN +1 A
e induced from the weak topology on A
e
on A
ogy.
R
e + is by construction the π–
E
R
2.6(3). Note that the topology
≤N
topolR coincides with the vE
R
For every S as in 2.3 define
¡
¢
e S := W E
eS ,
A
∞
∞
¡ + ¢
e + := W E
e
A
S∞
S∞ ,
¡
¢
¡ + ¢
e S 0 := W E
e S0
e +0 := W E
e 0 .
A
and
A
S
S∞
∞
∞
∞
¡ ¢
e
They are subrings of W E
R closed for the weak topology.
e (0,r] as the subring of elements
Overconvergent coefficients. For r ∈ Q>0 define A
R
∞
P
e such that
z=
pk [zk ] of A
R
k=0
lim rvE (zk ) + k = +∞;
k→∞
e† =
see [AB, Prop. 4.3]. Write A
R
S
r∈Q>0
(
wr (z) =
e (0,r] , put
e (0,r] . For z = P pk [zk ] ∈ A
A
R
R
k∈Z
∞
inf (rvE (zk ) + k)
k∈Z
12
if z = 0;
otherwise.
e (0,r] → R ∪ {∞} satisfies
Thanks to [AB, Prop. 4.3] one knows that the map wr : A
R
(i) wr (x) = +∞ ⇔ x = 0;
(ii) wr (xy) ≥ wr (x) + wr (y);
(iii) wr (x + y) ≥ min(wr (x), wr (y));
(iv) wr (p) = 1 and wr (px) = wr (p) + wr (x).
eS ∩ A
e (0,r] and A
e † := A
eS ∩ A
e † . Similarly, define
e (0,r] := A
For every S∞ define A
∞
∞
S∞
S∞
R
R
(0,r]
(0,r]
†
†
e
e
e
e
e
e
A 0 := AS 0 ∩ A
and A 0 := AS 0 ∩ A . By [AB, Prop. 4.3] they are wr –adically
S∞
R
∞
S∞
R
∞
e .
complete and separated subrings of A
R
Noetherian
coefficients. Let S be as in §2.1. In [An, Appendix II] a ring AS ⊂
¡
¢
eS
W E
has
been
constructed, functorially in S, with the following properties:
∞
i. it is complete and separated for the weak topology. In particular, it is p–adically
complete and separated.
¡
¡ ¢¢
e
ii. AS ∩ pW E
= pAS ;
R
iii. AS /pAS ∼
= ES . In particular, it is noetherian and regular.
iv. it is endowed with continuous commuting actions of ΓS and of an operator ϕ lifting
those defined on ES ;
v. AR contains the Teichmüller lifts of ², x1 , . . ., xd ;
vi. AS is the unique finite and étale AR -algebra lifting the finite and étale extension ER ⊂ ES .
+
One also requires the existence of a subring A+
S lifting ES , with suitable properties, so
(0,r]
e (0,r]
that AS is unique. We refer to [AB, Prop. 4.51] for details. Define AS := AS ∩ A
R
†
†
0
e
e
e
and A := AS ∩ A . Define A to be the closure of the image of AS ⊗A AW in A
V
S
R
†
0†
0
e .
for the weak topology. Put AS := AS ∩ A
R
Eventually, let AR (resp. A0R ) be the completion for the p–adic topology of ∪S∞ AS
(resp. ∪S∞ A0S£), where
the union is taken £over¤ all normal R∞ –subalgebras S∞ ⊂ R
¤
−1
e † (resp. A0† :=
such that S∞ p
is finite étale over R∞ p−1 . Write A†R := AR ∩ A
R
R
0
e † ).
AR
∩A
R
S
R
e † and A† ⊂ A† are finite and étale.
e† ⊂ A
2.11 Proposition. The extensions A
S
S∞
R
R∞
e
e
Their reduction modulo p coincide with ER∞ ⊂ ES∞ and ER ⊂ ES respectively.
e † coincides with E
e S modulo p since it contains A
e + and
Proof: It is clear that A
∞
S∞
S∞
e ∩A
e † = pA
e † . The fact that A
e† ⊂ A
e † is finite and étale is proven in [AB,
pA
S∞
S∞
R∞
S∞
R
Prop. 4.9]. See [AB, Prop. 4.35] for the statements regarding A†R ⊂ A†S .
2.12 Lemma. The following hold:
13
¡ 0 ¢HS
0
S
eS = A
e HS , and A
e S0 = A
e HS . The same equali=
AR
, A
a) AS = AH
,
A
∞
S
∞
R
R
R
¡ (0,r] ¢HS
(0,r]
e (0,r] =
ties hold considering overconvergent coefficients i. e., AS = AR
,A
S∞
¡ (0,r] ¢HS
¡ (0,r] ¢HS
¡ † ¢HS
¡ † ¢HS
(0,r]
†
†
e
e
e
e
e
AR
, A S∞
= AR
and AS = AR
, A S∞ = A R
, A†S∞
=
0
0
¡ † ¢HS
e
A
.
R
e R ⊗A AS → A
e S , A0 ⊗A AS → A0 and A
e R 0 ⊗ A AS →
b) The natural maps A
∞
∞
R
R
R
R
S
∞
e
0
A S∞
are isomorphisms. Similarly, considering overconvergent coefficients, the maps
†
e † , A0† ⊗ † A† → A0† and A
e † 0 ⊗ † A† → A
e † 0 are isomore
AR∞ ⊗A† A†S → A
S∞
R AR
S
S
R∞ A R
S
S∞
R
phisms.
e † 0 /pA
e†0 =
c) We have A† /pA† = AS /pAS = ES , A0† /pA0† = A0 /pA0 = E0 and A
S
S
S
S
S
S
S
e S 0 /pA
e S0 = E
e S0 .
A
∞
∞
∞
eS ⊗
d) The maps A
∞
e
A
V∞
S∞
S∞
eW → A
e S 0 and AS ⊗A A
e W → A0 are injective and have
A
V
S
∞
e (0,r] ⊗ (0,r] A
e (0,r] → A
e (0,r]
dense image for the weak topology. The image of A
is
0
S∞
W
S∞
e
A
V∞
dense for the wr –adic topology for every r ∈ Q>0 .
¡
¢ HS
HS
eS ⊂ A
e HS , A
e S0 ⊂
,A
, A0S ⊂ A0R
Proof: (a)&(b) We have inclusions AS ⊂ AR
∞
∞
R
HS
0
0
e
e
e
e
e
0 ⊗A
0 .
AR and maps AR∞ ⊗AR AS → AS∞ , AR ⊗AR AS → AS and AR∞
A S → A S∞
R
The extension AR → AS is finite and étale and, hence, AS is projective as AR –module.
e R , A0 and A
e R0 are p–adically complete and separated and p is a not a zero
Since A
∞
R
∞
e R ⊗A AS , A0 ⊗A AS and A
e R0 ⊗A AS are p–adically
divisor in these rings, A
∞
R
R
R
R
∞
HS
complete and separated and p is a non–zero divisor. The same holds for AS , AR
,
H
H
e S , A0 and A
e S 0 . To check that all the inclusions and all the maps
eS , A
e S, A
A
∞
R
R
S
∞
above are isomorphisms it then suffices to show it modulo p. This follows from 2.9 if we
e S 0 /pA
e S0 = E
e S 0 . Once this is established the other
prove that A0S /pA0S = E0S and A
∞
∞
∞
statements in (a) and the first part of (b) follow.
0† e †
†
†
e
e
e†
e
(c) Since by construction A0†
0 ∩ pAR = pAS 0 , AS ∩ pAR = pAS
S ∩ pAR = pAS , AS∞
∞
e = pA0 , A0 ∩ pA
e = pA
e S 0 , AS ∩ pA
e = pAS the maps A† /pA† →
and A0S ∩ pA
S
S∞
S
S
R
R
R
∞
¡
¢
0†
0†
†
†
HS
0
0
0 HS
0
e
e
AS /pAS → ER = ES , AS /pAS → AS /pAS → ER
= ES and AS∞
→
0 /pAS 0
∞
H
S
e S 0 /pA
e S0 → E
e
e S 0 are injective. It follows from [AB, §4.11(e)&Lem. 4.20]
A
= E
∞
∞
∞
R
n αo
+
+
e
e (0,r] , for suitable α, β ∈ N and r ∈ Q>0 , so
that there exists AS ⊂ AS∞ πp β ⊂ A
S∞
n αo
p
+ pα
+ ∼
+
+
e+
that AS / πβ AS = ES and AS is complete for the weak topology. Here, A
S∞
πβ
h i
e + pαβ with respect to the weak topology. In particular,
denotes the completion of A
S∞ π
n αo
+
e
since π–adic convergence in AS∞ πp β implies convergence for the weak topology, A+
S
+
+
+
+
e
e
e
is π–adically complete. Note that AS ⊗A+ AW and AS∞ ⊗A
e + AW map to the π–adic
V
n Vα∞o
n αo
e+ p
e+ ⊗ + A
e+ p
completion of the image of A
S∞ A
e V∞ W πβ in AR πβ and that the latter is
e (0,r]
e † 0 /pA
e † 0 contain the
contained in A
. We conclude that A† /pA† , A0† /pA0† and A
0
S∞
S
S
14
S
S
S∞
S∞
+
e+
e+
e+
π–adic completion of the image of E+
S , ES ⊗E+ EW and ES∞ ⊗E
e+ EW respectively.
V
V∞
Claim (c) follows then from 2.9.
†
†
e † ⊗ † A† ∼
e†
(b) The fact that A
R∞ A
S = AS∞ follows from 2.11. Since AR ⊂ AS is fiR
nite and étale by 2.11 there is a unique idempotent eS/R of A†S ⊗A† A†S such that for
R
¢
¡
every x ∈ A†S ⊗A† A†S we have m(x) = TrA† /A† xeS/R . Here, m is the multipliR
R
S
Pu
(0,s]
cation map. Write eS/R =
for some s ∈ Q>0 .
i=1 ai ⊗ bi with ai and bi ∈ AS
0†
†
0†
e
e † 0 . By the first
Then, eS/R is an idempotent of AS ⊗A0† AS and of AS∞
0 ⊗ †
e A S∞
A
R
0
S∞
e R0 ⊂ A
e S 0 are finite and étale and
⊂
and A
part of (b) the extensions
∞ ¡
∞
¡
¢
¢
m(x) = TrA0S /A0R xeS/R and m(x) = TrA
xe
S/R . We then get that for eve S∞
e R0∞
0 /A
e † 0 ) we have x = m(x ⊗ 1) = Pu TrA0 /A0 (xai )bi (resp. x =
ery x ∈ A0†
(resp.
A
SP
S∞
i=1
S
R
u
0 /A0
m(x ⊗ 1) = i=1 TrA
(xa
)b
).
Since
Tr
and
Tr
i
i
A
e 0 /A
e 0
e 0 /A
e 0 send overconA
S
R
A0R
S∞
A0S
R∞
S∞
R∞
vergent elements to overconvergent elements, we conclude that the maps in the second
part of (b) are surjective.
Since the extension A†R ⊂ A†S is finite and étale, A†S is projective as A†R –module.
†
e † 0 ⊗ † A† and those
In particular, p is not a zero divisor in A0†
R ⊗A†R AS and in AR∞
S
AR
rings are p–adically separated. Thus, to prove that the maps in the second part of (b)
are injective, and hence are isomorphisms, it suffices to prove that they are injective
modulo p. This follows from (c).
eV ⊂ A
e W are extensions of DVR’s, they are flat.
(d) Since the extensions AV ⊂ A
∞
e S and AS , it is not a zero divisor in A
eS ⊗
e
Since p is not a zero divisor in A
∞
∞
e V ∞ AW
A
e W . Thus, to check the injectivity in (d) we may reduce modulo p. The
and AS ⊗A A
V
density can be checked modulo pn for every n ∈ N and, using induction, it suffices in
fact to prove it for n = 1. Then, the first claim of (d) follows from 2.11 and 2.9(c).
We prove the second claim of (d). Suppose that r = a/b with a· and b ∈ ¸N and let
pa
e+
0 ,(a,b) ) denote the p–adic completion of A
(resp. of
AS∞ ,(a,b) (resp. AS∞
S∞
b
( p−1
p )
[π]
¸
·
pa
+
e
A S∞
), where [π] is the Teichmüller lift of π. Arguing as in the proof of [AB,
0
p−1
b
[π]( p )
Lem. 4.20] one has
· ¸
n
o
(0,r]
e 0 |wr (x) ≥ 0 ⊆ AS 0 ,(a,b) 1 .
0 ,(a,b) ⊆
AS∞
x∈A
S∞
∞
[π]
h i
1
e (0,r]
0 ,(a,b)
for the wr –adic
Since wr ([π]) > 0, we conclude that AS∞
0
[π] is dense in AS∞
e + is contained in the wr –
topology. Since the [π]–adic completion of AS∞ ,(a,b) ⊗A
A
W
e+
V∞
(0,r]
(0,r]
`
e
e
0 ,(a,b) contains π AS 0 ,(a,b) by 2.7,
adic closure of A
A
and its image in AS∞
S∞ ⊗ A
W
e (0,r]
∞
V∞
the conclusion follows.
e R0 ⊂ A
e S0 , A
e† 0 ⊂ A
e † 0 , A0 ⊂ A0 and A0† ⊂ A0†
2.13 Corollary. The extensions A
R
S
R∞
S∞
R
S
∞
∞
15
e R0 ⊂ E
e S 0 and E0 ⊂ E0
are finite and étale. Their reduction modulo p coincide with E
R
S
∞
∞
respectively.
¡ ¢
2.14 (ϕ, Γ)–modules and Galois representations. Let S be as in 2.3. Let Rep GS be
the abelian tensor category of finitely generated Zp –modules endowed with a continuous
linear action of GS .
eS , A
e † , AS , A† , A
e S0 , A
e † 0 , A0 and A0† and let Γ be
Let A be one of the rings A
S∞
∞
S
∞
S
S∞
S
respectively ΓS or Γ0S . Let (ϕ, Γ) − ModA (resp. (ϕ, Γ) − Modet
A ) be the tensor category
of finitely generated A-modules D endowed with
i. a semi-linear action of Γ;
ii. a semi-linear homomorphism ϕ commuting with Γ (resp. so that ϕ ⊗ 1: D ⊗ϕ
AA→D
is an isomorphism of A-modules).
Note that if A = AS , then AS is noetherian and (ϕ, ΓS ) − ModAS (resp. (ϕ, ΓS ) −
Modet
AS ) is an abelian category.
¡ ¢
For any object M in Rep GS , define
³
,
³
´HS
0
D(M ) := AR ⊗ M
´HS
³
e
e
,
D(M ) := AR ⊗ M
³
´HS
e
e
D(M ) := AR ⊗ M
.
D(M ) := AR ⊗ M
´HS
Zp
Zp
Zp
Zp
e
e S –module) endowed with
Note that D(M ) (resp. D(M
)) is an AS -module (resp. A
∞
e
a semi-linear action of ΓS . Analogously, D(M ) (resp. D(M
)) is a A0S –module (resp. a
e S 0 –module) endowed with a semi-linear action of Γ0 . Analogously, define
A
S
∞
,
³
´HS
0†
D (M ) := AR ⊗ M
³
´HS
†
†
e
e
D (M ) := AR ⊗ M
,
³
´HS
†
†
e
e
D (M ) := AR ⊗ M
.
†
D (M ) :=
³
A†R ⊗
Zp
M
´HS
†
Zp
Zp
Zp
e † (M )) is an A† -module (resp. A
e † –module) endowed with a
Then, D† (M ) (resp. D
S
S∞
e † (M )) is a A0† –module (resp. a
semi-linear action of ΓS . Analogously, D† (M ) (resp. D
S
e † 0 –module) endowed with a semi-linear action of Γ0 .
A
S
S∞
†
e
e
The homomorphism ϕ on A and A defines a semi-linear action of ϕ on all these
R
R
modules commuting with the action of ΓS (resp. of Γ0S ).
2.15 Theorem. ¡The¢ functor D defines an equivalence of abelian tensor categories from
the category Rep GS to the category (ϕ, ΓS ) − Modet
AS . Let M be a finitely generated
Zp –module endowed with a continuous action of GS . The inverse is defined associating
´ϕ=1
³
.
to an étale (ϕ, ΓS )–module D the GR –module V(D) := AR ⊗AS D
16
Proof: See [An, Thm. 7.11].
2.16 Lemma. Let M be a finitely generated Zp –module endowed with a continuous
action of GS . Then,
e
e
(i) D(M ) (resp. D(M ), D(M
), D(M
)) is an étale (ϕ, ΓS )–module over AS (resp. A0S ,
eS , A
e S 0 );
A
∞
∞
e † (M ), D
e † (M )) is an étale (ϕ, ΓS )–module over A† (resp. A0† ,
(i’) D† (M ) (resp. D† (M ), D
S
S
e† , A
e † 0 );
A
S∞
S∞
n
e
e
(ii) D(M ) = lim D(M/pn M ), D(M ) = lim D(M/pn M ), D(M
) = lim D(M/p
M ),
∞←n
∞←n
∞←n
n
e
e
D(M
) = lim D(M/p
M ) where the limits are inverse limits with respect to n ∈
∞←n
N. More precisely, D(M )/pn D(M ) ∼
= D(M/pn M ), D(M )/pn D(M ) ∼
= D(M/pn M ),
n
n
e
e
e
e
e
e
D(M
)/pn D(M
)∼
M ) and D(M
)/pn D(M
)∼
M ) for every n ∈ N.
= D(M/p
= D(M/p
e † (M ) = D(M
e
(ii’) if M is torsion, then D† (M ) = D(M ), D† (M ) = D(M ), D
) and
†
e (M ) = D(M
e
D
);
(iii) the natural maps
D(M ) ⊗ A0S −−→ D(M ),
D(M ) ⊗ AR −−→ M ⊗ AR
AS
AS
and
e
e S −−→ D(M
D(M ) ⊗ A
),
∞
AS
Zp
e S 0 −−→ D(M
e
D(M ) ⊗ A
)
∞
AS
are isomorphisms;
(iii’) the natural maps
†
D† (M ) ⊗ A0†
S −−→ D (M ),
†
D† (M ) ⊗ AR
−−→ M ⊗ A†R
and
Zp
A†S
A†S
e † (M ),
e † −−→ D
D† (M ) ⊗ A
S∞
A†S
e † 0 −−→ D
e † (M )
D† (M ) ⊗ A
S∞
A†S
are isomorphisms;
(iv) the natural maps
D† (M ) ⊗ AS −−→ D(M ),
A†S
and
e † (M ) ⊗ A
e
e S −−→ D(M
D
),
∞
†
e S∞
A
are isomorphisms.
17
D† (M ) ⊗ A0S −−→ D(M )
A0†
S
e S 0 −−→ D(M
e † (M ) ⊗ A
e
D
)
∞
†
e S0
A
∞
Proof: We refer the reader to [An, Thm. 7.11] and [AB, Thm 4.42] for the proofs
that D(M ) and D† (M ) are étale (ϕ, ΓS )–modules and that D(M ) = D† (M ) ⊗A† AS ,
S
† ∼
†
†
∼
that D(M ) ⊗A A = M ⊗Z A and D (M ) ⊗ † A = M ⊗Z A . Claims (i), (i’)
S
R
p
R
AS
p
R
R
and (iv) follow from this and the displayed isomorphisms. We prove the other statements.
Due to 2.11, 2.12 and 2.13 to prove (ii’), (iii) and (iii’) one may pass to an extension
¡
¢HT
S∞ ⊂ T∞ in R finite, étale and Galois after inverting p. For example, M ⊗Zp AR
=
¡
¡
¡
¢HS
¢ HT
¢ HS
M ⊗Zp AR
⊗AS AT and M ⊗Zp AR
= M ⊗Zp AR
⊗A0S A0T by étale de¢HT
¡
¢
¡
HT
⊗AT A0T → M ⊗Zp AR
is an isomorphism, takscent. Hence, if M ⊗Zp AR
¡
¢HS
ing the HS –invariants, we get the claimed isomorphism M ⊗Zp AR
⊗AS A0S →
¢HS
¡
.
M ⊗Zp AR
Suppose first that there exists N ∈ N such that pN M = 0. Then, there exists an
extension S∞ ⊂ T∞ such that HT ⊂ HS acts trivially on M . Replacing S∞ with T∞ ,
we may then assume that HS acts trivially on M . By 2.12 we have A†S /pN A†S =
N e
N 0†
0
N 0
N e†
e†
e
AS /pN AS , A0†
S /p AS = AS /p AS , AS∞ /p AS∞ = AS∞ /p AS∞ and eventually
e†0 = A
e S 0 /pN A
e S 0 . Furthermore, we have in this case D(M ) = M ⊗Z AS ,
e † 0 /pN A
A
p
S∞
S∞
∞
∞
e
e S , D(M
e S 0 and similarly for the
e
D(M ) = M ⊗Z A0 , D(M
) = M ⊗Z A
) = M ⊗Z A
p
S
p
∞
p
∞
overconvergent (ϕ, ΓS )–modules. Then, the claims follow from 2.12.
Assume next that M is free of rank n. It follows from [AB, Thm. 4.42] that there
exists an extension R∞ ⊂ T∞ in R finite, étale and Galois after inverting p such
that D† (M ) ⊗A† A†T is a free A†T –module of rank n. As we have seen above we may and
S
will replace S∞ with T∞ so that D† (M ) (resp. D(M )) is a free A†S –module (resp. AS –
module). Fix a basis {e1 , . . . , en } of D† (M ). It is also a AR –basis of D(M ). Hence, it
e† , A , A
e ) of M ⊗Z A† (resp. M ⊗Z A
e † , M ⊗Z A ,
is a basis over A†R (resp. A
p
p
p
R
R
R
R
R
R
e
M ⊗Zp AR ). Since HS and HS act trivially on {e1 , . . . , es }, we get claims (iii) and (iii’).
¡
¢
e
e HS = A
e S e1 ⊕ · · · ⊕ A
e S en = D(M ) ⊗A A
eS .
For example, D(M
) = M ⊗Zp A
∞
∞
∞
S
R
We are left to prove (ii). We may assume that M is torsion free, since the claim for
e
e
the torsion part is trivial. Note that D(M ), D(M
), D(M ) and D(M
) are submodules of
invariants¡ of free modules
¢HS over p–adically complete and separated rings. For example,
D(M ) = M ⊗Zp AR
⊂ M ⊗Zp AR . Hence, they are themselves p–adically complete
and separated. It
every
from¢ their quotient
¡ suffices
¢ to show ¡that for
¢ ¡ n ∈ nN the
¢ map
¡
n
n
n
e
e
modulo p to D M/p M (resp. D M/p M , D M/p M , D M/pn M ) is an isomorphism. Due to (iii) it suffices to show it for D(M ) and in this case it follows from the
fact that D is an exact functor by 2.15.
Qm
2.17 The weak topology on the (ϕ, Γ)–modules. Suppose that M ∼
= Znp × i=1 Z/psi Z
e is isomorphic to A
e n × Qm A
e as A
e –module
e /psi A
as a Zp –module. Then, M ⊗ A
i=1
R
R
R
R
R
e
and, in particular, the product topology defines a topology on M ⊗ A . It is independent
R
18
of the choice of the presentation of M as Zp –module and the action of GS is continuous
e
e † (M ), D† (M ) and
e
for such topology. Note that D(M ), D(M
), D(M ), D(M
), D† (M ), D
e . They are then endowed with the
e † (M ) are by construction submodules of M ⊗ A
D
R
e . We call it the weak topology.
topology induced from the one just defined on M ⊗ A
R
We state the following theorem relating the cohomology (continuous for the weak
topology) of the various (ϕ, Γ)–modules introduced above.
2.18 Theorem. The natural maps
n
H
n
H
¡
¶
µ
¢
¡
¢
n
n
e
e
ΓS , D(M ) −−→ H ΓS , D(M ) −−→ H GS , M ⊗ AR ,
Zp
¡
¢
Γ0S , D(M )
µ
¶
¡ 0
¢
n
e
e
−−→ H ΓS , D(M ) −−→ H GS , M ⊗ AR ,
n
Zp
µ
¶
¡
¢
¡
¢
†
†
n
†
n
e
e
H ΓS , D (M ) −−→ H ΓS , D (M ) −−→ H GS , M ⊗ AR
n
Zp
and
µ
¶
¡ 0
¢
¡ 0
¢
†
†
n
†
n
e
e
H ΓS , D (M ) −−→ H ΓS , D (M ) −−→ H GS , M ⊗ AR
n
Zp
are all isomorphisms.
Proof: See 7.10.6.
3 Galois cohomology and (ϕ, Γ)–modules.
In this section we show how, given a finitely generated Zp –module M with continuous
action of GS , one can compute the cohomology groups Hn (GS , M ) and Hn (GS , M ) in
e
e † (M ), D(M ), D(M
e
terms of the associated (ϕ, ΓS )–modules D(M ), D(M
), D† (M ), D
),
†
†
e
D (M ) and D (M ). We start with the following crucial:
e ,A
e † , A , A† A0 and A0† is surjective and
3.1 Proposition. The map ϕ − 1 on A
R
R
R
R
R
R
its kernel is Zp . Furthermore, the exact sequence
ϕ−1
e −−→
e −−→ 0
0 −−→ Zp −−→ A
A
R
R
¡
¢
e → A
e (as sets) so that σ A
admits a continuous right splitting σ: A
⊂ AR ,
R
R
R
¡ †¢
¡ †¢
¡ 0 ¢
¡ 0† ¢
†
†
0†
0
e
e
σ AR ⊂ AR , σ AR ⊂ AR , σ AR ⊂ AR and σ AR ⊂ AR .
Proof: See 8.1.
e S or A† or A
e†
3.2 Definition. Let D be a coniunous (ϕ, ΓS )-module over AS or A
∞
S
S∞
e S 0 or A0† or A
e † 0 ). Define C • (ΓS , D) (resp. C • (Γ0 , D)) to be the
(resp. over A0S or A
S
S
S∞
∞
complex of continuous cochains with values in D.
19
•
Let T • (D) (resp. T 0 (D)) be the mapping cone associated to ϕ − 1: C • (ΓS , D) →
•
C (ΓS , D) (resp. to ϕ − 1: C • (Γ0S , D) → C • (Γ0S , D)).
3.3 Theorem. There are isomorphisms of δ-functors from the category of Rep(GS ) to
the category of abelian groups:
¡
¢ ∼ i
¡ •
¢ ∼ i
ρi : Hi T • (D( )) −→H
(GS , ),
ρ0i : Hi T 0 (D( )) −→H
(GS , ),
¡
¢ ∼ i
¡ •
¢ ∼ i
e )) −→H
e )) −→H
ρei : Hi T • (D(
(GS , ),
ρe0i : Hi T 0 (D(
(GS , ),
¡
¢ ∼ i
¡ 0• †
¢ ∼ i
i
ρ†i : Hi T • (D† ( )) −→H
(GS , ),
ρ0†
i : H T (D ( )) −→H (GS , ),
¡
¢ ∼ i
¡ 0• †
¢ ∼ i
i
e † ( )) −→H
e
ρe†i : Hi T • (D
(GS , ),
ρe0†
i : H T (D ( )) −→H (GS , )
The isomorphisms ρ0i , ρe0i , ρ0†
e0†
i and ρ
i are Gal(M/K)–equivariant.
Furthermore, all the maps in the square
¡
¢
¡ • †
¢
†
i
e ( ))
T
(
D
Hi T • (D
(
))
−−→
H


y
y
¡
¢
¡
¢
e )) ,
Hi T • (D( ))
−−→ Hi T • (D(
e
induced by the natural inclusions of (ϕ, ΓS )–modules D† (W ) ⊂ D(W ) ⊂ D(W
) and
†
†
e
e
D (W ) ⊂ D (W ) ⊂ D(W ), for W ∈ Rep(GS ), are isomorphisms and they are compatible with the maps ρ†i , ρe†i , ρi and ρei . Similarly, all the maps in the square
¡ • †
¢
¡ • †
¢
e ( ))
(D ( )) −−→ Hi T 0 
(D
Hi T 0 
y
y
¢
¡
¢
¡
• e
•
−−→ Hi T 0 (D(
))
Hi T 0 (D( ))
0
are isomorphisms and are compatible with the maps ρ0†
e0†
e0i
i , ρ
i , ρi and ρ
Proof: First of all we exhibit in 3.4 the maps ρi and ρ0i (with or withouteor †) so that
they are compatible with the displayed squares and ρ0i they are compatible with the
residual action of GV (if one exists). We then prove that they are isomorphisms in 3.5.
Eventually, we show that they are isomorphisms of δ–functors in 3.6.
3.4 The maps. Let W be a Zp -representation of GS . Let D(W ) and A be (1) D(W )
e and A,
e † (W ) and A
e (3) D† (W ) and A† or (4) D
e † . Since in each
and A, (2) D
case D(W ) ⊗AS AR ∼
= W ⊗Zp AR by 2.16 and since the sequence (8.1.1) admits a right
continuous splitting, we have exact sequences of GS –modules
ϕ−1
0 −−→ W −−→ D(W ) ⊗ AR −−→ D(W ) ⊗ AR −−→ 0
AS
AS
(3.4.1)
e (3) D† (W ) and A0†
e and A,
Similarly, let D0 (W ) and A0 be (1) D(W ) and A0 , (2) D
e † . In each case D0 (W ) ⊗A0 A0 ∼
e † (W ) and A
or (4) D
= W ⊗Zp A0R by 2.16. Thanks to 8.1
S
R
20
we get exact sequences of GS –modules
ϕ−1
0 −−→ W −−→ D0 (W ) ⊗ A0R −−→ D0 (W ) ⊗ A0R −−→ 0
A0S
A0S
(3.4.2).
The maps in both exact sequences are continuous for the weak topology by 2.16.
Let (α, β) be an n-cochain of T • (D(W )) i. e., in C n−1 (ΓS , D(W )) × C n (ΓS , D(W )).
Define
¡
¢
cnα,β := β + (−1)n d σ(α) ∈ C n (GS , W ⊗ AR ),
Zp
where d is the differential operator on C n (ΓS , D(W )) and σ is the left inverse of ϕ − 1
defined in 8.1 (for each of the four possibilities for A).
¡
¢ ¡
•
Recall
that
the
derivation
on
T
(D(W
))
is
given
by
d
(α,
β)
= (−1)n (ϕ − 1)(β) +
¢
dα, dβ . Thus, (α, β) is an n–cocycle if and only if it satisfies (−1)n (ϕ − 1)β + dα = 0
and dβ = 0. In this case, dcnα,β = 0 and (ϕ − 1)cnα,β = (ϕ − 1)β + (−1)n d(ϕ − 1)σ(α) =
(ϕ − 1)β + (−1)n dα = 0. Thus, cnα,β is an n-cocycle in C n (GS , W ) by (3.4.1).
Choose
left inverse σ 0 of ϕ − 1.¡ Then,¢ (ϕ − 1)(σ 0 − σ)
¡ 0 a ¢different continuous
¡ = 0¢ so
n−1
n
0
n
that σ − σ (α) lies in C
(GS , W ). Thus, β + (−1) d σ (α) − β − (−1) d σ(α) =
(−1)n d(σ 0 − σ)(α). In particular, cnα,β depends on the choice of σ up to a continuous
coboundary with values in W .
Let (α, β) = ((−1)n−1 (ϕ − 1)b + da, db) ∈ C n−1 (ΓS , D(W )) × C n (ΓS , D(W )) be an n2n−1
n
coboundary¡ in T • (D(W )). Then,
cnα,β¡ = db+(−1)
¢
¢
¡
¢ d(σ◦(ϕ−1))(b)+(−1) d(σ(d(a))).
Note that 1 − (σ ◦ (ϕ − 1) b and σ d(a) − d σ(a) are annihilated
¡ ¡by ϕ¢− 1.¡Hence,
¢¢
n
n
cα,β is the image via the differential of (1 − (σ ◦ (ϕ − 1))(b) + (−1) σ d(a) − d σ(a)
which lies in C n−1 (GS , W ). In particular, it is a continuous coboundary.
We thus get a map
¡
¢
riW : Hi T • (D(W )) −−→ Hi (GS , W ).
By construction it is functorial in W . In case (1) we get the map ρi , in case (2) we
get ρei , in case (3) we get the map ρ†i and in case (4) we get ρe†i . By construction they
are compatible with the first commutative displayed square appearing in 3.3.
0
Analogously, using (3.4.2), one gets the claimed map riW . In case (1) we get the
map ρ0i , in case (2) we get ρe0i , in case (3) we get the map ρ0†
e0†
i and in case (4) we get ρ
i .
They are compatible with the second commutative displayed square appearing in 3.3.
Furthermore, we also have actions of GV and we claim that ri0 is GV –equivariant.
0
0
Indeed,
in C n−1 (Γ0S , D0 (W )) × C n (Γ
S , D (W¢)). Let g ∈ GV .
¡ let (α,
¢ β)¡be an n–cocycle
¢
¡
Then, g (α, β) = g(α), g(β) and cng((α,β)) = g(β) + (−1)n d σ(g(α)) . On the other
¡
¢
¡
¢
hand, g cnα,β = g(β)+(−1)n g d(σ(a)) . Note that g◦d = d◦g and (ϕ−1)(σ◦g−g◦σ) = 0
21
¡
¢
¡
¢
since ϕ is ΓV –equivariant. Thus, cng((α,β)) − g cnα,β = (−1)n d (σ ◦ g − g ◦ σ)(α) is a
coboundary in C n (GS , W ).
3.5 Proposition. The maps ρi , ρei , ρ†i , ρe†i , ρ0i , ρe0i , ρ0†
e0†
i and ρ
i are isomorphisms.
•
Proof: We use the notation of 3.4. Since T • (D) and T 0 (D0 ) are mapping cones, we get
exact sequences
(−1)n (ϕ−1)
δ̂
n
n
Hn−1 (ΓS , D(W ))−→H
(T • (D(W )) −→ Hn (ΓS , D(W )) −−−−−−→ Hn (ΓS , D(W ))
(3.5.1)
and
δ̂ 0
(−1)n (ϕ−1)
•
n
n
Hn−1 (Γ0S , D0 (W ))−→H
(T 0 (D0 (W )) −→ Hn (Γ0S , D0 (W )) −−−−−−→ Hn (Γ0S , D0 (W )).
(3.5.2)
†
e
They are compatible with respect to the natural inclusions D (W ) ⊂ D(W ) ⊂ D(W
)
†
†
†
†
†
e
e
e
e
and D (W ) ⊂ D (W ) ⊂ D(W ) (resp. D (W ) ⊂ D(W ) ⊂ D(W ) and D (W ) ⊂ D (W ) ⊂
e
D(W
)). Thanks to 7.10.6 we deduce that the horizontal arrows in the two displayed
squares of 3.3 are isomorphisms. We are then left to prove that ρei , ρe†i , ρe0i and ρe0†
i are
isomorphisms.
From the exactness of (3.4.1) and (3.4.2) we get the exact sequences
ϕ−1
δ
n
Hn−1 (GS , W ⊗ AR ) −−→
Hn (GS , W ) −−→ Hn (GS , W ⊗ AR ) −−→ Hn (GS , W ⊗ AR )
Zp
Zp
Zp
(3.5.3)
and
δ0
ϕ−1
n
0
0
Hn−1 (GS , W ⊗ A0R ) −−→
Hn (GS , W ) −−→ Hn (GS , W ⊗ AR
) −−→ Hn (GS , W ⊗ AR
)
Zp
Zp
Zp
(3.5.4).
Thanks to 7.10.6 the inflation maps
Inf n : Hn (ΓS , D(W )) −−→ Hn (GS , D(W ) ⊗ AR ) = Hn (GS , W ⊗ AR )
AS
Zp
and
Inf 0n : Hn (Γ0S , D0 (W )) −−→ Hn (GS , D0 (W ) ⊗ A0R ) = Hn (GS , W ⊗ A0R )
A0S
Zp
in cases (2) and (4)
Take a cocycle τ in C n−1 (ΓS , D(W )).
¡ of 3.4 ¢are isomorphisms.
¡ ¡
¢¢
One constructs δn Inf n−1 (τ ) as d σ Inf n−1 (τ ) . On
) = (τ, 0) in
¡ the¢ other hand, δ̂n (τn−1
n
n−1
n
n
C
(ΓS , D(W ))×C (ΓS , D(W )) and cτ,0 = (−1) d σ(τ ) . Thus, δn ◦(−1)
Inf n−1 =
W
•
n
ρn ◦ (−1)δ̂n . If (α, β) is an n–cocycle in T (D(W )), its image in H (ΓS , D(W
¡ )) ¢is the
n
n
n
class of β. The image of cα,β in H (GS , W ⊗Zp AR ) is the class of β + (−1) d σ(α) i. e.,
of β. We conclude that the exact sequences (3.5.1) and (3.5.3) are compatible via rnW
and the inflation maps Inf n and Inf n−1 i. e., the following diagram commutes (the rows
22
continue on the left and on the right):
−δ̂
n
Hn−1 (ΓS, D(W ))
−→
Hn (T •
(D(W )) −→


ρW
y(−1)n−1 Inf n−1
n y
δ
n
Hn−1 (GS , W ⊗Zp AR ) −→
Hn (GS , W )
Hn (ΓS 
, D(W ))
Inf
y n
−→ Hn (GS , W ⊗Zp AR )
(−1)n (ϕ−1)
−→
ϕ−1
−→.
An analogous argument implies that the exact sequences (3.5.2) and (3.5.4) are com0
patible via rnW and the inflation maps Inf 0n and Inf 0n−1 . The proposition follows.
3.6 Proposition. The functors ρi , ρei , ρ†i , ρe†i , ρ0i , ρe0i , ρ0†
e0†
i and ρ
i are isomorphisms of
δ–functors.
Proof: We use the notation of 3.4. We prove the proposition for ri . The proof for ri0 is
similar. Let 0 → W1 → W2 → W3 → 0 be an exact sequence of G¡S –representations.
¢
n
•
We need
to
prove
that
one
has
a
connecting
homomorphism
δ:
H
T
(D(W
))
−→
3
¡ •
¢
¡ •
¢
n+1
n
H
T (D(W1 )) , making H T (D( )) into a δ–functor, and that the diagram
¡
¢
¡ •
¢
δ
n+1
Hn T • (D(W
))
−−→
H
T
(D(W
))
3
1


 W1
W 
ri 3 y
yri
Hi (GS , W3 )
δ0
−−→
Hi+1 (GS , W1 ),
where δ 0 is the connecting homomorphisms, commutes.
We start defining δ. Let (α, β) be an n–cocycle in T n (D(W3 )) = C n−1 (ΓS , D(W3 )) ×
n
C (ΓS , D(W3 )). Due to 7.10.7 there exist a ∈ ¡C n−1 (ΓS , D(W2 )) and b¢∈ C n (ΓS , D(W2 ))
lifting α and β respectively. Then, d(a, b) = (−1)n (ϕ − 1)b + da, db is an element of
•
C n (ΓS , D(W1 )) × C n+1 (ΓS , D(W
¡ 1 )) and
¢ it is a n + 1–cocycle of T (D(W1 )). Its cohomology class is, by definition, δ (α, β) . Note that
¡
¢
¡
¢
2n+1
n+1
cn+1
=
db
+
(−1)
d
σ
◦
(ϕ
−
1)(b)
+
(−1)
d
σ(da)
.
d(a,b)
¡
¢
¡
¢
On the other hand, cn(α,β) = β + (−1)n d σ(α) . Consider cn(a,b) = b + (−1)n d σ(a)
¡
¡
¢¢
in C n (GS , W2 ⊗Zp AR ). Then, γ := cn(a,b) − σ (ϕ − 1) cn(a,b) lies in C n (GS , W2 ). Further¡
¢
more, it lifts cn(α,β) since σ (ϕ − 1)cn(α,β) = 0 because (α, β) is a cocycle. Then, the class
¡
¢
of δ 0 cn(α,β) is dγ. To compute it we may take the differential in C n (GS , W2 ⊗Zp AR )
i. e.,
¡
¡
¢¢
¡
¢
¡
¢
dcn(a,b) − dσ (ϕ − 1) cn(a,b) = db − d σ ◦ (ϕ − 1)(b) + (−1)n+1 d σ(da) .
¡
¡
¢¢
¡
¢
Here, we used that dσ (ϕ − 1) d(σ(a)) = d σ(da) . The conclusion follows.
Global Theory.
23
4 Étale cohomology and relative (ϕ, Γ)–modules.
As in the Introduction, let X denote a smooth, geometrically irreducible and proper
scheme over Spec(V ). Fix a field extension K ⊂ M ⊂ K. In this section we review
a Grothendieck topology on X, introduced by Faltings in [Fa3] and denoted XM , and
its relation to étale cohomology; see 5.8. We also define the analogue Grothendieck
b M on the formal completion X of X along its special fiber. In this section
topology, X
we study p-power torsion sheaves for these Grothendieck topologies and compare their
cohomology theories. The main result of this section is the following. Let L be an étale
local system of Z/pn Z-modules on XK , for some n ≥ 1 and let Lrig be the corresponding
rig
étale local system on the rigid space XK
attached to XK . For every pointed étale open
(U , s) of X (see 4.5) such that U = Spf(RU ) is affine and RU is a small V -algebra (see
rig
2.4), let L be the fiber of Lrig at the geometric generic point of UK
defined by s (see
e U (L),
4.8). As the notation suggests it is independent of U and s. Let DU (L), DU (L), D
e U (L) denote the respective (ϕ, Γ)–modules over RU . For each i ≥ 0 the associations
D
¡
¢
¡
¢
¡
¢
e U (L)) ,
(U , s) −→ Hi T • (DU (L)) , (U , s) −→ Hi T • (DU (L)) , (U, s) −→ Hi T • (D
¡
¢
e U (L)) are functorial and we denote by Hi,ar (L), Hi,ge (L), Hi,t,ar (L)
U −→ Hi T • (D
•
and Hi,t,ge (L) respectively the associated sheaves on the pointed étale site Xet
. We have
4.1 Theorem. There are spectral sequences
³
³
´
¡ ¢´
•
i) E2p,q = Hq Xet
, Hp,∗,ge L =⇒ Hp+q XK,et , L .
³
´
³
¡ ¢´
•
, Hp,∗,ar L =⇒ Hp+q XK,et , L .
ii) E2p,q = Hq Xet
where ∗ stands for nothing or t. Moreover, the spectral sequence i) is compatible with
the residual GV -action on all of its terms.
The proof of theorem 4.1 will take the rest of this section. In particular, see 4.8.
4.2 Some Grothendieck topologies and associated sheaves. Following [Fa3, §3, p. 214]
we define the following site:
Let X be a scheme flat over V . We denote by XM,et the small étale site of XM and
by Sh(XM,et ) the category of sheaves of abelian groups on XM,et .
¡
¢
The site XM . The objects consists of pairs U, W where
(i) U → X is étale;
(ii) W → U ⊗V M is a finite étale cover.
The maps are compatible maps of pairs and the coverings of a pair (U, W ) are families {(Uα , Wα )}α over (U, W ) such that qα Uα → U and qα Wα → W are surjective.
It is easily checked that we get a Grothendieck topology in the sense of [Ar, I.0.1].
It is noetherian if X is noetherian; see [Ar, II.5.1]. Note that one has a final object,
namely (X, XM ). Let Sh(XM ) be the category of sheaves of abelian groups in XM .
24
Let X be a formal scheme, flat over Spf(V ) and with ideal of definition generated
by p. Denote by Xet the small étale site on X and by Sh(Xet ) the category of sheaves
of abelian groups on Xet .
The sites U M,fet and UM,fet . Let U → X be an étale
¡ map
¢ of formal schemes. DeM,fet
fine U
to be the category whose objects are pairs W, L where
(i) L is a finite extension of K contained in M ;
(ii) W → U rig ⊗K L is a finite étale cover of L–rigid analytic spaces; here U rig denotes
the K–rigid analytic space associated to U.
¡
¢
Define HomU M,fet (W 0 , L0 ), (W, L) to be empty if L 6⊂ L0 and to be the set of morphisms
g: W 0 → W ⊗L L0 of L0 –rigid analytic spaces if L ⊂ L0 . The coverings of a pair (W, L)
in U M,fet are families of pairs {(Wα , Lα )}α over (W, L) such that qα Wα → W is
surjective. Define the fiber product of two pairs (W 0 , L0 ) and (W 00 , L00 ) over a pair
(W, L) to be (W 0 ×W W 00 , L000 ) with L000 equal to the composite of L0 and L00 . It is the
fiber product in the category U M,fet
Let U2 → U1 be a map of formal schemes over X . Assume that they are étale over X .
M,fet
We then have a morphism
→ U2M,fet given
³ of Grothendieck
´ topologies ρU1 ,U2 : U1
on objects by (W, L) 7→ W ×U rig U2rig , L . It is clear how to define such a map for
1
morphisms and that it sends covering families to covering families.
Let SU be the system of morphisms in U M,fet of pairs (W 0 , L0 ) → (W, L) such that
g: W 0 → W ⊗L L0 is an isomorphism. Then,
4.2.1 Lemma. The following hold:
i) the composite of two composable elements of SU is in SU ;
¡
¢
ii) given a map U2 → U1 of formal schemes étale over X , we have ρU1 ,U2 SU1 ⊂ SU2 ;
iii) the base change of an element of SU via a morphism in U M,fet is again an element
of SU ;
iii) if f : (W1 , L1 ) → (W, L) and g: (W2 , L2 ) → (W, L) are morphisms lying in SU and
if h: (W1 , L1 ) → (W2 , L2 ) is a morphism in U M,fet such that f = g ◦ h, then h is
in SU ;
Proof: Left to the reader.
Thanks to 4.2.1 the category U M,fet localized with respect to SU exists and we denote
it by UM,fet . Note that the fiber product of two pairs over a given one exists in UM,fet and
it coincides with the fiber product in U M,fet . The coverings of a pair (W, L) in UM,fet are
families of pairs {(Wα , Lα )}α over (W, L) such that qα Wα → W is surjective. By 4.2.1
the category UM,fet and the given families of covering define a Grothendieck topology. It
is a noetherian topology if the topological space X is noetherian. By abuse of notation
we will simply write W for an object (W, L) of UM,fet .
25
We recall that, given pairs (W1 , L1 ) and (W2 , L2 ) in UM,fet , one defines the set of
homomorphisms
¡
¢
¡
¢
HomUM,fet (W1 , L1 ), (W2 , L2 ) := 0 0 lim
HomU M,fet (W 0 , L0 ), (W2 , L2 ) ,
(W ,L )→(W1 ,L1 )
where the direct limit is taken over all morphisms (W 0 , L0 ) → (W1 , L1 ) in SU . Equivalently, due to 4.2.1, it is the set of classes of morphisms (W1 , L1 ) ← (W 0 , L0 ) → (W2 , L2 ),
where (W 0 , L0 ) → (W1 , L1 ) is in SU , and two such diagrams (W1 , L1 ) ← (W 0 , L0 ) →
(W2 , L2 ) and (W1 , L1 ) ← (W 00 , L00 ) → (W2 , L2 ) are equivalent if and only if there is
a third one (W1 , L1 ) ← (W 000 , L000 ) → (W2 , L2 ) mapping to the two. If (W1 , L1 ) ←
(W 0 , L0 ) → (W2 , L2 ) and (W2 , L2 ) ← (W 00 , L00 ) → (W3 , L3 ) are two homomorphisms,
the composite (W1 , L1 ) ← (W 000 , L000 ) → (W3 , L3 ) is defined by taking (W 000 , L000 ) to be
the fiber product of (W 0 , L0 ) and (W 00 , L00 ) over (W2 , L2 ).
Let U2 → U1 be a map of formal schemes étale over X . Due to 4.2.1, the map
ρU1 ,U2 : U1M,fet → U2M,fet extends to the localized categories and defines a morphism of
fet
fet
Grothendieck topologies U2,M
→ U1,M
which, by abuse of notation, we write W 7→
rig
W ×U rig U2 .
1
b M . Define X
b M to be the category of pairs (U, W) where U → X is an étale map
The site X
of formal schemes and W is an object of UM,fet . A morphism of pairs (U 0 , W 0 ) → (U, W)
0
is defined to be a morphism U 0 → U as schemes over X and a map W 0 → W ×U rig U rig
0
in UM,fet
. A family {(Uα , Wα,β ,¡Lα,β )}α → ¢(U, W, L) is a covering if {Uα }α is an étale
fet
covering of U and, for every α, Wα,β , Lα,β β is a covering of W ×U rig Uαrig in Uα,M
.
000
000
0
0
00
00
Remark that the fiber product (U , W ) of two pairs (U , W ) and (U , W ) over a
000
pair (U, W) exists putting U 000 := U 0 ×U U 00 and W 000 to be the fiber product in UM,fet
000
000
000
of W 0 ×U 0 rig U rig and W 00 ×U¡00 rig U¢ rig over W ×U rig U rig . The pair (X , (X rig , K)) is a
b M . We let Sh X
b M be the category of sheaves of abelian groups on X
bM .
final object in X
We remark that in all the categories Sh( ) introduced above AB3∗ and AB5 hold and
the representable objects provide families of generators. In particular, one has enough
injectives; see [Ar, Thm. II.1.6 & § II.1.8].
4.3 Morphisms of Grothendieck topologies. One has natural functors:
(I) uX,M : XM −→ (X ⊗V M )et with uX,M (U, W ) := W ;
¡
¢
(II.a) vX,M : Xet −→ XM given by vX,M (U ) := U, U ⊗V M ;
b M given by vbX ,M (U) := (U , (U rig , K));
(II.b) vbX ,M : Xet −→ X
Assume that X is the formal scheme associated to X i. e., that it is the formal completion
of X along its special fiber. We then have:
¡
¢
b M given by µX,M U, W := (U, (W, L)) where U is the formal
(III) µX,M : XM −→ X
scheme associated to U and if the cover W → U ⊗V M is defined over a finite exten26
sion K ⊂ L, contained in M , then W → ULrig is the pull–back via ULrig → ULrig of the
associated finite and étale cover of rigid analytic spaces W rig → ULrig .
(IV) νX : Xet → Xet given by νX (U ) = U where U is the formal scheme associated to U .
Let K ⊂ M1 ⊂ M2 ⊂ K be field extensions. Define
¡
¢
(V.a) βM1 ,M2 : XM1 → XM2 by βM1 ,M2 (U, W ) = U, W ⊗M1 M2 .
¡
¢
bM → X
b M by βbM ,M (U, W) = U, W .
(V.b) βbM ,M : X
1
2
1
2
1
2
b M , the functors µX,M and βbM ,M are well defined. More
Due to the definition of X
1
2
¡
¢
precisely, given (U, W ), the image µX,M U, W does not depend on the subfield L ⊂ M
to which W descends. Analogously, given U ∈ Xet , then βbM1 ,M2 sends the multiplicative
b M , to the multiplicative system SU used to define X
bM .
system SU , used to define X
1
2
It is clear that the above functors send covering families to covering families and
commute with fiber products. In particular, they are morphisms of topologies, see [Ar,
Def. II.4.5]. Given any such functor g, we let g∗ and g ∗ be the induced morphisms of the
associated category of sheaves for the given topologies; see [Ar, p. 41–42]. Note that the
functors above preserve final objects and commute with finite fibred products. Therefore,
the induced functor g ∗ on the categories of sheaves is exact by [Ar, Thm. II.4.14].
b M , then µX,M,∗ (F) is the sheaf on XM
We work out an example.
If¡ F is ¢¢
a sheaf on X
¡
defined by (U, W ) 7→ F µX,M U, W . If F is a sheaf on XM , then µ∗X,M (F) is the sheaf
associated to the separated presheaf defined by (U, W) 7→ lim(U 0 ,W 0 ) F(U 0 , W 0 ) where
the limit
direct limit taken over all pairs (U 0 , W 0 ) in XM and all maps (U, W) →
¡ 0 is the
¢
bM .
µX,M U , W 0 in X
Notation: If F is a sheaf on Xet or is in Sh(Xet )N (see section 5 for the definition),
∗,N
∗
(F).
we write F form for νX
(F), respectively for νX
If L is a locally constant sheaf on XM,et , by abuse of notation we denote L its push
forward uX,M,∗ (L) ∈ Sh(XM ). It is a locally constant sheaf on XM .
If F is a sheaf on XM or is in Sh(XM )N , we denote by F rig the pull–back µ∗X,M (F).
Note that if F ∈ Sh(XM ) is locally constant, then F rig is also a locally constant sheaf
bM .
on X
4.4 Stalks [Fa3, p. 214]. Let Kx be a finite field extension of K contained in K and
denote by Vx its valuation ring.
Fix a map x: Spec(Vx ) → X of V –schemes and denote by x̄: Spec(V ) → X the
sh
composite of x with the natural map Spec(V ) → Spec(Vx ). Let OX,x
be the the direct
limit limi Ri taken over all pairs {(Ri , fi )}i where
i ) is étale over X and fi : Ri →
¡ Spec(R
¢
V defines a¡ point¢over x̄. Let F be a sheaf in Sh Xet . The stalk
Fx of ¢F at x is defined
¡
sh
as Fx = F OX,x
by which we mean the direct limit limi F Spec(Ri ) .
0
0
0
Define OX,x,M as the direct limit limi,j Ri,j
over the pairs {(Ri,j
, Ri,j
→ V )}i,j where
0
0
(1) Ri,j is an integral Ri –algebra and is normal as a ring, (2) Ri,j ⊗V K is a finite and
0
étale extension of Ri ⊗V M , (3) the composite Ri ⊗V M → Ri,j
⊗V K → K is r ⊗ ` 7→
27
¡
¢
¡
¢
,
we
then
write
F
or
equivalently
F
fi (r) · `. If F is a sheaf in Sh
X
O
⊗
K
x
M
X,x,M
V
¡
¢
0
for the direct limit limi,j F Spec(Ri ), Spec(Ri,j
⊗V K) . We call it the stalk of F at x.
sh
Let Gx,M be the Galois group of OX,x,M ⊗V K over OX,x
⊗V M . Then, Fx is endowed
with an action of Gx,M .
Let V ⊂ Vx̂ be a finite extension of DVRs. Let x̂: Spf(Vx̂ ) → X be a map of V –
b̄ Spf(Vb ) → X be the composite of x̂ with Spf(Vb ) → Spf(Vx ).
formal schemes and let x:
sh
Define OX
,x̂ be the direct limit limi Si over all pairs {(Si , gi )}i∈I such that Si is p–
adically complete and separated V –algebra, Spf(Si ) → X is an étale map of formal
¡ ¢
b̄ If F is a sheaf in Sh Xet , the
schemes and gi : Si → Vb defines a formal point over x.
¡ sh ¢
¡
¢
stalk Fx̂ of F at x̂ is defined to be the direct limit F OX ,x̂ := limi∈I F Spf(Si ) .
Write O
for the direct limit lim S 0 over all triples {(S 0 , S 0 → Vb , L )}
X ,x̂,M
i,j
i,j
i,j
0
Si,j
i,j
i,j
i,j
where (1) Li,j is a finite extension of K contained in M , (2)
is an integral ex0
tension of Si and is normal as a ring, (3) Si,j ⊗V K is a finite and étale Si ⊗V Li,j –
b is a ⊗ ` 7→ g (a) · `. Given
0
algebra, (4) the composite Si ⊗V Li,j → Si,j
⊗V K → K
i
¡
¢
¡
¢
b
a sheaf F in¡ Sh XM , denote by Fx̂ , or equivalently
F
O
⊗
K
, the direct
X
,x̂,M
V
¢
0
limit limi,j F Spf(Si ), (Spm(Si,j ⊗V K), Li,j ) . We call it the stalk of F at x̂.
sh
Denote by Gx̂,M the Galois group of OX ,x̂,M ⊗V K over OX
,x̂ · M . Then, Fx̂ is
endowed with an action of Gx̂,M .
4.4.1 Lemma. Let k(x) (resp. k) be the residue field of Vx (resp. V ) and denote
by xk : Spec(k(x)) → Xk (resp. x̄k : Spec(k) → Xk ) the points induced by x or x̂ (resp. x̄
b̄ on the special fiber Xk of X or of X . Then,
or x)
sh
sh
i. OX,x
coincides with the strict henselization of OX,xk and OX
,x̂ coincides with the
strict formal henselization of OX ,xk .
Assume that X is the formal scheme associated to X and that x̂ is the map of formal
schemes defined by x. Then,
¡ sh
¢
¡ sh
¢
ii. OX,x
, (p) and OX
,x̂ , (p) are noetherian henselian pairs and the natural map
sh
sh
OX,x
→ OX
,x̂ is an isomorphism after taking p–adic completions;
sh
iii. the base change functor from the category of finite extensions of OX,x
, étale after
sh
inverting p, to the category of finite extensions of OX ,x̂ , étale after inverting p, is an
equivalence of categories;
sh
sh
sh
sh
iv. the maps OX,x
/pOX,x
→ OX
,x̂ /pOX ,x̂ and O X,x,M /pO X,x,M → O X ,x̂,M /pO X ,x̂,M
are isomorphisms.
1
v. Frobenius on OX ,x̂,M /pOX ,x̂,M is surjective with kernel p p OX ,x̂,M /pOX ,x̂,M .
Proof: (i) The strict henselization of OX,xk is defined as the direct limit limj Tj over
all pairs {(Tj , tj )}j where Spec(Tj ) is étale over X and tj : Tj → k is a point over x̄k .
sh
sh
In particular, we get a map OX,x
= limi Ri → OX,x
= limj Tj by associating to a
¢
¡
¢ k
¡
pair Ri , fi : Ri → V the pair Ri , Ri → V → k . To conclude that such a map is
an isomorphism it suffices to show that for any pair (Tj , tj ) there is a unique map of
28
V –algebras Tj → V lifting tj and inducing the point x̄. The base change of Tj via x̄
defines an étale V –algebra Aj and tj induces a map of V –algebras Aj → k. By étalness
of A the latter lifts uniquely to a map of V –algebras A → Vb which, since T is of finite
j
i
j
type over V , factors via V .
The strict formal henselization of OX ,Xk is defined as the direct limit limj Qj over all
pairs {Qj , qj }j where Qj is a p-adically complete and separated V –algebra, Spf(Qj ) →
X is an étale map of formal schemes and qj : Qj → k is a point over x̄k . The proof
sh
that OX
,x̂ is the strict formal henselization of OX ,Xk is similar to the first part of the
proof and left to the reader.
sh
sh
(resp. OX
(ii) It follows from (i) that OX,x
,x̂ ) is a local ring with residue field k and
maximal ideal mx (resp. mx̂ ) generated by the maximal ideal of OX,xk . In particular,
sh
sh
sh
sh
and grmx̂ OX
the graded rings grmx OX,x
,x̂ are noetherian so that OX,x and OX ,x̂ are
noetherian.
sh
We claim that (OX
,x̂ , mx̂ ) is a henselian pair; see [El, §0.1]. This amounts to prove that
sh
sh
sh
any étale map OX ,x̂ → B, such that k = OX
,x̂ /mx̂ OX ,x̂ → B/mx B is an isomorphism,
admits a section. Note that there exists i and an étale extension Si → A such that B
sh
∼
is obtained by base change of A via Si → OX
,x̂ . Via a: A/mx A → B/mx B = k the
b a), where A
b is the p–adic completion of A, appears in the inductive system
pair (A,
used to define the strict formal henselization of OX ,xk so that, thanks to (i), we get a
sh
sh
sh
natural map A → OX
,x̂ and, hence base–changing, a map of OX ,x̂ –algebras B → OX ,x̂ .
sh
Analogously, one proves that (OX,x
, mx ) is a henselian pair.
¡ sh
¢
¡ sh
¢
Note that p is contained in mx , so that OX,x
, (p) and OX
,x̂ , (p) are henselian
sh
pairs. Let OX
⊗ Fp ,xk be the strict henselization of the local ring of X ⊗V V /pV at xk . By
sh
sh
sh
sh
sh
construction we have natural injective maps OX,x
/pOX,x
→ OX
,x̂ /pOX ,x̂ → OX ⊗ Fp ,xk .
We claim that such maps are isomorphisms. It suffices to show that the composite is
surjective. Using (i) this is equivalent to prove that the map from the strict henselization
of OX,xk to the strict henselization of OX ⊗ Fp ,xk is surjective. This amounts to show that
given an étale map f : Spec(R) → OX ⊗ Fp ,xk , there exists an étale map g: Spec(S) →
OX,xk reducing to f modulo p. By the Jacobian criterion of étalness we have R =
d
OX ⊗ Fp ,xk [T1 , . . . , Td ]/(h1 , . . . , hd ) with det (∂hi /∂Tj )i,j=1 invertible in R. Then, S :=
h
i
−1
OX,xk [T1 , . . . , Td ]/(q1 , . . . , qd ) det (∂qi /∂Tj )
, with qi lifting hi , is an étale OX,xk –
sh
sh
algebra and lifts R as wanted. Since p is not a zero divisor in OX,x
and in OX
,x̂ , we
sh
sh
conclude that the graded rings grp OX,x and in grp OX ,x̂ are isomorphic, concluding the
proof of (ii).
d
sh (resp. O
sh ) be the p–adic completion of O sh (resp. O sh ). Thanks
(iii) Let Od
X,x
X,x
X ,x̂
X ,x̂
sh
to [El, Thm. 5] one knows that the category of finite extensions of OX,x
, étale after
sh
inverting p (resp. the category of finite extensions of OX ,x̂ , étale after inverting p), is
d
sh = O
sh , étale after inverting p.
equivalent to the category of finite extensions of Od
X,x
X ,x̂
The claim follows from (ii).
(iv) The first claim follows from (ii). The second follows from the first and (iii).
29
(v) Note that pα ∈ OX ,x̂,M for every α ∈ Q>0 . It follows from [Fa3, §3, Lemma 5]
that Frobenius is surjective on OX ,x̂,M /pα OX ,x̂,M for every 0 < α < 1. Let a ∈ OX ,x̂,M .
1
1
Write a = bp + p p c with b and c ∈ OX ,x̂,M . Write c = dp + p1− p e with e ∈ OX ,x̂,M .
1
Then, a ≡ (b + p p2 d)p modulo pOX ,x̂,M .
µ
p
Let a ∈ OX ,x̂,M be such that a ∈ pOX ,x̂,M . Then,
Since the latter is a normal ring, this implies that
a
1
pp
ap
p
=
¶p
a
1
pp
lies in OX ,x̂,M .
∈ OX ,x̂,M as claimed.
4.4.2 Proposition. The notation is as above (in (2), (3) & (5) below we also assume
that X is the formal scheme associated to X and that x̂ is the map of formal schemes
associated to x: Spec(Vx ) → X).
1) Suppose that X (resp. X ) is locally (topologically) of finite type over V and that
every closed point of X maps to the closed point of Spec(V ). Then, a sequence of
b M ) is exact if and only
sheaves F → G → H on Xet (resp. XM , resp. Xet , resp. X
if for every point x of X (resp. x̂ of X ) as above the induced sequence of stalks
Fx → Gx → Hx (resp. Fx̂ → Gx̂ → Hx̂ ) is exact;
2) let F be in Sh(Xet ). Then, ν ∗ (F)x̂ ∼
= Fx ;
X
3) if F is in Sh(XM ), then,
µ∗X,M (F)x̂
∼
= Fx ;
∗
∗
4) fix field extensions K ⊂ M1 ⊂ M2 ⊂ K. Then, βM
(resp. βbM
) of a flasque
1 ,M2
¡
¢ 1 ,M2
¡
¢
b
sheaf is flasque. Furthermore, if F is in Sh XM1 (resp. F is in Sh XM1 ), then one
has natural identifications:
∗
∗
a) βM
(F)x = Fx (resp. βbM
(F)x̂ = Fx̂ );
1 ,M2
1 ,M2
¡
¢G
∗
b) if M1 ⊂ M2 is Galois with group G, then H0 (XM1 , F) = H0 XM2 , βM
(F)
1 ,M2
´G
³
∗
0 b
0 b
b
(resp. H (XM1 , F) = H XM2 , βM1 ,M2 (F) ).
Assume that Kx is contained in M . Then,
5) we have a natural isomorphisms Gx̂,M ∼
= Gx,M and, if F is in XM , the isomorphism
∗
∼
µX,M (F)x̂ = Fx is compatible with the actions of Gx̂,M and Gx,M
¡
¢
6) let F be a sheaf in XM . Then, (Rq vX,M,∗ (F))x ∼
= Hq Gx,M , Fx ;
¡
¢
q
b M . Then, (Rq vbX ,M,∗ (F)) ∼
7) let F be a sheaf in X
H
G
,
F
.
=
x̂,M
x̂
x̂
Proof: (1) In each case it suffices to prove that a sheaf is trivial if and only if all its
stalks are.
We give a proof for a sheaf on XM and leave the other cases to the reader. Let F ∈
Sh(XM ) such that for every point x of X, defined over a finite extension of K, we
have Fx = 0. Let (U, W ) ∈ XM and let α ∈ F(U, W ). Then, for every x: Spec(Vx ) → U
and every point y: Spec(Ky ) → W over¡ x ⊗V K,
¢ which exists since W → UM is finite,
there exists (Ux , Wy ) ∈ XM and a map Ux , Wy → (U, W ) such¡that (1)¢x ⊗V V factors
via Ux , (2) y ⊗K K factors via Wy and (3) the image of α in F Ux , Wy is 0.
30
Due to the assumption, the set of points x (resp. y) as above are dense in U (resp. W )
so that there exist points xi and yi such that qi (Uxi , Wyi ) → (U, W ) is
a covering
¢
Q ¡
of (U, W ) in XM . Since F is a sheaf, the homomorphism F(U, W ) → i F Uxi , Wyi is
injective. Hence, α = 0 to start with.
(2) Since any sheaf is the direct limit of representable sheaves and direct limits
∗
commute with νX
and with taking stalks, we may assume that F is represented by an
∗
étale X–scheme Y → X. In particular, νX
(F) is represented by the formal scheme Y
associated to Y . Let Yx (resp. Yx̂ ) be the pull back of Y (resp. Y) to Spec(OX,x )
(resp. Spec(OX ,x̂ )). We then have the following diagram
Fx
k
¡
¢
sh
HomOX,x OYx , OX,x
∗
(F)x̂
νX
³k
sh
−→ HomOX ,x̂ OYx̂ , OX
,x̂
´
¡
¢
−→ Homk OYx ⊗V k, k .
By 4.4.1(i) these maps are bijective as claimed.
∗
(F) is the sheaf in XM2 associated to the presheaf
(4) If F is in Sh(XM1 ), then βM
1 ,M2
−1
βM1 ,M2 (F) defined by (U, W ) 7→ lim F(U 0 , W 0 ) where the limit is taken over all the
pairs (U 0 , W 0 ) in XM1 and all the maps (U, W ) → (U 0 , W 0 ⊗M1 M2 ). This is equivalent
to take the direct limit over all pairs (U, W 0 ) in XM1 and over all map (U, W ) → (U, W 0 )
as UM1 –schemes. If M1 ⊂ M2 is finite, there exists an initial pair, namely (U, W ) itself, viewed in XM1 via¡ the finite
and étale map W → U ⊗V M2 → U ⊗V M1 , so that
¢
−1
βM1 ,M2 (F)(U, W ) = F U, W . In general, there exists a finite extension M1 ⊂ L contained in M2 and a pair (U, WL ) in XL such that W = WL ⊗L M2 . Since any morphism of
−1
pairs
to a finite extension of M1 , we ¡conclude that¢βM
(F)(U, W ) =
1 ,M2
¡ in XM2 descends
¢
0
F U, WL ⊗L M2 , defined as the direct limit lim¡L0 F U, WL ⊗¢L L taken over all finite
extensions L ⊂ L0 contained in M2 , considering U, WL ⊗L L0 in XM1 via the finite and
−1
étale map WL ⊗L L0 → U ⊗V L → U ⊗V K. In any case, we conclude that βM
(F) is
1 ,M2
−1
∗
∗
already a sheaf i. e., βM1 ,M2 (F) = βM1 ,M2 (F). Furthermore, βM1 ,M2 preserves flasque
objects and satisfies (a).
For (b), recall that XM1 and XM2 have final objects so that global sections can be
computed using the final objects. Since XM2 → XM1 is a limit of finite and étale covers
with Galois group G and F is a sheaf on XM1 , one has F(X, XM1 ) = F(X, XM2 )G .
¡
¢G
∗
Then, H0 (XM1 , F) = F(X, XM1 ) = F(X, XM2 )G = H0 XM2 , βM
(F)
and (b)
1 ,M2
follows.
∗
A similar argument works for βbM
. Details are left to the reader.
1 ,M2
(3)&(5) The first claim in (5) follows from 4.4.1(iii). To get the second claim and (3),
one argues as in (2) reducing to the case of a sheaf represented by a pair (U, W ), so
that µ∗X,M (U, W ) = (U, W, L), and using 4.4.1(iii).
(6) Consider the functor Sh(XM ) → (Gx,M -Modules),¡ associating to a sheaf F its
¢
0
stalk Fx . It is an exact functor. Recall that F
=
lim
F
Spec(R
),
Spec(R
⊗
K)
.
x
i,j
i
V
i,j
¡
¢
∗
Thus, the continuous Galois cohomology H Gx,M , Fx ¡ is the direct limit over i and
¢ j
0
of the Chěch cohomology of F relative to the covering Spec(Ri ), Spec(Ri,j ⊗V K) . In
31
¡
¢
q
= 0 for q ≥ 1.
particular,
if
F
is
injective,
it
is
flasque
and
H
G
,
F
x,M
x
© q
ª
© q¡
¢ª
Both (R vX ,M,∗ (F))x q and H Gx,M , Fx q are δ–functors from Sh(XM ) to the
category of abelian groups. Also (Rq vX ,M,∗ (F))x is zero for q ≥ 1 and F injective.
For q = 0 we have
¡ 0
¢
¡
¢
¡
¢Gx,M
R vX ,M,∗ (F) x = lim F Spec(Ri ), Spec(Ri ⊗ K) = F OX,x,M ⊗ K
.
i
V
V
This proves the claim.
(7) The proof is similar to the proof of (6) and left to the reader.
4.4.3 Lemma. Assume that X is the formal scheme associated to X, that X is locally of finite type over V and that every closed point of X maps to the closed point
of Spec(V ). We then have the following equivalences of δ–functors :
¡ ∗
¢
¡
¢ ¡
¢
∗
i. Rq νX
◦ vX,M,∗ = νX
◦ Rq vX,M,∗ and Rq vbX ,M,∗ ◦ µ∗X,M = Rq vbX ,M,∗ ◦ µ∗X,M ;
¢
∼ ¡ q
∗
ii. νX
◦ Rq vX,M,∗ −→
R vbX ,M,∗ ◦ µ∗X,M .
∗
Proof: (i) Since νX
and µ∗X,M are exact and vX,M,∗ and vbX ,M,∗ are left exact, the derived
∗
◦ vX,M,∗ and vbX ,M,∗ ◦ µ∗X,M exist. By 4.4.2 we have
functors of νX
∗
(Rq vX,M,∗ (F))x̂ ∼
νX
= (Rq vX,M,∗ (uX,M,∗ (F)))x ∼
= Hq (Gx,M , Fx )
and
¡
¢
¡
¢
Rq vbX ,M,∗ µ∗X,M (F) x̂ ∼
= Hq Gx̂,M , µ∗X,M (F)x .
¡ ∗
¢
∗
q
q
This implies that if F is¡ injective, νX
(R
v
(F))
=
0
and
R
v
b
µ
(F)
X,M,∗
X
,M,∗
X,M
¢
¡
¢ =
q
∗
∗
p
q
∗
0 for q ≥ 1. Hence, R νX ◦ vX,M,∗ = νX ◦ R vX,M,∗ (resp. R vbX ,M,∗ ◦ µX,M =
∗
Rq vbX ,M,∗ ◦ µ∗X,M ). Indeed, they are both δ–functors since νX
(resp. µ∗X,M ) is exact,
they are both erasable and they coincide for q = 0.
¡
¢
∗
(ii) We construct a map γF : νX
(vX,M,∗ (F)) −→ vbX ,M,∗ µ∗X,M (F) functorial in F.
∗
The sheaf νX
(vX,M,∗ (F)) is the sheafification of the presheaf F1 which associates to
an object U in Xet the direct limit lim F(U, UK ) taken over all U ∈ Xet and all maps
from U¡to the formal
¢ ∗ scheme associated to U . On the other hand, the presheaf F2 :=
∗
vbX ,M,∗ µX,M (F) (µX,M (F) is taken as presheaf) associates to U ∈ Xet the direct limit
lim F(U, W ) over all (U, W ) in XM and all maps from U to the formal scheme associated
to U and from U rig to the rigid analytic space defined by W rig ×U rig U rig . We thus get
a morphism at the level of presheaves F1 → F2 . Passing to the associated sheaves we
get the claimed map.
¡ ∗
¢
¡
¢
The map γF induces Rq γF : Rq νX
◦vX,M,∗ (F) −→ Rq vbX ,M,∗ ◦µ∗X,M (F). Using (i),
we get a natural transformation of δ–functors as claimed in (ii). We are left to prove that
it is an isomorphism. This can be checked on stalks and, as explained ¡in the proof of (i), ¢it
amounts to prove that for any sheaf F one has Hq (Gx,M , Fx ) ∼
= Hq Gx̂,M , µ∗X,M (F)x̂ .
The conclusion follows since µ∗X,M (F)x̂ ∼
= Fx and Gx,M ∼
= Gx̂,M thanks to 4.4.2.
For later purposes we introduce the following variants of the topologies introduced
above:
32
4.5 Pointed étale sites. Let K be a separable closure of the field of fractions of X ⊗V k.
Let WK be a Cohen ring for K i. e., a complete DVR such that WK /pWK ∼
= K.
Let CX be the p–adic completion of an algebraic closure of the fraction field of WK
containing K.
•
•
The site Xet
. Denote by Xet
the following Grothendieck topology. As a category it
consists of pairs¡(U, s) where U
¢ → X is an étale morphism of formal schemes and s is
a morphism Spf WK ⊗W(k) V → U of V –formal schemes inducing a geometric generic
point of Uk . A map of pairs (U, s) → (U 0 , s0 ) is a map of X –schemes U → U 0 such
that the composite with s is s0 . A covering qi∈I (Ui , si ) → (U, s) is defined to a map of
pairs (Ui , si ) → (U, s) for every i such that qi Ui → U is an étale covering.
sh
Fix x̂: Spf(Vx ) → X as in 4.4 and choose a homomorphism ηx̂ : OX
,x̂ → WK ⊗W(k) V
•
inducing¡the geometric
generic
point
K
on
X
⊗
k.
Given
a
sheaf
F
on Xet
define Fx̂
V
¢
sh
sh
to be F OX ,x̂ as in 4.4 i. e., if OX ,x̂ is the direct limit limi Si as in loc.
¡ cit.
¢and
sh
if si : Spf(W
OX ,x̂ :=
¡ K ⊗W(k) V¢ ) → Spf(Si ) is defined composing with ηx̂ , then F
•
limi∈I F (Spf(Si ), si ) . One then proves that a sequence of sheaves on Xet is exact if
and only if the associated sequence of stalks is exact for every x̂: Spf(Vx ) → X .
b • . Define X
b • to be the following Grothendieck topology. Its objects
The site X
¡M
¢M
b M and (U, s) is an obare the pairs (U , s), W, L where (U , W, L) is an object of X
¡
¢
¡
¢
•
b • is a morphism
. A morphism (U, s), W, L → (U 0 , s0 ), W 0 , L0 in X
ject of Xet
M
b • such that the induced map U → U 0 arises from a
(U , W, L) → (U 0 , W 0 , L0 ) in X
M
¡
¢
¡
¢
•
map (U, s) → (U 0 , s0 ) in X¡et
. A covering ¢qi∈I ¡(Ui , si ), Wi , Li ¢ → (U 0 , s0 ), W 0 , L0 is
the datum of morphisms (Ui , si ), Wi , Li → (U 0 , s0 ), W 0 , L0 for every i ∈ I such
bM .
that qi (Ui , Wi , Li ) → (U 0 W 0 , L0 ) is a covering in X
sh
Fix x̂: Spf(Vx ) → X as in 4.4. Choose a map ηx̂ : OX
,x̂ → WK ⊗W(k) V inducb•
ing
¡ the geometric
¢ generic point K on X ⊗V k. Given a sheaf F ¡on XM put F¢x̂ :=
F OX ,x̂,M ⊗V K where, using the notation of 4.4, we write F OX ,x̂,M ⊗V K :=
¢
¡
0
b • is exact
limi,j F (Spf(Si ), si ), Spm(Si,j
⊗V K), Li,j . Then, a sequence of sheaves on X
M
if and only if the associated sequence of stalks is exact for every x̂: Spf(Vx ) → X .
We have functors
¡
¢
•
i) a: Xet
→ Xet given by a U, s = U;
¡
¢
b• → X
b M given by b (U, s), W, L = (U , W, L);
(ii) b: X
M
¡
¢
•
b • given by vbX ,M (U, s) = (U, s), U rig , K .
(iii) vbX ,M : Xet
→X
M
As in 4.4.2(7) one proves that for every point x̂: Spf(Vx ) → X ,
¡ i
¢
¡
¢
sh
R vbX ,M,∗ (F) x̂ ∼
Gx̂,M := Gal OX ,x̂,M ⊗ K/OX
= Hi (Gx̂,M , Fx̂ ) ,
,x̂ · M .
V
Then:
•
b • admit final objects. Furthermore, a (resp. b,
4.5.1 Lemma. The categories Xet
and X
M
resp. vbX ,M ) send final object to final object and a (resp. b) are surjective.
33
Proof: Since X is formally smooth over Spf(V ), for every étale map U → X also U
is formally smooth over V . Thus, if U irreducible and
¡ if we fix a ¢geometric generic
point sU : Spec(K) → Uk , there exists a map sU : Spf WK ⊗W(k) V → X lifting sU .
This proves that a, and hence b, are surjective.
¡
•
We claim that (X , sX ) is a final object in Xet
. This implies that (X , sX ), X rig , K)
b • and that a, b and vbX ,M preserve final objects. To prove the
is a final object in X
M
¡
¢
claim it suffices to show that given two maps s, s0 : Spf WK ⊗W(k) V → X as V –
formal schemes, inducing the generic point of Xk , there exists an automorphism ρ of
WK ⊗W(k) V (as V –algebra) such that s0 = s ◦ ρ. Let R be the p–adic completion of the
localization of X at its generic point. It is a DVR. Write Z for WK ⊗W(k) V . Then s
and s0 induce maps f and f 0 : R → Z such that, considering f or f 0 , the maximal ideal
of Z is generated by the maximal ideal of R and the residue field of Z is a separable
closure of the residue field of R. By uniqueness of étale extensions, there exists an
automorphism h of Z, as V –algebra, such that g = h ◦ f .
b M ). We have a natural isomorphism
4.5.2 Corollary.
Let F be¢ a sheaf on Xet (resp. X
¡
¡ •
¢
•
b , b∗ (F) ∼
b M , F)).
of δ–functors Hi Xet
, a∗ (F) ∼
= Hi (Xet , F) (resp. Hi X
= Hi (X
M
Proof: We have functors
¡ ¢
¡ •¢
a∗ : Sh Xet −−→ Sh Xet
,
¡
¢
¡ • ¢
b M −−→ Sh X
b ,
b∗ : Sh X
M
which send flasque objects ¡to flasque objects.
Since a and b are
a∗ and
¢
¡ surjective ¢by 4.5.1,
0
0 b•
0 b
•
∼
, a∗ (F) ∼
H
(X
,
F)
and
H
X
,
b
(F)
H
(
X
b∗ are also exact. Since H0 Xet
=
=
et
M , F),
M ∗
the lemma follows.
This allows to work with pointed sites, which are better suited for Galois cohomology
rig
•
computations. Indeed, let (U, s) ∈ Xet
with U connected and let η s : Spm(CX ) → UK
be
rig
the composite of sK and the morphism WK ⊗W(k) K → CX chosen in 4.5. It induces
a map RU ⊂ CX . Let RU ⊂ RU be the union of all finite and normal RU –subalgebras
rig
of C¡X , which are étale after
π1alg
¢ inverting p. Define
¡
¢ (UM , η s ), or simply π1 (UM ), to be
Gal RU ⊗V K/RU ⊗V M and let Repdisc π1 (UM ) be the category of abelian groups,
with the discrete topology, endowed with a continuous action of π1 (UM ).
4.5.3 Lemma. The category UM,fet is equivalent, as Grothendieck
to the
³
htopology,
i
´
1
category of finite sets with continuous action of π1 (UM ) := Gal RU p /RU ⊗V M .
In particular,
1) the functor
¡
¢
Sh (UM,fet ) −−→ Repdisc π1 (UM ) ,
F 7→ F(RU ⊗ K),
V
with F(RU ⊗V K) := lim(U ,W) F(U, W) where the direct limit is over all elements
of UM,fet , defines an equivalence of categories;
34
¡
¢
¡
¢
i
2) for F ∈ Sh (UM,fet ) we have Hi UM,fet
,
F
=
H
π
(U
),
F(R
⊗
K)
, where the
1
M
U
V
¡
¢
π1 (UM )
latter is the derived functor of Repdisc π1 (UM ) 3 A 7→ A
(the Galois invariants
of A);
Proof: The first claim follows noting that UM,fet is the category of finite and étale covers
of RU ⊗V M . By Grothendieck’s reformulation of Galois theory the latter is equivalent
to the category of finite sets with continuous action of π1 (UM ).
¡
¢
An inverse of the functor given in (1) is given as follows. Let G ∈ Repdisc π1 (UM ) .
b • with Wi = Spm(Si ) and Si ⊗L M a domain and fix an
Let (U, qi (Wi , Li )) ∈ X
i
M
embedding fi : Si ⊗Li M → RU ⊗V K. Let
H
:=
Gal(R
⊗
i ⊗L M ) ⊂ π1 (UM )
¡ i
¢ U QV K/S
Hi
which is independent of fi . Then, define G U, qi (Wi , Li ) = i G . One verifies that G
is a sheaf and that the two functors are the inverse one of the other.
For claims (2) we note that the cohomology groups appearing are universal δ–functors
coinciding for i = 0.
We next show that the sites introduced above are very useful in order to compute
étale cohomology:
4.6 Proposition. (Faltings) Assume that X is locally of finite type over V and that
every closed point of X maps to the closed point of Spec(V ). Let L be a finite locally
constant étale sheaf on XM annihilated by ps . For every i the map Hi (XM , L) −→
Hi (XM,et , L), induced by pull–back along uX,M , is an isomorphism.
Proof: [Fa3, Rmk. p. 242] Put GM := Gal(K/M ). We have a spectral sequence
¡
¢
¡
¢
Hp GM , Hq (XK,et , L) =⇒ Hp+q XM,et , L
and, thanks to 4.4.2(4.b), a spectral sequence
¡
¢
¡
¢
Hp GM , Hq (XK , L) =⇒ Hp+q XM , L .
Hence, it suffices to prove the proposition for M = K. Let zX,K : Xet → XK,et be the
map U → U ⊗V K. We can factor it via the maps vX,K : Xet → XK and uX,K : XK →
XK,et . This induces a spectral sequence Rp vX,K,∗ ◦ Rq uX,K,∗ (L) =⇒ Rp+q zX,K,∗ (L)
¡
¢
which provides with a map Rp vX,K,∗ uX,K,∗ (L) → Rp zX,K,∗ (L). As usual we write L
for uX,K,∗ (L). Furthermore, since L is a finite locally constant étale sheaf on XK ,
¡
¢
for every point x: Spec(Vx ) → X the stalk uX,K,∗ (L) x is isomorphic to LxK . One
¡ sh
¢
K
is K(π,¡1). This implies
knows¡ from [Fa1, Cor.
II.2.2]
that
Spec
O
⊗
V
X,x
¢
¡
¢
¢ that the
q
q
q
stalk R zX,K,∗ (L) x is H Gx,K , Lx . By 4.4.2 also the stalk R vX,K,∗ (L) x coincides
¡
¢
with Hq Gx,K , Lx . Hence, Rq uX,K,∗ (L) ∼
= Rq zX,K,∗ (L). Using the spectral sequences
³
´
¢
¡
Hp Xet , Rq zX,K,∗ (L) =⇒ Hp+q XK,et , L
and
¡
¢
¡
¢
Hp Xet , Rq vX,K,∗ (L) =⇒ Hp+q XK , L
35
the proposition follows.
4.7 Comparison between algebraic cohomology and formal cohomology. Assume that X
is locally of finite type over V and that every closed point of X maps to the closed point
∗
of Spec(V ). Let X be the associated formal scheme. Since νX
is an exact functor, given
•
∗
∗
•
an injective resolution I of F, then 0 → νX (F) → νX (I ) is exact so that given an
∗
injective resolution J • of F form = νX
(F) we can extend the identity map on F to a
∗
•
•
morphism of complexes νX (I ) → J . Since νX sends the final object X of Xet to the
∗
final object X of Xet , one has a natural map I • (X) → νX
(I • )(X ). Then,
4.7.1 Definition. One has natural maps of δ–functors
¡
¢
¡
¢
ρqX,X (F): Hq Xet , F → Hq Xet , F form ,
ρq
bM
XM , X
¡
¢
¡
¢
b M , F rig
(F): Hq XM , F → Hq X
Note that one has spectral sequences
¡
¢
Hq Xet , Rp vX,M,∗ (F) =⇒ Hp+q (XM , F),
(4.7.1)
and
¡
¢
¡
¡
¢¢
¡
¢
∗ p
b M , F rig .
Hq Xet , νX
R vX ,M,∗ (F) = Hq Xet , Rp vbX ,M,∗ F rig =⇒ Hp+q X
(4.7.2)
where the equality on the left hand side is due to 4.4.3.
4.7.2 Proposition. The following hold:
a. If F in Sh(Xet ) is torsion, the map ρqX,X (F) is an isomorphism.
b. the spectral sequences (4.7.1) and (4.7.2) are compatible via ρqX,X and ρp+q
bM
XM ,X
c. if F is a torsion sheaf on XM , the map ρq
bM
XM , X
;
(F) is an isomorphism.
Proof: (a) Let Xk := X ⊗V k and denote by ι: Xet → Xk,et and b
ι: Xet → Xk,et the
functors induced by the closed immersions Xk ⊂ X and Xk ⊂ X respectively. In fact,
b
ι is an equivalence, since the étale sites of X and of Xk coincide,
ι ◦ νX = ι.
¡ formand
¢ b
∗
∗
For¡ any sheaf
ι F
. We then have
¢ F qon
¡ Xet denote
¢ Fk q:=
¡ ι (F) or,
¢ equivalently, b
q
form ∼
H Xet , F → H Xet , F
= H Xk,et , Fk where the first map is ρX,X (F)q . The
composite is defined by restriction and is an isomorphism if F is a torsion sheaf due
to [Ga, Cor. 1] and the fact that X is proper over V . The conclusion follows.
(b) Left to the reader.
(c) The left hand side of the spectral sequences (4.7.1) and (4.7.2) are isomorphic
by (a) since Rp vX,M,∗ sends a torsion sheaf to a torsion sheaf. The conclusion follows
from (b).
36
4.7.3 Corollary. Let L be a locally constant sheaf on XM annihilated by ps . Then,
the two sides of the Leray spectral sequences
³
³
´
¡
¢´
b M , Lrig
Hj Xet , Ri vbX ,M,∗ Lrig =⇒ Hi+j X
and
³
¡ ¢´
Hj Xet , Ri vX,M,∗ L =⇒ Hi+j (XM , L) ,
b M → Xet and vX,M : XM → Xet , are
obtained from the morphisms of topoi vbX ,M : X
naturally isomorphic.
Proof: The statements follow from 4.7.2.
4.8 Theorem 4.1. Let L be an étale local system of Z/pn Z-modules on XK , for some
rig
n ≥ 1 and let Lrig be the corresponding étale local system on the rigid space XK
•
attached
to X¢K . Let (U, s) ∈ Xet
with U = Spf(RU ) small affine; see 2.4. Put LU :=
¡
rig
L RU ⊗V K ; the notation is as in 4.5.3. It has a continuous action of the algerig
braic fundamental group π1alg (UK
, η s ). Since Lrig is finite and locally constant there
exists (U , W) ∈ UK,fet such that Lrig (U, W) is trivial and then LU := Lrig (U, W). As a
Z–module it is independent of U and W and we simply write L by abuse of notation. It
i,∗,ge
•
(L) is the sheaf on Xet
associated to the following functor
follows from 3.3 that HK
¡
¢
alg
rig
i,∗,ar
i
•
(U , s) −→ H π1 (UK , η s ), L . Analogously, HK (L) is the sheaf on Xet
associated
¡
¢
rig
, η s ), L .
to the following functor (U , s) −→ Hi π1alg (UK
We come to the proofs of (i) and (ii) of 4.1. Since they are very similar we prove only
b̄ Spf(Vb ) → X be the composite
(i). Consider a point x̂: Spf(Vx̂ ) → X of X and let x:
of x̂ with Spf(Vb ) → Spf(Vx̂ ). Then, Gx̂,K is the direct limit of the fundamental groups
¡ rig
¢
b̄ which is cofinal
π1alg Uα,K
, η sα over a set (Uα , sα ) of affine small neighborhoods of x,
among such étale neighborhoods and is totally ordered with
to morphisms in
¡ respect
¢
•
. As remarked above one has a canonical identification Lrig x̂ ∼
L
Xet
= (as groups). We
i,∗,ge
conclude that the stalk of HK (L) at x̂ can be described as follows:
¡ i,∗,ge ¢
¡
¢
HK (L) x̂ ∼
= Hi Gx̂,K , L .
•
On the other hand for every (U, s) ∈ Xet
with U = Spf(RU ) small, the map of
•
b
Grothendieck topologies ι: UK,fet → XK (see 4.2) induces a spectral sequence
¡
¢
Hi UK,fet , Rj ι∗ (Lrig ) =⇒ Ri vbX ,K,∗ (Lrig )(U, s).
The category of sheaves on UK,fet is equivalent to the category of discrete groups with
¡ rig ¢
continuous action of π1alg UK
, η s by 4.5.3. Via this equivalence ι∗ (Lrig ) = L as repre¡ alg rig
¢
¢
¡
¢
¡
rig
sentation of π1alg UK
, η s . Furthermore, Hi UK,fet , ι∗ (Lrig ) ∼
= Hi π1 (UK , η s ), L by
37
loc. cit. We then obtain a natural, functorial map
rig
αUi : Hi (π1alg (UK
, η s ), L) −→ Ri vbX ,K,∗ (Lrig )(U, s),
•
which induces a morphism of sheaves on Xet
i,∗,ge
(L) −→ Ri vbX ,K,∗ (Lrig ).
αi : HK
For every point x̂ of X the map αx̂i induced by αi on stalks is the canonical morphism
¡ i,∗,ge ¢
¢
¡
¡
¢
αx̂i : HK
(L) x̂ ∼
= Hi (Gx̂,K , Lrig x̂ ) −→ Ri vbX ,K,∗ (Lrig ) x̂ ,
which by proposition 4.4.2(7) is an isomorphism. Therefore, αi induces an isomorphism
i,∗,ge
•
(L) ∼
. Thus, the Leray spectral sequence takes
HK
= Ri vbX ,K,∗ (Lrig ) of sheaves on Xet
the form
³
´
p,q
p,∗,ge
q
•
b • , Lrig ).
E2 = H Xet , HK (L) =⇒ Hp+q (X
K
b , Lrig ). Furthermore, by 4.7.3 we
b • , Lrig ) ∼
It follows from 4.5.2 that Hp+q (X
= Hp+q (X
K
K
p+q
b , Lrig ) ∼
have ¡Hp+q (X
H
(X
,
L)
and,
thanks
to
4.6, we know that Hp+q (XK , L) ∼
=
=
K¢
K
p+q
H
XK,et , L . All these isomorphisms are equivariant for the residual action of GV .
This proves the claim.
5 A geometric interpretation of classical (ϕ, Γ)–modules.
Let the notations be as in the previous section and fix as before M an algebraic extension of K contained in K. In this section we work with continuous sheaves on
all¡ our topologies
¢ + ¡ (see
¢ §4). We define families of continuous sheaves denoted OXM ,
R OXM ³, Ainf ´OXM ³and call
´ them algebraic Fontaine sheaves on XM (respectively
+
b
OX
bM , R O X
bM , Ainf OX
bM called analytic Fontaine sheaves on XM ) and study their
properties. In this section we compare the cohomology on XM of an étale local system
L of Z/ps Z-modules on XK tensored by one of the algebraic Fontaine sheaves with the
b M of its analytic analogue. As a consequence we derive the following
cohomology on X
result.
¡
¢
Let us fix M = K∞ = K(µp∞ ) and consider the sheaf F∞ := Lrig ⊗ A+
inf O X
bK∞ on
b K∞ .
X
5.1 Theorem. Let L be an étale local system L of Z/ps Z-modules on XK . We have
natural isomorphisms of classical (ϕ, ΓV )–modules
³ ¡
¢´
b K , F∞ ) ∼
e V Hi X
,
L
,
Hi (X
D
=
∞
K,et
for all i ≥ 0.
The proof of theorem 5.1 will take the rest of this section.
38
5.2 Categories of inverse systems. We review some of the results of [Ja] which will
be needed in the sequel. Let A be an abelian category. Denote by AN the category
of inverse systems indexed by the set of natural numbers. Objects are inverse systems
{An }n := . . . → An+1 → An . . . A2 → A1 , where the Ai ’s are objects of A and the
arrows denote morphisms in A. The morphisms in AN are commutative diagrams
...
...
→
An+1
↓
→ Bn+1
→
An
↓
→ Bn
...
...
A2
↓
B2
→
→
A1
↓
B1 ,
where the vertical arrows are morphisms in A. Then, AN is an abelian category with
kernels and cokernels taken componentwise and if A has enough injectives, then AN
also has enough injectives; see [Ja, Prop. 1.1]. Furthermore, there is a fully faithful and
exact functor A → AN sending an object A of A to the inverse system {A}n := . . . →
A → A . . . → A with transition maps given by the identity and a morphism f : A → B
of A to the map of inverse systems {A}n → {B}n defined by f on each component.
By [Ja, Prop. 1.1] such map preserves injective objects.
Let h: A → B be a left exact functor of abelian categories. It induces a left exact
functor hN : AN → B N which, by abuse of notation and if no confusion is possible, we
denote again by h. If A has enough injectives, then also AN does and the injective
objects of AN are of the form (In , dn ) where In ∈ A is injective and dn is a split surjection; see [Ja, Prop. 1.1]. One can derive the functor hN . It is proven in [Ja, Prop. 1.2]
¡ ¢ ¡
¢N
that Ri hN = Ri h .
If inverse limits over N exist in B, define the left exact functor lim h: AN → B as the
composite of h
N
and the inverse limit functor lim: B
←
N
→ B. Assume that A and B have
←
enough injectives. For every A = {An }n ∈ AN one then has a spectral sequence
¡
¢
lim(p) Rq h(An ) =⇒ Rp+q lim h(A) ,
←
←
where lim(p) is the p–th derived functor of lim in B. If in B infinite products exist and
←
←
are exact functors, then lim(p) = 0 for p ≥ 2 and the above spectral sequence reduces
←
to the simpler exact sequence
¡
0 −→ lim(1) Ri−1 h(An ) −→ Ri lim h)(A) → lim Ri h(An ) −→ 0;
(5.2.1)
←
←
←
see [Ja, Prop. 1.6]. In particular, if A is injective, then lim← (p) Rq h(An ) = 0 for q ≥ 1
and for q = 0 and p ≥ 1 by the structure of the injective objects in AN . In a particular,
injective objects of AN are acyclic for Rp+q lim← h.
¡
¢
Note that via the map A → AN given above, if A ∈ A then Ri hN {A}n =
{Ri h(A)}n and Ri lim h ({A}n ) = Ri h(A).
←
5.3 Example [Ja, §2]. Let G be a profinite group. Let A be the category of discrete
39
modules with continuous
groups. For
¡
¢ N action of G and let B be the category of abelian
0
j
every j let H G, : A → B be the j–th derived functor of lim H (G, ) on AN . By
←
loc. cit. for every inverse system T = {Tn }n ∈ AN we have an exact sequence
¡
¢
¡
¢
¡
¢
0 −→ lim(1) Hj−1 G, Tn −→ Hj G, T −→ lim Hj G, Tn −→ 0.
←
←
(5.3.1)
Moreover given {(Nn , dn )}n ∈ B N , one computes lim(1) Nn as the cokernel of the map
←
Y
n
(Id − dn ):
Y
Nn −−→
n
Y
Nn .
(5.3.2)
n
For later use we remark the following. Assume that each Nn is a module over a ring C and
that dn : Nn+1 → Nn is a homomorphism of C–modules. Suppose that for every n there
exists an element cn ∈ C annihilating the cokernel of dn . One then proves by induction
Q
Q
Q
on m ∈ N that the cokernel of n (Id − dn ): n≤m Nn → n≤m Nn is annihilated
1
by c1 · · · cm . In particular, if C is a complete local domain and cn = c pn ∈ C for every n
p
Q
for some c in the maximal ideal of C so that the product m cm converges to c p−1 in C,
p
then c p−1 annihilates lim(1) Nn .
←
³
´
For every {(Tn , dn )} ∈ AN one defines Hjcont G, lim Tn as the continuous coho∞←n
mology defined by continuous cochains modulo continuous coboundaries with values
in lim Tn endowed with the inverse limit topology considering on each Tn the dis∞←n
crete topology. As explained in [Ja, Pf. of Thm. 2.2] there exists a canonical complex D• (G, Tn ) whose G–invariants define the continuous cochains C • (G, Tn ) of G with
values in Tn and such that each Di (G, Tn ) is G–acyclic. This resolution is functorial so
that we get a resolution
(Tn , dn ) ⊂ (D1 (G, Tn ), d1n ) → (D2 (G, Tn ), d2n ) → · · · .
¡
¢
The continuous cohomology Hicont G, lim Tn is obtained by applying lim H0 (G, ) to
∞←n
this resolution and taking homology. Due to (5.3.1), since D
have

0
¡
¢ 
i
j
j
j
Hcont G, lim(D (G, Tn ), dn ) = lim(1)
← C (G, Tn )
←

lim∞←n C j (G, Tn )
i
∞←n
(G, Tn ) is G–acyclic, we
if i ≥ 2
for i = 1
if i = 0.
(5.3.3)
In particular, if the system {Tn }n is Mittag–Leffler, then (Dj (G, Tn ), djn ) is acyclic for
every j and we obtain
¡
¢ ∼ i
Hicont G, lim Tn −→H
(G, T ).
∞←n
Next, assume as before that there exists a complete local domain C such that Tn is a
C–module and dn is a homomorphism of C–modules. Suppose also that there is c in the
1
1
maximal ideal of C such that c pn ∈ C and c pn annihilates the cokernel of dn . Then,
40
p
1
also the ¢cokernel of C i (G, Tn+1 ) → C i (G, Tn ) so that c p−1 annihilates
c pn annihilates
¡
p
1
j
Hcont G, (D (G, Tn ), d1n ) . This implies that if we invert c p−1 we have an isomorphism
¡
¢£
£
p ¤
p ¤
∼
i
Hicont G, lim Tn c− p−1 −→H
(G, T ) c− p−1 .
(5.3.4)
∞←n
b M . We now come to the definition of a family of
5.4 Fontaine sheaves on XM and X
b M which will play a crucial role in the sequel. See 5.8.
sheaves on XM and X
5.4.1 Definition. [Fa3, p. 219–221] The notation is as in 4.2. Let OXM be the sheaf of
rings on XM defined requiring that
(U,¢W ) in XM the ring OXM (U, W )
¡ for every
¢ object
¡
consists of the normalization of Γ U, OU in Γ W, OW .
¡
¢
Denote
by
R
O
the sheaf of rings in Sh(XM )N given by the inverse system
X
M
©
ª
OXM /pOXM , where the transition maps are given by Frobenius.
¡
¢
For every s ∈ N define the sheaf of rings A+
OXM in Sh(XM )N as the ininf,s
©
¡
¢ª
¡
¢
¡
¢s
verse system Ws OXM /pOXM . Here, Ws OXM /pOXM is the sheaf OXM /pOXM
with ring operations defined by Witt polynomials and
¡ the¢ transition maps in the in+
verse system
Define Ainf OXM to be the inverse system
© are¡ defined by Frobenius.
¢ª
of sheaves Wn OXM /pOXM ¡ n where the¢transition
as the com¡ maps are defined
¢
posite
Wn+1 OXM /pOXM → Wn OXM /pOXM and Frobenius on
¡ of the projection
¢
Wn OXM /pOXM .
b
bM
to an object (U, W, L) in X
Similarly, OX
bM is the sheaf of rings on XM associating
¡
¢
¡
¢
the ring O(U , W) defined as the normalization of Γ U, OU in Γ W, OW ⊗L M .
´
n¡
³
¢o
N
b
Let R OX
bM be the sheaf of rings in Sh(XM ) given by OX
bM /pOX
bM , where
the inverse system is taken using Frobenius as³transition
map.
´
+
b M )N as the inverse sysFor s ∈ N define the sheaf of rings Ainf,s OX
in Sh(X
b
M
n
¢o
¡
/pO
with transition maps given by Frobenius. Eventually, let
tem Ws OX
bM
X
´ bM
n
³
¡
¢o
+
N
b
Ainf OX
where the transition maps
bM in Sh(XM ) be the sheaf Wn OX
bM /pOX
bM
are defined as the composite
¡
¢
¡
¢
¡
¢
Wn+1 OX
/pO
−→
W
O
/pO
−→
W
O
/pO
n
n
bM
bM
bM
bM
bM
bM ;
X
X
X
X
X
here, the first map is the natural projection and the second is Frobenius.
We denote by ϕ the Frobenius operator acting on the sheaves, or inverse systems of
sheaves, introduced above. One can define analogous sheaves for the pointed sites X•M
b • ; we leave the details to the reader.
and X
M
¢
¡
e+,
5.4.2 Remark. Note that if X = V and M = K, one has H0cont (V, K), R(OK ) = E
V
´
³
¡
¢
¡ +¢
¡
¡
¢¢
+
+
0
e
e
=
W
E
and
H
(V,
=
A
H0cont (V, K), A+
O
K),
A
O
.
s
cont
inf,s
inf
K
K
V
V
41
e + where ε is the
For later use, we recall that we denote by π the element [ε] − 1 of A
V
element (1, ζp , ζp2 , · · ·) ∈ E+
and
[ε]
is
its
Teichmüller
lift.
V
b M )N ) write Hi (XM , F) (respectively
5.4.3 Notation. If F is in Sh(XM )N (resp. Sh(X
cont
0
i
b
b M , ))
Hcont (XM , F)) for the i–th derived functor of lim H (XM , ) (resp. lim H0 (X
←
←
b M )), then
applied to F. Note that if F = {G}n with G ∈ Sh(XM ) (resp. in Sh(X
b M , F) = Hi (X
b M , G)).
Hicont (XM , F) = Hi (XM , G) (resp. Hicont (X
b M )N ), we have a natural isoOne proves as in 4.5.2 that ¡if F ∈ Sh(X¢et )N (resp. Sh(X
¢
¡ • N
•
i
∼
∼ i
b
morphism of δ–functors Hicont Xet
, aN
∗ (F) = Hcont (Xet , F) (resp. Hcont XM , b∗ (F) =
b M , F)).
Hicont (X
From now on we assume that X is locally of finite type over V and that every
closed point of X maps to the closed point of Spec(V ). We let X be the formal scheme
associated to X.
¡
¢rig ∼ + ¡
¢
where ∗ = s ∈ N or ∗ = ∅.
5.4.4 Lemma. One has A+
−→Ainf,∗ OX
inf,∗ O XM
b
M
Proof: Consider a pair (U, W ) in XM , with W defined over some finite extension K ⊂
L contained in M . Recall from section
4 ´that µX,M (U, W ) := (U , W, L). We have a
³
natural map OXM (U, W ) → µX,M,∗ OX
bM (U, W ) i. e., a map from the normalization
of Γ(U, OU ) in Γ(W, OW ) ⊗L M to the normalization³of Γ(U,
´ OU ) in Γ(W, OW ) ⊗L M .
This induces a natural morphism OXM → µX,M,∗ OX
bM and, hence, a morphism
¡
¢
∗
µ∗X,M OXM −→ OX
bM , coming from adjunction of µX,M,∗ and µX,M . We then get a
homomorphism
³
¢
¡
¢´
¡
+
+
µ∗,N
A
O
→
A
O
X
M
X,M
inf,∗
inf,∗
bM .
X
We claim that these maps are isomorphisms. It ¡suffices to prove
¢ it componentwise and
∗
by devissage it is enough to show that µX,M OXM /pOXM −→ OX
bM /pOX
bM is an
isomorphism. Due to 4.4.2(3) this amounts to prove that, for every point x of X as
in 4.4, the natural map OX,x,M /pOX,x,M → OX ,x̂,M /pOX ,x̂,M is an isomorphism. This
follows from 4.4.1(iv).
5.4.5 Lemma. We have the following equivalences of δ–functors :
¡ N
¡ q N
¢
¢
¡ ∗,N
∗,N ¢
∗,N
N
N
◦ Rp vX,M,∗
and Rq vbX
bX ,M,∗ ◦ µ∗,N
i. Rq νX
◦ vX,M,∗
= νX
,M,∗ ◦ µX,M = R v
X,M ;
¡
¢
∗,N
∗,N
∼
N
N
ii. νX
◦ Rq vX,M,∗
−→
Rq vbX
,M,∗ ◦ µX,M .
Proof: The result follows for lemma 4.4.3 and 5.2.
5.5 Comparison between algebraic and formal cohomology of continuous sheaves. Since
∗,N
νX
is an exact functor, as in section 4.7, given an injective resolution I • of a continuous
∗,N
∗,N •
sheaf F, then 0 → νX
(F) → νX
(I ) is exact so that given an injective resolution J •
42
∗,N
of F form = νX
(F) we can extend the identity map on F to a morphism of com∗,N •
plexes νX (I ) → J • . Since νX sends the final object X of Xet to the final object X
∗,N •
of Xet , one has a natural map I • (X) → νX
(I )(X ). Then,
5.5.1 Definition. One has natural maps of δ–functors
¡
¢
¡
¢
q
q
form
(F):
H
X
,
F
→
H
X
,
F
,
ρcont,q
et
et
cont
cont
X,X
and
¡
¢
¡
¢
b M , F rig .
ρcont,q (F): Hqcont XM , F → Hqcont X
bM
X M ,X
Note that one has spectral sequences
¡
¢
N
(F) =⇒ Hp+q
Hqcont Xet , Rp vX,M,∗
cont (XM , F),
(5.5.1)
and
¡
¢
¡
¡ rig ¢¢
¡
¢
∗ p N
N
rig
b
Hqcont Xet , νX
R vX ,M,∗ (F) = Hqcont Xet , Rp vbX
=⇒ Hp+q
,
cont XM , F
,M,∗ F
(5.5.2)
where the equality on the left hand side is due to 5.4.5.
5.5.2 Proposition. The following hold:
a. If F is a torsion sheaf on Sh(Xet )N , then ρcont,q
X,X (F) is an isomorphism.
cont,p+q
;
b. the spectral sequences (5.5.1) and (5.5.2) are compatible via ρcont,q
X,X and ρX ,X
M bM
c. if F is a torsion sheaf in Sh(XM )N , the map ρcont,q (F) is an isomorphism.
bM
XM ,X
Proof: (a) follows from 4.7.2 (a) and the exact sequence (5.2.1) noting that the inverse
limit of a torsion inverse system of sheaves is itself torsion; (b) is left to the reader; (c)
is proven similarly to 4.7.2 (c).
5.5.3 Corollary. Let L be a locally constant sheaf on XM annihilated by ps . Then,
the two sides of the Leray spectral sequences
³
³
¡
¢´´
j
+
i N
rig
Hcont Xet , R vbX ,M,∗ L ⊗ Ainf,s OX
=⇒
bM
³
¡
¢´
b M , Lrig ⊗ A+ O
Hi+j
X
cont
inf,s
b
X
M
and
³
³
³
¡
¢´´
¡
¢´
i+j
+
N
Hjcont Xet , Ri vX,M,∗
L ⊗ A+
O
=⇒
H
X
,
L
⊗
A
O
XM
M
XM
cont
inf,s
inf,s
are isomorphic.
Proof: The statements follow from 5.5.2 and 5.4.4.
43
5.6 Proposition. (Faltings) Let L be a finite locally constant étale sheaf on XK an¡ +¢
e –
nihilated by ps . For every i the kernel and the cokernel of the induced map of Ws E
V
modules
³
´
¡
¢
¡ +¢
¡
¢
+
i
e
Hi XK , L ⊗ Ws E
−−→
H
X
,
L
⊗
A
O
X
cont
inf,s
K
K
V
e+.
are annihilated by the Teichmüller lift of any element in the maximal ideal of E
V
Proof: By devissage one reduces to the case s = 1. The statement follows then from [Fa3,
§3, Thm. 3.8].
5.7 Proposition. We have a commutative square
³
´
³
¡ +¢
¡
¢´
+
i b
rig
i
rig
b
e
H XK , L
⊗ Ws EV
−−→
Hcont XK , L ⊗ Ainf,s OX
bK




ky
y
³
´
³
´´
³
¡
¢
rig
i b
b , Lrig ⊗ Ws V /pV
lim← Hi X
,
L
⊗
W
/pO
,
−−→
lim
H
X
O
s
←
K
K
b
b
X
X
K
K
where the inverse limits are taken with respect to Frobenius. The kernel and the cokernel
of any two maps in the square are annihilated by the Teichmüller lift of any element in
e + . Furthermore, each map
the maximal ideal of E
V
³
´
³
³
´´
¡
¢
b , Lrig ⊗ Ws V /pV −−→ Hi X
b , Lrig ⊗ Ws O /pO
Hi X
(5.7.1)
K
K
bK
bK ,
X
X
appearing in the inverse limits in the displayed square, has kernel and cokernel annihie+.
lated by the Teichmüller lift of any element in the maximal ideal of E
V
Proof: We first of all construct the maps ³in the ´square. The top horizontal map is defined
by the natural map Lrig → Lrig ⊗ A+
the lower horizontal arrow is
inf,s O X
bK . Similarly,
´
³
¡
¢
0
b
induced by the map Lrig → Lrig ⊗ Ws OX
bK /pOX
bK . Note that Hcont XK , (Fn )n is the
¡
¢
b , Fn . This gives a spectral sequence in which the
composite of the functors lim H0 X
K
n
(i)
derived functors lim
←
∞←n
of lim on the category of abelian groups appear. Since lim(i) = 0
←
←
for i ≥ 2, see [Ja, §1], we get an exact sequence
³
³
´´³
´´
(1) i−1 b
rig
0 → lim HN XK , L ⊗ Ws OX
OX
−→
bK /pOX
b
bK
←
³K
³
´´
b , Lrig ⊗ A+
−→ Hicont X
O
−→
inf,s
K
bK
X
³
³
´´
b , Lrig ⊗ Ws O /pWs O
−→ lim HiN X
−→ 0.
K
bK
bK
X
X
←
This provides the right vertical map in the square. Clearly the square commutes. The
fact that the top horizontal arrow has kernel and cokernel annihilated by the Teichmüller
e + follows by 5.5.3 and 5.6. The equality
lift of any element in the maximal ideal of E
V ´
³
b , Lrig is a finite group being isomorphic
on the left hand side follows since Hi X
K
44
i
to H
³
´
XK,et , L by 4.7.3 and 4.6.
To conclude the proof, it suffices to show that the kernel and cokernel of (5.7.1) are
e + . We may
annihilated by the Teichmüller lift of any element in the maximal ideal of E
V
reduce to the case s = 1 i. e., to prove that the map
³
´
³
³
´´
b , Lrig ⊗ E
b , Lrig ⊗ R O
e + −−→ Hj
fj : Hj X
X
cont
K
K
b
X
V
K
e + . For
has kernel and cokernel annihilated by any any element in the maximal ideal of E
³
´≥m
n
oV
any integer m ≥ 1 let OX
be the inverse system OX
bK /pOX
bK
bK /pOX
bK where
the transition maps are the identity in degree ≥ m and are Frobenius in degree < m.
³
´
³
´≥m
Let βm : R OX
/pO
−→
O
be the map of inverse systems whose n–th
bK
bK
bK
X
X
component is ϕn−m : OX
b /pOX
b → OK /pOK for n > m and is the identity for n < m.
K
K
We claim that βm is surjective. It suffices to check it componentwise and, for each com/pOX
ponent, to check surjectivity of ϕn : OX
bK → O X
bK /pOX
bK on stalks. This³follows
³
´bK
´
1
m
1
pm
p , p p2 , · · ·). Then, π
from 4.4.1(v). Consider π0p R OX
with
π
:=
(p,
p
R
O
0
0
bK
bK
X
1
/pO b }n with transition map given by Frobenius. We
is the inverse system {p pn−m OX
K
³ bK ´ X
m
p
claim that Ker(βn ) = π0 R OX
bK . This also can be checked componentwise, for each
component it can be checked on stalks and it follows from 4.4.1(v). Note that
µ
³
´≥m ¶
³
´
i
i b
∼
b M , O /pO
Hcont X
/pO
H
X
,
O
=
M
b
b
b
b .
X
X
X
X
K
K
K
K
≥m
is given by an
Indeed, by [Ja, Prop. 1.1] an injective resolution of (OX
bK /pOX
bK )
injective resolution of each component of this inverse system³which´is constant in deb , associated to the
gree n ≥ m. Take the long exact sequence of the groups Hicont X
K
short exact sequence
³
³
´ 1 ⊗ πpm
´ 1⊗β
³
´≥m
m
0
rig
rig
0 −→ Lrig ⊗ R OX
−
−
−
−
−
−
→
L
⊗
R
O
−
−
−
−
−
−
→
L
⊗
O
−→ 0.
bK
bK
bK
X
X
We get the exact sequence
³
³
´´ πpm
³
³
´´
0
i
b , Lrig ⊗ R O
b , Lrig ⊗ R O
Hicont X
−−→
H
X
−−→
cont
K
K
bK
bK
X
X
³
´
³
³
´´
δi
i+1
b , Lrig ⊗ O /pO
b , Lrig ⊗ R O
Hi X
−−→
H
X
cont
K
K
bK
bK
bK
X
X
X
´´
³
³
pm
π0
rig
b
−−→
Hi+1
⊗ R OX
cont XK , L
bK
45
which we will compare with the exact sequence
m
³
´
´
³
´ ¡
³
¢ 0
π0p
rig
i b
b , Lrig ⊗ E
b , Lrig ⊗ V /pV −→
e + −→H
e + −→ Hi X
Hi X
,
L
⊗
E
X
K
K
K
V
V
³
³
´
´
pm
+ π0
i+1 b
rig
b , Lrig ⊗ E
e
e+
XK , L
−−→ H
⊗ EV −→Hi+1 X
K
V
³
´
³
´
b , Lrig ⊗ V /pV −→ Hi
b , Lrig ⊗ O /pO
via the map gi : Hi X
X
cont
K
K
bK
bK , defined by
X
X
(5.7.1), and via fj for j = i or j = i + 1.
Set δ−1 = 0 and let us denote for the rest of the proof F := Lrig ⊗ OX
bK /pOX
bK , G :=
e+
Lrig ⊗ R(OX
b ) and E := EV . Fix m ≥ 1 and i ≥ 0 and consider the (not necessarily
K
commutative) diagram
³
´
b , Lrig ⊗ V /pV
Hi X
K
g³i ↓
´
i
b ,F
Hcont X
K
m
³
´
³
´
π0p
0
i+1 b
rig
b , Lrig ⊗ E −→
H
X
,
L
⊗E
−→ Hi+1 X
K
K
fi+1
fi+1
m
³ ↓ ´
³ ↓ ´
π0p
δi
i+1
i+1
b
b ,G
−→
Hcont XK , G
−→
Hcont X
K
Let us remark that the right square of the diagram is commutative and that the rows
are exact. We claim that the image of δi is annihilated by every element of the maximal
²
ideal of E, i.e. that δi is “almost zero”. For every ² ∈ Q with ² > 0 let us denote
³ by π´0
b ,F .
any element r of E such that vE (r) = ². Let us fix any such ² and let x ∈ Hicont X
K
m
Denote by y = δi (x) ∈ Ker(π0p ). As the cokernel of fi+1 is ³annihilated
´ by any element of
²/2
i+1 b
rig
the maximal ideal of E, π y = fi+1 (t) for some t ∈ H
X ,L
⊗ E and therefore
m
²/2
π0p (π0 y)
0
pm
π0 fi+1 (t)
K
m
fi+1 (π0p t).
0=
=
=
As the kernel of fi+1 is also annihilated
m
m
²/2
²/2
by every element of the maximal ideal of E we have 0 = π0 (π0p t) = π0p (π0 t) and
m
because multiplication by π0p is injective on the top row of the diagram, we deduce
²/2
²/2
²/2
π0 t = 0. Thus π0² δi (x) = π0 (fi+1 (t)) = fi+1 (π0 t) = 0, which proves the claim.
Now we consider the diagram.
m
³
´
³
´
π0p
i b
rig
b , Lrig ⊗ E
0 → H XK , L
−→ Hi X
⊗E
K
³f i ↓ ´
b , G /Mi−1
0 → Hicont X
K
³
´
b , Lrig ⊗ V /pV → 0
→ Hi X
K
³ gi ↓ ´
δi
b , F −→M
−→
→
Hicont X
i
K
³
´
b , G and f is the
where for every i ≥ 0 we denoted by Mi the image of δi in Hi+1
X
i
cont
K
composition of fi with the natural projection. It is clear that the diagram is commutaive
and the rows are exact. Moreover, the snake lemma and the fact that δi ◦ gi = 0 give
the following exact sequence of E-modules.
m
π0p
f³i ↓
´
i
b
Hcont XK , G
Ker(fi ) → Ker(gi ) → Coker(f i ) → Coker(fi ) → Coker(gi ) → Mi .
As Coker(f i ) is a quotient of Coker(fi ), we deduce that the modules Ker(fi ), Coker(f i ),
Coker(fi ) and Mi are annihilated by every element of the maximal ideal of E, and
46
therefore the same holds for Ker(gi ) and Coker(gi ). This finishes the proof of Proposition
5.7.
5.8 Theorem. Let L be a locally constant étale sheaf on XM annihilated by ps . We
have a first quadrant spectral sequence:
j
H
³
cont
Xet , Ri vbX
,M,∗
³
rig
L
⊗ A+
inf,s
³
´´´
OX
b
=⇒
M
Hi+j
cont
³
XM , L ⊗ A+
inf,s
³
´´
O XM
.
¡ +¢
e –modules
If M = K, there is a map of Ws E
V
³
¢
¡ +¢
¡
¢´
¡
+
n
e
−−→
H
X
,
L
⊗
A
O
Hn XK,et , L ⊗ Ws E
,
X
cont
inf,s
K
K
V
which is an isomorphism after inverting π.
´´
³
+
rig
b
Proof: The first spectral sequence abuts to
XM , L ⊗ Ainf,s OX
bM . The first
statement follows then from 5.5.3. The second one is the content of 5.6.
Hi+j
cont
Hncont
³
³
b K , F∞
X
∞
´£
¤
π −1 are modules for the
5.9 Proof of theorem 5.1. The groups
¡
¢
¡ ¢HV
eV
e
= W E
and have residual action of ΓV and ϕ. By 4.4.2 the
ring W E
∞
V
¡
¢N
¡ ¢N
∗,N
bK
b
functor βK ,K : Sh X
is exact, sends flasque objects to flasque ob→ Sh X
∞
K
∞
³
¡
¢´
b K , F) is equal to lim H0 HV , H0 X
b , β ∗,N (F) for every F in
jects and H0cont (X
N
∞
K
K ,K
←
¡
¢
¡
¢N ∞
¢N
0 b
b
b
Sh XK∞ . Here, HN (XK , is the functor from Sh XK
to the category of inverse
¡
¢
0 b
systems of HV –modules mapping {Gn }n 7→ {H XK , Gn }. We then get a spectral sequence
³
³
³
´´´
³
´
i+j
b , Lrig ⊗ A+
b K , F∞ .
Hj HV , HiN X
O
=⇒
H
X
(5.9.1)
cont
∞
inf,s
K
bK
X
¡
¢
Here, Hj HV , stands for the j–th derived functor of lim H0 (HV , ) on the category of
←
inverse systems of ³HV –modules.
³
´´
b , Lrig ⊗ A+
Put M := HiN X
O
inf,s
K
bK . ´Then, M is the inverse system {Mn }n
X
³
¡
¢
b , Lrig ⊗ Ws O /pO
with Mn := Hi X
and transition maps dn : Mn+1 → Mn
K
bK
bK
X
X
given by Frobenius. By 5.7 each dn has cokernel annihilated the Teichmüller lift of any
e + . Let C • (HV , Mn ) be the complex of continuous
element in the maximal ideal of E
V
cochains with values in Mn . For every i ∈ N the transition maps in {C i (HV , Mn )}n
are given by Frobenius and their cokernels are also annihilated the Teichmüller lift of
e + . We deduce from (5.3.4) and the following
any element in the maximal ideal of E
V
discussion that we have a canonical isomorphism
¡
£
¤
¢£
¤ ∼ i
Hicont HV , lim Mn π −1 −→H
(HV , M ) π −1 ,
∞←n
47
¡
¢
where Hicont HV , lim Mn is continuous cohomology. Eventually, we conclude from 5.8
∞←n
that
³
£
¤
¡ ¢´
e
Hj (HV , M ) π −1 ∼
.
= Hj HV , Hi (XK,et , L) ⊗ W E
V
By 7.8 the latter is zero for j ≥ 1 and is equal to the invariants under HV for j = 0. In
particular, the spectral sequence (5.9.1) degenerates if we invert π. Since L is defined
on XK the isomorphism one gets is compatible with respect
to the
¡
¢ residual
¡ ¢action of ΓV
n
e
and the action of Frobenius. The HV –invariants of H XK,et , L ⊗ W E
V coincide by
¢
¡ n
e V H (X
, L) .
definition with D
K,et
6 The cohomology of Fontaine sheaves.
In this section X denotes a formal scheme topologically of finite type, smooth and
rig
rig
geometrically irreducible over V and let XK
be its generic fiber. Let XK
be the
Grothendieck topology defined by étale and quasi–compact maps. We refer to [JP,
§3.1&3.2] for generalities about étale morphisms of rigid analytic spaces. We study the
b M of continuous sheaves satisfying certain assumptions (see 6.4.3). For
cohomology on X
example, it follows from 6.5 that these sheaves F can be taken to be of the following
form:
³
´
rig
+
1) If L is a p-power torsion étale local system on XK we set F := L ⊗ Ainf OX
bM .
rig
2) If L is an étale sheaf on XK
such that L = lim Ln , with each Ln a locally constant
←
ˆ
Z/pn Z-module and we set F := L⊗O
bM .
X
¡
¢£ −1 ¤
i b
Then
the
cohomology
groups
H
X
,
F
π
can be calculated as follows (here π is
M
£ ¤
² − 1 ∈ A+
(V
)
if
F
is
of
the
first
type
and
π
is p if F is of the second).
inf
Let us fix a geometric generic point η = Spm(CX ) as in §5 and for each small
formal scheme U = Spf(RU ) (see 6.4.1) with a map U −→ X which is étale, define RU to be the union of all finite, normal RU -algebras contained in CX , which
are étale after inverting p. Denote by F(RU ⊗ K) the inductive limit of the sections
fet
F(U, W), where W runs over all objects of UK
. Then F(R ⊗V K) is a continuous rep£ −1 ¤
alg
b £p−1 ¤ and
∼
resentation of π1 (UK , η). Moreover (see 6.7.3) OX
(R
⊗
K)
p
R
=
U
V
U
bM
¡
¢
£ −1 ¤
+
+
Ainf OX
is isomorphic to the relative Fontaine ring Ainf (in which
bM (RU ⊗V K) π
π was inverted) constructed
, RU ). ¤For any such U = Spf(RU ), the
¡ alg using the pair (RU ¢£
i
association U −→ H π1 (UK , η), F(RU ⊗V K) π −1 is functorial and we denote by
i
HGal
(F) the sheaf on Xet associated to it. Then the main result of this section is:
M
6.1 Theorem. Assume that the above assumption holds. Then, there exists a spectral
sequence
¡
¢
p
b M , F).
E2p,q = Hq Xet , HGal
(F) =⇒ Hp+q (X
M
48
As mentioned in the Introduction, theorem 6.1 is the main technical tool needed to
rig
prove comparison isomorphisms relating different p-adic cohomology theories on XK
.
The proof of theorem 6.1 will take the rest of the section.
6.2 Remarks on various Grothendieck topologies. Denote by XZar the Zariski topology
on X .
b M,Zar . Let the underlying category of X
b M,Zar be the full subcategory of
The site X
b M defined in 4.2 whose objects are pairs (U, W) with (U , W) ∈ X
bM
the category of X
b
and U → X is a Zariski open formal subscheme. We define a family of maps in XM,Zar
b M . We let
to be a covering family if it is a covering family when considered in X
b M,Zar −−→ X
bM
ι: X
be the natural functor. We also denote by
b M,Zar
vbX ,M : XZar −−→ X
rig
the map of Grothendieck topologies given by vbX ,M (U) := (U,
¡ (U ¢ , K)). ¡Since ι ¢sends
b M → Sh X
b M,Zar and
covering families to covering families, it is clear that ι∗ : Sh X
¡
¢
¡
¢
b M N → Sh X
b M,Zar N send flasque objects to flasque objects.
ιN : Sh X
∗
Stalks. Let x̂: Spf(Vx̂ ) → X be a closed immersion of formal schemes with V ⊂ Vx̂ (⊂
K) a finite extension of discrete valuation rings. Let OX ,x̂ be the local ring of OX at x̂.
Zar
Define O
to be the limit lim S over all quadruples (R , S , S → Vb , L )
X ,x̂,M
i,j
i,j
i
i,j
i,j
i,j
where (1) Spf(Ri ) ⊂ X is a Zariski open neighborhood of x̂, (2) Li,j is a finite extension of K contained in M , (3) Ri ⊂ Si,j is an integral extension with Si,j normal,
(4) Si,j ⊗V K is a finite and étale Ri ⊗V Li,j –algebra, (5) the composite Ri ⊗V Li,j →
b is a ⊗ ` 7→ x̂∗ (a) · `. If F is a sheaf on X
b
S ⊗ K→K
, define the stalk of F at x̂
i,j
to be
V
M,Zar
¡
¢
Zar
Fx̂ = F(OX ,x̂,M ) := lim F Spf(Ri ), (Spm(Si,j ⊗ K), Li,j ) .
i,j
V
b M,Zar is exact if and only if the induced sequence of stalks is
A sequence of sheaves on X
exact for every closed immersion x̂: Spf(Vx̂ ) → X as above. As in 4.4.2 one proves that
¢
¡
¢
¡ Zar
(Rq vbX ,M,∗ (F))x̂ ∼
= Hq Gx̂,M , Fx̂ where Gx̂,M := Gal OX ,x̂,M /OX ,x̂ ⊗V K .
•
The site UM,fet . Let U ⊂ X be a Zariski open formal subscheme or an object of Xet
.
Let UM,fet be the Grothendieck topology UM,fet introduced in 4.2. It is a full subcategory
b M,Zar (resp. X
b M ). If U 0 → U is a morphism in XZar (resp. X • ), we have a map of
of X
et
Grothendieck topologies
0
ρU ,U 0 : UM,fet −−→ UM,fet
¡
¢
0
letting ρU ,U 0 U, W be the pair (U 0 , W 0 ) where W 0 := W ×U rig U rig ; see 4.2.
Assume that U = Spf(RU ) is affine. By 4.5.1 we have an inclusion RU ⊂ CX (this
•
way we work with Xet
instead of Xet ). Let RU ⊂ RU be the union of all finite and
49
normal RU –subalgebras of CX , which are étale after inverting p. If U = qi Ui , with Ui
Q
of the type above for every i, define RU := i RUi .
¢
¡
¢
¡
Define π1 (UM ) to be Gal RU ⊗V K/RU ⊗V M and let Repdisc π1 (UM ) be the category of abelian groups, with the discrete topology, endowed with a continuous action
of π1 (UM ). We have proved in 4.5.3 that the functor F 7→ F(RU ⊗V K) ¡defines an
¢
equivalence of categories from the category Sh (UM,fet ) to the category Repdisc π1 (UM ) .
Taking continuous sheaves we get:
6.2.1 Lemma. 1) The functor
Sh (UM,fet )
N
¡
¢N
−−→ Repdisc π1 (UM ) ,
{Fn } → {Fn (RU ⊗ K)}
V
is an equivalence of categories;
N
2) for every F ∈ Sh (UM,fet ) we have
¡
¢
¡
¢
Hicont UM,fet , F = Hi π1 (UM ), F(RU ⊗ K) ,
V
¡
¢N
where the latter is the i–th derived functor of Repdisc π1 (UM )
→ AbGr given by
π1 (UM )
{An } 7→ lim An
.
∞←n
¡
¢
¡
¢
b M,Zar (or in Sh(X
b • ), or in Sh X
b M,Zar N , or
6.2.2 Definition. Let F be in Sh X
M
¡ • ¢N
¡
¢
b
in Sh X
). We define F(RU ⊗V K) as the image of F in Repdisc π1 (UM ) (or
M
¡
¢N
b M,Zar ) → Sh(UM,fet ) ∼
in Repdisc π1 (UM ) ) of F via the pull–back maps Sh(X
=
¡
¢
¡
¢
•
b ) → Rep
Repdisc π1 (UM ) (respectively via the pull–back Sh(X
disc π1 (UM ) , etc.).
M
b M,∗ for X
b M,Zar or X
b • and X∗ for XZar
Convention: From now on we simply write X
M
•
.
or, respectively, Xet
¡ ¢
i
b M,∗ ). Let U 0 → U be a map in X∗ with U 0
6.3 The sheaf HGal
F . Let F ∈ Sh(X
M
and U affine. We then get an induced map
¡
¢
¡
¢
0
Hi π1 (UM ), F(RU ⊗ K) −−→ Hi π1 (UM
), F(RU 0 ⊗ K) .
V
V
¡
i
¢
In particular, U → H π1 (UM ), F(RU ⊗V K) is a contravariant functor on the category
of affine objects of X∗ .
i
6.3.1 Definition. Define HGal
¡ M (F) to be the sheaf¢ on X∗ associated to the contravarii
ant functor given by U → H π1 (UM ), F(RU ⊗V K) for U affine.
50
b M,∗ . For i ∈ N and for U =
6.3.2 The standard resolution. Let G be a presheaf on X
Spf(RU ) an affine object of X∗ , define
µ
¶
£
¤
i
i+1
E (G)U := HomZ Z π1 (UM )
, G(RU ⊗ K) .
V
It is endowed with an action of π1 (UM ) defined as follows.¡For every γ, g0¢, . . . , gi ∈
π1 (UM ) and every f ∈ E i (G)U put γ · f (g0 , . . . , gi ) = γ −1 f (γg0 , . . . , γgi ) . Denote
by C i (G)U ⊂ E i (G)U the subgroup of invariants for the action of π1 (UM ). Consider the
map
¤
£
¤
£
di : Z π1 (UM )i+1 → Z π1 (UM )i ,
(g0 , . . . , gi ) 7→
i
X
(−1)j (g0 , . . . , gj−1 , gj+1 , . . . , gi )
j=0
for i ≥ 1 and given by g0 7→ 1 for i = 0. We then get an exact sequence of π1 (UM )–
modules
£
¤
£
¤
· · · −→ Z π1 (UM )2 −→ Z π1 (UM )1 −→ Z −→ 0.
¢
¡
Taking HomZ , G(RU ⊗V K) we get an exact sequence of π1 (UM )–modules
0 −→ G(RU ⊗ K) −→ E 0 (G)U −→ E 1 (G)U −→ · · ·
V
(6.3.1)
which provides a resolution of G(RU ⊗V K) by acyclic π1 (UM )–modules. Using 4.5.3 we
define the sheaf W 7→ E i (G)(U, W) on the category UM,fet associated to E i (G)U . Furthermore, (U, W) 7→ E i (G)(U, W) is a contravariant functor defined on the subcategory
b M,∗ of pairs (U, W) with U affine.
of X
b M,∗ ). For every i ∈ N define Ei (F) to be the sheaf
6.3.3 Definition. Let F ∈ Sh(X
b M,∗ associated to the contravariant functor (U , W) → E i (F)(U , W) for U affine.
on X
Define C i (F) to be the sheaf on X∗ associated to the contravariant functor associating
to an affine U the continuous i–th cochains of π1 (UM ) with values in F(RU ⊗V K) i. e.,
π (U )
C i (F)(U) = E i (F)U1 M .
6.3.4 Proposition. The following hold:
b M,∗
i) the differentials di of 6.3.2 define an exact sequence of sheaves on X
0 −−→ F −−→ E0 (F) −−→ E1 (F) −−→ E2 (F) −−→ · · · ;
ii) for every j ≥ 1 and every i one has Rj vbX ,M,∗ Ei (F) = 0;
iii) for every i one has vbX ,M,∗ Ei (F) = C i (F).
b M,∗ with U affine. Suppose that W = Spm(S) with S ⊗L M
Proof: (i) let (U, (W, L)) ∈ X
¡
¢
an integral domain. Write GalM (W) := Gal RU ⊗V K/S ⊗L M . Then, E i (F)(U, W) is
Gal (W)
E i (F)U M
. In particular, using (6.3.1), it follows that the kernel of E 0 (F)(U, W) →
51
¡
¢GalM (W)
. This coincides with F(U, W) since F is a sheaf
E 1 (F)(U, W) is F RU ⊗V K
thanks to 6.2.1. In particular, the kernel of E0 (F) → E1 (F) is F. To check the exactness
of the sequence in (i) it is enough to pass to the stalks. Given x̂: Spf(Vx̂ ) → X as
in 6.2 or 4.5, the stalk Ei (F)x̂ is the direct limit lim E i (F)(U, W) over all (U, W)
with U an affine neighborhood of x̂ and W = Spm(S) with S ⊗L M ⊂ RU ⊗V K. Hence,
Ei (F)x̂ = lim Ei (F)U where the limit is now taken over all affine open neighborhoods U
of x̂. Since for any such (6.3.1) is exact, we conclude that the stalk at x̂ of the sequence
in (i) is exact as well.
(ii) The claim can be
on stalks.
in 6.2 or¢ 4.5, given x̂: Spf(Vx̂ ) →
¡ checked
¢ As explained
¡
q
i
q
∼
X as before, one has R vbX ,M,∗ (E (F)) x̂ = H Gx̂,M , Ei (F)x̂ . But Ei (F)x̂ coincides
with the direct limit lim Ei (F)U taken over all affine neighborhoods U of x̂. Hence,
µ
¶
£
¤
i+1
, F(RU ⊗ K) =
E (F)x̂ = lim E (F)U = lim Hom Z π1 (UM )
x̂∈U
x̂∈U
V
¶
µ
³ h¡ ¢ i
´
£
¤
i+1
, Fx̂ ,
= Hom lim Z π1 (UM )i+1 , lim F(RU ⊗ K) = Hom Z Gx̂
i
i
←
→
V
Zar
b
b
b•
where
¡ q Gx̂ is iGx̂,M¢ or Gx̂,M depending whether XM,∗ is XM,Zar or XM . In particular,
R vbX ,M,∗ (E (F)) x̂ = 0 if q ≥ 1. Claim (ii) follows.
(iii) For every affine open U ⊂ X there exists a map from the group of i–th cochains
¡
¢
¡ ¢π1 (UM )
i
C π1 (UM ), F(RU ⊗V K) = EUi
to vbX ,M,∗ Ei (F)(U). This provides a natural
map ¡C i (F) → vbX ,M,∗
Ei (F). On the other hand, it follows from ¡the discussion
above
¢
¢
i
i
Zar
that vbX ,M,∗ (E (F)) x̂ is equal to the group of i–th cochains C Gx̂,M , Fx̂ i. e., the
stalk of C i (F). The claim follows.
i
b M,∗ ), then Ri vbX ,M,∗ (F) ∼
(F) functorially in F.
6.3.5 Corollary. If F ∈ Sh(X
= HGal
M
i
6.4 The sheaf HGal
. We wish to prove an analogue of 6.3.5 in the case of a
M ,cont
b M,∗ )N . We need some assumptions.
continuous sheaf F = {Fn }n ∈ Sh(X
6.4.1 Definition. Consider a small object U of X∗ . Define RU ,M,∞ to be the normalization of RU,∞ in the subring of RU ⊗V K generated
by M and RU ,∞ , where
RU ,∞ is
¡
¢
defined as in 2.1. Denote by ΓU ,M the
¢ group Gal RU,M,∞ ⊗V K/RU ⊗V M . Let HU ,M
be the kernel of the map π1 (UM → ΓU,M . Let us remark that the definitions of
RU ,∞ , RU ,M,∞ , HU ,M , ΓU,M depend on a choice of local parameters of RU and so are
not canonical.
b M,∗ (∞) . Let U be a small object of X∗ . For every map U 0 → U
6.4.2 The site U
with U 0 := Spf(RU 0 ) affine and RU 0 ⊗V K an integral domain, we let HU 0 ,M be the
0
kernel of π1 (UM
) → ΓU ,M . Note that such a map is surjective.
52
b M,∗ (∞) be the following full subcategory of U
b M,∗ . Let (U 0 , W 0 ) ∈ U
b M,∗ and
Let U
0
0
0
0
0
assume that U := qi Ui with Ui connected. Then, W lies in UM,fet which, via the
equivalence of 6.2.1, is equivalent to the category of finite sets with continuous acQ
0
0
b M,∗ (∞) if and only
tion of π1 (UM
) = i π1 (Ui,M
). We then say that (U 0 , W 0 ) lies in U
Q
0
if W lies in the subcategory of finite sets with continuous action of i ΓU (viewed
0
as a quotient of π1 (UM
)). We then have natural maps of Grothendieck topologies
β
α b
b
U∗ −→UM,∗ (∞)−→UM,∗ giving rise to maps on the category of sheaves
β∗
α
∗
b M,∗ ) −−→ Sh(U
b M,∗ (∞)) −−→
Sh(U
Sh(U∗ )
b M,∗ (∞)).
whose composite is vbU ,M,∗ . As in 6.2 or 4.5 one has a notion of stalks in Sh(U
For x̂: Spf(Vx̂ ) → X a point as in 4.4, let Hx̂,M be the kernel of the map Gx̂,M → ΓU ,M .
b M ) and Fx̂ is its stalk, one proves as in 4.4.2 that
If F ∈ Sh(U
¡
¢
Rq β∗ (F)x̂ ∼
= Hq Hx̂,M , Fx̂ .
b M (∞) depends on the choice of an extension RU ⊂ RU ,∞ . In particCaveat: The site U
b i,M (∞) do not necessarily
ular, if {Ui }i is a covering of X by small objects, the sites U
b M (∞) is not defined in general.
glue so that the site X
6.4.3 Assumption. We suppose that
n
b
i) {Fn }n∈N is a sheaf of A+
inf (V∞ )–modules (resp. of {V∞ /p V∞ }n –modules) on XM,∗ ;
ii) X
a)
b)
c)
admits
¡
¢
a covering S := {Wi }i in X∗ by small objects Wi := Spf RWi ,
a choice RWi ⊂ RWi ,∞ as in 2.1,
for every i a basis Ti := {Ui,j }j of Wi by small objects such that, putting RUi,j ,∞
to be the normalization of RUi,j ⊗RWi RWi ,∞ , condition (RAE) holds for RUi,j ,∞ .
Furthermore, for every i, j and n ∈ N, putting U := Ui,j , the following hold:
iii) the cokernel of Fn+1 (RU¡ ⊗V K) → Fn (RU ⊗V K) is annihilated by any element of
the maximal ideal of W V /pV ) (resp. V );
¡
¡
¢¢
iv) for every q ≥ 1 the group ¡Hq HU,M , Fn RU ⊗V K is annihilated by any element
of the maximal ideal of W V /pV ) (resp. V );
v) the cokernel of the transition maps Fn+1 (RU,M,∞ ⊗V ¡K) → Fn (RU ,M,∞ ⊗V K) is
annihilated by any element of the maximal ideal of W V∞ /pV∞ ) (resp. V∞ );
vi) for every covering Z¡ → U by small obiects in
¢ X∗ and every q ≥ 1 the Chech coq
homology group H ¡Z → U , Fn (RZ,∞ ⊗V K) is annihilated by any element of the
maximal ideal of W V∞ /pV∞ ) (resp. V∞ ).
e + if {Fn }n∈N is a sheaf of A
e + –modules. Instead,
We write π for the element [²]−1 in A
V∞
V∞
we put π = p if {Fn }n∈N is a sheaf of {V∞ /pn V∞ }n –modules. It follows from (iii) and 5.3
53
that we have an isomorphism
¡
¢£
¤
¡
¢£
¤
Hi π1 (UM ), F(RU ⊗ K) π −1 ∼
= Hicont π1 (UM ), lim Fn (RU ⊗ K) π −1 .
∞←n
V
V
If U 0 → U is a map in X∗ with U 0 and U small objects in Ti , we then get an induced
map
¤
¡
¤
¡
¢£
¢£
0
Hi π1 (UM ), F(RU ⊗ K) π −1 −−→ Hi π1 (UM
), F(RU 0 ⊗ K) π −1 .
V
V
As in 6.3.1, we define
i
6.4.4 Definition. Assume that F satisfies the assumption above. Let HGal
(F)
M ,cont
be the sheaf on X∗ associated
to the contravariant
¡
¢£ −1 ¤ functor sending an object U of X∗ ,
i
with U ∈ ∪i Ti , to H π1 (UM ), F(RU ⊗V K) π
.
We want to prove the following:
b M,∗ )N be such that the conditions of 6.4.3 are fulfilled.
6.4.5 Theorem. Let
X
£ −1F¤ ∈ Sh(
i
i
∼
Then, R vbX ,M,∗ (F) π
= HGalM ,cont (F). The isomorphism is functorial in F.
Proof: It suffices
for every small object Wi ∈ S, we have an isomorphism
£ −1to
¤ prove that
i
i
∼
R vbX ,M,∗ (F) π
|Wi = HGalM (F)|Wi functorially in Wi and F. We construct the
isomorphism and leave it to the reader to check the functoriality in Wi and F.
We may and will, till the end of this section, assume that X = Wi is small. We
put T := Ti and we write Γ for ΓWi ,M . Consider the maps on the category of sheaves
βN
lim
α
∗
← ∗
b M,∗ )N −−→
b M,∗ (∞))N −−→
Sh(X
Sh(X
Sh(X∗ ),
introduced in 6.4.2. The composite is lim vbX ,M,∗ . Since α∗ and β∗ are left exact and β∗
←
sends injective to injective, we have a spectral sequence
¡
¢
¡
¢
Rp lim α∗ Rq β∗N (F) =⇒ Rp+q lim vbX ,M,∗ (F).
(6.4.1)
←
←
q
6.4.6 Lemma. For every
¡ q ≥ 1 the group R β∗ (F) is annihilated by any element of
the maximal ideal of W V∞ /pV∞ ) (resp. V∞ ).
¡
¢N
Proof: Since Rq β∗N = Rq β∗
as remarked in 5.2, it suffices to prove that for evq
ery n ∈ N and every
¡ q ≥ 1 the sheaf R β∗ (Fn ) is annihilated by any element of the
maximal ideal of W V∞ /pV∞ ) (resp. V∞ ). It suffices to prove the vanishing
on¢ stalks.
¡
q
q
But for x̂: Spf(Vx̂ ) → X a point as in 4.4, we have R β∗ (Fn )x̂ ∼
=
¡ H Hx̂,M¡, Fx̂ as ex¢¢
q
plained in 6.4.2. The latter coincides with the direct limit lim H HU ,M , Fn RU ⊗V K
taken over the small objects belonging to a basis of X∗ containing x̂. The claim then
follows from 6.4.3(i)&(iv).
54
Using 6.4.6 and (6.4.1) we conclude that
¡
¢£
¤
¡
¢
£
¤
Rp lim α∗ β∗N (F) π −1 ∼
= Rp lim vbX ,M,∗ (F) π −1 .
←
←
b M,∗ (∞).
We are left to compute Rp lim α∗ . For this we use the analogue of 6.3.2 on X
←
Given U in T , write RU ,M,∞ as the union ∪n RU ,M,n of finite RU –algebras such
that RU ⊗V K ⊂ RU ,M,n ⊗V K is finite and étale. Then, for every covering U 0 → U
with U 0 ∈ T , we have RU 0 ,M,∞ ⊗V K ∼
= ∪n RU 0 ⊗RU RU ,M,n ⊗V K by construction.
00
0
0
Let U → U ×U U be a covering with U 00 in T . Then, we also have RU 00 ,M,∞ ⊗V K ∼
=
∪n RU 00 ⊗RU RU,M,n ⊗V K. Since Fn is a sheaf, we conclude that the sequence
0−−−−−−→Fn (RU,M,∞ ⊗ K) −−→ Fn (RU 0 ,M,∞ ⊗ K) −−→ Fn (RU 00 ,M,∞ ⊗ K)
V
V
V
is exact i. e., U → Fn (RU ,M,∞ ⊗V K) satisfies the sheaf property with respect to coverings U 0 → U with U 0 and U small and lying in T . Then, the following makes sense:
6.4.7 Definition. For every small
U → X lying in¢ T and every i, n ∈ N
¡ £ object
¤
define E i (Γ, Fn )U to be HomZ Z Γi+1 , Fn (RU ,M,∞ ⊗V K) . Define Ei (Γ, Fn ) to be
b M,∗ (∞) characterized by the property that, for every small object U ∈ T ,
the sheaf on X
i
i
its
© irestriction
ª to UM,fet (see 4.2) is E (Γ, Fn )U as representation of ΓU . Let E (Γ, F) :=
E (Γ, Fn ) n .
Let C i (Γ, Fn )U ⊂ E i (Γ, Fn )U be the subgroup of invariants for the action of Γ i. e.,
the group of i–th cochains of Γ with values in Fn (RU,M,∞ ⊗V K). Denote by C i (Γ, Fn )
the unique sheaf
on X∗ whose
value for every small object U is C i (Γ, Fn )U . Eventually,
©
ª
let C i (Γ, F) := C i (Γ, Fn ) n .
6.4.8 Proposition. Assume that F satisfies 6.4.3. Then:
b M,∗ (∞))N
i) we have an exact sequence in Sh(X
0 −→ β∗N (F) −→ E0 (Γ, F) −→ E1 (Γ, F) −→ · · · ;
¡
¢£
¤
ii) Rq lim α∗ Ei (Γ, F) π −1 = 0 for every q ≥ 1 and every i;
←
¡
¢£
¤
iii) lim α∗ Ei (Γ, F) π −1 is the sheaf associated to the contravariant functor sending a
←
µ
¶
¡
£
¤
i
small object U to lim C Γ, Fn RU,M,∞ ⊗ K) π −1 .
∞←n
V
£ −1 ¤
q
N
is the q–th cohomology of the complex
In particular, R lim α∗ (β∗ (F)) π
←
£
¤
£
¤
£
¤
lim C 0 (Γ, Fn ) π −1 −→ lim C 1 (Γ, Fn ) π −1 −→ lim C 2 (Γ, Fn ) π −1 −→ · · · ,
∞←n
∞←n
∞←n
proving 6.4.5.
Proof: Claim (i) can be checked componentwise and then it follows as in the proof
of 6.3.4(i).
55
(ii)–(iii) We use the spectral sequence
¡
¢
lim(p) Rq α∗ (Ei (Γ, Fn )) =⇒ Rp+q lim α∗ (Ei (Γ, F))
←
given in 5.2.
Since each Fn is a sheaf we
have Ei (Γ, Fn )(RU,M,∞ ⊗V K) = E i (Γ, Fn )U .
¡
¢
Hence, Hq Γ, Ei (Γ, Fn )(RU,M,∞ ⊗V K) is 0 for every q ≥ 1 and it coincides with the
cochains C i (Γ, Fn )(RU ,M,∞ ⊗V K) = C i (Γ, Fn )U for q = 0.
¡
¢
Arguing as in 6.3.4(ii) we conclude that Rq α∗ β∗ (Ei (Γ, Fn )) = 0 for q ≥ 1. We are
¡
¢
left to compute lim(p) α∗ Ei (Γ, Fn ) = lim(p) C i (Γ, Fn ).
Due to 6.4.3(vi), for every small object U ∈ T and every n the Chech cohomology
group Hq (Z → U, Fn (RZ,∞ ⊗V K)), relative to every covering Z → U by¡ small objects lying in T , is annihilated by any element of the maximal ideal of W V∞ /pV∞ )
(resp. V∞ ). But we have
¡ Y
¢
C i (Γ, Fn (R ,∞ ⊗ K)) = lim C i (Γ/pm Γ, Fn (R ,∞ ⊗ K)) = lim
Fn (R ,∞ ⊗ K) .
m→∞
m→∞
Γ/pm Γ
(6.4.2)
As both inductive limit¡ and finite ¡products are exact functors
we deduce that the Chech
¢¢
q
i
cohomology group H Z → U, C Γ, Fn (RZ,∞ ⊗V K) relative to every covering
Z→
¡
U with Z and U ∈ T is annihilated by any¡ element of the maximal
ideal of W V∞ /pV∞ )
¢
i
(resp. V∞ ). Hence, the restriction of C Γ, Fn (R ,∞ ⊗V K) to U is flasque,
¡ see [Ar,
II.4.2], up to multiplication by
¡ any element
¢ of the maximal ideal of W V∞ /pV∞ )
(resp. V∞ ). In particular, Hq U , C i (Γ, Fn ) is almost zero for every q ≥ 1; see [Ar,
II.4.4]. Due to 6.4.3(v) the projective system {Fn (RU,M,∞ ⊗V K)}n is almost
© i ¡ Mittagª
Lefler and using once again (6.4.2) we also have that the projective system C Γ, Fn ) n
is almost Mittag–Leffler. Hence, lim(1) C i (Γ, Fn ) is almost zero.
By [Ja,
Lemma
3.12]
the
sheaf lim(q) C i (Γ, Fn ) is the sheaf associated to the presheaf
¡
¡
¢
¢
U 7→ Hq U, C i (Γ, Fn ) n . We have, for each q ≥ 1, exact sequences
¡
¢
¡ ¡
¢ ¢
¡
¢
0 → lim(1) Hq−1 U, C i (Γ, Fn ) −→ Hq U , C i (Γ, Fn ) n −→ lim Hq U, C i (Γ, Fn ) → 0.
∞←n
¡
¢
For q ≥ 2, using the fact proved above that Hs U, C i (Γ, Fn ) is almost zero for s ≥ 1,
¡ ¡
¢ ¢
we deduce that Hq U, C i (Γ, Fn ) n is almost zero. We¡ conclude that lim(q) C i (Γ, Fn ) is
annihilated by every element of the maximal ideal W V∞ /pV∞ ) (resp. V∞ ) for q ≥ 2.
Thus,
¡
¢£
¤
Rp lim α∗ Ej (Γ, F) π −1 = 0
←
for p ≥ 1 and all j ≥ 0, and
£
¤
£
¤
lim α∗ E• (Γ, F) π −1 ∼
= lim C • (Γ, Fn ) π −1 .
←
∞←n
The conclusion follows.
56
6.5 Theorem. Let X be formally smooth, topologically of finite type and geometrically
irreducible over V for which assumption ii) of 6.4.3 is satisfied. Let L = (Ln )n be a
projective system of sheaves such that Ln ∼
= Ln+1 /pn Ln+1 for every n. Let F ∈ Sh(X∗ )N
be a sheaf of one of the following types:
³
´
³
¡
¢´
A) F is Lrig ⊗ A+
O
:=
L
⊗
W
O
/pO
n
n
inf
bM
bM
bM n ;
X
X
X
³
¡
¢´
n
n+1
n
B) F := Ln ⊗ OX
/p
O
OX
bM
bM n where OX
bM /p
bM → O X
bM /p OX
bM is the natural
X
projection for each n ∈ N.
Then, the assumptions in 6.4.3 hold.
6.6 Remark. Assumption (ii) of 6.4.3 holds. Indeed, X is formally smooth and topologically of finite type over V . In particular, it is Zariski locally the p–adic completion
of a smooth scheme over Spec(V ). Thus, X admits a basis by affine subschemes satisfying (RAE) due to 2.5.
In case (A), assume further that L is a p–power torsion i. e., annihilated by ps
i
for some s. Then, one can compute the sheaf HGal
(F) introduced in 6.4.4 via
M ,cont
relative (ϕ, Γ)–modules. Indeed, assume that U = Spf(RU ) is small and that (RAE)
holds for RU ,∞ .
For M = K we have π1 (UM ) = GRU and by 7.10.6 the inflation
¢
¡
¢
¢£ −1 ¤
¡
i
∼
e
Hi (ΓRU , D(L) −−→ Hi π1 (UK ), L ⊗ A
⊗
K)
π
H
π
(U
),
F(R
=
U
1
K
RU
V
Zp
is an isomorphism.
Analogously, for M = K we have π1 (UM ) = GRU so that
¢ ∼ i¡
¢
¡
¢£ −1 ¤
∼ i
e
Hi (Γ0RU , DK (L) −→H
π1 (UK ), L ⊗ A
.
RU = H π1 (UK ), F(RU ⊗ K) π
V
Zp
Here, DK (L) is the (ϕ, Γ0R )–module associated to L and the field K as in §2.
For M = K∞ , the group π1 (UM ) is the subgroup of GRU generated by GRU and HV .
In this case
¢ ∼ i¡
¢
¡
¢£ −1 ¤
i
∼
e
Hi (Γ0RU , DK∞ (L) −→H
π1 (UK∞ ), L ⊗ A
⊗
K)
π
H
π
(U
),
F(R
=
U
1
K
∞
RU
Zp
V
where DK∞ (L) is the (ϕ, Γ0R )–module defined in §2 using the field K∞ .
6.7 Proof of 6.5. We start with some preliminary results.
6.7.1 Lemma. Let R be as in 2.1. Let S∞ ⊂ T∞ be integral extensions of R∞ such
that S∞ ⊗V K = T∞ ⊗V K and R∞ ⊂ S∞ is almost étale (see 2.4). Then, the cokernel
of S∞ ⊂ T∞ is annihilated by any element of the maximal ideal of V∞ .
57
£ −1 ¤
£ −1 ¤
p
⊂
S
p
=
Proof:
Let
e
be
the
canonical
idempotent
of
the
étale
extension
R
∞
∞
∞
£ −1 ¤
£ −1 ¤
α
T∞ p . Since R∞ ⊂ S∞ is almost étale, for every α ∈ Z p >0 we may write p e∞ as
£ −1 ¤
£ −1 ¤
£ −1 ¤
P
a finite sum i ai ⊗ bi with ai and bi in S£∞ . Let
m:
S
p
⊗
S
p
→
S
p
∞
R
∞
∞
∞
¤
£ −1 ¤
−1
be the multiplication map and let Tr: S∞ p
→ R∞ p ¡be the trace¢map. Then, e∞
is characterized by the property that
m(x
⊗
y)
= (Tr ⊗ Id) (x ⊗ y) · e∞ . In particular,
¡ α
¢ P
α
for every x ∈ T∞ we have p x = m p x ⊗ 1 = i Tr(ai x)bi . But Tr(ai x) ∈ R∞ since x
P
and ai are integral over R∞ and R∞ is integrally closed. Hence, pα x = i Tr(ai x)bi
lies in S∞ as claimed.
6.7.2 Lemma. Let U = Spf(RU ) be an affine small object of X∗ and let U 0 → U be a
covering with U 0 affine. Then, RU 0 ,M,∞ ∼
= RU ,M,∞ ⊗RU RU 0 .
Proof: Write the composite of M and K∞ (in K) as the union ∪n Mn where M0 = K ⊂
· · · ⊂ Mn ⊂ · · · and K ⊂ Mn is a finite extension for every n. Let Wn be the ring of
integers of Mn and let Fn be its residue filed. Let T1 , . . . , Td ∈ RU be parameters as
1 ¤
£ 1n
n
in 2.1. Since RU ⊗V k is a smooth k–algebra, then RU T1p , . . . , Tdp ⊗V Fn is a smooth
1 ¤
£ 1n
n
k–algebra. Hence, RU ⊗V Wn T1p , . . . , Tdp is a regular ring modulo the maximal ideal
of Wn and, hence, it is a regular ring itself. In particular, it is normal. This implies
1 ¤
£ 1n
n
that RU ,M,∞ ∼
= ∪n RU ⊗V Wn T p , . . . , T p .
1
d
Since U 0 → U is formally étale, then RU 0 ⊗V k is a smooth k–algebra. Reasoning as
1 ¤
£ 1n
n
above we conclude that RU 0 ,M,∞ ∼
= ∪n RU 0 ⊗V Wn T p , . . . , T p . The lemma follows.
1
d
Let U = Spf(RU ) be an affine small object of X∗ . Let A be the union of some
collection of almost étale, integral RU ,M,∞ –subalgebras of RU . Write
¢
¡
¢
¡
K) := lim OX
/pOX
(U, W, L)
OX
bM /pOX
bM (A ⊗
b
b
M
M
V
b M with W = Spm(SW ) such
where the direct limit is taken over all (U , (W, L)) ∈ X
that SW ⊗L M ⊂ A ⊗V K.
6.7.3 Proposition. Assume that RU is small over V . Then, the natural map
¡
¢
A/pA −−→ OX
/pO
b
b (A ⊗ K)
X
M
M
V
has kernel and cokernel annihilated by any element of the maximal ideal of V∞ .
0
0
0
Proof: The presheaf OX
bM /pOX
bM is separated i. e., if (U , W , L ) → (U , W, L) is a
covering map, the natural map
0
0
0
0
0
0
OX
b (U, W, L)/pOX
b (U, W, L) −→ OX
b (U , W , L )/pOX
b (U , W , L )
M
M
M
M
is injective. This implies that we have an injective map
¡
¢
A/pA = OX
(A
⊗
K)/pO
(A
⊗
K)
,−→
O
/pO
K).
bM
bM
bM
bM (A ⊗
X
X
X
V
V
V
58
We also get that the sheaf associated to the presheaf associating
¡ to a triple¢(U , W, L)
the ring OX
bM (U, W, L)/pOX
bM (U, W, L) is defined by taking OX
bM /pOX
bM (U , W, L)
to be the direct limit, over all coverings (U 0 , W 0 , L0 ) of (U , W, L) with U 0 affine, of the
0
0
0
0
0
0
elements b in the group OX
b (U , W , L )/pOX
b (U , W , L ) such that the image of b
M
M
00
00
00
00
00
00
00
00
00
in OX
bM (U , W , L )/pOX
bM (U , W , L ) is 0 where (U , W , L ) is the fiber product
of (U 0 , W 0 , L0 ) with itself over (U, W, L). Hence,
¡
¢
OX
K) = lim KerS,T
bM /pOX
bM (A ⊗
S,T
V
where the notation is as follows. The direct limit is taken over all normal RU ,M,∞ –
subalgebras S of A, finite and étale after inverting p over RU,M,∞ , all covers U 0 → U
and all normal extensions RU 0 ,M,∞ ⊗RU S → T , finite, étale and Galois after inverting p.
Eventually, we put U 00 := Spf(RU 00 ) to be the fiber product of U 0 with itself over U i. e.,
d
RU 00 := RU 0 ⊗
RU RU 0 . We let
³
´
−−→
KerS,T := Ker T /pT −→ TeS /pTeS ,
where TeS is the normalization of the base change to RU 00 of T ⊗(RU 0 ,M,∞ ⊗RU ,M,∞ S) T .
¡
¢
For every S and T as above, write GS,T := Gal T ⊗V K/S ⊗RU ,M,∞ RU 0 ,M,∞ ⊗V K .
Q
RU 00 where tilde stands for the normalization
Then, TeS is the product g∈GS,T T ⊗Rg
U0
(of T ⊗RU 0 RU 00 ) and we view RU 00 as RU 0 –algebra choosing the left action. Hence,


g
Y
00
T ⊗RU 0 RU
−−→
.
KerS,T = Ker T /pT −→
00
R
pT ⊗Rg
U
0
g∈GS,T
U
¡
¢
The two maps in the display are a 7→ (a, . . . , a) and a 7→ g(a) g∈G .
S,T
For the rest of this proof we make the following notations: if B is a RU ,M,∞ -algebra
we denote by B 0 := B ⊗RU ,M,∞ RU 0 ,M,∞ = B ⊗RU RU 0 , by B 00 := B ⊗RU 0 ,M,∞ RU 00 ,M,∞ =
e the normalization
B ⊗RU 0 RU£00 (the
second equalities above follow form 6.7.2) and by B
¤
of B in B p−1 . We then get a commutative diagram
0 →
S/pS

y
0 → KerS,T
0
−−→ S 0 /pS


αy
−−→
−−→
−−→
T /pT
−→
−→
00
S 00 /pS

β
y
¡
¢
Q
TeS /pTeS = g∈GS,T Te00 /pTe00 .
(6.7.1)
The top row is exact by étale descent and the bottom row is exact by construction. We
claim that the kernel and cokernel of the map S/pS −→ KerS,T are annihilated by any
element of the maximal ideal of V∞ . To do this we analyze the maps α and β.
Analysis of Ker(α) and Ker(β). Note that he extension RU ,M,∞ ⊂ S is integral and
almost étale by 2.5. Hence, the extensions RU 0 ,M,∞ ⊂ S 0 and RU 00 ,M,∞ ⊂ S 00 are integral
and almost étale as well. Since the extension RU → RU 0 (resp. RU → RU 00 ) is faithfully
flat, the rings S 0 and S 00 have no non–trivial p–torsion. In particular, S 0 (resp. S 00 )
59
injects into its normalization Se0 (resp. Se00 ) which is T GS,T . Thanks to Lemma 6.7.1 the
cokernel of S 0 → Se0 = T GS,T (resp. S 00 → Se00 ) is annihilated by any element of the
maximal ideal of V∞ .
Consider the following. If 0 −→ B −→ C −→ D −→ 0 is an exact sequence of abelian
groups then the kernel of the induced map B/pB → C/pC is the image in B/pB of
the group of p-torsion elements of D. In particular if B, C, D are V∞ -modules and D is
annihilated by an element a ∈ V∞ then Ker(B/pB → C/pC) is also annihilated by a.
It follows from this obvious fact that the kernel of the map S 0 /pS 0 → Se0 /pSe0 and the
kernel of the map S 00 /pS 00 → Se00 /pSe00 are annihilated by any element of the maximal
ideal of V∞ .
The map Se0 /pSe0 → T /pT (resp. Se00 /pSe00 → TeS /pTeS ) is injective since Se0 → T
(resp. Se00 → TeS ) is an integral extension of normal rings. Hence, the kernel of α and the
kernel of β are annihilated by any element of the maximal ideal of V∞ .
Analysis
of the image of Coker(S/pS
−→ KerS,T ) in Coker(α). Define Z as Z :=
¡
¢
Coker S 0 /pS 0 → (T /pT )GS,T ⊂ Coker(α). Since KerS,T is GS,T -invariant (by definition), the ¡image of Coker(S/pS
¢ −→ KerS,T ) in Coker(α) is contained in Z. Put
0
0
0
0
e
e
Y := Coker S /pS → S /pS . Let us remark that we have an exact sequence of
groups:
¢
¡
0 −→ Y −→ Z −→ Coker Se0 /pSe0 −→ (T /pT )GS,T −→ 0.
We know that Y is annihilated by any element of the maximal ideal
¡ of V∞ , so let us
examine the¢ last term of the sequence. This is the same as Coker T GS,T /pT GS,T −→
(T /pT )GS,T . Consider the exact sequence
¡
¢GS,T
¡
¢
0 −−→ T GS,T /pT GS,T −−→ T /pT
−−→ H1 GS,T , T .
¡
¢
Since RU 0 ,M,∞ → T is almost étale, the group H1 GS,T , T is annihilated by any element of the maximal ideal of V∞ ; see [Fa1, Thm. I.2.4(ii)]. Hence, the cokernel of
¡
¢GS,T
T GS,T /pT GS,T −→ T /pT
is annihilated by any element of the maximal ideal
of V∞ . We deduce that the same is true for the module Z above.
Now using the snake lemma applied to the commutative diagram (6.7.1), we get that
the kernel and cokernel of the map S/pS −→ KerS,T are annihilated by any element of
the maximal ideal of V∞ as claimed.
This concludes the proof in the case that A is the union of almost étale, integral and
normal RU,M,∞ –subalgebras of RU . In the general case, assume that Q is an almost étale,
integral RU,M,∞ –subalgebra of A and let S be its normalization. Then, the cokernel of
Q → S annihilated by any element of the maximal ideal of V∞ by Lemma 6.7.1. The
same then applies to the kernel and the cokernel of Q/pQ → S/pS. The conclusion
follows.
6.7.4 End of proof of 6.5. Assumption (i) clearly holds. We let {Wi }i = S be a covering
of X and let Ti := {Uij }j be a basis of Wi as in 6.4.3(ii). Let U ∈ Ti for some i.
60
(iii) The group Ln (RU ⊗V K) is constant on the connected components of U and does
not depend on U itself. It then suffices to verify assumption (iii) for Ln the constant
sheaf i. e, Ln = Z/ps Z for some s in case (A) or Ln = (Z/pn Z) in case (B). In this case
(iii) follows from 6.7.3 with A = RU .
¡
¢
q
(iv) Due
to
6.7.3
it
suffices
to
prove
that
H
H
,
L
(R
⊗
K)
⊗
W
(R
/pR
)
U
n
U
V
n
U
U
¡
¢
(resp. Hq H¡U , Ln (RU ⊗V K) ⊗ RU /pn RU is annihilated by any power of the maximal
ideal of Wn V∞ /pV∞ ) (resp. V∞ ) for every q ≥ 1. In both cases, one reduces by devissage to the case n = 1. The claim then follows from 7.8 and 7.5.
(v) Given n ∈ N, let RU ⊗V K ⊂ B be a finite and étale extension such that Ln+1
and Ln are constant on the étale site of BM . We may assume that B is defined over
a finite extension K ⊂ L contained in M and that RU ⊗V M ⊂ B ⊗L M is a Galois
extension of integral domains. Define AU as the normalization of RU in the subring
of RU ⊗V M generated by RU,M,∞ and B. Then, assumption (v), with AU in place
of RU,M,∞ , holds due to 6.7.3 since we may reduce to the case where L is trivial.
Let Dn be the kernel of Fn+1 (AU ⊗V K) → Fn (AU ⊗V K). Let Mn be the kernel of Ln+1 (B) → Ln (B). It is an Fp –vector space. In case (B), the sequence 0 −→
AU /pAU −→ AU /pn+1 AU −→ ¡AU /pn AU¢ → 0 is exact since AU is normal. Tensor it with
Ln+1 (B) and put En := Mn ⊗ AU /pAU . Since Ln (B) = Ln+1 (B)/pn Ln+1 (B), the sequence 0 −→ Mn ⊗ AU /pAU −→ Ln+1 (B) ⊗ AU /pn+1 AU −→ Ln (B) ⊗ AU /pn AU → 0
is exact. Thanks to 6.7.3 we get that the natural map En → Dn has kernel and cokernel
annihilated by any element of the maximal ideal of V∞ /pV∞ .
¡
¢
1
In¡ case (B)¢ consider the exact sequence 0 −→ AU /p p AU −→ Wn+1 AU /pAU −→
Wn AU /pAU where the last map is the natural projection composed with Frobe1
p
nius. Tensoring it¡with Ln+1
¢ (B) we get the exact
¡ sequence
¢ 0 −→ Mn ⊗ AU /p AU1 −→
Ln+1 (B) ⊗ Wn+1 AU /pAU −→ Ln (B) ⊗ Wn AU /pAU . Put Fn := Mn ⊗ AU /p p AU .
Also in this case the natural map Fn → Dn has kernel and cokernel annihilated by any
element of the maximal ideal of V∞ /pV∞ . It follows from 7.8 and 7.5 that Hq (HU , En )
and Hq (HU , Fn ) are annihilated by any element of the maximal ideal of V∞ /pV∞ .
Thus, the same applies to Hq (HU , Dn ) and, hence, to the cokernel of the map from
Fn+1 (AU ⊗V K)HU = Fn+1 (RU,M,∞ ⊗V K) to Fn (AU ⊗V K)HU = Fn (RU ,M,∞ ⊗V K).
This concludes the proof of (v).
(vi) For every covering Z → U in X∗ with Z ∈ Ti define HqFn (Z → U) as the Chech
cohomology group
¡
¢
HqFn (Z → U) := Hq Z → U, Ln (B) ⊗ Wn (AU ⊗ RZ /pRZ )
RU
respectively
¡
¢
HqFn (Z → U) := Hq Z → U, Ln (B) ⊗(AU ⊗ RZ /pn RZ ) .
RU
See the proof of (v) for the notation. For every q ≥ 1 the group HqFn (Z → U) is 0 since
the sheaves considered are quasi–coherent.
61
Due to 6.7.3 we conclude that assumption (vi) holds using AU ⊗RU RZ ⊗V K instead
of RZ,M,∞ . Let G be the Galois group of AU ⊗V K over RU ,M,∞ ⊗V K. Using the
spectral sequence
¡
¡
¢
¡
¢
Hp G, Hq Z → U, Fn (AU ⊗ (RZ ⊗ K) =⇒ Hp+q Z → U, Fn (RZ,∞ ⊗ K)
RU
V
V
¡
¢
we ¡deduce¡ that the group Hp Z → U, F¢¢
n (RZ,∞ ⊗V¡ K) is isomorphic
¢ to the group
p
0
p
H G, H Z → U , Fn (AU ⊗RU (RZ ⊗V K) i. e., H G, Fn (AU ⊗V K) . Let C be the
kernel of the surjective map HU → G. Consider the spectral sequence
¡
¢
¡
¢
Hp G, Hq (C, Ln (B) ⊗ RU /pRU =⇒ Hp+q HU , Ln (B) ⊗ RU /pRU .
¢
¢
Note that Hq (C, Ln (B) ⊗ RU /pRU and Hq (HU , Ln (B) ⊗¡RU /pRU are annihilated by
multiplication by any element of the maximal ideal of W V∞
¡ /pV∞ ) (resp. V∞ ) for
¢ q≥
1 due to 7.8 and 7.5. Hence, the same must hold for Hq G, Ln (B)
⊗
A
/pA
U
U . By
¡
¢
q
devissage and 6.7.3 one concludes that the same must hold for H G, Fn (AU ⊗V K) .
Thus, (vi) holds.
b M,∗ )N such that
6.8 Proof of theorem 6.1. By theorem 6.4.5 if F is a£ sheaf
of Sh(X
¤
i
the assumptions 6.4.3 are satisfied then Ri vbX ,M,∗ (F) π −1 ∼
(F). Using this
= HGal
M
isomorphism the Leray spectral sequence for the composition of functors H0 (X∗ , −) ◦
vbX ,M,∗ becomes
¡
¢
p
b M,∗ , F).
E2p,q = Hq X∗ , HGal
(F) =⇒ Hp+q (X
M
In particular, we obtain a spectral sequence for ∗ = •. Now theorem 6.1 follows as the
•
functors Hi (Xet
, −) and Hi (Xet , −) are canonically isomorphic, same as the functors
b • , −) and Hi (X
b M , −); see 4.5.2.
Hi (X
M
7 Appendix I: Galois cohomology via the Tate–Sen method.
The goal of this section is to prove Proposition 7.8 stating that, if M is a Zp e
e
representation of GS , then the groups Hi (HS , D(M ) ⊗AS AR ), Hi (HS , D(M
) ⊗A
e S AR ),
0
e ) are trivial for i ≥ 1. This is the
e
e
) and Hi (HS , D(M
) ⊗A
A
Hi (HS , D(M
) ⊗A0S AR
R
e S∞
0
key tool to compute the Galois cohomology of M in terms of the associated (ϕ, ΓS )–
modules.
To treat all the cases above, we follow the axiomatic approach started by Colmez in
[Co, §3.2&3.3] and developed in [AB, §2].
e be Zp -algebra which is an integral
7.1 The axioms. Let G be a profinite group and let Λ
e → R ∪ {+∞} such that:
domain and is endowed with a map v: Λ
(i) v(x) = +∞ ⇔ x = 0;
62
(ii) v(xy) ≥ v(x) + v(y);
(iii) v(x + y) ≥ min(v(x), v(y));
(iv) v(p) > 0 and v(px) = v(p) + v(x).
e with the (separated) topology induced by v. We assume that Λ
e is
We endow Λ
complete for this topology and that it is endowed with a continuous action of G such
e and for g ∈ G.
that v(g(x)) = v(x) for x ∈ Λ
Let H a closed normal subgroup of G such that Γ = G/H is endowed with a continuous
d
homomorphism χ: Γ → Z×
p with open image, with kernel isomorphic to Zp and such that
γgγ −1 = g χ(γ) for every g ∈ Ker(χ) and every γ ∈ Γ. Let γ0 ∈ Γ be such that Im(χ) =
e such
Zp χ(γ0 ) ⊕ F with F a finite group. Assume that there exist γ1 , . . . , γd ∈ Aut(Λ)
that Ker(χ) is an open subgroup of Zp γ1 ⊕ · · · ⊕ Zp γd and let m0 ∈ N be such that
pm0 ⊕di=1 Zp γi ⊂ Ker(χ). Let G ⊂ G be a closed normal subgroup, put H := G ∩ H and
∼
assume that Γ0 := G/H−→Ker(χ).
Assume that for every open normal subgroup H0 ⊂ H there exists an integer m0,H0 ≥
m0 such that for every i ∈ {0, . . . , d} one has
(a) a lifting pm0,H0 −m0 Γ ⊂ G/H0 as a normal subgroup centralizing H/H0 ;
³
´
(i)
e H0 ;
(b) an increasing sequence Λm,H0
of closed subrings of Λ
m≥m0,H0
³
´
(i)
e H0 → Λ(i) 0
(c) maps τm,H0 : Λ
m,H
m≥m0,H0
and the following axioms à la Tate-Sen hold:
(TS1) there exists c1 ∈ R>0 such that for every open normal subgroups H1 ⊆ H2 of H
P
e H1 such that v(α) > −c1 and
(resp. of H), there exists α ∈ Λ
τ (α) = 1;
τ ∈H2 /H1
(TS2) there exists c2,H0 ∈ R>0 such that for every i and j ∈ {0, . . . , d} and every m ≥ m0,H0 :
(i)
(i)
(i)
(i)
(a) τm,H0 is Λm,H0 -linear and τm,H0 (x) = x if x ∈ Λm,H0 ;
³
´
(i)
(i)
e H0 ;
(b) one has v τm,H0 (x) ≥ v(x) − c2,H0 and lim τm,H0 (x) = x for every x ∈ Λ
m→∞
(c)
(i)
τm,H0
commutes with
(j)
τm0 ,H0 ;
(i)
(i)
(d) the ring Λm,H0 is stable under G/H0 and τm,H0 commutes with the action of G/H0
for i ∈ {1, . . . , d};
m
m
0,H0
0,H0
(0)
(0)
and τm,H0 commutes with γ0p
and
(d’) the ring Λm,H0 is stable under γ0p
(0)
(1)
(d)
τm,H0 ◦ τm,H0 ◦ · · · ◦ τm,H0 commutes with the action of G/H0 ;
(i)
(i)
e for every open normal subgroup H00 ⊂
(e) we have Λm,H0 ⊂ Λm,H00 , as subrings of Λ,
H0 and the following diagram commutes
(i)
e H0
Λ

y
e H00
Λ
τm,H0
−−→
(i)
τm,H00
−−→
³
´ ³ 0´
(i)
(i)
e H . Then,
(TS3) let Xm,H0 = 1 − τm,H0 Λ
63
(i)
Λm,H
 0
y ;
(i)
Λm,H00
(a) there exists c3,H0 ∈ R>0 such that for every m ≥ m0,H0 and every i ∈ {0, . . . , d},
m
(i)
(i)
the map 1 − γip is invertible on Xm,H0 and for every x ∈ Xm,H0 , one has
µ³
¶
´−1
pm
v
1 − γi
(x) ≥ v(x) − c3,H0 .
(a) There exists c4,H0 ∈ R>0 such that
every
³³ for
´ m´≥ m0,H0 and every i ∈ {1, . . . , d}
m
(i)
and every x ∈ Λm,H0 , one has v γip − 1 (x) ≥ v(x) + c4,H0 .
(TS4) Let H0 ⊂ H be an open normal subgroup. Assume that there exists an integer m0,H0 ≥
m0 and a lifting pm0,H0 −m0 Γ0 ⊂ G/H0 as a normal subgroup
H/H0 and
³ centralizing
´
(i)
that for every i ∈ {1, . . . , d} one has an increasing sequence Λm,H0
of closed
m≥m0,H0
´
³
e H0 stable under G/H0 and maps t(i) 0 : Λ
e H0 → Λ(i) 0
such that
subrings of Λ
m,H
m,H
the analogues of (TS2) and (TS3) hold.
m≥m0,H0
We followed closely the formalism of [AB, §2] with the differences that we added (TS2)(e)
and (TS4).
e
e a . We consider it
7.2 Notation. Let W be a free Λ–module
of finite rank a i. e., W := Λ
as a topological module with respect to the (separated) topology defined as the product
e the v–adic topology. Note that such topology is independent
topology considering on Λ
e
e ≥n for the subgroup
of the choice of Λ–basis
of W . For every positive n ∈ Q write Λ
e consisting of elements x such that v(x) ≥ n. They are a fundamental system of
of Λ
e for n → ∞. Let W≥n be the image of Λ
e a in W ;
neighborhoods for the topology on Λ
≥n
they form a fundamental system of neighborhoods for the given topology on W . Assume
that W is endowed with a continuous action of G. We consider¡ continuous
cohomology
¢
0
r
0
of a closed subgroup H of G with values in W . If f ∈ C H , W is a continuous
©
0
cochain, with r ≥ 0 and with the profinite
topology
on
H
,
write
v(f
)
:=
min
ª
¡ 0
¢
¡ 0
¢n ∈
0
r
r+1
N|(∀g1 , . . . , gr ∈ H )f (g1 , . . . , gr ) ∈ W≥n . We write ∂: C H , W → C
H , W for
the boundary map.
7.3 Lemma. [Ta, §3.2] Let H0 be an open subgroup of H (resp. of H) and let f be
an r–cochain of H0 with values in W for r ≥ 1.
(1) Assume that there exists an open normal subgroup H1 ⊂ H0 such that f factors
via an r–cochain of H0 /H
¡ 1 . Then,
¢ there exists an (r − 1)–cochain h of H0 /H1 with
values in W such that v f − ∂h > v(∂f ) − c1 and v(h) > v(f ) − c1 .
(2) There exists
a sequence
¡
¢ of open normal subgroups Hn ⊂ H0 and continuous cochains
r
fn ∈ C H0 /Hn , W for n ∈ N such that f ≡ fn modulo W≥n for n → ∞.
Proof: We work out the case of H0 ⊂ H. For H0 ⊂ H the argument is analogous and
the details are left to the reader.
e H1 be an element satisfying (TS1). Define the (r − 1)–cochain α ∪ f
(1) Let α ∈ Λ
64
of H0 /H1 with values in W by
¡
¢¡
¢
α ∪ f g1 , . . . , gr−1 := (−1)r
X
¡
¢
g1 · · · gr−1 t(α) · f g1 , . . . , gr−1 , t .
t∈H0 /H1
One computes that
r−1
¡
¢ X
∂(α ∪ f )(g1 , . . . , gr ) = g1 (α ∪ f )(g2 , . . . , gr ) +
(−1)j (α ∪ f )(. . . , gj gj+1 , . . .)+
j=1
r
+ (−1) (α ∪ f )(g1 , . . . , gr−1 ) =
X
¡
¢
= (−1)r
g1 · · · gr t(α) · g1 f g2 , . . . , gr , t +
t∈H0 /H1
+ (−1)r
+
X
r−1
X
X
¡
¢
(−1)j g1 · · · gr t(α) · f g1 , . . . , gj gj+1 , gr , t +
j=1 t∈H0 /H1
¡
¢
g1 · · · gr−1 t(α) · f g1 , . . . , gr−1 , t
t∈H0 /H1
and
¡
¢
α ∪ ∂f (g1 , . . . , gr ) = (−1)r+1
X
¡
¢
g1 · · · gr t(α) · ∂f g1 , . . . , gr , t =
t∈H0 /H1
= (−1)r+1
X
g1 · · · gr t(α) · g1 f (g2 , . . . , gr , t)+
t∈H0 /H1
r+1
+ (−1)
−
X
X
g1 · · · gr t(α) ·
t∈H0 /H1
¡
r−1
X
¡
¢
(−1)j f g1 , . . . , gj gj+1 , gr , t +
j=1
¢
g1 · · · gr t(α) · f g1 , . . . , gr t +
t∈H0 /H1
+
X
g1 · · · gr t(α) · f (g1 , . . . , gr )
t∈H0 /H1
¡
¢
¡
¢
P
Since t∈H0 /H1 t(α) = 1, we have α ∪ ∂f = f − ∂ α ∪ f . Put h = α ∪ f . Then,
¡
¢
¡
¢
v(h) > v(f ) − c1 and v f − ∂h = v α ∪ ∂f ≥ v(α) + v(∂f ).
(2) Since f is continuous there exists an open¡ normal¢subgroup Hn such that the
r
composite f¯n : H0r → W → W/W≥n factors via H0 /Hn . Let fn be the composite
of f¯n with a splitting W/W≥n → W (as sets). Then, fn is a continuous cochain and
v(f − fn ) ≥ n.
¡
¢
¡
¢
r
r
7.4 Proposition. [Ta, Prop.
10]
We
have
H
H,
W
=
0
for
r
≥
1
and
H
H,
W
=0
¡
¢
¡ 0
¢
¡
¢
¡
¢
r
r
H
r
r
H
for r ≥ 1. In particular, H G, W = H Γ , W
and H G, W = H Γ, W .
65
Proof: The last statement follows from the first one and from the spectral sequences
Hj (Γ, Hj (H, )) =⇒ Hi+j (G, ) and Hj (Γ0 , Hj (H, )) =⇒ Hi+j (G, ). Let H0 := H or H.
Let f be an r–th cochain of H0 , for r ≥ 1, with values in W . Let {Hn , fn }n be as
in 7.3(2)
¡ and, for¢ each n, write hn for the continuous (r − 1)-cochain satisfying 7.3(1)
i. e., v fn − ∂hn > v(∂fn ) − c1 and v(hn ) > v(fn ) − c1 . Then, {hn } is Cauchy so that
it converges to a continuous (r − 1)–cochain h. Furthermore, v(fn − ∂hn ) ≥ n − c1 for
every n so that ∂hn → f for n → ∞. We conclude that f = ∂h as claimed.
e ≥n be the subset of Λ
e consisting of elements b such that v(b) ≥ n. Then, Λ
e ≥0
Let Λ
e ≥n is an ideal for every n ≥ 0 due to the properties of v. We write Λn
is a ring and Λ
e ≥0 /Λ
e ≥n . Assume that the following strengthening of (TS1) holds:
for the quotient Λ
(TS1’) for every c ∈ R>0 and for every open
H1 ⊆ H2 of H (resp. of H),
à normal subgroups
!
P
e H1 such that v
there exists α ∈ Λ
τ (α) ≤ c;
≥0
τ ∈H2 /H1
One then has the following variant of 7.4:
7.5 Proposition. Let W be a free Λn –module of finite rank a endowed with a continuous action of G. Then, for every c ∈ R>0 and every¡ integer
an
¢ r ≥ 1 there rexists
¡
¢ eleH
r
e
ment γc ∈ Λ≥0 of valuation v(γc ) < c such that γc ·H H, W = 0 and γc ·H H, W = 0.
7.6 Decompletion. The notation is as in 7.2. Write D(W ) := W H and D(W ) := W H .
They are closed subgroups of W endowed with the topology induced from W .
It is proven in [AB, Cor. 2.3] that (TS1) implies that there exists an open normal sube
e HW e1 ⊕ · · · ⊕ Λ
e HW ea .
group HW ⊂ H and a Λ–basis
e1 , . . . , ea of W such that W HW ∼
=Λ
(i)
For every i = 0, . . . , d and every m ≥ mW = m0,HW define the map τm,HW : W HW →
Pa
Pa
(i)
W HW by i=1 βi ei 7→ i=1 τm,HW (βi )ei . Due to (TS2) such map is independent of HW
and the basis e1 , . . . , ea and it descends to a map on D(W ) = W H . Due to (TS2)(b) it
is continuous for the topology on W HW induced from W . We then drop the index HW
and we write simply
(i)
τm
: D(W ) −−→ D(W ) for i = 0, . . . , d,
m ≥ mW .
Using (TS4) and repeating the construction above, we get similarly continuous maps
t(i)
m : D(W ) −−→ D(W )
for i = 1, . . . , d,
m ≥ mW .
For every m ≥ mM due to (TS2) we have a decomposition
(d)
D(W ) := Dm (W ) ⊕ D(0)
m (W ) ⊕ · · · ⊕ Dm (W ),
³
´
¡
¢
¡
(d) ¢¡
(d−1)
(d−1) ¢
where
:= 1 − τm D(W ) , Dm (W ) := 1 − τm
D(W )τ (d) =1 , . . .,
m
³
´
¡
¢
(0)
(0)
Dm (W ) := 1−τm
D(W )τ (d) =1,...,τ (1) =1 and Dm (W ) = D(W )τ (d) =1,...,τ (0) =1 . They
(d)
Dm (W )
m
m
m
66
m
are closed Γ–submodules of D(W ). We endow them with the induced topology. By (TS2)
the decomposition above is an isomorphism of topological Γ–modules. Similarly, we have
an isomorphism of topological Γ0 –modules
(d)
D(W ) := Dm (W ) ⊕ D(1)
m (W ) ⊕ · · · ⊕ Dm (W ),
³
´
¡
¢
¡
(d)
(d) ¢¡
(1)
(1) ¢
D(W )t(d) =1,...,t(2) =1
where Dm (W ) := 1 − tm D(W ) , . . ., Dm (W ) := 1 − tm
and Dm (W ) = D(W )t(d) =1,...,t(1) =1 are closed Γ0 –submodules of D(W ).
m
m
m
m
7.7 Proposition. There exists an integer N ≥ mW such that for every n ≥ N ,
n
n
(i)
if γip ∈ Γ the map γip −1 is bijective with continuous inverse on Dm (W ) for i = 0, . . . , d
(i)
(resp. Dm (W ) for i = 1, . . . , d).
¡
¢
¡
¢
j
j
Then,
the
maps
of
continuous
cohomology
groups
H
Γ,
D
(W
)
→
H
Γ,
D(W
)
m
¡
¢
¡
¢
and Hj Γ0 , Dm (W ) → Hj Γ0 , D(W ) are isomorphisms.
³
´
n
(i)
Proof: We deduce from the first statement that Hj Zp γip , Dm (W ) = 0 and that
³
´
n
(i)
Hj Zp γip , Dm (W ) = 0 for every j ≥ 0. We get from the Hochschild—Serre spectral
³
´
³
´
(i)
(i)
j
j
0
sequence that H Γ, Dm (W ) = 0 and H Γ , Dm (W ) = 0 for i ≥ 1. For i = 0 the
n
first statement implies that γ0p − 1 is bijective
with continuous
inverse on the group
´
³
(i)
(0)
j
j
0
H (Γ , Dm (W )). By Hochschild–Serre H Γ, Dm (W ) = 0 for i = 0 as well. The
second statement follows.
¡Ppt−s −1 ps j ¢
t
s
t
Since γip −1 = (γip −1)
γi
for t ≥ s, if γip −1 is bijective with continuous
j=0
s
(i)
(i)
m
inverse on Dm (W ) (resp. Dm (W )) also γip −1 is. Hence, it suffices to prove that γip −1
is invertible with continuous inverse.
(i)
(i)
We prove the statement for Dm (W ). The proof for Dm (W ) is similar and the dee HW e1 ⊕ · · · ⊕ Λ
e HW ea as in 7.6 and write
tails are left to the reader.
Write W HW ´∼
= Λ
³
¢
¡
H ,(i)
(i)
W
W H(i−1)
Wm W
:= 1−τm
. Due to the assumptions in 7.1 we have a lift(1)
τm
m0,HW −m0
=1,...,τm =1
m0,HW −m0
(i)
ing p
Γ ⊂ G/HW so that p
Γ and H/HW commute. Since Dm (W ) =
¡ HW ,(i) ¢H
m
by (TS2) it then suffices to prove that γip − 1 is invertible with continuous
Wm
H ,(i)
inverse on Wm W .
¡
¢
e 1 ⊕ · · · ⊕ Λe
e a by v Pa zj ej := inf{v(zj )|j = 1, . . . , a}. It defines
Extend v on Λe
j=1
e 1 ⊕ · · · ⊕ Λe
e a . Since the action of G/HW on W HW is continuous,
the weak topology on Λe
N
∼
there exists an integer N ≥ m0,H such that γ p acts trivially on W HW /W HW
=
i
W
⊕aj=1
e HW
Λ
e HW
/Λ
≥c3,HW +1 ej .
HW ,(i)
fi : Wm
≥c3,HW +1
Take m ≥ N . Following [CC1, Prop. II.6.4] define
−−→
HW ,(i)
Wm
,
fi
a
¡X
j=1
¢
zj ej :=
a
X
¡
1 − γip
m
¢−1
(zj )ej .
j=1
It is well defined, continuous, bijective and with continuous inverse (for the weak topol67
³P
´
¡
¢
¡
a
pm
pm
pm ¢
ogy) due to (TS3). Then, z − fi (1 − γi )(z) = −fi
(ej ) .
j=1 γi (zj ) 1 − γi
Write
³¡
´
m¢
HW ,(i)
HW ,(i)
gi,z : Wm
−−→ Wm
, gi,z (y) := y − fi 1 − γip (y) − z .
¡
¢
¡
¢
¡
¢
pm
Then, v fi (z) ≥ v(z)−c3,HW by (TS3)¡and v gi,0 (y) ≥ v(y)+inf{v
(1−γ
)(e
)
|i =
i
i
¢
¡
¢
1, . . . , a} − c3,HW ≥ v(y) + 1. Hence, v gi,z (y1 ) − gi,z (y2 ) = v gi,0 (y1 − y2 ) ≥ v(y1 −
y2 ) + 1. This implies that gi,z is a contracting operator for the v–adic topology so
that there exists¡a unique¢ fixed point yz . Since fi is bijective, we get that yz is the
m
m
H ,(i)
only solution of 1 − γip (y) = z. We deduce that 1 − γip is bijective on Wm W .
¢
m
n
Furthermore, since yz is the limit of the sequence gi,z
(z) and gi,z (z)−z = fi (γip (z) , we
¡
¢
¡ m ¢
¡
m ¢−1
have v(yz − z) ≥ v gi,z (z) − z ≥ v γip (z) − c3,HW . Hence, 1 − γip
is continuous
HW ,(i)
on Wm
.
We are ready to apply the considerations above in the cases of interest to us. Let S
be as in 2.3. Let M be a Zp –representation of GS . Let M ∼
= Zap ⊕bi=1 Zp /pci Zp . For A =
¡
¢
b ⊗ K, A , A† , A
e or A
e † , then M ⊗ A = Aa ⊕b A/pci A . We consider M ⊗ A
R
V
R
R
R
R
Zp
i=1
Zp
as topological module for the product topology considering on A the topology induced
b ⊗ K or form the weak topology on A and considering
from the p–adic topology on R
V
R
on each A/pci A the quotient topology.
7.8 Proposition. We have:
n
o
b ⊗ K with v(b) := min α ∈ Q| b ∈ R
b satisfies (TS1). Furthere := R
1) the ring Λ
V
pα
more, the following holds
(TS1’) for every c ∈ R>0 and every open normal subgroups H1 ⊆ H2 of H (resp. of H),
P
b H1 such that
there exists α ∈ R
τ (α) is an element of V of valuation ≤ c;
τ ∈H2 /H1
e (0,r] with v = wr satisfies (TS1);
e := A
2) for every r ∈ Q>0 the ring Λ
R
e /pN +1 A
e with v = v≤N satisfies (TS1);
e := A
3) for every N ∈ N the ring Λ
E
R
R
¡
¢
b ⊗ K) = 0 for every i ≥ 1;
4) Hi HS , M ⊗Zp (R
V
´
³
¡
¢
†
i
e
e
= 0 for every i ≥ 1;
5) (a) Hi HS , M ⊗Zp A
=
0,
(b)
H
H
,
M
⊗
A
S
Zp
R
R
³
´
¡
¢
i
e
e † = 0 for every i ≥ 1.
6) (a) Hi HS , M ⊗Zp A
=
0,
(b)
H
H
,
M
⊗
A
S
Z
p
R
R
Proof: For open normal subgroups H1 ⊂ H2 of H claim (1) follows from [AB, Prop 3.4
& Rmk. 3.5] and claim (2) follows from [AB, Prop. 4.6].
0
0 2
Let H1 ⊂ H2 be normal subgroups of H.£ They
correspond to extensions R∞
⊂ S∞
⊂
¤
0 1
0
−1
S∞ which are finite and Galois over R∞ p
of degree d1 and d2 respectively. In par\
ticular, there exists an extension V∞ ⊂ V∞ , finite and Galois after inverting p, such that
68
they arise by taking the normalization of the base change of extensions of R∞ ⊗V∞ V∞ \
finite and Galois after inverting p of degree d1 and d2 respectively. This is equivalent to
require that there exist open normal subgroups of H1 ⊂ H2 of H such that H1 ∩H = H1 ,
H2 ∩ H = H2 and H1 /H2 ∼
= H1 /H2 . Then, (TS1) (resp. (TS1’)) for H1 ⊂ H2 implies
(TS1) (resp. (TS1’)) for H1 ⊂ H2 . Hence, (1) and (2) follow.
(3) Let H1 ⊂ H2 be open normal subgroups of H (resp. H) and let αr be an element
e (0,r] satisfying (TS1). If we write αr := P pk [zk ] with zk ∈ E
e , since wr (αr ) =
of A
k
R
R
inf{rvE (zk ) + k} ≥ −c1 , we get that vE (zk ) ≥ −c1r−k . Hence, for every N ∈ N we have
≤N
vE
(αr ) ≥ −c1r−N . In particular, the claim follows.
(4) It follows from 7.4.
e † /pN +1 A
e† = A
e /pN +1 A
e , claims (5) and (6) follow from 7.4 for M
(5)–(6) Since A
R
R
R
R
N +1
a GS –representation which is free as Zp /p
Zp –module. In particular, (5) and (6) hold
for torsion representations.
e or A
e † and let H = HS
We may then assume that M is torsion free. Let A := A
R
R
or HS . Let f be an i–th cocycle of H with values in M ⊗Zp A, continuous for the weak
e † , then A
e † = ∪r A
e (0,r] is open in A
e for the weak
e (0,r] and A
topology. If A = A
R
R
R
R
R
e + . In particular, since H is compact in this case f takes
topology since it contains A
R
e (0,r] for some r.
values in M ⊗Zp A
R
Since f is continuous, for every n ∈ N there exists an open
¡ normal subgroup Hn¢
i
¯
of H such that the composite fn : H → M ⊗Zp A → M ⊗Zp A/(Un+1,[ c1 +n ]+n ∩ A)
r
¡
¢i
factors via H0 /Hn ; see 2.10 for the notation Un,h . Here, for u ∈ Q we write [u]
for the smallest positive¡ integer bigger or equal¢ to u. Let fn be the composite of f¯n
with a splitting M ⊗Zp A/(Un+1,[ c1 +n ]+n ∩ A) → M ⊗Zp A (as sets). Then, fn is
r
a continuous
i–cochain
and
we
also
¡
¢ have fn ≡ f , if viewed as cochains with values
in M ⊗Zp A/(Un+1,[ c1 +n ]+n ∩ A) . For every n let hn := αr ∪ fn be the continuous
r
(i − 1)–cochain defined as in the proof of 7.3(1). The computations
in ¢loc. cit. show
¡
that fn − ∂hn ≡ α ∪ ∂fn ≡ 0 and hn+1 ≡ hn in M ⊗Zp A/(Un+1,n ∩ A) . Then, {hn }
is Cauchy for the weak topology and {∂hn }n converges to f for the weak topology. In
e
particular, hn converges to a continuous (i − 1)–cochain h with values in M ⊗Z A
p
R
and ∂h = f .
e , this concludes the proof. If A = A
e † , since pn A
e ∩A
e (0,r] = pn A
e (0,r]
If A = A
R
R
R
R
R
pr
and since wr (p) = 1 and wr (π) ≥ p−1
by [AB, Prop. 4.3(d)], we conclude that {hn }
e (0,r] is complete and separated for
is Cauchy for the wr –adic topology as well. Since A
R
the wr –adic topology by [AB, Prop. 4.3(c)], we conclude that h in fact takes values
e (0,r] . The conclusion follows.
in M ⊗Zp A
R
b £p−1 ¤. Before passing to the (ϕ, Γ)–modules, we first show that
7.9 Sen’s theory for R
b £p−1 ¤–representations. These results are due to Sen
our theory applies in the case of R
[Se], in the classical case of a DVR with perfect residue field, and are due to [Br]
69
for a DVR with imperfect residue field. The key point is of course to show that 7.7
applies. This follows essentially from results proven in [AB]. We review some of the
basic definitions and properties from loc. cit.
d
L
Let S be a R–algebra as in 2.1. Fix m0,S ∈ N such that pm0,S
Zp γi ⊆ ΓS . Then, for
i=1
every m ≥ m0,S , the ring Sm+1 [p−1 ] is a free Sm [p−1 ]-module of rank pd+1 (resp. Sm+1 ·
W [p−1 ] is a free Sm · W [p−1 ]-module of rank pd ). For every i ∈ {1, . . . , d} and every
n ∈ N, define
1
1
1 ¤
£ 1n
n
(i)
pn
pn
, Ti+1
, . . . , Tdp .Vn
Sn,∗ = S T1p , . . . , Ti−1
and
1
1 ¤
1
0
£ 1n
n
(i)
pn
pn
, Ti+1
, . . . , Tdp .
Sn,∗ = S 0 T1p , . . . , Ti−1
1 ¤
£ 1n
S (i)
n
(i)
(i)
For i = 0, one puts Sn,∗ = S T1p , . . . , Tdp . Eventually, let S∞,∗ =
Sn,∗ and
n∈N
0
S 0 (i)
(i)
S∞,∗ =
Sn,∗ . For every i ∈ {0, . . . , d} and m ∈ N, one defines
n∈N
(i)
Sb∞,m,K
³
´
(0)

−1
 Sd
·
V
] if i = 0,
∞,∗
m [p
= ³
´
(i)

−1
 Sd
] if i ∈ {1, . . . , d},
∞,∗ · Sm [p
where the hat stands for p–adic completion. Similarly, for i ∈ {1, . . . , d} and m ∈ N,
put
´
³d
0
0 (i)
(i)
Sb
= S∞,∗ .Sm [p−1 ].
∞,m,K
0
(i)
(i)
Note that Sb∞,m,K ⊂ Sb∞ [p−1 ] for every i ∈ {0, . . . , d} and m ∈ N and that Sb∞,m,K ⊂
−1
0
d
S
]. For n ≥ m ≥ m0 and x ∈ Sn [p−1 ], one puts
∞ [p
(
(i)
τm
(x)
=
1
(x)
(0)
pn−m TrSn /Sn,∗
·Vm
1
(x)
(i)
pn−m TrSn /Sn,∗
·Sm
if i = 0,
if i ∈ {1, . . . , d}.
For n ≥ m ≥ m0 and x ∈ Sn0 [p−1 ] and every i = 1, . . . , d define
t(i)
m (x) =
1
pn−m
TrS 0 /S 0 (i) .S (x).
n
n,∗
m
Such maps do not depend on n for n À 0 so that they are defined on S∞ [p−1 ]
0
(resp. S∞
[p−1 ]).
(i)
7.9.1 Proposition. For every i = 0, . . . , d and every m ≥ m0 the map τm is continu(i)
ous for the p–adic topology so that it extends to a unique Sb∞,m,K –linear map
(i)
(i) d −1
τm
: S∞ [p ] −−→ Sb∞,m,K .
70
(i)
Analogously, for every i = 1, . . . , d and every m ≥0 m0 the map tm is continuous for the
(i)
p–adic topology so that it extends to a unique Sb∞,m,K –linear map
0
(i)
−1
0
d
t(i)
] −−→ Sb∞,m,K .
m : S∞ [p
(i)
(i)
(i)
Proof: The claim for τm follows from [AB, Lem. 3.8]. Since tm is obtained from τm by
(i)
base–change from V∞ to W , the claim for tm follows as well.
£ −1 ¤
¡ b £ −1 ¤¢HS
£ −1 ¤
¡ b £ −1 ¤¢HS
0
d
d
p
=S
and R
p
=S
due to 2.9. FurtherNote that R
∞ p
∞ p
more,
(i)
(i)
7.9.2 Proposition. The rings Sb∞,m,K and the applications τm satisfy (TS2) and
0
(i)
(i)
with the applications tm satisfy (TS4).
(TS3). The rings Sb
∞,m,K
Proof: The fact that (TS2) and (TS3) hold is proven in [AB, Prop. 3.9] and in [AB,
(i)
(i)
Prop. 3.11]. Axiom (TS4) follows from this since tm is obtained from τm by base–change
c
from Vc
∞ to W and taking p–adic completions.
´
£ −1 ¤
S ³T b(i)
7.9.3 Lemma. We have m
S
=
S
p . Analogously, we also have
∞
i ∞,m,K
´ S
£ −1 ¤
S ³T b0 (i)
0
c
.
m Sm p
i S∞,m,K =
m
Proof: The first claim follows from [AB, Lem. 3.12]. The second is proven as in loc. cit.
b £p−1 ¤. Due to 7.8 we
Let M be a Zp –representation of GS and let Q := M ⊗Zp R
know that the natural maps
¡
¢
¡
¢
Hn ΓS , QHS −→ Hn (GS , Q) and Hn Γ0S , QHS −−→ Hn (GS , Q)
are isomorphisms. Furthermore,
£
¤
7.9.4 Theorem. There exists a finitely generated, projective S∞ p−1 –submodule
∼ HS and the natural map
d
N ⊂ QHS , stable under ΓS , such that N ⊗S∞ S
∞ =Q
¡
¢
Hn (ΓS , N ) −−→ Hn ΓS , QHS
´
³S
0
HS
0
0
0 ∼
c
d
is an isomorphism. Furthermore, if N := N ⊗S∞
S
∞ =Q
0
m Sm , then N ⊗∪m S
c
m
and the natural map
¡
¢
¡
¢
Hn Γ0S , N 0 −−→ Hn Γ0S , QHS
is an isomorphism.
Proof: Put N to be the
base
as defined in 7.6, via the natural
£ −1
¤ change of Dm (Q),
T b(i)
0
map i S∞,m,K → S∞ p . Similarly, put N to be the base change of Dm (Q) via
£
¤
T b(0 i)
S c
S
→
S 0 p−1 .
i
∞,m,K
m
m
71
b £p−1 ¤–basis e , . . . , e of M ⊗ R
b £p−1 ¤
Due to [AB, Thm. 3.1] there exists an R
1
a
Zp
stable under an open subgroup HQ of HS , normal in GS . Let S∞ [p−1 ] ⊂ T∞ [p−1 ] be
the corresponding Galois extension. Then Dm (Q) (resp. Dm (Q)) is by construction
T (i)
the set of HS /HQ –invariants (resp. HS /HQ –invariants) of the free i Tb∞,m,K –module
T b0 (i)
−1 ∼
(resp.
T
–module) with basis e1 , . . . , ea . By [An, Cor. 3.11] we have Tc
]=
∞ [p
i
∞,m,K
−1
−1
−1
d
d
S
] ⊗S∞ T∞ so that the extension S
] ⊂ Tc
] is finite, étale and Galois
∞ [p
∞ [p
∞ [p
−1 ∼ d
0
0 [p−1 ] ⊗
c
with group HS /HQ . Similarly, one proves that T∞ [p ] = S∞
S∞ T∞ so that the
−1
−1
0
0
d
c
extension S∞ [p ] ⊂ T∞ [p ]³is also finite,
étale and Galois with group HS /HQ . Then,
´
S c
0
0
0
the claims that N = N ⊗S∞
m Sm and that N and N satisfy the requirements of
the theorem follow as in the proof of 7.7 and étale descent.
e † . We recall some facts proven in [AB, §4] needed in
e and A
7.10 Sen’s theory for A
R
R
e and A
e † . Let S be a
order to prove that (TS2) and (TS3) hold also for the rings A
R
R
R–algebra as in 2.1. For every i = 0, . . . , d let
h£ ¤ 1
£
¤ 1n £
¤ 1n
£ ¤ 1n i
(i)
n
AS (∞) := ∪n AS x0 p , . . . , xi−1 p , xi+1 p , . . . , xd p
(i)
(i)
e
and let AS (∞) be the closure of AS£ (∞)
¤ in AS∞ for the weak topology. Here, we
write x0 for the element ² and we write xi for the Teichmüller lift of xi . By construction
s
it is stable under GR if i 6= 0 and it is stable under γ0p , for s À 0, if i = 0. Then,
7.10.1 Proposition. For every m ≥ 0 and every i = 0, . . . , d there exists a homomorphism
h
i
1
(i)
(i)
(i)
m
e
p
τm = τS,m : AS∞ −→ AS (∞) [xi ]
,
called the generalized trace à la Tate, such that
h
i
h
i
1
1
(i)
(i)
m
m
p
p
(i) it is AS (∞) [xi ]
-linear and it is the identity on AS (∞) [xi ]
;
(ii) it is continuous for the weak topology;
(iii) it commutes with the action of Gal (S∞ /R) if i 6= 0 and it commutes with the action
Z
(j)
(i)
(i)
(j)
of γ0 p if i = 0. Furthermore, τm ◦ τn = τn ◦ τm for m, n ∈ N and i, j ∈ {0, . . . , d}
(d)
(0)
(1)
and τm ◦ τm ◦ · · · ◦ τm commutes with the action of GR ;
(i)
(i)
(iv) for every n ∈ Z such that m + n ≥ 0 we have ϕn ◦ τm+n = τm ◦ ϕn ;
(v) it is compatible for varying S i. e., given a map of R–algebras S → T as in 2.1 we
(i)
e S coincides with τ (i) ;
have that τm,T restricted to A
∞
m,S
e
(vi) there exists rS ∈ Q>0 such that (TS2) and (TS3) hold for every 0 < r < rS with
h Λ1:=i
(0,r]
e (0,r] ∩A(i) (∞) [xi ] pm
e (i)
A
and v = wr , taking Λ
m for m ≥ 0 to be the closure of A
S∞
R
in
e (0,r]
A
R
and taking
(i)
τS,m
S
for m ≥ 0 to be the restriction of the maps defined in (i);
72
e := A /pN +1 A and v = v≤N ,
(vii) for every N ∈ N (TS2) and (TS3) hold for Λ
E
Ri
R
h
1
(i)
(i)
(i)
N +1 (i)
m
e
p
taking Λm for m ≥ 0 to be AS (∞)/p
AS (∞) [xi ]
and taking τS,m for m ≥
0 to be the reduction modulo pN +1 of the maps defined in (i);
m
(viii) there exists mS ∈ N such that for m ≥ mS the map γip − 1 is an isomorphism
¡
¢
(i) ¢¡ e
on 1 − τm A
S∞ with continuous inverse (for the weak topology);
Proof: Claims (i)–(v) follow from [AB, Prop. 4.15]. The verification of (TS2) (resp. of
(TS3)) in (vi) follows from [AB, Prop. 4.24] (resp. [AB, Prop. 4.30&Prop. 4.32]). The
fact that (TS2) holds in (vii) follows from (ii) and the fact that the weak topology
≤N
–adic topology.
on AR /pN +1 AR is the vE
¡
¢
m
(i) ¢¡ e
Since 1 − τm AS
is p–adically complete and separated, the fact that γ p − 1
i
∞
is bijective can be verified modulo p and follows
from
[AB,¢ Prop. 4.30&Prop. 4.32].
¡
¡ In
(i) ¢¡ e
pm
−1
N +1
particular, (γi − 1) is bijective on p
1 − τm AS∞ and, consequently, on 1 −
¢
(i) ¢¡ e
e + is
eS
τm AS∞ /pN +1 A
for every N ∈ N. Note that for every h ∈ N the group π h A
∞
S∞
e (0,r] such that wr (x) ≥ hwr (π). By (TS3)
contained in the subgroup of elements x ∈ A
S∞
e (0,r] there exist constants c3,S and c4,S such that for every element z in π h (1 −
for A
R
h
i
1
(i) e +
(i)
e + ) one has
τm )AS∞ (resp. AS (∞) [xi ] pm ∩ π h A
S∞
¡
¢
¡
¢
m
m
rvE (zk )≤N (1 − γip )−1 (z) ≥ wr (1 − γip )−1 (z) − N ≥ wr (z) − c3,S − N
and, respectively,
¡
¢
¡
¢
m
m
rvE (zk )≤N (1 − γip )(z) ≥ wr (1 − γip )(z) − N ≥ wr (z) + c4,S − N.
¡
¢
m
c
+N
≤N
≤N
Since wr (z) ≥ rvE
(z), we conclude that vE (zk )≤N (1−γip )−1 (z) ≥ vE
(z)− 3,Sr
¡
¢
m
c
−N
≤N
and vE (zk )≤N (1 − γip )(z) ≥ vE
(z) + 4,Sr . Hence, (vii) and (viii) follow.
(i)
Similarly,
S as in 2.1, for every
i = 1, . . . , d let A0S (∞) :=
h£ ¤ 1given £a R–algebra
1
1
1 i
¤
£
¤
£
¤
n
n
n
n
∪n A0S x1 p , . . . , xi−1 p , xi+1 p , . . . , xd p and let A0S (i) (∞) be the closure of
(i)
e S 0 for the weak topology. Note that since i ≥ 1 the ring A (∞) contains
(∞) in A
S
∞
h£ ¤ 1 i
pn
e V by [AB, Cor. 4.13].
the closure for the weak topology of ∪n AV x0
which is A
∞
A0S
(i)
(i)
0 (i)
e
Then, AS (∞) ⊗A
e V∞ AW maps to AS (∞) and the image is dense for the weak topoleS ⊗
e
e 0 and AS ⊗A A
e W injects
ogy. Recall that A
∞
V
e AW injects and is dense in AS∞
A
V∞
(i)
A0S
and is dense in
by 2.12. Hence, for every i = 1, . . . , d we may base–change τS,m
e
via ⊗A
e V∞ AW and complete with respect to the weak topology. We obtain a map
i
h
1
(i)
(i)
(i)
m
0
e
p
0
.
tm = tS,m : AS∞ −→ AS (∞) [xi ]
73
e S0 ,
7.10.2 Proposition. The analogues of the statements (i)–(viii) of 7.10.1 hold for A
∞
h
i
1
(i)
(i)
m
0
p
the rings AS (∞) [xi ]
and the maps tm .
(i)
Proof: The proposition follows from 7.10.1, from the construction of tm and density
e (0,r] maps to A
e S 0 (0,r] and has dense image
e (0,r] ⊗ (0,r] A
arguments. For (vi) note that A
W
S∞
∞
e
A
for the wr –adic topology by 2.12(d).
(0)
V∞
(d)
e τ0 =1,...,τ0
7.10.3 Lemma. We have A
S∞
=1
(d)
¡
¢ (1)
e S 0 t0 =1,...,t0 =1 = A0 .
= AS and A
S
∞
£ ¤ αn1
£ ¤ αnd
Proof: By [AB, Cor. 4.13] the monomials { x0 p · · · , xd p }0≤αi <pn form an AS –
h£ ¤ 1
£ ¤ 1n i
n
e S for the
basis of ϕ−n (AS ) = AS x0 p , . . . , xd p and ∪n ϕ−n (AS ) is dense in A
∞
(0)
weak topology. In particular, ∪n ϕ−n (AS )τ0
(d)
=1,...,τ0
=1
(1)
(d)
e τ0 =1,...,τ0
is dense in A
S∞
=1
and
(1)
(d)
¢
¡
eW)
e S 0 t0 =1,...,t0 =1 respectively. From
∪n ϕ−n (AS ⊗AV A
is dense in A
∞
α ¢
(i)
(i) ¡
the fact that τ0 [xi ] pn = 0 for 0 < α < pn , we get that ∩di=0 AS (∞) = AS is dense
(d)
(0)
(d)
¡
¢ (1)
e τ0 =1,...,τ0 =1 and AS ⊗A A
e W is dense in A
e S 0 t0 =1,...,t0 =1 respectively. The
in A
V
S∞
∞
conclusion follows from 2.12.
(0)
(d)
t0 =1,...,t0 =1
(i)
(i)
7.10.4 The operators τm and tm on (ϕ, Γ)–modules. Let S be as in 2.3. Let M be a Zp –
p
ci e
ea b e
e ∼
representation of GS . If M ∼
= Zap ⊕i=1 Zp /pci Zp , then M ⊗Zp A
R = AR ⊕i=1 AR /p AR
e the weak topology.
and on the latter we have the product topology considering on A
R
¡
¢HS
¡
¢
e
e
e HS .
e
Recall that we have defined D(M ) = M ⊗Zp AR
and D(M ) := M ⊗Zp A
R
e
e
We then define the weak topology on D(M
) and D(M
) to be the topology induced from
e
e .
e
the inclusions D(M
) ⊂ D(M
) ⊂ M ⊗Zp A
R
e /pN +1 A
e satisfies (TS1)–(TS4)
e := A
Assume first that M is killed by pN +1 . Since Λ
R
R
(i)
(i)
due to 7.10.1&7.10.2(vii), we may apply 7.6 and define the operators τm (resp. tm )
e
e
on D(M
) and on D(M
) and we get decompositions
e
e m (M ) ⊕ D
e (0) (M ) ⊕ · · · ⊕ D
e (d) (M )
D(M
) := D
m
m
and
e
e m (M ) ⊕ D
e (1)
e (d)
D(M
) := D
m (M ) ⊕ · · · ⊕ Dm (M ).
(i)
(i)
e
e
) and on D(M
) and the
By devissage we get the operators τm (resp. tm ) on D(M
decomposition above for any torsion GS –representation M .
n
n
e
e
e
e
If M is torsion free, D(M
) := lim D(M/p
M ) and D(M
) := lim D(M/p
M)
∞←n
∞←n
by 2.16. Using the construction for the torsion case and passing to the limit, we get the
(i)
(i)
e
e
operators τm (resp. tm ) on D(M
) and on D(M
) and the decomposition above.
74
7.10.5 Proposition. Let S be as in 2.3 and let M be a Zp –representation of GS .
Then,
e 0 (M ) = D(M ) and D
e 0 (M ) = D(M );
1) D
(i)
(i)
e
e
2) the operators τm (resp. tm ) on D(M
) (resp. D(M
)) are continuous for the weak
topology;
(i)
(i)
e † (M ) (resp. D
e † (M )). In particular, we have
3) the operators τm (resp. tm ) preserve D
e † (M ) := D
e † (M ) ⊕ D
e †,(0) (M ) ⊕ · · · ⊕ D
e †,(d) (M )
D
m
m
m
and
e † (M ) := D
e †,(1)
e †,(d)
e †m (M ) ⊕ D
D
m (M ) ⊕ · · · ⊕ Dm (M ),
where the modules on the right hand side are defined as in 7.6;
n
n
e (i)
e (i)
4) if γip ∈ ΓS (resp. Γ0S ) then γip − 1 is bijective on D
m (M ) (resp. Dm (M )) with
continuous inverse (for the weak topology);
n
n
e †,(i)
e †,(i)
5) if γip ∈ ΓS (resp. Γ0S ) then γip − 1 is bijective on D
m (M ) (resp. Dm (M )) with
continuous inverse (for the weak topology);
e † (M ) = D
e m (M ) ∩ D
e † (M ), D
e †,(i)
e (i)
e†
e †,(i)
e (i)
6) D
m (M ) = Dm (M ) ∩ D (M ), Dm (M ) = Dm (M ) ∩
m
e † (M ) = D† (M ) and
e † (M ) and D
e †m (M ) = D
e m (M ) ∩ D
e † (M ). In particular, D
D
0
e † (M ) = D† (M ).
D
0
¡Ppt−s −1 ps j ¢
t
s
for t ≥ s, it suffices to prove the bijectivity
Proof: Since γip −1 = (γip −1)
γi
j=0
and the existence of a continuous inverse in (4) and (5) for n À 0. Assuming (3), we
e†
e (i)
e †,(i)
e†
e (i)
e †,(i)
have D
m (M ) = Dm (M ) ∩ D (M ) and Dm (M ) = Dm (M ) ∩ D (M ). Then, (4) and
the bijectivity in (5) imply the existence of a continuous inverse in (5). Claim (6) follows
from the others. For every m and n ∈ N the maps
ϕn
∼ e
e
ϕ ⊗ 1: D(M
) ⊗ AS −→
D(M ),
n
AS
and
ϕn
∼ e†
e † (M ) ⊗ A† −→
ϕ ⊗ 1: D
D (M )
S
n
A†S
ϕn
ϕn
∼ e†
e † (M ) ⊗ A0† −→
D (M )
ϕn ⊗ 1: D
S
∼ e
e
ϕn ⊗ 1: D(M
) ⊗ A0S −→
D(M ),
A0S
A0†
S
¡
¢ (i)
are isomorphisms by 2.16(i). It follows from 7.10.1 and 7.10.2(iv) that ϕn ⊗ 1 ◦τm+n =
¡
¢
¡
¢
¡
¢
(i)
(i)
(i)
τm ◦ ϕn ⊗ 1 and ϕn ⊗ 1 ◦ τm+n = tm ◦ ϕn ⊗ 1 and that ϕn ⊗ 1 defines an ison
n
e m+n (M ) ⊗ϕ AS (respectively from D
e (i) (M ) ⊗ϕ AS , respectively
morphism from D
m+n
AS
AS
n
n
e m (M ) (respectively
e m+n (M ) ⊗ϕ 0 A0 , respectively from D
e (i) (M ) ⊗ϕ AS ) to D
from D
S
(i)
e m (M )).
D
AS
e (i)
D
m (M ),
m+n
AS
e m (M ),
Hence, it suffices to prove claims (2), (4) and (5) for m À 0
D
to deduce it for every m ∈ N.
n
n
e
e
e
e
Since D(M
) := lim D(M/p
M ) and D(M
) := lim D(M/p
M ) by 2.16 and the operators
(i)
τm
(resp.
∞←n
(i)
tm ) are
∞←n
n
n
e
e
constructed on each D(M/p
M ) (resp. D(M/p
M )) passing
75
to the limit, to prove (1), (2) and (4) one may assume that M is a torsion representation.
By devissage one may also assume that M is a free Z/pN +1 Z–module for some N ∈ N.
(i)
(i)
Note that τm and tm commute with the Galois action and are compatible with exten0
0
sions S∞ ⊂ T∞ and S∞
⊂ T∞
by 7.10.1 and 7.10.2. Due to 2.11, 2.12 and 2.13 and
étale descent, it then suffices to prove (1), (2) and (4) passing to an extension S∞ ⊂ T∞
¡
¢
e
e HT instead of D(M
in R finite, étale and Galois after inverting p i. e., for M ⊗Zp A
)
R
¡
¢HT
e
e
and M ⊗Z A
instead of D(M
). We may then assume that HT , and hence HT ,
R
p
act trivially on M . Claim (1) follows then from 7.10.3. Claim (2) follows from 7.10.1(ii)
e /pN +1 A
e satise := A
and 7.10.2(ii). Claim (4) for m À 0 follows from 7.7 since Λ
R
R
fies (TS1)–(TS4). This concludes the proof of (1), (2) and (4).
If M is a torsion representation, then (3) and the bijectivity in (5) follow from 2.16(ii’).
Assume that M is free of rank n. Thanks to 2.11, 2.12 and 2.13 and étale descent we may
pass to an extension S∞ ⊂ T∞ in R finite, étale and Galois after inverting p. By [AB,
Thm. 4.42] there exists such an extension S∞ ⊂ T∞ so that D† (M ) ⊗A† A†T is a free A†T –
S
module of rank n. Fix a basis {e1 , . . . , en } and choose r ∈ Q>0 such that these elements
e (0,r] . By 2.16(iii’) we have M ⊗Z A
e† = A
e † e1 ⊕ · · · A
e † en . For every s <
lie in M ⊗Zp A
p
R
R
R
R
e (0,s] (M ) := W HT
e (0,s] . Define D
min{r, rT }, see 7.10.1(vi)&7.10.2, let Ws := M ⊗Z A
p
s
R
e † (M ) = ∪s D
e (0,s] (M ) and D
e † (M ) = ∪s D
e (0,s] (M ) := WsHT . Then, D
e (0,s] (M ).
and D
(i)
e (0,s] satisfies (TS1)–(TS4) by 7.10.1(vi)&7.10.2. Hence, the operators τm
Note that A
R
(i)
e (0,s] (M ) (resp. D
e (0,s] (M )) and we further have decompositions
(resp. tm ) preserve D
e (0,s],(d) (M )
e (0,s],(0) (M ) ⊕ · · · ⊕ D
e (0,s] (M ) := D
e (0,s] (M ) ⊕ D
D
m
m
m
and
e (0,s] (M ) := D
e (0,s]
e (0,s],(1) (M ) ⊕ · · · ⊕ D
e (0,s],(d)
D
(M )
m (M ) ⊕ Dm
m
n
by 7.6. This proves (3) in the overconvergent case. It follows from 7.7 that if γip ∈ ΓS
n
e (0,s],(i)
e (0,s],(i)
(resp. Γ0S ) then γip − 1 is bijective on D
(M ) (resp. D
(M ) for m À 0. We
m
m
conclude that the bijectivity in (5) holds. Claim (5) follows.
We deduce from 7.7, 7.8 and 7.10.5 the following theorem which summarizes the
results proven so far:
7.10.6 Theorem. The natural maps
n
H
n
H
¡
µ
¶
¢
¡
¢
n
n
e
e
ΓS , D(M ) −−→ H ΓS , D(M ) −−→ H GS , M ⊗ AR ,
Zp
¡
¢
Γ0S , D(M )
µ
¶
¡ 0
¢
n
e
e
−−→ H ΓS , D(M ) −−→ H GS , M ⊗ AR ,
n
Zp
µ
¶
¡
¢
¡
¢
†
†
n
†
n
e
e
H ΓS , D (M ) −−→ H ΓS , D (M ) −−→ H GS , M ⊗ AR
n
Zp
76
and
¶
µ
¡ 0
¢
¡ 0
¢
†
†
n
†
n
e
e
H ΓS , D (M ) −−→ H ΓS , D (M ) −−→ H GS , M ⊗ AR
n
Zp
are all isomorphisms.
7.10.7 The structure of δ–functors. The cohomology groups appearing in 7.10.6 are δ–
functors i. e., given an exact sequence 0 → W1 → W2 → W3 → 0 of GS –representations
we get an associated long exact sequence of cohomology groups.
For the cohomology groups on the right hand side it suffices to construct a left inverse
e → W3 ⊗Z A
e which is continuous for the weak
(as sets) of the surjection W2 ⊗Zp A
p
R
R
e † is contained in W2 ⊗Z A
e† .
topology and has the property that the image of W3 ⊗Zp A
p
R
R
We consider two cases.
The first case is that W3 is a free Zp –module. Any left inverse W2 → W3 as Zp –
modules induces by extension of scalars the claimed inverse.
The second case is that W3 is a torsion Zp –module. By devissage one can suppose
that W3 is a free Z/pn Z–module. The splitting is defined lifting a basis of W3 to W2 and
e† → A
e /pn A
e . Since A
e /pn A
e =
constructing a left inverse ζn to the projection A
R
R
R
¡R
¢ R
e ) we define such inverse sending (a0 , . . . , an ) 7→ a0 , . . . , an , 0, . . . .
Wn (E
R
For the cohomology groups of (ϕ, Γ)–modules we argue as follows. Due to 7.10.5
e ∗ (Wi ) and D∗ (Wi ) ⊂
we have continuous right inverses of the inclusions D∗ (Wi ) ⊂ D
e ∗ (Wi ), where ∗ stands for nothing or †. Such inverses are compatible with the morD
phisms of (ϕ, Γ)–modules induced by the map W2 → W3 . Thus, it suffices to construct
e 2 ) → D(W
e 3 ) sending D
e † (W3 ) to D
e † (W2 ) (resp. to the map
a left inverse to the map D(W
†
†
e 2 ) → D(W
e 3 ) sending D
e (W3 ) to D
e (W2 )). As before we distinguish two cases.
D(W
The first is when W3 is a free Zp –module. Due to [AB, Thm. 4.42] there exists an
extension S ⊂ T , finite and étale after inverting p, so that D† (W3 ) ⊗A† A†T is a free
S
A†T –module. Since A†S ⊂ A†T is finite and étale by 2.11 we deduce that D† (W3 ) is a
projective A†S –module so that we can find a continuous left inverse to the surjection
D† (W2 ) → D† (W3 ) as A†S –modules. Thanks to 2.16 we conclude the construction in
this case simply extending scalars.
The second case is when W3 is a torsion Zp –module. By devissage we may assume
that W3 is a free Z/pn Z–module. Let S ⊂ T be an extension such that GT acts trivially
on W3 . Then, D(W3 ) ⊗AS AT = W3 ⊗ AT is a free AT /pn AT –module. Due to 2.12
and 2.16 the various (ϕ, Γ)–modules associated to Wi as GT –representations, for i = 2
and 3, are defined by the corresponding (ϕ, Γ)–modules as GS –representations extending
scalars via the finite and étale extension A†S ⊂ A†T . A splitting as A†S –modules to
the inclusion A†S ⊂ A†T produces at the level of (ϕ, Γ)–modules a left inverse ζT /S
to the process of extending scalars. To conclude it suffices to construct the inverse
77
considering GT –representations composing then with ζT /S . One gets the seeked for map
e /pn A
e† → A
e
lifting a basis of W3 to W2 and using the left inverse to the projection A
R
R
R
ne
e
e
constructed above. This map has the required properties since ζn : AR /p AR → AR
e T /pn A
e T to A
e † and A
e T 0 /pn A
e T 0 to A
e † 0.
sends A
∞
∞
T∞
∞
∞
T∞
8 Appendix II: Artin–Schreier theory.
The aim of this section is to prove the following:
e ,A
e † , A , A† A0 and A0† is surjective and
8.1 Proposition. The map ϕ − 1 on A
R
R
R
R
R
R
its kernel is Zp . Furthermore, the exact sequence
ϕ−1
e −−→
e −−→ 0
0 −−→ Zp −−→ A
A
R
R
(8.1.1)
¡
¢
e →A
e (as sets) so that so that σ A ⊂ A ,
admits a continuous right splitting σ: A
R ¡
R
R
¡ †¢
¡
¢
¡
¢ R
¢
e
e † , σ A† ⊂ A† , σ A0 ⊂ A0 and σ A0† ⊂ A0† .
σ A
⊂
A
R
R
R
R
R
R
R
R
¡ (0,r] ¢
¡ (0,r] ¢
(0,r/p]
e
Proof: Note that by [AB, Prop. 4.3] we have ϕ AR
⊂ AR
⊂
and ϕ A
R
(0,r/p]
e† ) ⊂ A
e † . We know from 2.10
e
A
so that (ϕ − 1)(A† ) ⊂ A† and (ϕ − 1)(A
R
R
R
R
R
e and that A /pA = E ,
and 2.12 that p is a regular element of AR , A0R and A
R
R
R
R
e /pA
e = E
e and that, thanks to 2.9, the image of E ⊗E E
e W → A0 /pA0 is
A
V
R
R
R
R
R
R
e , A and A0 is
dense for the π–adic topology. In particular, to prove that ϕ − 1 on A
R
R
R
e is Fp
surjective and its kernel is Zp , it suffices to prove that the kernel of ϕ − 1 on E
R
e and on the completion of E ⊗E E
e W for
and that ϕ − 1 is surjective on ER , on E
V
R
R
e is an integral domain by 2.6(5), the kernel of ϕ − 1 is Fp .
the π–adic topology. Since E
R
The other claim follows from 8.1.1.
e † , it
e † , to conclude that ϕ − 1 is surjective on A† and on A
Since A†R = AR ∩ A
R
R
R
†
e
e
e† .
suffices to prove that for every z ∈ A the solutions y ∈ A of (ϕ − 1)(y) = z lie in A
R
R
R
e† ,
Since any such solutions differ by an element of Zp and the latter is contained in A
R
e † and choose r ∈ Q>0 so
e † . Let z ∈ A
it suffices to show that ϕ − 1 is surjective on A
R
R
P
(0,r]
k
e
e
that z ∈ AR . Write z = k [zk ]p with zk ∈ ER . Then, putting c = min{−1, wr (z)},
we have rvE (zk ) + k ≥ c for every k ∈ N i. e., vE (zk ) ≥ c−k
r . By 8.1.1 there exists yk ∈
e such that (ϕ − 1)(yk ) = zk and 0 ≤ vE (yk ) ≤ vE (zk ) if vE (zk ) ≥ 0 or vE (yk ) =
E
R
P k
vE (zk )
e (0,pr]
if vE (zk ) < 0. In any case, vE (yk ) ≥ c−k
k p [yk ] lies in AR
p
pr . Hence, y :=
and (ϕ − 1)(y) = z.
e and E
e + . Furthermore,
8.1.1 Lemma. The map ϕ − 1 is surjective on ER , E+
, E
R
R
R
e such that ap − a = b we have
given a and b ∈ E
R
½
vE (a) =
0 ≤ vE (a) ≤ vE (b) if vE (b) ≥ 0;
vE (b)/p
if vE (b) < 0.
78
Proof: Recall that ER := ∪S∞ ES (resp. E+
:= ∪S∞ E+
S ) and the union is taken
R
+
over a maximal chain
of finite
normal extensions
of E
R (resp. E
R ), étale after invert¡
¢
¡
¢
¡
¢
+
1
1
1
ing π. Then, Het ER , Z/pZ = 0 and Het ER , Z/pZ = 0, Het ϕ−∞ (ER ), Z/pZ = 0
¡
¢
+
and H1et ϕ−∞ (ER
), Z/pZ = 0. By Artin–Schreier theory ER /(ϕ − 1)ER injects in
¡
¢
+
H1et ER , Z/pZ and, hence, it is zero. Analogously, E+
/(ϕ − 1)ER
= 0. This imR
¡
¢
¡ −∞ + ¢
+
−∞
−∞
−∞
plies that ϕ (ER )/(ϕ − 1) ϕ (ER ) = 0 and ϕ (ER )/(ϕ − 1) ϕ (ER ) = 0.
£
¤
e + is the π–adic completion of ϕ−∞ (E+ ) and E
e = E
e + π −1 . In
By 2.6 the ring E
R
R
R
R
e = ϕ−∞ (E ) + π E
e + and we are left to prove that given a power separticular, E
R
R
R
P+∞
ries b = n=1 bn π n with {bn }n in ϕ−∞ (E+
), we can solve the equation (ϕ − 1)(a) = b.
R
+
It suffices to find {an }n in ϕ−∞ (ER
) such that π (p−1)n apn − an = bn . Indeed, if we
P∞
put a := n=1 an π n , then (ϕ − 1)(a) = b. Given bn ∈ ϕ−∞ (E+
) there exists S∞ and m
R
¡ +¢
¡ +¢
+
−m
−m
such that bn ∈ ϕ
ES . But ES and ϕ
ES are π–adically complete and sep(p−1)n p
arated, the equation π
X − X = bn in the variable X has 1 as derivative and
admits ¡bn as
¢ solution modulo π. By Hensel’s lemma it admits a unique solution an
+
−m
in ϕ
ES . The first part of the lemma follows.
≤1
Assume that ap − a = b. Then, the properties of vE
recalled in 2.10 imply that if
p
vE (a) < 0 we have vE (a ) = pvE (a) < vE (a) and vE (b) = vE (ap −a) = pvE (a). On the
other hand, if vE (a) > 0 we have vE (ap ) = pvE (a) > vE (a) and vE (b) = vE (ap − a) =
vE (a). If vE (a) = 0, then vE (ap − a) ≥ 0 and vE (b) ≥ vE (a). The second claim follows.
¡
¡ +¢
¡ + ¢¢
e + pm W E
e
8.1.2 Lemma. For every m and n ∈ N we have (ϕ − 1) [π]n W E
=
R
R
¢
¡
¢
¡
e + where [π] is the Teichmüller lift of π. In particular, the map
e + + pm W E
[π]n W E
R
R
e is open for the weak topology.
e →A
ϕ − 1: A
R
R
¡ +¢ m ¡ +¢
e }m,n is a fundamental system of neighe +p W E
Proof: By construction {[π]n W E
R
R
e . Since ϕ − 1 is linear, the first claim implies
borhoods for the weak topology on A
R
¡ ¢
e , since ϕ − 1 is
the second. Since (ϕ − 1)(pm a) = pm (ϕ − 1)(a) for every a ∈ W E
¡ +¢
¡ + ¢ m ¡R + ¢
+
e
e /p W E
e , it is enough
e ) by 8.1.1 and since Wm E
=W E
surjective on W(E
R
¡ n R ¡ + ¢¢ R n
¡ +R¢
e
e . Indeed, (ϕ −
to prove that for every n we have (ϕ − 1) [π] Wm E
= [π] Wm E
R
R
¡ n
¡ + ¢¢ ¡ n
¡ + ¢¢
n
pn p
e
e
1) [π] Wm E
⊂
[π]
remarking
that
(ϕ
−
1)([π]
W
E
a)
=
[π]
a − [π]n a =
m
R
R
¢
¡
¡
¢
¡
¡ + ¢¢
e + ⊂ (ϕ − 1) [π]n Wm E
e
[π]n [π](p−1)n ap − a . On the other hand, [π]n Wm E
since
R
R
¡ +¢
(p−1)n
p
e
for every b ∈ Wm E
the equation [π]
X − X = b admits a solution modulo p
R
¡ +¢
e
by Hensel’s lemma. The lemma follows.
(cf. proof of 8.1.1) and, hence, in Wm E
R
e
8.1.3 Lemma. There exists a left inverse ρ as Zp –modules of the inclusion ι: Zp → A
R
of (8.1.1), which is continuous for the weak topology.
Proof: Let R∗ be the p–adic completion of the localization of R at the generic point
e →A
e ∗ , which is continuous for the weak topology,
of R ⊗V k. We then have a map A
R
R
79
so that it suffices to construct ρ for R∗ . We may then assume that R = R∗ is a complete
discrete valuation ring with residue field L. In particular, ER is a discrete valuation field
with valuation ring E+
R and AR is a complete discrete valuation ring with uniformizer p
and residue field ER .
Recall that ER is the union ∪S ES over all finite normal extensions R ⊂ S ⊂ R, étale
after inverting p. Let R ⊂ S be any such. Since R is a complete discrete valuation ring,
also S is a complete discrete valuation ring. Then, E+
S is a complete discrete valuation
ring. For S ⊂ T ⊂ R finite normal extensions, étale after inverting p of degree nS,T ,
+
we get that E+
T is a finite and torsion free as ES –module, of rank nS,T ; see 2.6. By
+
−m
−m
loc. cit. the choice of a E+
(E+
(E+
S –basis of ET determines a ϕ
S )–basis of ϕ
T ) for
every m ∈ N and maps
π
`S,T
pm
e + → ϕ−m (E+ )
E
T∞
T
⊗
ϕ−m (E+
)
S
¡ + ¢nS,T
e+ ∼
e
e+ ,
E
→E
S∞ = E S∞
T∞
e nS,T →
where `S,T is a constant depending on S ⊂ T . We thus get an isomorphism E
S∞
e T as topological groups (for the π–adic topology). Since E+ is integrally closed
E
∞
S
+
+
in E+
,
we
may
assume
that
the
given
E
–basis
of
E
contains
1.
Suppose furthermore
T
S
T
`S,S 0
eS ⊂ E
e T as E
e S –modules
that pm < 1. We then get a splitting of the inclusion E
∞
∞
∞
+
+
e
e
such that π E
T∞ is mapped to ES∞ . Consider the set F of pairs (A, t) where A is a normal
e R –algebra of ∪S E
e S and t: A → E
e R is a splitting of the inclusion E
eR ⊂ A
sub–E
∞
∞ ³
∞ ´
∞
¡
¢
+
+
e
e
e
as ER –modules such that t A ∩ π · ∪S E
⊂ E . It is an ordered set in which
S∞
∞
R∞
every chain has a maximal element. Zorn’s lemma implies that F has a maximal element
e S . We conclude that there
which, by the discussion above, must coincide with ∪S E
∞
e
e
e S such that π E
e+
exists a left inverse ξ as ER∞ –modules of the inclusion ER∞ ⊂ ∪S E
∞
S∞
+
+
e
e
e
e
is mapped to ER∞ for every S. Since ER (resp. ER ) is the π–adic completion of ∪S ES∞
e + ) and since E
e R (resp. E
e + ) is π–adically complete and separated, ξ
(resp. ∪S E
∞
S∞
R∞
e
eR ⊂ E
e mapping π E
e+
extends to a left inverse ζ as ER –modules of the inclusion E
∞
R
∞
e + . In particular, ζ is continuous for the π–adic topology.
to E
R∞
R
e+
On the other hand, recall from 2.6 that E+
R = L ⊗k k∞ [[πK ]] and that ER∞ is the
· 1
¸
1
1
pn
pn
pn
completion of ∪n E+
π
,
x
,
.
.
.
,
x
for the topology defined by the fundamental
1
R
K
d
½
µ
· 1
¸¶¾
1
1
pn
pn
pn
+
m
system of neighborhoods πK ∪n L ⊗k EV [πK ] x1 , . . . , xd
. Define
m
· 1
¸
1
1
pn
pn
pn
δ: ∪n L ⊗ k∞ ((πK )) πK , x1 , . . . , xd
→ L ⊗ k∞
k
k
i0 i1
as the L–linear map sending πK
x1 · · · xidd to 0 for every (i0 , i1 , . . . , id ) ∈ Qd+1 such
that (i1 , . . . , id ) is not equal to 0 in (Q/Z)d . It is well defined since {πK , x1 , . . . , xd } is
an absolute p–basis of E+
R . Furthermore, δ is continuous for the πK –topology and, hence,
80
eR
it extends to a continuous left inverse ν as L–modules of the inclusion L ⊗k k∞ ⊂ E
∞
e
considering the π–adic topology on ER∞ and the discrete topology on L. Finally, choose
a left splitting τ as Fp –vector spaces of Fp ⊂ L ⊗k k∞ .
¡ ¢
e → Zp be the map sending a Witt vector (a0 , . . . , an , . . .) of A
e =W E
e
Let δ: A
R
R
R
¡
¢
e
to τ ◦ ν ◦ ζ(a0 ), . . . , τ ◦ ν ◦ ζ(an ), . . . . It is a left inverse of the inclusion Zp ⊂ AR and it
e and on Zp . Note that the topology induced
is continuous for the weak topology on A
R
e is the p–adic topology. The lemma follows.
on Zp from the weak topology on A
R
e →A
e . It is a
End of the proof 8.1. With the notations of 8.1.3, let e := ι ◦ ρ: A
R
R
continuous homomorphism of Zp –modules and e2 = e. Thus, if M := Ker(e) = Im(e−1),
e and A
e = Zp ⊕ M . Then, (ϕ − 1)|M : M → A
e is
we have that M is closed in A
R
R
R
bijective. It is open thanks to 8.1.2. Hence, its
inverse
is
a
continuous
homomorphism
of
¡
¢−1
e .
Zp –modules. We let σ be the composite of (ϕ − 1)|M
and the inclusion M ⊂ A
R
It satisfies the requirements of 8.1.
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Fabrizio Andreatta
DIPARTIMENTO DI MATEMATICA “FEDERIGO ENRIQUES”,
STATALE DI MILANO, VIA C. SALDINI 50, MILANO 20133, ITALIA
E-mail address: andreat@mat.unimi.it
UNIVERSITÁ
Adrian Iovita
DEPARTMENT OF MATHEMATICS AND STATISTICS, CONCORDIA UNIVERISTY
1455 DE MAISONNENEUVE BLV. WEST,
H3G 1MB, MONTREAL, QUEBEC, CANADA
E-mail address: iovita@mathstat.concordia.ca
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