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Φ-Γ-Modules for Families of Galois Representations
Journal of Algebra 235, 636᎐664 Ž2001. doi:10.1006rjabr.1999.8272, available online at http:rrwww.idealibrary.com on
⌽-⌫-Modules for Families of Galois Representations Jonathan Dee California Institute of Technology, Pasadena, California 91125 E-mail: [email protected] Communicated by Jan Saxl Received May 20, 1999
0.1. Introduction 0.2. Notations and conventions 1. Definitions and commutati¨ e algebra 1.1. Some types of rings 1.2. Completed tensor products 1.3. Lifting fields to characteristic zero 2. An equi¨ alence of categories 2.1. The characteristic p case 2.2. The characteristic zero case ŽVersion 1. 2.3. The characteristic zero case ŽVersion 2. 2.4. Change of the coefficient ring 3. Galois cohomology 3.1. Comparing ⌽-⌫-modules with cohomology groups 3.2. The operator 3.3. Iwasawa Theory Key Words: ⌽-⌫-modules; local fields; Galois representations; complete Noetherian local rings; completed tensor product; Galois cohomology: Iwasawa Theory
0.1. INTRODUCTION Fontaine has found a new approach to understanding the category of ⺪ p-adic representations of GK s G Ž KrK ., where K is a field complete with respect to a discrete valuation whose residue field is perfect of characteristic p Žsee wFonx.. Here a ⺪ p-adic representation of GK is a ⺪ p-module of finite rank with a continuous linear action of GK and p is a prime number. In the equal characteristic case he constructed the category of ´ etale ⌽-modules over K Žsee below for the definition. which a priori seems much simpler than the category of ⺪ p-adic representations of GK . 636 0021-8693r01 $35.00 Copyright 䊚 2001 by Academic Press All rights of reproduction in any form reserved.
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He then proved in an elementary way an equivalence of this category with the category of all ⺪ p-adic representations of GK . The mixed characteristic case is then deduced from the equal characteristic case via the field of norms machine due to Fontaine and Wintenberger Žsee wWinx.. The category of ⺪ p-adic representations of GK in this case is equivalent to the category of ⌽-⌫-modules over K. Suppose for simplicity that p / 2. Then define O to be the p-adic completion of W ww T xxw Ty1 x, K ⬁ to be the cyclotomic ⺪ p-extension of K in K, GK s GŽ KrK ., HK s GŽ KrK ⬁ ., and ⌫ s GK rHK . One can define an action of ⌫ and also the action of a Frobenius ⌽ on O Žsee below for details.. An ´ etale ⌽-⌫-module is then a finite rank O-module with semilinear action of ⌫ and a bijective semilinear operator commuting with the action of ⌫. In this paper we extend Fontaine’s results to give an understanding of the category of R-modules of finite type with a continuous R-linear action of GK , where now R may be any complete Noetherian local ring whose residue field is a finite extension of ⺖p . We construct a category of ´ etale families of ⌽-⌫-modules over K parameterised by R and prove that it is equivalent to the category of R-linear representations of GK defined above. Such a family is defined to be a module of finite type over the ˆ⺪ p R with actions of ⌽ and ⌫ as before. We completed tensor product O m show that our functor is compatible with the extension of scalars for an arbitrary local homomorphism of complete Noetherian local rings R ª S. The proof of this result uses the equivalence of categories we prove in Section 4 in a crucial way. If S is further supposed to be finite over R then our functor is compatible with restriction of scalars. Our method for proving the equivalence of categories referred to above is to use the results of Fontaine for the case where the representation V has finite length and then extend it to the general case by taking inverse limits. Given this equivalence of categories, it is clear that given an object functorially associated to an R-representation of GK , one should be able to describe this object purely in terms of the corresponding ⌽-⌫-module. We show in Section 3 that this is the case for the continuous Galois cohomology groups of V and the inverse limit
6
lim H 1 Ž K n , V .
where K n is the extension of K of degree p n in the cyclotomic ⺪ p-extension of K. Actually, we work over the tower K Ž p⬁ . and use a slight variant of ⌽-⌫-modules Ždue to Colmez and Cherbonnier wCCx. defined using this tower, but the method is identical. It is possible that this
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approach may lead to the construction of a Perrin᎐Riou homomorphism for families of Galois representations. I thank Jan Nekovar ´ˇ for his help and encouragement. 0.2. NOTATIONS AND CONVENTIONS All rings will be commutative with a 1. If R is a ring we shall write ᒊ R for its radical, the intersection of all maximal ideals. Unless specified to the contrary all semilocal rings R will be considered as topological rings with the ᒊ R-adic topology. Write R= for the group of units of R. If R is local then we write k R s Rrᒊ R . If F is a local or global field then OF will denote its ring of integers. We refer to a ring equipped with a linear topology as complete if the natural homomorphism to its completion is an isomorphism Ži.e., it is separated and complete in the sense of Bourbaki.. We shall say that an R-module is finite if it is finitely generated as a module. All Galois cohomology groups in this paper will be continuous Galois cohomology groups in the sense of wTatex. We fix a prime number p for the rest of this paper. 1. DEFINITIONS AND COMMUTATIVE ALGEBRA 1.1. Some Types of Rings We first define some types of local rings which we shall use later on. The following may be found in wMazx. DEFINITION 1.1.1. A coefficient ring R is a Noetherian complete local ring with finite residue field k R of characteristic p. Remark 1.1.2. A coefficient ring R is a W Ž k R .-algebra in a natural way. The structure map W Ž k R . ª R need not be injective, of course. The following is what we shall mean by p-rings for the purposes of this paper. DEFINITION 1.1.3. A p-ring is a complete discrete valuation ring whose valuation ideal is generated by p. 1.2. Completed Tensor Products We shall need to understand the completed tensor product of a p-ring with a coefficient ring. Let R and S be arbitrary rings and I ; R, J ; S be two ideals. Suppose that R and S are both T-algebras for some third ring T.
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ˆT S is defined DEFINITION 1.2.1. The completed tensor product R m as the completion of R mT S with respect to the Ž I m S q R m J .-adic topology. Remark 1.2.2. We should of course carry the ideals I and J in the notation, but in practice there will be no ambiguity. The following is part of wEGA, 0.19.7.1.2x and gives us a large part of the properties of completed tensor products that we require. PROPOSITION 1.2.3. Let A be a ring, let B and C be local Noetherian A-algebras. Suppose that C is complete and that k B is finite as an A-module. ˆA C is a complete Noetherian semilocal ring. Then B m In more specialised circumstances we may prove more. LEMMA 1.2.4. Let A, B, and C be as abo¨ e. Assume further that all three rings are complete Noetherian local rings with residue characteristic p and that k B is a finite separable extension of k A . Assume the structural homomorphism ˆA C is equal to the A ª B is a local homomorphism. Then the radical of B m ideal generated by the image of ᒊ B m C q B m ᒊ C .
ˆA C. Since B m ˆA C Proof. Write ᒊ s ᒊ B m C q B m ᒊ C and E s B m is Noetherian, ᒊ generates the kernel of ˆA C ª k B mk A k C . : Bm The right-hand side is a finite separable k C -algebra, so it equals a finite product of finite separable extensions of k C . It follows immediately that ᒊ E contains the radical. On the other hand, E is complete with respect to the ᒊ-adic topology, so ᒊ is contained in the radical. Remark 1.2.5. It follows from the proof of Lemma 1.2.4 that the local ˆA C correspond to the field components of k B mk A k C . components of B m Next we introduce some notation. From now on we shall usually use O to refer to a p-ring, and capital letters R, S, etc., to refer to coefficient rings or general rings. If O is a p-ring and R a coefficient ring then we write
ˆ⺪ p R. OR s O m Note that by Proposition 1.2.3 OR is a complete Noetherian semilocal ring, although its residue fields need not be finite as we have made no restriction on the residue field of O , and indeed later k O will not even be perfect.
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Now suppose R and S are two coefficient rings and O is a p-ring Žor indeed any local ring with residue field of characteristic p .. If : R ª S is a ring homomorphism then it induces
: O m⺪ p R ª O m⺪ p S. If is local, then
Ž O m ᒊ R q ᒊ O m R . ; O m ᒊ S q ᒊ O m S, so is continuous Žwith respect to the obvious topologies.. It thus induces a semilocal homomorphism
: OR ª OS . Finally, in this section we show a flatness result that we shall need later. More precisely, we have: PROPOSITION 1.2.6.
Let
: O1 ª O2 be a local homomorphism of p-rings and let R be a coefficient ring. If is flat then the induced homomorphism
R : O1, R ª O2, R is faithfully flat. Proof. First recall that a local homomorphism is flat if and only if it is faithfully flat. Next, the flatness result for R is a simple extension of the proof in wEGA, 0.19.7.1.2x. Note that Ž ᒊ R Oi, R . n s ᒊ Rn Oi, R . The local criterion for flatness wMat, Theorem 22.3x thus implies that it is enough to check the flatness of O2, Rrᒊ Rn O2, R over O1, Rrᒊ Rn O1, R for all n. We have that Oi , Rrᒊ Rn Oi , R s Oi m⺪ p Rrᒊ Rn for i s 1, 2, and flatness is preserved under base extension. To see faithful flatness we need to show that every maximal ideal of O1, R is in the image of the induced map Spec Ž O2, R . ª Spec Ž O1, R . .
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On dividing by the radicals of O1, R , and O2, R we obtain the commutative diagram with surjective vertical arrows O2, R 6
k O 2 m⺖ p k R .
6
6
k O1 m⺖ p k R
6
O1, R
The lower horizontal homomorphism is faithfully flat since it is the base change of a field extension, and faithful flatness is preserved by base change. It follows that R is faithfully flat. 1.3. Lifting Fields to Characteristic Zero Let E be a field of characteristic p. Following wFonx, assume we have a p-ring O of characteristic zero with fraction field E and residue field E. Fix a choice of E . In general O is unique up to a non-unique isomorphism. If OE n r is a strict Henselisation of O with field of fractions En r , then OE n r has a valuation induced from O which is discrete and p generates the valuation ideal. The valuation ring in the completion Eˆn r of En r is a p-ring with residue field E sep , a separable closure of E. Write Oˆn r for this ring. There is an identification of Galois groups ;
E., GE s G Ž E seprE . ª G Ž En rrE and GE acts by continuity on Eˆn r . From now on R will always denote a coefficient ring, unless expressly stated otherwise. Define the ring
ˆ⺪ p R OR s O m as in Section 1.2. In particular there is a decomposition OR (
Ł OR , i , igI
for some finite index set I, where the OR, i are complete local Noetherian rings. Similarly, define the ring
ˆ⺪ p R, OˆRn r s Oˆn r m with decomposition OˆRn r (
Ł OˆRn,r j . jgJ
It is an OR-algebra and is faithfully flat over OR by Proposition 1.2.6.
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The action of GE on En r induces an action on Oˆn r and hence an action on Oˆn r m⺪ p R, via the trivial action on R. Being continuous on En r this action is continuous on Oˆn r m⺪ p R, so it induces a GE -action on OˆRn r , continuous with respect to the ᒊ R OˆRn r-adic topology, which we shall use in the following. Remark 1.3.1. It follows from Section 1.2 that OR and OˆRn r are Noetherian semilocal rings, complete with respect to the ᒊ R-adic topology, and that ᒊ R generates the radical of these rings. 2. AN EQUIVALENCE OF CATEGORIES 2.1. The Characteristic p Case Recall that E is an arbitrary field of characteristic p. We certainly do not want to assume that E is perfect. As before let O be a p-ring of characteristic zero with residue field E and field of fractions E . From now on we suppose that O is equipped with a lift of Frobenius: a ring homomorphism Žnot necessarily bijective. such that
Ž x. ' x p
mod p.
We shall assume that is flat. By tensoring with R we deduce an R-linear homomorphism Žalso denoted by .:
: O m⺪ p R ª O m⺪ p R. Since the ideal ᒊ O m⺪ p R q O m⺪ p ᒊ R in O m⺪ p R is generated by ᒊ R it is immediate that maps this radical to itself. We have LEMMA 2.1.1. The induced homomorphism
: OR ª OR is flat. Proof. This follows immediately from Proposition 1.2.6. The Frobenius on O extends uniquely by functoriality and continuity to a Frobenius on Oˆn r , and then as above to a flat homomorphism from OˆRn r to itself. We remark that the fixed ring of Frobenius acting on Oˆn r is just ⺪ p .
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DEFINITION 2.1.2. An R-representation of GE is a finitely generated R-module with continuous, R-linear action of GE . Write Rep R for the category of such representations of GE . DEFINITION 2.1.3. Write ⌽ Mod R for the following category. The objects of ⌽ Mod R are ⌽-modules: they are OR-modules M, equipped with a -semilinear homomorphism
M : M ª M, or equivalently a OR-linear homomorphism ⌽M : M ª M, where M s M mO , O is the base change of M by OR via . Morphisms in ⌽ Mod R or OR-linear homomorphisms commuting with . Remark 2.1.4. It is clear that ⌽ Mod R is abelian. Indeed, it is just the category of modules over the noncommutative ring OR w x, where the addition in this ring is as for the polynomial ring, but
x s Ž x. . We next define a functor from Rep R to ⌽ Mod R . Let V be any R-representation of GE . Define ⺔R Ž V . s Ž OˆRn r mR V .
GE
,
where GE acts diagonally. Then ⺔R Ž V . carries the structure of an OR-module since multiplication by OR on OˆRn r m V is GE -equivariant. The Frobenius on OˆRn r acts GE -equivariantly, and thus induces a -semilinear homomorphism
⺔ R ŽV . : ⺔R Ž V . ª ⺔R Ž V . . We shall often write V rather than ⺔ R ŽV . and this will hopefully cause no confusion. The association V ¬ ⺔R Ž V . clearly induces a functor ⺔R : Rep R ª ⌽ Mod R . LEMMA 2.1.5.
Suppose that ᒊ R V s 0. Then as an OR-module ⺔R Ž V . s ⺔ k R Ž V . .
Proof. By the assumption on V we have OˆRn r mR V ( Ž OˆRn rrᒊ R . mk R V .
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We know from Lemma 1.2.4 that ᒊ R OˆRn r is the radical of OˆRn r , and that OˆRn rrᒊ R OˆRn r ( E sep m⺪ p k R ( OˆknRr .
If V is killed by ᒊ Rn then our construction agrees with that contained in wFonx by the following lemma. If ᒊ Rn V s 0 for some n then as O⺪ p-modules
LEMMA 2.1.6.
⺔ ⺪ p Ž V . s ⺔R Ž V . . The operator is the same on both sides. Proof. Since V is finite as an R-module and is killed by ᒊ Rn it must be finite as a ⺪ p-module Žrecall that R has finite residue field.. Hence
ˆR R m ˆ⺪ p Oˆn r s V m ˆ⺪ p Oˆn r s V m⺪ p Oˆn r . V mR OˆRn r s V m
ž
/
Remark 2.1.7. The previous two results show that the family of functors ⺔ commutes with restriction of scalars in some special cases. We shall show in Section 2.4 that ⺔ commutes with a more general restriction of scalars. We shall now show that ⺔R commutes with inverse limits. This result will be very useful in deducing our results from those in wFonx. If V is any object in Rep R , write Vn s Vrᒊ Rn V. Here ᒊ Rn V is the submodule of V generated by elements of the form m¨ where m g ᒊ Rn and ¨ g V. PROPOSITION 2.1.8.
Let V be an R-representation of GE . Then ;
6
⺔R Ž V . ª lim ⺔R Ž Vn . . Proof. It is immediate that taking GE -invariants commutes with inverse limits, so it suffices to prove that the tensor product with OˆRn r commutes with inverse limits. But 6
6
lim Ž OˆRn r mR Vn . s lim Ž OˆRn r mR Ž Vrᒊ Rn V . . 6
s lim Ž OˆRn r mR V . rᒊ Rn s OˆRn r mR V .
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The last equality in the equation follows from the fact that OˆRn r mR V is complete with respect to the ᒊ R OˆRn r-adic topology. This in turn follows from Ž1.3.1. and the fact that V is finitely generated as an R-module. Next we show that the functor ⺔R preserves a lot of the structure of the category Rep R : PROPOSITION 2.1.9. The functor ⺔R is exact and faithful. Proof. First note that ⺔R is left exact by the flatness of OˆRn r over R. We know from Lemma 2.1.6 that if V is of finite length then ⺔R Ž V . s ⺔ ⺪ pŽ V . . Hence, by the exactness of ⺔⺪ p proved by Fontaine, if A, B, and C are finite length R-representations sitting in a sequence 0ªAªBªCª0 then 0 ª ⺔R Ž A . ª ⺔R Ž B . ª ⺔R Ž C . ª 0 is exact. Now suppose A, B, and C are arbitrary, still sitting in the above sequence. On tensoring with Rrᒊ Rn and using the exactness for finite length representations we deduce exact sequences ⺔R Ž Arᒊ Rn . ª ⺔R Ž Brᒊ Rn . ª ⺔R Ž Crᒊ Rn . ª 0 for all n. Let K n denote the kernel of the map ⺔R Ž Brᒊ Rn . ª ⺔R Ž Crᒊ Rn . , so that there is a surjective homomorphism ⺔R Ž Arᒊ Rn . ª K n . Using Lemma 2.1.6 again we see that ⺔R Ž Arᒊ Rn . is finite over the Artin ring ORrᒊ Rn . Hence the inverse system Ž⺔R Ž Arᒊ Rn .. n satisfies the Mittag-Leffler condition, and thus so does K n . Taking limits we deduce that ⺔R Ž B . ª ⺔R Ž C . is surjective, as required. From this we deduce the following handy fact. COROLLARY 2.1.10.
Let I be any ideal of R. Then I ⭈ ⺔R Ž V . s ⺔R Ž I ⭈ V .
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and ;
⺔R Ž V . rI ⭈ ⺔R Ž V . ª ⺔R Ž VrI ⭈ V . . Proof. Let x 1 , . . . , x n4 denote a set of generators for I. Consider the map
: V n ª V
Ý ¨ i ¬ Ý x i¨ i . It is clear that the image of is I ⭈ V. By the exactness of ⺔R , the image of ⺔R Ž . is ⺔R Ž IV .. On the other hand, identifying ⺔R Ž V n . with ⺔R Ž V . n, the map n
⺔R Ž . : ⺔R Ž V . ª ⺔R Ž V . is just
Ý di ¬ Ý di xi , since the Galois action is R-equivariant. The image of this map is just I ⭈ ⺔R Ž V ., and so we deduce I ⭈ ⺔R Ž V . s ⺔R Ž I ⭈ V . . We conclude by applying ⺔R to the short exact sequence 0 ª I ⭈ V ª V ª VrI ⭈ V ª 0.
LEMMA 2.1.11. ;
6
⺔R Ž V . ª lim Ž ⺔R Ž V . rᒊ Rn ⺔R Ž V . . . Proof. This is immediate from Proposition 2.1.8 and Corollary 2.1.10. PROPOSITION 2.1.12. If V is an R-representation of GE then ⺔R Ž V . is finitely generated as an OR-module. Proof. We may appeal to wFonx together with Lemma 2.1.6 for the case of V killed by ᒊ R . It follows from Lemma 2.1.11 that ⺔R Ž V . is separated for the ᒊ R-adic topology. Furthermore, using exactness we know ⺔R Ž V .rᒊ R s ⺔R Ž V1 . is finite as an OR-module since it is finite as an O-module. It thus follows from Theorem 8.4 in wMatx that ⺔R Ž V . is finite over OR . We next compute the ⌽-⌫-module corresponding to the trivial one-dimensional representation. The answer is reassuring.
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LEMMA 2.1.13. ⺔R Ž R . s OR . Proof. First note that OR ¨ OˆRn r. Indeed, let C denote the cokernel of the inclusion O ¨ Oˆn r. It is easy to see that C is torsion free as a ⺪ p-module, and so is flat. Hence Tor1⺪ p Ž C, R . s 0. Hence there is an inclusion O m⺪ p R ¨ Oˆn r m⺪ p R. Taking completions with respect to the ᒊ R-adic topology we deduce the desired inclusion. We next show that this inclusion has image equal to Ž OˆRn r . G E . In order to prove this it suffices by Nakayama’s lemma to divide by ᒊ R . We have seen that ⺔R Ž R . rᒊ R s ⺔R Ž Rrᒊ R . s ⺔k RŽ Rrᒊ R . , so we may assume without loss of generality that R s k R . In this case ⺔R Ž R . s Ž E sep m⺖ p k R .
GE
s E m⺖ p k R s OR .
PROPOSITION 2.1.14. If V is an R-representation of GE then the canonical OˆRn r-linear homomorphism of GE -modules OˆRn r mO R ⺔R Ž V . ª OˆRn r mR V
Ž 1.
is an isomorphism. Proof. To show that Ž1. is an isomorphism, note that if V is killed by ᒊ Rn for some n then this reduces after Lemma 2.1.6 to Proposition 1.2.6 in wFonx. We have 6
6
lim OˆRn r mO R ⺔R Ž Vn . ( lim OˆRn r mO R ⺔R Ž V . rᒊ Rn ( OˆRn r mO R ⺔R Ž V .
ž
/
since ⺔R commutes with taking quotients by Corollary 2.1.10 and OˆRn r mO R Ž V . is complete with respect to the ᒊ R-adic topology. Also, 6
lim Ž OˆRn r mR Vn . ( OˆRn r mR V ,
so the result holds for general V.
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We now introduce a full subcategory of ⌽ Mod R , which we shall subsequently show to be the essential image of ⺔R . DEFINITION 2.1.15. An object M in ⌽ Mod R is said to be ´ etale if ⌽M is an isomorphism and M is finitely generated as OR-module. We remark that if is bijective then ⌽ is an isomorphism if and only if is bijective. etale Write ⌽ ModetR for the full subcategory of ⌽ Mod R consisting of ´ ⌽-modules. A morphism of ´ etale ⌽-modules is a morphism of the underlying ⌽-modules. It is not immediately obvious that ⌽ ModetR R is an abelian category. However, this is true by the following lemma from wFonx Žsince we already know ⌽ Mod R is abelian .. LEMMA 2.1.16. In ⌽ Mod R , the kernel and cokernel of a map of ´ etale ⌽-modules are ´ etale. Proof. This follows immediately from the commutative diagram of abelian groups 0
6
6 K
0
6
6
L
K
k
6
6
M
⌽L
6
6
N
6
6
0
⌽M
L
6
⌽N
M
6
N
6
6
0
where the middle two vertical arrows are isomorphisms. In fact, the full subcategory ⌽ ModetR of ⌽ Mod R is stable under subobject and quotient. This is the substance of the following lemma. LEMMA 2.1.17.
Suppose we ha¨ e a short exact sequence in ⌽ Mod R , 0 ª M ª N ª P ª 0.
Then N is an object of ⌽ ModetR if and only if both M and P are. Proof. Assuming M and P to be ´ etale we deduce that N is ´ etale by the Snake Lemma Žthe fact that it is finite as an OR-module is obvious.. For the converse, we have an exact sequence Mn ª Nn ª Pn ª 0. Since N is ´ etale, so is Nn for all n. Since Mn , Nn , and Pn are finite O-modules this sequence is naturally a sequence in ⌽ Mod ⺪ p. By Proposition 1.1.6 in wFonx, both Pn and K n are ´ etale, where K n s ker Ž Nn ª Pn . .
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Now since ORrᒊ Rn is Artinian for all n it follows that the inverse limit functor is exact on the category of finite torsion ⌽-modules. It follows that 6
6
M s lim Mn s lim K n is ´ etale. Remark 2.1.18. Note that ⌽ ModetR is also stable under tensor product. Indeed, if M and N are objects in ⌽ ModetR , then M mO R N is clearly a finite OR-module, and the bijectivity of ⌽MmN follows from the fact that ⌽M and ⌽N are bijective since ⌽MmN s ⌽M mO R ⌽N . PROPOSITION 2.1.19. is ´ etale.
If V is any R-representation of GE , then ⺔R Ž V .
Proof. After Proposition 2.1.12 it remains to prove that ⌽ V is bijective. This holds for V of finite length by Lemma 2.1.6 and Proposition 1.2.6 in wFonx, then for general V by the isomorphism of ⌽-modules 6
⺔R Ž V . ( lim ⺔R Ž Vn . .
We next introduce an inverse functor to ⺔R . DEFINITION 2.1.20. Let M be an ´ etale ⌽-module. Then write ⺦R Ž M . s OˆRn r mO R M
ž
/
s1
.
The association M ¬ ⺦R Ž M . extends in a natural way to a functor from ⌽ ModetR to the category of R-modules with R-linear action of GE . First we show the analogue of Proposition 2.1.8. Write Mn s Mrᒊ Rn M. PROPOSITION 2.1.21. phism of OR-modules
Suppose that M is ´ etale. Then the natural homomor6
⺦R Ž M . ª lim ⺦R Ž Mn . is an isomorphism of GE -modules. Proof. We proceed as we did in the proof of Proposition 2.1.8. It is clear that taking invariants commutes with inverse limits. The OˆRn r-module OˆRn r mO R M is finitely generated since M is ´ etale. Hence 6
lim OˆRn r mO R Ž Mrᒊ Rn M . ( OˆRn r mO R M
ž
and we conclude as before.
/
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JONATHAN DEE
To continue we need the analogue of Lemma 2.1.6. A glance at the proof, however, shows that no extra work is required. If ᒊ Rn M s 0 then as O⺪ p-modules,
LEMMA 2.1.22.
⺦⺪ pŽ M . s ⺦R Ž M . . Our next task is to show the exactness of ⺦R . We first prove a preliminary result which will also be useful in other circumstances. etale ⌽-module and let V be an R-representaLEMMA 2.1.23. Let M be an ´ tion of GE . Then 1 y is a surjecti¨ e homomorphism of abelian groups acting on both OˆRn r mR V and OˆRn r mR M. Proof. We first show the surjectivity for OˆRn r mR V. Suppose that V is killed by ᒊ R . The map 1 y is surjective on E sep since for all g E sep the polynomial x p y x y is separable. Since V is free over k R and acts on E sep m⺖ p V via its action on E sep it follows that 1 y is surjective Žusing the on E sep m⺖ p V. The result for general V follows by devissage ´ Snake Lemma. and passage to the limit Žusing the fact that OˆRn r mR Vn satisfies the Mittag-Leffler condition.. Now suppose M is an ´ etale ⌽-module. If M has finite length as an OR-module then we know from the proof of Proposition 1.2.6 in wFonx and Lemma 2.1.22 that ;
OˆRn r mR ⺦R Ž M . ª OˆRn r mR M, and that ⺦R Ž M . is an R-representation of GE . This isomorphism respects the action of , so by the result for V we deduce that 1 y is surjective on OˆRn r mR M. The general case proceeds by passage to the limit. PROPOSITION 2.1.24. The functor ⺦R is exact. Proof. Let 0ªAªBªCª0 be a short exact sequence of ´ etale ⌽-modules. We apply the Snake Lemma to the diagram
1y
0.
6
6
OˆRn r mR C
6
6
OˆRn r mR B
6
6
OˆRn r mR A
6
0
1y
0
6
1y
OˆRn r mR C
6
OˆRn r mR B
6
OˆRn r mR A
6
0
⌽-⌫-MODULES
651
We obtain an exact sequence 0 ª ⺦R Ž A . ª ⺦R Ž B . ª ⺦R Ž C . ª Ž OˆRn r mR A . r Ž 1 y . . By lemma 2.1.23 the last term vanishes. We now compute ⺦R Ž M . when M is the simplest ⌽-module of rank 1. LEMMA 2.1.25. ⺦R Ž OR . s R, with the usual action of on OR and tri¨ ial GE action on R. Proof. There is a short exact sequence of ⺪ p-modules 1y
0 ª ⺪ p ª Oˆn r ª Oˆn r ª 0. If we tensor with R the sequence remains exact since Tor1⺪ p ŽR, Oˆn r . s 0. On taking completions we deduce the result. PROPOSITION 2.1.26. If M is an ´ etale ⌽-module then ⺦R Ž M . is finitely generated as an R-module and the homomorphism of OˆRn r-modules OˆRn r mR ⺦R Ž M . ª OˆRn r mO R M
Ž 2.
is an isomorphism. Proof. Given the exactness of ⺦R and the compatibility with inverse limits, the finiteness of ⺦R Ž M . proceeds exactly as for ⺔R Ž V .. Similarly, we may show Ž2. just as we proved Proposition 2.1.14. THEOREM 2.1.27. The functor ⺔R : Rep R ª ⌽ ModetR is an equi¨ alence of categories, with quasi-in¨ erse ⺦R . Proof. Our task is to construct functorial isomorphisms ⺦R Ž ⺔R Ž V . . ª V and ⺔R Ž VR Ž M . . ª M for R-representations V and ´ etale ⌽-modules M respectively. Consider the isomorphism ⺔R Ž V . mO R OˆRn r ª V mR OˆRn r .
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JONATHAN DEE
On taking invariants of both sides we deduce an isomorphism ⺦R Ž ⺔R Ž V . . ª Ž V mR OˆRn r .
s1
.
Since acts trivially on V there is a map V ª Ž V mR OˆRn r .
s1
.
This map is an isomorphism if V is of finite length Žsee the proof of Lemma 2.1.6. by wFonx. We may pass to the limit as in the proof of Lemma 2.1.8. We similarly find that the canonical map M ª M mO R OˆRn r
ž
/
GE
is an isomorphism. The isomorphism ⺔R Ž ⺦R Ž M . . s Ž ⺦R Ž M . mR OˆRn r .
GE
ª M mO R OˆRn r
ž
/
GE
then provides the desired isomorphism. It follows formally from what we have proved so far that ⺔R and ⺦R commute with tensor product. Note in the following that M mO R N is ´ etale Žsee Remark 2.1.18.. COROLLARY 2.1.28. Let V and W be R-representations of GE . Let M and N be ´ etale ⌽-modules. The natural homomorphism of ⌽-modules
Ž V , W . : ⺔R Ž V . mO R ⺔R Ž W . ª ⺔R Ž V mR W . and the natural homomorphism of R-representations of GE
Ž M, N . : ⺦R Ž M . mR ⺦R Ž N . ª ⺦R Ž M mO R N . are isomorphisms. Proof. We have a commutative diagram ⺦R Ž⺔R Ž V . mO R ⺔R ŽW ..
6
⺦R Ž .
6
6
VmW
Ž⺔ R Ž V ., ⺔ R ŽW ..
6
⺦R Ž⺔R Ž V .. mR ⺦R Ž⺔R ŽW ..
⺦R Ž⺔R Ž V mR W ...
The left vertical arrow and bottom horizontal arrow are isomorphisms. Hence is injective for all M and N Žusing the fact that ⺔R is essentially surjective on objects., and ⺦R Ž . is surjective for all V and W. Since ⺦R is
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653
an equivalence of categories, we deduce that is surjective. Switching the roles of ⺦R and ⺔R we see that in fact and are isomorphisms, as required. 2.2. The Characteristic Zero Case Ž Version 1. Let K be a field complete with respect to a discrete valuation and suppose that K has perfect residue field. We show how one may use the equivalence of categories proven in previous sections to deduce an equivalence between the category of R-modules of finite rank with a continuous R-linear action of GK and a certain category of ⌽-⌫-modules. In the paragraph following this we shall see a slight variant which is often found in the literature Žsee for instance wCCx or wHerrx.. Since we have already dealt with the case of a field of characteristic p we assume from now on that K has characteristic 0. Let K 0 denote the fraction field of W Ž k . where k is the residue field of K. Let K be an algebraic closure of K and ⺓ p its p-adic completion on which GK acts by continuity. We first follow Fontaine in defining the ring 6
R s lim O⺓ p , where the transition maps are x ¬ x p , addition is given by
Ž x q y.
Ž n.
s lim
mª⬁
Ž x Ž nqm. q y Ž nqm. .
pm
,
and multiplication by
Ž xy .
Ž n.
s x Ž n. y Ž n. .
The ring R is known to be a complete valuation ring whose fraction field is algebraically closed of characteristic p with a continuous action of GK . The group R= contains a copy of ⺪ p Ž1., and we choose a generator, in other words, a system of roots of unity in K ⑀ s Ž ⑀ Ž n. . in R, where ⑀ Ž0. s 1 and ⑀ Ž1. / 1. Denote by the Frobenius on R given by x ¬ x p. Define
˜0 s
Ý
⑀ w ax ,
ag⺖ p
where w ax denotes a Teichmuller lift of a. Define the ring ¨ E0 s k Ž Ž ˜0. . inside R and let E0sep be the separable close of E0 inside FracŽ R .. It is stable under GK . To define E s E Ž K . we need some more notation. Let
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JONATHAN DEE
K ⬁ be the cyclotomic ⺪ p-extension of K 0 in K if p / 2, K ⬁ s D K 0 Ž 2 ⬁ . if p s 2, and let HK s GK K ⬁ . Here KK ⬁ is the compositum of K and K ⬁ . Note that KK ⬁ is the cyclotomic ⺪ p-extension of K Žif p / 2.. Put ⌫K s GK rHK s G Ž KK ⬁ rK .. Write p r K s w K ⬁ l K : K 0 x. We may now define E s E Ž K . s yr K ŽŽ E0sep . H K ., a field of characteristic p which is in fact independent of the choice of ⑀ and is complete with respect to a discrete valuation. In order to apply the theory we developed in the previous sections we need a p-ring with residue field E. We work inside the field WK 0Ž Frac R . s W Ž Frac R . w 1rp x and mirror the construction of E Ž K .. First define
0 s yp q
w x Ý w⑀ x a .
ag⺖ p
Let O0 denote the p-adic completion of W ŽŽ 0 ... Since W Ž R . is p-adically complete we may consider O0 ; W Ž R . ; WK 0Ž Frac R . . Note that O0 is a p-ring of characteristic zero with residue field E0 . Write E0 for the fraction field of O0 , a subfield of WK 0ŽFrac R .. The Frobenius on Frac R extends by functoriality and semilinearity to WK 0ŽFrac R ., and we define E s E Ž K . s yr K Ž E0 . . Then E receives a discrete valuation from O0 . Write O for its ring of integers, a p-ring with residue field E. The strict Henselisation O hs of O may Žand will. be identified with the ring of integers in the maximal unramified extension of E in WK 0ŽFrac R .. We may thus take Oˆn r ; WK 0ŽFrac R ., where Oˆn r is the completion of O hs . The field of norms construction due to Fontaine and Wintenberger Žsee wWinx. gives us a canonical isomorphism ;
HK ª G E . The group GK acts on R, then acts by functoriality on WK 0ŽFrac R ., and Oˆn r is stable by this action. On the other hand, GE acts on Oˆn r by continuity and functoriality, and these actions are compatible with the above identification of Galois groups. If V is a finite-dimensional R-representation of GK we may thus consider the ⌽-module ⺔R Ž V . s Ž V mR OˆRn r .
HK
s Ž V mR OˆRn r .
GE
.
⌽-⌫-MODULES
655
The action of GK on V mR OˆRn r induces a semilinear action of GK rHK s ⌫K s ⌫ on ⺔R Ž V .. We are thus led to introduce the category of Ž R-linear. ⌽-⌫ modules Žin the sense of Fontaine. over K. Objects in this category are Ž R-linear. ⌽-modules equipped with an OR-semilinear action of ⌫ commuting with the action of . We say that a ⌽-⌫-module is ´ etale if its underlying ⌽-module is ´ etale. By the above considerations ⺔R yields a functor from the category of continuous R-linear representations of GK to the category of ⌽-⌫-modules. If M is a ⌽-⌫-module we may consider the GK -representation ⺦R Ž M . s Ž M mR Oˆn r . s1. Here GK acts on Oˆn r as before, and acts via ⌫ s GK rHK on M. The diagonal action on M mR Oˆn r is -equivariant, so it induces a GK -action on ⺦R Ž M .. If V is an R-representation of GK then there is a canonical R-linear homomorphism of representations of GK , V ª ⺦R Ž ⺔R Ž V . . . Since this is an isomorphism when restricted to HK Žby Theorem 2.1.27. it must already be an isomorphism of GK -representations. Similarly, if M is an ´ etale ⌽-⌫-module, the canonical homomorphism of ⌽-⌫-modules M ª ⺔R Ž ⺦R Ž M . . is an isomorphism. Indeed, the underlying map of ⌽-modules is an isomorphism after Theorem 2.1.27. We have shown THEOREM 2.2.1. The functor ⺔R yields an equi¨ alence of categories between the category of continuous R-linear representations of GK and the category of ´ etale ⌽-⌫-modules. The functor ⺦R is a quasi-in¨ erse functor. We finish with a description of OR . Recall that as a ring O is Žnoncanonically. the p-adic completion of W ww T xxw Ty1 x, where W s W Ž k .. It is easy to see that in fact Os
½ Ý a T ; a g W , a ª 0 in W as n ª y⬁ 5 ; W w T , T n
n
n
n
y1
x .
ng⺪
We show that OR is what one would hope it was. PROPOSITION 2.2.2. Write S s R m⺪ p W. Then OR s
½ Ý a T ; a g S, a ª 0 as n ª y⬁ 5 ; S w T , T n
n
n
n
y1
x . Ž 3.
ng⺪
Here the limit a n ª 0 means with respect to the ᒊ R-adic topology of S.
ˆ⺪ p ŽW ww T xxw Ty1 x.. It is clear that the Proof. First note that OR s R m ring on the right-hand side of Ž3. is the ᒊ R-adic completion of Sww T xxw Ty1 x,
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JONATHAN DEE
so it suffices to show that R m⺪ p ŽW ww T xxw Ty1 x. is dense in S ww T xxw Ty1 x for the ᒊ R-adic topology. Given a series f s Ýa nT n in Sww T xxw Ty1 x, write M Ž f . for the W-submodule of S generated by the a n . It is clear that f g R m⺪ p ŽW ww T xxw Ty1 x. if and only if M Ž f . is a finite W-module. We now show that we may approximate an arbitrary f g Sww T xxw Ty1 x by an element of R m⺪ p ŽW ww T xxw Ty1 x.. Let t 1 , . . . , t r denote a set of Teichmuller lifts in R of ¨ elements of k R , and let m1 , . . . , m n denote a set of generators in R of the ideal ᒊ R . Any x g S may be written as a finite sum xs
Ý x i m⺪
p
zi
where x i g R and z i g W. On the other hand, given a g R, we may write it aswq
Ý wi m i q Ý wi , j m i m j q ⭈⭈⭈ q Ý i
i, j
wi1 , . . . , i k m i1 ⭈⭈⭈ m, i k q m,
i1 , . . . , i k
where the w, wi , etc., are in W Ž k R ., and m g ᒊ Rkq 1. Hence there exists g g ᒊ Rkq 1S ww T xxw Ty1 x such that M Ž f y g . is a subset of the W-submodule of S generated by t 1 m⺪ p 1, . . . , t r m⺪ p 1 and m i1 ⭈⭈⭈ m i j for j s 1, . . . , k, which is finite. Hence f y g g R m ŽW ww T xxw Ty1 x. and we are done. 2.3. The Characteristic Zero Case Ž Version 2. In this section we discuss a variant of the previous section where we work relative to the extension K Ž p⬁ .rK. This fits more naturally with Iwasawa theory and Galois cohomology and is proved in the same manner as was used in the previous section. For this paragraph put K n s K Ž p n . for n s 1, 2, . . . , ⬁, HK s Ž G KrK ⬁ ., G⬁ s G Ž K ⬁ rK ., and ⌫K s GK rHK . Recall that K 0 is the field of fractions of W Ž k . where k is the residue field of K. We must again define a field E and rings O and Oˆn r. Define R as before, containing ⑀ s Ž ⑀ Ž n. .. Let E Ž K 0 . s k Ž Ž ⑀ y 1 . . ; Frac Ž R . and define EŽ K . s Ž EŽ K0 .
sep H K
.
,
where again the separable closure of E Ž K 0 . is taken inside FracŽ R .. Put s w ⑀ x y 1 in W ŽFracŽ R ... Let O Ž K 0 . be the p-adic completion of W Ž k .ww xxwy1 x which is a subring of W ŽFracŽ R .., and let E Ž K 0 . be the fraction field of O Ž K 0 .. Let Eˆ be the completion of the maximal unrami-
⌽-⌫-MODULES
657
fied extension of E contained within W⺡ pŽFracŽ R .., and Oˆn r s Eˆ l W ŽFracŽ R ... Finally, put O Ž K . s Ž Oˆn r . H K and E s Eˆ H K . A ⌽-⌫-module in the sense of Cherbonnier and Colmez wCCx is now formally the same Žwith our new definitions. as it was before, and the result is the same. THEOREM 2.3.1. The functor V ¬ ⺔R Ž V . s Ž V mR OˆRn r .
HK
induces a functor between the category of R-representations of GK and the category of ´ etale ⌽-⌫-modules Ž in the sense of Cherbonnier and Colmez .. 2.4. Change of the Coefficient Ring In this section we discuss extension and restriction of scalars for ⌽-⌫modules. Since these results are based on facts established for ⌽-modules it does not matter whether we work with Fontaine’s ⌽-⌫-modules or those of Cherbonnier᎐Colmez. For the rest of this section let R and S be two coefficient rings and let
: RªS be a local homomorphism of coefficient rings. We show that the families of functors ⺔ and ⺦ behave well with respect to extension of scalars and via , and with restriction of scalars if S is finite over R. First, recall that induces homomorphisms of rings, also denoted ,
: OR ª OS
and
: OˆRn r ª OˆSn r .
We may thus use to change rings for ⌽-⌫-modules. Let V be an R-representation of GK . Write VS s V mR S. There is a natural GK -equivariant homomorphism of OˆRn r-modules OˆRn r mR V ª OˆSn r mS Ž VS . , and this induces a homomorphism of ⌽-⌫-modules Žwith OS coefficients . ⌰: OS mO R ⺔R Ž V . ª ⺔S Ž VS . .
Ž 4.
PROPOSITION 2.4.1. ⌰ is an isomorphism of ⌽-⌫-modules. Proof. We show that ⌰ mOS OˆSn r is an isomorphism. We then appeal to Proposition 1.2.6 which tells us that OˆSn r is faithfully flat over OS . We first observe that OˆSn r mOS Ž OS mO R ⺔R Ž V . s OˆSn r mO R ⺔R Ž V . s OˆSn r mOˆRn r Ž OˆRn r mR ⺔R Ž V . . .
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JONATHAN DEE
We now use Proposition 2.1.14 which yields a commutative diagram OˆSnrm⌰
OˆSn r mOS ⺔S Ž VS .
6
OˆSn r mOˆRn r Ž OˆRn r mO R ⺔R Ž V ..
6
6
6
OˆSn r mOˆRn r Ž OˆRn r mR V .
OˆSn r mS VS
in which the vertical arrows are isomorphisms. On the other hand, OˆSn r mOˆRn r Ž OˆRn r mR V . s OˆSn r mR V , OˆSn r mS VS s OˆSn r mS Ž S mR V . s OˆSn r mR V , and the lower horizontal map in the diagram is just the canonical isomorphism. COROLLARY 2.4.2. Let : R ª S be a local homomorphism of coefficient rings. Let V be an R representation of GE and M on ´ etale ⌽-⌫-module with coefficients in OR . Then ⺔R Ž V . mOR OS ( ⺔S Ž V mR S . ,
⺦R Ž M . mR S ( ⺦S Ž M mOR OS . .
Proof. The fact that ⺔ commutes with extension of scalars is the substance of Proposition 2.4.1. The result for ⺦ follows from the fact that it is a quasi inverse for ⺔. If S is a finite R module under then one has restriction of scalar functors Rep S ª Rep R ,
⌽ ModetS ª ⌽ ModetR .
One may prove formally that ⺔ and ⺦ commute with restriction of scalars. COROLLARY 2.4.3.
Suppose S is a finite R-module. The map
V mR OˆRn r s V mS Ž S mR OˆRn r . ª V mS OˆSn r induces an isomorphism ⌳ : ⺔R Ž V . ª ⺔S Ž V . . Proof. Let V be an R-representation of GK and W an S-representation of GK . Consider the equalities Hom ⌽ Mod etR Ž ⺔R Ž V . , ⺔R Ž W . . s Hom Rep RŽ V , W . s Hom Rep SŽ V mR S, W . s Hom ⌽ Mod etS Ž ⺔S Ž VS . , ⺔S Ž W . . s Hom ⌽ Mod etS Ž ⺔R Ž V . mO R OS , ⺔S Ž W . . s Hom ⌽ Mod etR Ž ⺔R Ž V . , ⺔S Ž W . . .
⌽-⌫-MODULES
659
Since ⺔R is an equivalence of categories, every object in ⌽ ModetR is of the form ⺔R Ž V . for some V. Hence by the full faithfulness of the Yoneda embedding we are done.
3. GALOIS COHOMOLOGY 3.1. Comparing ⌽-⌫-Modules with Cohomology Groups In this section K will be a finite extension of ⺡ p . This condition is equivalent to assuming that the residue field k is finite. Until we say otherwise we shall assume that K contains p if p / 2, and K contains 4 if p s 2. Hence K Ž p⬁ . is the cyclotomic ⺪ p-extension of K and ⌫ is cyclic. Fix a topological generator ␥ of ⌫. This section extends work of Fontaine and Herr Žsee wHerrx or wCCx.. Let u: ⺔R Ž V . ª ⺔R Ž V . be any R-linear map commuting with the action of ⌫. Let ␥ be a topological generator for ⌫. Then there is a complex 
␣
CuR, ␥ Ž K , V . : 0 ª ⺔R Ž V . ª ⺔R Ž V . [ ⺔R Ž V . ª ⺔R Ž V . ª 0.
Ž 5.
Here ␣ Ž x . s ŽŽ u y 1. x, Ž␥ y 1. x . and  Ž x, y . s Ž␥ y 1. x y Ž u y 1. y. Write Hui Ž V . s H i Ž CuR, ␥ Ž K , V . . . Note that the complex CR, ␥ Ž K, V . is functorial in V. By the exactness of ⺔R an exact sequence 0ªVªWªXª0 will thus induce an exact sequence of complexes 0 ª CR, ␥ Ž K , V . ª CR, ␥ Ž K , W . ª CR, ␥ Ž K , X . ª 0. We deduce that V ¬ Ž Hi Ž V . . i is a ␦-functor from Rep R to the category of systems of R-modules. PROPOSITION 3.1.1. ␦-functors.
If V is any object in RepR , there is an isomorphism of ;
Ž Hi Ž V . . i ª Ž H i Ž K , V . . i .
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JONATHAN DEE
Proof. For V of finite length morphisms, Hi Ž V . ª H i Ž K , V . are constructed in wHerrx using the fact that H i Ž K, V . are universal ␦-functors on the category of discrete representations of GK . One then proves this map to be an isomorphism, and it remains for us to show how to pass to the limit. First recall that since Vn is finite and K is a finite extension of ⺡ p it is known that H i Ž K, Vn . is finite. It follows from wTatex that 6
H i Ž K , V . s lim H i Ž K , Vn . , where we recall that the left-hand side is computed using continuous cochains. On the other hand, we may show 6
Hi Ž V . s lim Hi Ž Vn . . Indeed, the inverse limit functor is exact on the category of torsion ´ etale ⌽-⌫-modules with coefficients in R, since these modules are finite over the Artin ring ORrᒊ Rn . Also, recall that we showed in Lemma 2.1.8 that 6
⺔R Ž V . s lim ⺔R Ž Vn . . The result follows formally from these two facts. 3.2. The Operator One may define a one-sided inverse to , denoted by . Recall that since the residue field of O is not perfect, will not be an automorphism but will be injective. The field Eˆ, which we recall is the fraction field of Oˆn r , is an extension of degree p of Ž Eˆ.. Define
: Eˆ ª Eˆ by
Ž x. s
1 p
y1 Ž Tr Eˆr Ž Eˆ. Ž x . . .
Note that maps Oˆn r to itself and O to itself. Indeed, this follows from the fact that the residue extensions E sepr Ž E . sep and Er Ž E . are totally inseparable. The trace map defined by Trace Eˆr Ž Eˆ. Ž x . s Trace Ž Eˆ. Ž y ¬ xy .
⌽-⌫-MODULES
661
is then trivial for these extensions. Hence if x g Oˆn r then Trace Eˆr Ž Eˆ.Ž x . g p Oˆn r. It is easy to verify that commutes with the action of GK . Furthermore, Tr Eˆr Ž Eˆ. Ž Ž x . . s p Ž x . implies that
Ž Ž x . . s x. We may extend to Oˆn r m⺪ p R by making it act trivially on R, and then it will map ᒊ Oˆn r m R q Oˆn r m R to itself. We may thus extend to an R-linear map
: OˆRn r ª OˆRn r . Making act on OˆRn r m V via its action on OˆRn r we obtain an operator on DŽ V . Ž acts Galois equivariantly.. It is clear that if V has finite length then the on ⺔R Ž V . agrees with the on ⺔ ⺪ pŽ V . under the identification ⺔R Ž V . s ⺔ ⺪ pŽ V . . Now suppose
Ž x, y . g Z 1 Ž CR, ␥ Ž K , V . . is a 1-cocycle. We claim that
Ž y Ž x . , y . g Z 1 Ž CR, ␥ Ž K , V . . . Indeed, y Ž ␥ y 1 . Ž x . y Ž y 1 . Ž y . s y Ž y 1 . Ž y . y Ž y 1 . Ž y . s yy q Ž y . y y 1 Ž y . s 0, since Ž␥ y 1.Ž x . s Ž y 1.Ž y .. Further,
Ž y Ž y 1 . Ž x . , Ž ␥ y 1 . Ž x . . s Ž Ž y 1 . x, Ž ␥ y 1 . y . , so we have a well-defined induced homomorphism H 1 Ž CR, ␥ Ž K , V . . ª H 1 Ž CR, ␥ Ž K , V . . . The following extends Proposition 4.1 in wHerrx. PROPOSITION 3.2.1. The map Ž x, y . ¬ Žy Ž x ., y . induces an isomorphism H 1 Ž CR, ␥ Ž K , V . . ª H 1 Ž CR, ␥ Ž K , V . . .
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JONATHAN DEE
Proof. If V has finite length then this follows from Proposition 4.1 in wHerrx and the compatibility of the operator with the isomorphism ⺔ ⺪ pŽ V . ( ⺔R Ž V .. If V is arbitrary, then the result holds for Vn for arbitrary n. We may pass to the limit as in the proof of Proposition 3.1.1. The following exact sequence follows formally from the definition of the complex C , ␥ Ž K, V .. 0 ª ⺔R Ž V .
s1
⌫
r Ž ␥ y 1 . ª H1 Ž V . ª ⺔R Ž V . r Ž y 1 . ª 0. Ž 6 .
where the maps are y ¬ Ž0, y . and Ž x, y . ¬ x. 3.3. Iwasawa Theory We still assume for the moment that ⌫ is cyclic. We work in this paragraph with ⌽-⌫-modules in the sense of Cherbonnier and Colmez wCCx Žnote that there is no difference between the two approaches while we still assume ⌫ to be cyclic, a condition we shall drop presently.. First, if V is an R-representation of GK we write 6
HIiw Ž K , V . s lim H i Ž K n , V . , where K n s K ⬁⌫n and ⌫n s ⌫ p and the transition maps are corestriction. We have immediately: n
LEMMA 3.3.1.
For any V, HI0w Ž K, V . s 0.
Proof. Since V is finitely generated as an R-module it is in particular Noetherian. The sequence V K n is an ascending chain of sub R-modules, so must be constant for large n. But then the transition maps for suitably large n are just multiplication by the degree. On the other hand, p⬁ N w K ⬁ : K x, and p is in the maximal ideal of R. Hence F p n V s 0, and the proof is complete. It is the HI1w that we shall be particularly interested in. First note that it follows immediately from the definitions that if V is an R-representation of GK , then ⺔R Ž V . is canonically isomorphic to ⺔R Ž V⬘. as ⌽-⌫n-modules, where V ⬘ is the restriction of V to GK n for any n. Combining the exact sequence Ž6. and the isomorphisms of cohomology groups in Proposition 3.2.1 and Proposition 3.1.1 we deduce a diagram 6
H 1Ž K n , V . cores
id
H K ny1 , V ..
6
6
⺔R Ž V .
1Ž
6
⺔R Ž V .
Ž7.
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Here the complexes over K n should be constructed using the generator n n ␥ p of ⌫ p . This diagram is commutative when R s ⺪ p by unpublished work of Fontaine Žbut see the proof given in wCCx.. It follows that the diagram is commutative when R is arbitrary but V has finite length. It is easy to see from this that we have a commutative diagram for all m and n, s1
ª
H 1 Ž K n , Vm . cores x
s1
ª
H 1 Ž K ny1 , Vm . .
⺔R Ž V . id x ⺔R Ž V .
The diagram Ž7., being the inverse limit over all m of these diagrams, is therefore also commutative. We deduce from this a natural homomorphism of R-modules ⺔R Ž V .
s1
ª HI1w Ž K , V . .
Observe that this will still exist even if K does not contain p , and possibly p s 2. Indeed, ⺔R Ž V . is the same when computed over K or K n as an R-module for any n. Similarly, HIiw Ž K , V . s HIiw Ž K n , V . . We may thus work over K 2 , and ⌫K 2 is cyclic. For R s ⺪ p this homomorphism is known, after Fontaine, to be an isomorphism Žsee wCCx.. By using the result for Vm and passing to the limit we deduce the following. PROPOSITION 3.3.2. There is a canonical isomorphism of R-modules ⺔R Ž V .
s1 ;
ª HI1w Ž K , V . .
Remark 3.3.3. If we work with ⌽-⌫-modules in the sense of Fontaine then the same methods yield an isomorphism ⺔R Ž V .
s1 ;
ª HI1w Ž K , V . ,
where now the right-hand side is defined using the cyclotomic ⺪ p-extension of K. We have already computed HI0w Ž K, V . s 0 for arbitrary V. An argument as above, using the method of wCCx for the case of finite length V, shows that ;
⺔R Ž V . r Ž y 1 . ª HI2w Ž K , V . .
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Putting these results together we deduce the following. THEOREM 3.3.4. The groups HIiw Ž K, V . are computed by the complex y1
⺔R Ž V . ª 0
6
0 ª ⺔R Ž V . for all i.
REFERENCES wCCx
F. Cherbonnier and P. Colmez, Theorie d’Iwasawa des representations p-adiques ´ ´ d’un corps local, J. Amer. Math. Soc. 12 Ž1999., 241᎐268. wEGAx A. Grothendieck and J. Dieudonne, de geometrie algebrique 0 IV , Publ. ´ Elements ´ ´ ´ ´ Math. Inst. Hautes Etudes Sci. 20 Ž1964.. wFonx J.-M. Fontaine, Representations p-adiques des corps locaux, I, in ‘‘The Grothendieck ´ Festschrift,’’ pp. 249᎐310, Birkhauser, Basel, 1990. ¨ wFon2x J.-M. Fontaine, Groupes p-divisible sur les corps locaux, Asterisque 47–48 Ž1977.. ´ wFTx A. Frohlich and M. J. Taylor, ‘‘Algebraic Number Theory,’’ Cambridge Univ. Press, ¨ Cambridge, UK, 1991. wHerrx L. Herr, Sur la cohomologie galoisienne des corps p-adique, Bull. Soc. Math. France 126 Ž1998., 563᎐600. wMatx H. Matsumura, ‘‘Commutative Ring Theory,’’ Cambridge Univ. Press, Cambridge, UK, 1989. wMazx B. Mazur, Deformation theory of Galois representations, in ‘‘Modular Forms and Fermat’s Last Theorem,’’ pp. 243᎐313, Springer, New York, 1997. wTatex J. Tate, Relations between K 2 and Galois cohomology, In¨ ent. Math. 36 Ž1976., 257᎐274. wWeix C. Weibel, ‘‘An Introduction to Homological Algebra,’’ Cambridge Univ. Press, Cambridge, UK, 1994. wWinx J.-P. Wintenberger, Le corps des normes de certaines extensions infinies des corps locaux; applications, Ann. Sci. Ecole Norm. Super. 16 Ž1983., 59᎐89.
⌽-⌫-Modules for Families of Galois Representations Jonathan Dee California Institute of Technology, Pasadena, California 91125 E-mail: [email protected] Communicated by Jan Saxl Received May 20, 1999
0.1. Introduction 0.2. Notations and conventions 1. Definitions and commutati¨ e algebra 1.1. Some types of rings 1.2. Completed tensor products 1.3. Lifting fields to characteristic zero 2. An equi¨ alence of categories 2.1. The characteristic p case 2.2. The characteristic zero case ŽVersion 1. 2.3. The characteristic zero case ŽVersion 2. 2.4. Change of the coefficient ring 3. Galois cohomology 3.1. Comparing ⌽-⌫-modules with cohomology groups 3.2. The operator 3.3. Iwasawa Theory Key Words: ⌽-⌫-modules; local fields; Galois representations; complete Noetherian local rings; completed tensor product; Galois cohomology: Iwasawa Theory
0.1. INTRODUCTION Fontaine has found a new approach to understanding the category of ⺪ p-adic representations of GK s G Ž KrK ., where K is a field complete with respect to a discrete valuation whose residue field is perfect of characteristic p Žsee wFonx.. Here a ⺪ p-adic representation of GK is a ⺪ p-module of finite rank with a continuous linear action of GK and p is a prime number. In the equal characteristic case he constructed the category of ´ etale ⌽-modules over K Žsee below for the definition. which a priori seems much simpler than the category of ⺪ p-adic representations of GK . 636 0021-8693r01 $35.00 Copyright 䊚 2001 by Academic Press All rights of reproduction in any form reserved.
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He then proved in an elementary way an equivalence of this category with the category of all ⺪ p-adic representations of GK . The mixed characteristic case is then deduced from the equal characteristic case via the field of norms machine due to Fontaine and Wintenberger Žsee wWinx.. The category of ⺪ p-adic representations of GK in this case is equivalent to the category of ⌽-⌫-modules over K. Suppose for simplicity that p / 2. Then define O to be the p-adic completion of W ww T xxw Ty1 x, K ⬁ to be the cyclotomic ⺪ p-extension of K in K, GK s GŽ KrK ., HK s GŽ KrK ⬁ ., and ⌫ s GK rHK . One can define an action of ⌫ and also the action of a Frobenius ⌽ on O Žsee below for details.. An ´ etale ⌽-⌫-module is then a finite rank O-module with semilinear action of ⌫ and a bijective semilinear operator commuting with the action of ⌫. In this paper we extend Fontaine’s results to give an understanding of the category of R-modules of finite type with a continuous R-linear action of GK , where now R may be any complete Noetherian local ring whose residue field is a finite extension of ⺖p . We construct a category of ´ etale families of ⌽-⌫-modules over K parameterised by R and prove that it is equivalent to the category of R-linear representations of GK defined above. Such a family is defined to be a module of finite type over the ˆ⺪ p R with actions of ⌽ and ⌫ as before. We completed tensor product O m show that our functor is compatible with the extension of scalars for an arbitrary local homomorphism of complete Noetherian local rings R ª S. The proof of this result uses the equivalence of categories we prove in Section 4 in a crucial way. If S is further supposed to be finite over R then our functor is compatible with restriction of scalars. Our method for proving the equivalence of categories referred to above is to use the results of Fontaine for the case where the representation V has finite length and then extend it to the general case by taking inverse limits. Given this equivalence of categories, it is clear that given an object functorially associated to an R-representation of GK , one should be able to describe this object purely in terms of the corresponding ⌽-⌫-module. We show in Section 3 that this is the case for the continuous Galois cohomology groups of V and the inverse limit
6
lim H 1 Ž K n , V .
where K n is the extension of K of degree p n in the cyclotomic ⺪ p-extension of K. Actually, we work over the tower K Ž p⬁ . and use a slight variant of ⌽-⌫-modules Ždue to Colmez and Cherbonnier wCCx. defined using this tower, but the method is identical. It is possible that this
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approach may lead to the construction of a Perrin᎐Riou homomorphism for families of Galois representations. I thank Jan Nekovar ´ˇ for his help and encouragement. 0.2. NOTATIONS AND CONVENTIONS All rings will be commutative with a 1. If R is a ring we shall write ᒊ R for its radical, the intersection of all maximal ideals. Unless specified to the contrary all semilocal rings R will be considered as topological rings with the ᒊ R-adic topology. Write R= for the group of units of R. If R is local then we write k R s Rrᒊ R . If F is a local or global field then OF will denote its ring of integers. We refer to a ring equipped with a linear topology as complete if the natural homomorphism to its completion is an isomorphism Ži.e., it is separated and complete in the sense of Bourbaki.. We shall say that an R-module is finite if it is finitely generated as a module. All Galois cohomology groups in this paper will be continuous Galois cohomology groups in the sense of wTatex. We fix a prime number p for the rest of this paper. 1. DEFINITIONS AND COMMUTATIVE ALGEBRA 1.1. Some Types of Rings We first define some types of local rings which we shall use later on. The following may be found in wMazx. DEFINITION 1.1.1. A coefficient ring R is a Noetherian complete local ring with finite residue field k R of characteristic p. Remark 1.1.2. A coefficient ring R is a W Ž k R .-algebra in a natural way. The structure map W Ž k R . ª R need not be injective, of course. The following is what we shall mean by p-rings for the purposes of this paper. DEFINITION 1.1.3. A p-ring is a complete discrete valuation ring whose valuation ideal is generated by p. 1.2. Completed Tensor Products We shall need to understand the completed tensor product of a p-ring with a coefficient ring. Let R and S be arbitrary rings and I ; R, J ; S be two ideals. Suppose that R and S are both T-algebras for some third ring T.
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ˆT S is defined DEFINITION 1.2.1. The completed tensor product R m as the completion of R mT S with respect to the Ž I m S q R m J .-adic topology. Remark 1.2.2. We should of course carry the ideals I and J in the notation, but in practice there will be no ambiguity. The following is part of wEGA, 0.19.7.1.2x and gives us a large part of the properties of completed tensor products that we require. PROPOSITION 1.2.3. Let A be a ring, let B and C be local Noetherian A-algebras. Suppose that C is complete and that k B is finite as an A-module. ˆA C is a complete Noetherian semilocal ring. Then B m In more specialised circumstances we may prove more. LEMMA 1.2.4. Let A, B, and C be as abo¨ e. Assume further that all three rings are complete Noetherian local rings with residue characteristic p and that k B is a finite separable extension of k A . Assume the structural homomorphism ˆA C is equal to the A ª B is a local homomorphism. Then the radical of B m ideal generated by the image of ᒊ B m C q B m ᒊ C .
ˆA C. Since B m ˆA C Proof. Write ᒊ s ᒊ B m C q B m ᒊ C and E s B m is Noetherian, ᒊ generates the kernel of ˆA C ª k B mk A k C . : Bm The right-hand side is a finite separable k C -algebra, so it equals a finite product of finite separable extensions of k C . It follows immediately that ᒊ E contains the radical. On the other hand, E is complete with respect to the ᒊ-adic topology, so ᒊ is contained in the radical. Remark 1.2.5. It follows from the proof of Lemma 1.2.4 that the local ˆA C correspond to the field components of k B mk A k C . components of B m Next we introduce some notation. From now on we shall usually use O to refer to a p-ring, and capital letters R, S, etc., to refer to coefficient rings or general rings. If O is a p-ring and R a coefficient ring then we write
ˆ⺪ p R. OR s O m Note that by Proposition 1.2.3 OR is a complete Noetherian semilocal ring, although its residue fields need not be finite as we have made no restriction on the residue field of O , and indeed later k O will not even be perfect.
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Now suppose R and S are two coefficient rings and O is a p-ring Žor indeed any local ring with residue field of characteristic p .. If : R ª S is a ring homomorphism then it induces
: O m⺪ p R ª O m⺪ p S. If is local, then
Ž O m ᒊ R q ᒊ O m R . ; O m ᒊ S q ᒊ O m S, so is continuous Žwith respect to the obvious topologies.. It thus induces a semilocal homomorphism
: OR ª OS . Finally, in this section we show a flatness result that we shall need later. More precisely, we have: PROPOSITION 1.2.6.
Let
: O1 ª O2 be a local homomorphism of p-rings and let R be a coefficient ring. If is flat then the induced homomorphism
R : O1, R ª O2, R is faithfully flat. Proof. First recall that a local homomorphism is flat if and only if it is faithfully flat. Next, the flatness result for R is a simple extension of the proof in wEGA, 0.19.7.1.2x. Note that Ž ᒊ R Oi, R . n s ᒊ Rn Oi, R . The local criterion for flatness wMat, Theorem 22.3x thus implies that it is enough to check the flatness of O2, Rrᒊ Rn O2, R over O1, Rrᒊ Rn O1, R for all n. We have that Oi , Rrᒊ Rn Oi , R s Oi m⺪ p Rrᒊ Rn for i s 1, 2, and flatness is preserved under base extension. To see faithful flatness we need to show that every maximal ideal of O1, R is in the image of the induced map Spec Ž O2, R . ª Spec Ž O1, R . .
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On dividing by the radicals of O1, R , and O2, R we obtain the commutative diagram with surjective vertical arrows O2, R 6
k O 2 m⺖ p k R .
6
6
k O1 m⺖ p k R
6
O1, R
The lower horizontal homomorphism is faithfully flat since it is the base change of a field extension, and faithful flatness is preserved by base change. It follows that R is faithfully flat. 1.3. Lifting Fields to Characteristic Zero Let E be a field of characteristic p. Following wFonx, assume we have a p-ring O of characteristic zero with fraction field E and residue field E. Fix a choice of E . In general O is unique up to a non-unique isomorphism. If OE n r is a strict Henselisation of O with field of fractions En r , then OE n r has a valuation induced from O which is discrete and p generates the valuation ideal. The valuation ring in the completion Eˆn r of En r is a p-ring with residue field E sep , a separable closure of E. Write Oˆn r for this ring. There is an identification of Galois groups ;
E., GE s G Ž E seprE . ª G Ž En rrE and GE acts by continuity on Eˆn r . From now on R will always denote a coefficient ring, unless expressly stated otherwise. Define the ring
ˆ⺪ p R OR s O m as in Section 1.2. In particular there is a decomposition OR (
Ł OR , i , igI
for some finite index set I, where the OR, i are complete local Noetherian rings. Similarly, define the ring
ˆ⺪ p R, OˆRn r s Oˆn r m with decomposition OˆRn r (
Ł OˆRn,r j . jgJ
It is an OR-algebra and is faithfully flat over OR by Proposition 1.2.6.
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The action of GE on En r induces an action on Oˆn r and hence an action on Oˆn r m⺪ p R, via the trivial action on R. Being continuous on En r this action is continuous on Oˆn r m⺪ p R, so it induces a GE -action on OˆRn r , continuous with respect to the ᒊ R OˆRn r-adic topology, which we shall use in the following. Remark 1.3.1. It follows from Section 1.2 that OR and OˆRn r are Noetherian semilocal rings, complete with respect to the ᒊ R-adic topology, and that ᒊ R generates the radical of these rings. 2. AN EQUIVALENCE OF CATEGORIES 2.1. The Characteristic p Case Recall that E is an arbitrary field of characteristic p. We certainly do not want to assume that E is perfect. As before let O be a p-ring of characteristic zero with residue field E and field of fractions E . From now on we suppose that O is equipped with a lift of Frobenius: a ring homomorphism Žnot necessarily bijective. such that
Ž x. ' x p
mod p.
We shall assume that is flat. By tensoring with R we deduce an R-linear homomorphism Žalso denoted by .:
: O m⺪ p R ª O m⺪ p R. Since the ideal ᒊ O m⺪ p R q O m⺪ p ᒊ R in O m⺪ p R is generated by ᒊ R it is immediate that maps this radical to itself. We have LEMMA 2.1.1. The induced homomorphism
: OR ª OR is flat. Proof. This follows immediately from Proposition 1.2.6. The Frobenius on O extends uniquely by functoriality and continuity to a Frobenius on Oˆn r , and then as above to a flat homomorphism from OˆRn r to itself. We remark that the fixed ring of Frobenius acting on Oˆn r is just ⺪ p .
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DEFINITION 2.1.2. An R-representation of GE is a finitely generated R-module with continuous, R-linear action of GE . Write Rep R for the category of such representations of GE . DEFINITION 2.1.3. Write ⌽ Mod R for the following category. The objects of ⌽ Mod R are ⌽-modules: they are OR-modules M, equipped with a -semilinear homomorphism
M : M ª M, or equivalently a OR-linear homomorphism ⌽M : M ª M, where M s M mO , O is the base change of M by OR via . Morphisms in ⌽ Mod R or OR-linear homomorphisms commuting with . Remark 2.1.4. It is clear that ⌽ Mod R is abelian. Indeed, it is just the category of modules over the noncommutative ring OR w x, where the addition in this ring is as for the polynomial ring, but
x s Ž x. . We next define a functor from Rep R to ⌽ Mod R . Let V be any R-representation of GE . Define ⺔R Ž V . s Ž OˆRn r mR V .
GE
,
where GE acts diagonally. Then ⺔R Ž V . carries the structure of an OR-module since multiplication by OR on OˆRn r m V is GE -equivariant. The Frobenius on OˆRn r acts GE -equivariantly, and thus induces a -semilinear homomorphism
⺔ R ŽV . : ⺔R Ž V . ª ⺔R Ž V . . We shall often write V rather than ⺔ R ŽV . and this will hopefully cause no confusion. The association V ¬ ⺔R Ž V . clearly induces a functor ⺔R : Rep R ª ⌽ Mod R . LEMMA 2.1.5.
Suppose that ᒊ R V s 0. Then as an OR-module ⺔R Ž V . s ⺔ k R Ž V . .
Proof. By the assumption on V we have OˆRn r mR V ( Ž OˆRn rrᒊ R . mk R V .
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We know from Lemma 1.2.4 that ᒊ R OˆRn r is the radical of OˆRn r , and that OˆRn rrᒊ R OˆRn r ( E sep m⺪ p k R ( OˆknRr .
If V is killed by ᒊ Rn then our construction agrees with that contained in wFonx by the following lemma. If ᒊ Rn V s 0 for some n then as O⺪ p-modules
LEMMA 2.1.6.
⺔ ⺪ p Ž V . s ⺔R Ž V . . The operator is the same on both sides. Proof. Since V is finite as an R-module and is killed by ᒊ Rn it must be finite as a ⺪ p-module Žrecall that R has finite residue field.. Hence
ˆR R m ˆ⺪ p Oˆn r s V m ˆ⺪ p Oˆn r s V m⺪ p Oˆn r . V mR OˆRn r s V m
ž
/
Remark 2.1.7. The previous two results show that the family of functors ⺔ commutes with restriction of scalars in some special cases. We shall show in Section 2.4 that ⺔ commutes with a more general restriction of scalars. We shall now show that ⺔R commutes with inverse limits. This result will be very useful in deducing our results from those in wFonx. If V is any object in Rep R , write Vn s Vrᒊ Rn V. Here ᒊ Rn V is the submodule of V generated by elements of the form m¨ where m g ᒊ Rn and ¨ g V. PROPOSITION 2.1.8.
Let V be an R-representation of GE . Then ;
6
⺔R Ž V . ª lim ⺔R Ž Vn . . Proof. It is immediate that taking GE -invariants commutes with inverse limits, so it suffices to prove that the tensor product with OˆRn r commutes with inverse limits. But 6
6
lim Ž OˆRn r mR Vn . s lim Ž OˆRn r mR Ž Vrᒊ Rn V . . 6
s lim Ž OˆRn r mR V . rᒊ Rn s OˆRn r mR V .
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The last equality in the equation follows from the fact that OˆRn r mR V is complete with respect to the ᒊ R OˆRn r-adic topology. This in turn follows from Ž1.3.1. and the fact that V is finitely generated as an R-module. Next we show that the functor ⺔R preserves a lot of the structure of the category Rep R : PROPOSITION 2.1.9. The functor ⺔R is exact and faithful. Proof. First note that ⺔R is left exact by the flatness of OˆRn r over R. We know from Lemma 2.1.6 that if V is of finite length then ⺔R Ž V . s ⺔ ⺪ pŽ V . . Hence, by the exactness of ⺔⺪ p proved by Fontaine, if A, B, and C are finite length R-representations sitting in a sequence 0ªAªBªCª0 then 0 ª ⺔R Ž A . ª ⺔R Ž B . ª ⺔R Ž C . ª 0 is exact. Now suppose A, B, and C are arbitrary, still sitting in the above sequence. On tensoring with Rrᒊ Rn and using the exactness for finite length representations we deduce exact sequences ⺔R Ž Arᒊ Rn . ª ⺔R Ž Brᒊ Rn . ª ⺔R Ž Crᒊ Rn . ª 0 for all n. Let K n denote the kernel of the map ⺔R Ž Brᒊ Rn . ª ⺔R Ž Crᒊ Rn . , so that there is a surjective homomorphism ⺔R Ž Arᒊ Rn . ª K n . Using Lemma 2.1.6 again we see that ⺔R Ž Arᒊ Rn . is finite over the Artin ring ORrᒊ Rn . Hence the inverse system Ž⺔R Ž Arᒊ Rn .. n satisfies the Mittag-Leffler condition, and thus so does K n . Taking limits we deduce that ⺔R Ž B . ª ⺔R Ž C . is surjective, as required. From this we deduce the following handy fact. COROLLARY 2.1.10.
Let I be any ideal of R. Then I ⭈ ⺔R Ž V . s ⺔R Ž I ⭈ V .
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and ;
⺔R Ž V . rI ⭈ ⺔R Ž V . ª ⺔R Ž VrI ⭈ V . . Proof. Let x 1 , . . . , x n4 denote a set of generators for I. Consider the map
: V n ª V
Ý ¨ i ¬ Ý x i¨ i . It is clear that the image of is I ⭈ V. By the exactness of ⺔R , the image of ⺔R Ž . is ⺔R Ž IV .. On the other hand, identifying ⺔R Ž V n . with ⺔R Ž V . n, the map n
⺔R Ž . : ⺔R Ž V . ª ⺔R Ž V . is just
Ý di ¬ Ý di xi , since the Galois action is R-equivariant. The image of this map is just I ⭈ ⺔R Ž V ., and so we deduce I ⭈ ⺔R Ž V . s ⺔R Ž I ⭈ V . . We conclude by applying ⺔R to the short exact sequence 0 ª I ⭈ V ª V ª VrI ⭈ V ª 0.
LEMMA 2.1.11. ;
6
⺔R Ž V . ª lim Ž ⺔R Ž V . rᒊ Rn ⺔R Ž V . . . Proof. This is immediate from Proposition 2.1.8 and Corollary 2.1.10. PROPOSITION 2.1.12. If V is an R-representation of GE then ⺔R Ž V . is finitely generated as an OR-module. Proof. We may appeal to wFonx together with Lemma 2.1.6 for the case of V killed by ᒊ R . It follows from Lemma 2.1.11 that ⺔R Ž V . is separated for the ᒊ R-adic topology. Furthermore, using exactness we know ⺔R Ž V .rᒊ R s ⺔R Ž V1 . is finite as an OR-module since it is finite as an O-module. It thus follows from Theorem 8.4 in wMatx that ⺔R Ž V . is finite over OR . We next compute the ⌽-⌫-module corresponding to the trivial one-dimensional representation. The answer is reassuring.
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LEMMA 2.1.13. ⺔R Ž R . s OR . Proof. First note that OR ¨ OˆRn r. Indeed, let C denote the cokernel of the inclusion O ¨ Oˆn r. It is easy to see that C is torsion free as a ⺪ p-module, and so is flat. Hence Tor1⺪ p Ž C, R . s 0. Hence there is an inclusion O m⺪ p R ¨ Oˆn r m⺪ p R. Taking completions with respect to the ᒊ R-adic topology we deduce the desired inclusion. We next show that this inclusion has image equal to Ž OˆRn r . G E . In order to prove this it suffices by Nakayama’s lemma to divide by ᒊ R . We have seen that ⺔R Ž R . rᒊ R s ⺔R Ž Rrᒊ R . s ⺔k RŽ Rrᒊ R . , so we may assume without loss of generality that R s k R . In this case ⺔R Ž R . s Ž E sep m⺖ p k R .
GE
s E m⺖ p k R s OR .
PROPOSITION 2.1.14. If V is an R-representation of GE then the canonical OˆRn r-linear homomorphism of GE -modules OˆRn r mO R ⺔R Ž V . ª OˆRn r mR V
Ž 1.
is an isomorphism. Proof. To show that Ž1. is an isomorphism, note that if V is killed by ᒊ Rn for some n then this reduces after Lemma 2.1.6 to Proposition 1.2.6 in wFonx. We have 6
6
lim OˆRn r mO R ⺔R Ž Vn . ( lim OˆRn r mO R ⺔R Ž V . rᒊ Rn ( OˆRn r mO R ⺔R Ž V .
ž
/
since ⺔R commutes with taking quotients by Corollary 2.1.10 and OˆRn r mO R Ž V . is complete with respect to the ᒊ R-adic topology. Also, 6
lim Ž OˆRn r mR Vn . ( OˆRn r mR V ,
so the result holds for general V.
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JONATHAN DEE
We now introduce a full subcategory of ⌽ Mod R , which we shall subsequently show to be the essential image of ⺔R . DEFINITION 2.1.15. An object M in ⌽ Mod R is said to be ´ etale if ⌽M is an isomorphism and M is finitely generated as OR-module. We remark that if is bijective then ⌽ is an isomorphism if and only if is bijective. etale Write ⌽ ModetR for the full subcategory of ⌽ Mod R consisting of ´ ⌽-modules. A morphism of ´ etale ⌽-modules is a morphism of the underlying ⌽-modules. It is not immediately obvious that ⌽ ModetR R is an abelian category. However, this is true by the following lemma from wFonx Žsince we already know ⌽ Mod R is abelian .. LEMMA 2.1.16. In ⌽ Mod R , the kernel and cokernel of a map of ´ etale ⌽-modules are ´ etale. Proof. This follows immediately from the commutative diagram of abelian groups 0
6
6 K
0
6
6
L
K
k
6
6
M
⌽L
6
6
N
6
6
0
⌽M
L
6
⌽N
M
6
N
6
6
0
where the middle two vertical arrows are isomorphisms. In fact, the full subcategory ⌽ ModetR of ⌽ Mod R is stable under subobject and quotient. This is the substance of the following lemma. LEMMA 2.1.17.
Suppose we ha¨ e a short exact sequence in ⌽ Mod R , 0 ª M ª N ª P ª 0.
Then N is an object of ⌽ ModetR if and only if both M and P are. Proof. Assuming M and P to be ´ etale we deduce that N is ´ etale by the Snake Lemma Žthe fact that it is finite as an OR-module is obvious.. For the converse, we have an exact sequence Mn ª Nn ª Pn ª 0. Since N is ´ etale, so is Nn for all n. Since Mn , Nn , and Pn are finite O-modules this sequence is naturally a sequence in ⌽ Mod ⺪ p. By Proposition 1.1.6 in wFonx, both Pn and K n are ´ etale, where K n s ker Ž Nn ª Pn . .
⌽-⌫-MODULES
649
Now since ORrᒊ Rn is Artinian for all n it follows that the inverse limit functor is exact on the category of finite torsion ⌽-modules. It follows that 6
6
M s lim Mn s lim K n is ´ etale. Remark 2.1.18. Note that ⌽ ModetR is also stable under tensor product. Indeed, if M and N are objects in ⌽ ModetR , then M mO R N is clearly a finite OR-module, and the bijectivity of ⌽MmN follows from the fact that ⌽M and ⌽N are bijective since ⌽MmN s ⌽M mO R ⌽N . PROPOSITION 2.1.19. is ´ etale.
If V is any R-representation of GE , then ⺔R Ž V .
Proof. After Proposition 2.1.12 it remains to prove that ⌽ V is bijective. This holds for V of finite length by Lemma 2.1.6 and Proposition 1.2.6 in wFonx, then for general V by the isomorphism of ⌽-modules 6
⺔R Ž V . ( lim ⺔R Ž Vn . .
We next introduce an inverse functor to ⺔R . DEFINITION 2.1.20. Let M be an ´ etale ⌽-module. Then write ⺦R Ž M . s OˆRn r mO R M
ž
/
s1
.
The association M ¬ ⺦R Ž M . extends in a natural way to a functor from ⌽ ModetR to the category of R-modules with R-linear action of GE . First we show the analogue of Proposition 2.1.8. Write Mn s Mrᒊ Rn M. PROPOSITION 2.1.21. phism of OR-modules
Suppose that M is ´ etale. Then the natural homomor6
⺦R Ž M . ª lim ⺦R Ž Mn . is an isomorphism of GE -modules. Proof. We proceed as we did in the proof of Proposition 2.1.8. It is clear that taking invariants commutes with inverse limits. The OˆRn r-module OˆRn r mO R M is finitely generated since M is ´ etale. Hence 6
lim OˆRn r mO R Ž Mrᒊ Rn M . ( OˆRn r mO R M
ž
and we conclude as before.
/
650
JONATHAN DEE
To continue we need the analogue of Lemma 2.1.6. A glance at the proof, however, shows that no extra work is required. If ᒊ Rn M s 0 then as O⺪ p-modules,
LEMMA 2.1.22.
⺦⺪ pŽ M . s ⺦R Ž M . . Our next task is to show the exactness of ⺦R . We first prove a preliminary result which will also be useful in other circumstances. etale ⌽-module and let V be an R-representaLEMMA 2.1.23. Let M be an ´ tion of GE . Then 1 y is a surjecti¨ e homomorphism of abelian groups acting on both OˆRn r mR V and OˆRn r mR M. Proof. We first show the surjectivity for OˆRn r mR V. Suppose that V is killed by ᒊ R . The map 1 y is surjective on E sep since for all g E sep the polynomial x p y x y is separable. Since V is free over k R and acts on E sep m⺖ p V via its action on E sep it follows that 1 y is surjective Žusing the on E sep m⺖ p V. The result for general V follows by devissage ´ Snake Lemma. and passage to the limit Žusing the fact that OˆRn r mR Vn satisfies the Mittag-Leffler condition.. Now suppose M is an ´ etale ⌽-module. If M has finite length as an OR-module then we know from the proof of Proposition 1.2.6 in wFonx and Lemma 2.1.22 that ;
OˆRn r mR ⺦R Ž M . ª OˆRn r mR M, and that ⺦R Ž M . is an R-representation of GE . This isomorphism respects the action of , so by the result for V we deduce that 1 y is surjective on OˆRn r mR M. The general case proceeds by passage to the limit. PROPOSITION 2.1.24. The functor ⺦R is exact. Proof. Let 0ªAªBªCª0 be a short exact sequence of ´ etale ⌽-modules. We apply the Snake Lemma to the diagram
1y
0.
6
6
OˆRn r mR C
6
6
OˆRn r mR B
6
6
OˆRn r mR A
6
0
1y
0
6
1y
OˆRn r mR C
6
OˆRn r mR B
6
OˆRn r mR A
6
0
⌽-⌫-MODULES
651
We obtain an exact sequence 0 ª ⺦R Ž A . ª ⺦R Ž B . ª ⺦R Ž C . ª Ž OˆRn r mR A . r Ž 1 y . . By lemma 2.1.23 the last term vanishes. We now compute ⺦R Ž M . when M is the simplest ⌽-module of rank 1. LEMMA 2.1.25. ⺦R Ž OR . s R, with the usual action of on OR and tri¨ ial GE action on R. Proof. There is a short exact sequence of ⺪ p-modules 1y
0 ª ⺪ p ª Oˆn r ª Oˆn r ª 0. If we tensor with R the sequence remains exact since Tor1⺪ p ŽR, Oˆn r . s 0. On taking completions we deduce the result. PROPOSITION 2.1.26. If M is an ´ etale ⌽-module then ⺦R Ž M . is finitely generated as an R-module and the homomorphism of OˆRn r-modules OˆRn r mR ⺦R Ž M . ª OˆRn r mO R M
Ž 2.
is an isomorphism. Proof. Given the exactness of ⺦R and the compatibility with inverse limits, the finiteness of ⺦R Ž M . proceeds exactly as for ⺔R Ž V .. Similarly, we may show Ž2. just as we proved Proposition 2.1.14. THEOREM 2.1.27. The functor ⺔R : Rep R ª ⌽ ModetR is an equi¨ alence of categories, with quasi-in¨ erse ⺦R . Proof. Our task is to construct functorial isomorphisms ⺦R Ž ⺔R Ž V . . ª V and ⺔R Ž VR Ž M . . ª M for R-representations V and ´ etale ⌽-modules M respectively. Consider the isomorphism ⺔R Ž V . mO R OˆRn r ª V mR OˆRn r .
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JONATHAN DEE
On taking invariants of both sides we deduce an isomorphism ⺦R Ž ⺔R Ž V . . ª Ž V mR OˆRn r .
s1
.
Since acts trivially on V there is a map V ª Ž V mR OˆRn r .
s1
.
This map is an isomorphism if V is of finite length Žsee the proof of Lemma 2.1.6. by wFonx. We may pass to the limit as in the proof of Lemma 2.1.8. We similarly find that the canonical map M ª M mO R OˆRn r
ž
/
GE
is an isomorphism. The isomorphism ⺔R Ž ⺦R Ž M . . s Ž ⺦R Ž M . mR OˆRn r .
GE
ª M mO R OˆRn r
ž
/
GE
then provides the desired isomorphism. It follows formally from what we have proved so far that ⺔R and ⺦R commute with tensor product. Note in the following that M mO R N is ´ etale Žsee Remark 2.1.18.. COROLLARY 2.1.28. Let V and W be R-representations of GE . Let M and N be ´ etale ⌽-modules. The natural homomorphism of ⌽-modules
Ž V , W . : ⺔R Ž V . mO R ⺔R Ž W . ª ⺔R Ž V mR W . and the natural homomorphism of R-representations of GE
Ž M, N . : ⺦R Ž M . mR ⺦R Ž N . ª ⺦R Ž M mO R N . are isomorphisms. Proof. We have a commutative diagram ⺦R Ž⺔R Ž V . mO R ⺔R ŽW ..
6
⺦R Ž .
6
6
VmW
Ž⺔ R Ž V ., ⺔ R ŽW ..
6
⺦R Ž⺔R Ž V .. mR ⺦R Ž⺔R ŽW ..
⺦R Ž⺔R Ž V mR W ...
The left vertical arrow and bottom horizontal arrow are isomorphisms. Hence is injective for all M and N Žusing the fact that ⺔R is essentially surjective on objects., and ⺦R Ž . is surjective for all V and W. Since ⺦R is
⌽-⌫-MODULES
653
an equivalence of categories, we deduce that is surjective. Switching the roles of ⺦R and ⺔R we see that in fact and are isomorphisms, as required. 2.2. The Characteristic Zero Case Ž Version 1. Let K be a field complete with respect to a discrete valuation and suppose that K has perfect residue field. We show how one may use the equivalence of categories proven in previous sections to deduce an equivalence between the category of R-modules of finite rank with a continuous R-linear action of GK and a certain category of ⌽-⌫-modules. In the paragraph following this we shall see a slight variant which is often found in the literature Žsee for instance wCCx or wHerrx.. Since we have already dealt with the case of a field of characteristic p we assume from now on that K has characteristic 0. Let K 0 denote the fraction field of W Ž k . where k is the residue field of K. Let K be an algebraic closure of K and ⺓ p its p-adic completion on which GK acts by continuity. We first follow Fontaine in defining the ring 6
R s lim O⺓ p , where the transition maps are x ¬ x p , addition is given by
Ž x q y.
Ž n.
s lim
mª⬁
Ž x Ž nqm. q y Ž nqm. .
pm
,
and multiplication by
Ž xy .
Ž n.
s x Ž n. y Ž n. .
The ring R is known to be a complete valuation ring whose fraction field is algebraically closed of characteristic p with a continuous action of GK . The group R= contains a copy of ⺪ p Ž1., and we choose a generator, in other words, a system of roots of unity in K ⑀ s Ž ⑀ Ž n. . in R, where ⑀ Ž0. s 1 and ⑀ Ž1. / 1. Denote by the Frobenius on R given by x ¬ x p. Define
˜0 s
Ý
⑀ w ax ,
ag⺖ p
where w ax denotes a Teichmuller lift of a. Define the ring ¨ E0 s k Ž Ž ˜0. . inside R and let E0sep be the separable close of E0 inside FracŽ R .. It is stable under GK . To define E s E Ž K . we need some more notation. Let
654
JONATHAN DEE
K ⬁ be the cyclotomic ⺪ p-extension of K 0 in K if p / 2, K ⬁ s D K 0 Ž 2 ⬁ . if p s 2, and let HK s GK K ⬁ . Here KK ⬁ is the compositum of K and K ⬁ . Note that KK ⬁ is the cyclotomic ⺪ p-extension of K Žif p / 2.. Put ⌫K s GK rHK s G Ž KK ⬁ rK .. Write p r K s w K ⬁ l K : K 0 x. We may now define E s E Ž K . s yr K ŽŽ E0sep . H K ., a field of characteristic p which is in fact independent of the choice of ⑀ and is complete with respect to a discrete valuation. In order to apply the theory we developed in the previous sections we need a p-ring with residue field E. We work inside the field WK 0Ž Frac R . s W Ž Frac R . w 1rp x and mirror the construction of E Ž K .. First define
0 s yp q
w x Ý w⑀ x a .
ag⺖ p
Let O0 denote the p-adic completion of W ŽŽ 0 ... Since W Ž R . is p-adically complete we may consider O0 ; W Ž R . ; WK 0Ž Frac R . . Note that O0 is a p-ring of characteristic zero with residue field E0 . Write E0 for the fraction field of O0 , a subfield of WK 0ŽFrac R .. The Frobenius on Frac R extends by functoriality and semilinearity to WK 0ŽFrac R ., and we define E s E Ž K . s yr K Ž E0 . . Then E receives a discrete valuation from O0 . Write O for its ring of integers, a p-ring with residue field E. The strict Henselisation O hs of O may Žand will. be identified with the ring of integers in the maximal unramified extension of E in WK 0ŽFrac R .. We may thus take Oˆn r ; WK 0ŽFrac R ., where Oˆn r is the completion of O hs . The field of norms construction due to Fontaine and Wintenberger Žsee wWinx. gives us a canonical isomorphism ;
HK ª G E . The group GK acts on R, then acts by functoriality on WK 0ŽFrac R ., and Oˆn r is stable by this action. On the other hand, GE acts on Oˆn r by continuity and functoriality, and these actions are compatible with the above identification of Galois groups. If V is a finite-dimensional R-representation of GK we may thus consider the ⌽-module ⺔R Ž V . s Ž V mR OˆRn r .
HK
s Ž V mR OˆRn r .
GE
.
⌽-⌫-MODULES
655
The action of GK on V mR OˆRn r induces a semilinear action of GK rHK s ⌫K s ⌫ on ⺔R Ž V .. We are thus led to introduce the category of Ž R-linear. ⌽-⌫ modules Žin the sense of Fontaine. over K. Objects in this category are Ž R-linear. ⌽-modules equipped with an OR-semilinear action of ⌫ commuting with the action of . We say that a ⌽-⌫-module is ´ etale if its underlying ⌽-module is ´ etale. By the above considerations ⺔R yields a functor from the category of continuous R-linear representations of GK to the category of ⌽-⌫-modules. If M is a ⌽-⌫-module we may consider the GK -representation ⺦R Ž M . s Ž M mR Oˆn r . s1. Here GK acts on Oˆn r as before, and acts via ⌫ s GK rHK on M. The diagonal action on M mR Oˆn r is -equivariant, so it induces a GK -action on ⺦R Ž M .. If V is an R-representation of GK then there is a canonical R-linear homomorphism of representations of GK , V ª ⺦R Ž ⺔R Ž V . . . Since this is an isomorphism when restricted to HK Žby Theorem 2.1.27. it must already be an isomorphism of GK -representations. Similarly, if M is an ´ etale ⌽-⌫-module, the canonical homomorphism of ⌽-⌫-modules M ª ⺔R Ž ⺦R Ž M . . is an isomorphism. Indeed, the underlying map of ⌽-modules is an isomorphism after Theorem 2.1.27. We have shown THEOREM 2.2.1. The functor ⺔R yields an equi¨ alence of categories between the category of continuous R-linear representations of GK and the category of ´ etale ⌽-⌫-modules. The functor ⺦R is a quasi-in¨ erse functor. We finish with a description of OR . Recall that as a ring O is Žnoncanonically. the p-adic completion of W ww T xxw Ty1 x, where W s W Ž k .. It is easy to see that in fact Os
½ Ý a T ; a g W , a ª 0 in W as n ª y⬁ 5 ; W w T , T n
n
n
n
y1
x .
ng⺪
We show that OR is what one would hope it was. PROPOSITION 2.2.2. Write S s R m⺪ p W. Then OR s
½ Ý a T ; a g S, a ª 0 as n ª y⬁ 5 ; S w T , T n
n
n
n
y1
x . Ž 3.
ng⺪
Here the limit a n ª 0 means with respect to the ᒊ R-adic topology of S.
ˆ⺪ p ŽW ww T xxw Ty1 x.. It is clear that the Proof. First note that OR s R m ring on the right-hand side of Ž3. is the ᒊ R-adic completion of Sww T xxw Ty1 x,
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JONATHAN DEE
so it suffices to show that R m⺪ p ŽW ww T xxw Ty1 x. is dense in S ww T xxw Ty1 x for the ᒊ R-adic topology. Given a series f s Ýa nT n in Sww T xxw Ty1 x, write M Ž f . for the W-submodule of S generated by the a n . It is clear that f g R m⺪ p ŽW ww T xxw Ty1 x. if and only if M Ž f . is a finite W-module. We now show that we may approximate an arbitrary f g Sww T xxw Ty1 x by an element of R m⺪ p ŽW ww T xxw Ty1 x.. Let t 1 , . . . , t r denote a set of Teichmuller lifts in R of ¨ elements of k R , and let m1 , . . . , m n denote a set of generators in R of the ideal ᒊ R . Any x g S may be written as a finite sum xs
Ý x i m⺪
p
zi
where x i g R and z i g W. On the other hand, given a g R, we may write it aswq
Ý wi m i q Ý wi , j m i m j q ⭈⭈⭈ q Ý i
i, j
wi1 , . . . , i k m i1 ⭈⭈⭈ m, i k q m,
i1 , . . . , i k
where the w, wi , etc., are in W Ž k R ., and m g ᒊ Rkq 1. Hence there exists g g ᒊ Rkq 1S ww T xxw Ty1 x such that M Ž f y g . is a subset of the W-submodule of S generated by t 1 m⺪ p 1, . . . , t r m⺪ p 1 and m i1 ⭈⭈⭈ m i j for j s 1, . . . , k, which is finite. Hence f y g g R m ŽW ww T xxw Ty1 x. and we are done. 2.3. The Characteristic Zero Case Ž Version 2. In this section we discuss a variant of the previous section where we work relative to the extension K Ž p⬁ .rK. This fits more naturally with Iwasawa theory and Galois cohomology and is proved in the same manner as was used in the previous section. For this paragraph put K n s K Ž p n . for n s 1, 2, . . . , ⬁, HK s Ž G KrK ⬁ ., G⬁ s G Ž K ⬁ rK ., and ⌫K s GK rHK . Recall that K 0 is the field of fractions of W Ž k . where k is the residue field of K. We must again define a field E and rings O and Oˆn r. Define R as before, containing ⑀ s Ž ⑀ Ž n. .. Let E Ž K 0 . s k Ž Ž ⑀ y 1 . . ; Frac Ž R . and define EŽ K . s Ž EŽ K0 .
sep H K
.
,
where again the separable closure of E Ž K 0 . is taken inside FracŽ R .. Put s w ⑀ x y 1 in W ŽFracŽ R ... Let O Ž K 0 . be the p-adic completion of W Ž k .ww xxwy1 x which is a subring of W ŽFracŽ R .., and let E Ž K 0 . be the fraction field of O Ž K 0 .. Let Eˆ be the completion of the maximal unrami-
⌽-⌫-MODULES
657
fied extension of E contained within W⺡ pŽFracŽ R .., and Oˆn r s Eˆ l W ŽFracŽ R ... Finally, put O Ž K . s Ž Oˆn r . H K and E s Eˆ H K . A ⌽-⌫-module in the sense of Cherbonnier and Colmez wCCx is now formally the same Žwith our new definitions. as it was before, and the result is the same. THEOREM 2.3.1. The functor V ¬ ⺔R Ž V . s Ž V mR OˆRn r .
HK
induces a functor between the category of R-representations of GK and the category of ´ etale ⌽-⌫-modules Ž in the sense of Cherbonnier and Colmez .. 2.4. Change of the Coefficient Ring In this section we discuss extension and restriction of scalars for ⌽-⌫modules. Since these results are based on facts established for ⌽-modules it does not matter whether we work with Fontaine’s ⌽-⌫-modules or those of Cherbonnier᎐Colmez. For the rest of this section let R and S be two coefficient rings and let
: RªS be a local homomorphism of coefficient rings. We show that the families of functors ⺔ and ⺦ behave well with respect to extension of scalars and via , and with restriction of scalars if S is finite over R. First, recall that induces homomorphisms of rings, also denoted ,
: OR ª OS
and
: OˆRn r ª OˆSn r .
We may thus use to change rings for ⌽-⌫-modules. Let V be an R-representation of GK . Write VS s V mR S. There is a natural GK -equivariant homomorphism of OˆRn r-modules OˆRn r mR V ª OˆSn r mS Ž VS . , and this induces a homomorphism of ⌽-⌫-modules Žwith OS coefficients . ⌰: OS mO R ⺔R Ž V . ª ⺔S Ž VS . .
Ž 4.
PROPOSITION 2.4.1. ⌰ is an isomorphism of ⌽-⌫-modules. Proof. We show that ⌰ mOS OˆSn r is an isomorphism. We then appeal to Proposition 1.2.6 which tells us that OˆSn r is faithfully flat over OS . We first observe that OˆSn r mOS Ž OS mO R ⺔R Ž V . s OˆSn r mO R ⺔R Ž V . s OˆSn r mOˆRn r Ž OˆRn r mR ⺔R Ž V . . .
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JONATHAN DEE
We now use Proposition 2.1.14 which yields a commutative diagram OˆSnrm⌰
OˆSn r mOS ⺔S Ž VS .
6
OˆSn r mOˆRn r Ž OˆRn r mO R ⺔R Ž V ..
6
6
6
OˆSn r mOˆRn r Ž OˆRn r mR V .
OˆSn r mS VS
in which the vertical arrows are isomorphisms. On the other hand, OˆSn r mOˆRn r Ž OˆRn r mR V . s OˆSn r mR V , OˆSn r mS VS s OˆSn r mS Ž S mR V . s OˆSn r mR V , and the lower horizontal map in the diagram is just the canonical isomorphism. COROLLARY 2.4.2. Let : R ª S be a local homomorphism of coefficient rings. Let V be an R representation of GE and M on ´ etale ⌽-⌫-module with coefficients in OR . Then ⺔R Ž V . mOR OS ( ⺔S Ž V mR S . ,
⺦R Ž M . mR S ( ⺦S Ž M mOR OS . .
Proof. The fact that ⺔ commutes with extension of scalars is the substance of Proposition 2.4.1. The result for ⺦ follows from the fact that it is a quasi inverse for ⺔. If S is a finite R module under then one has restriction of scalar functors Rep S ª Rep R ,
⌽ ModetS ª ⌽ ModetR .
One may prove formally that ⺔ and ⺦ commute with restriction of scalars. COROLLARY 2.4.3.
Suppose S is a finite R-module. The map
V mR OˆRn r s V mS Ž S mR OˆRn r . ª V mS OˆSn r induces an isomorphism ⌳ : ⺔R Ž V . ª ⺔S Ž V . . Proof. Let V be an R-representation of GK and W an S-representation of GK . Consider the equalities Hom ⌽ Mod etR Ž ⺔R Ž V . , ⺔R Ž W . . s Hom Rep RŽ V , W . s Hom Rep SŽ V mR S, W . s Hom ⌽ Mod etS Ž ⺔S Ž VS . , ⺔S Ž W . . s Hom ⌽ Mod etS Ž ⺔R Ž V . mO R OS , ⺔S Ž W . . s Hom ⌽ Mod etR Ž ⺔R Ž V . , ⺔S Ž W . . .
⌽-⌫-MODULES
659
Since ⺔R is an equivalence of categories, every object in ⌽ ModetR is of the form ⺔R Ž V . for some V. Hence by the full faithfulness of the Yoneda embedding we are done.
3. GALOIS COHOMOLOGY 3.1. Comparing ⌽-⌫-Modules with Cohomology Groups In this section K will be a finite extension of ⺡ p . This condition is equivalent to assuming that the residue field k is finite. Until we say otherwise we shall assume that K contains p if p / 2, and K contains 4 if p s 2. Hence K Ž p⬁ . is the cyclotomic ⺪ p-extension of K and ⌫ is cyclic. Fix a topological generator ␥ of ⌫. This section extends work of Fontaine and Herr Žsee wHerrx or wCCx.. Let u: ⺔R Ž V . ª ⺔R Ž V . be any R-linear map commuting with the action of ⌫. Let ␥ be a topological generator for ⌫. Then there is a complex 
␣
CuR, ␥ Ž K , V . : 0 ª ⺔R Ž V . ª ⺔R Ž V . [ ⺔R Ž V . ª ⺔R Ž V . ª 0.
Ž 5.
Here ␣ Ž x . s ŽŽ u y 1. x, Ž␥ y 1. x . and  Ž x, y . s Ž␥ y 1. x y Ž u y 1. y. Write Hui Ž V . s H i Ž CuR, ␥ Ž K , V . . . Note that the complex CR, ␥ Ž K, V . is functorial in V. By the exactness of ⺔R an exact sequence 0ªVªWªXª0 will thus induce an exact sequence of complexes 0 ª CR, ␥ Ž K , V . ª CR, ␥ Ž K , W . ª CR, ␥ Ž K , X . ª 0. We deduce that V ¬ Ž Hi Ž V . . i is a ␦-functor from Rep R to the category of systems of R-modules. PROPOSITION 3.1.1. ␦-functors.
If V is any object in RepR , there is an isomorphism of ;
Ž Hi Ž V . . i ª Ž H i Ž K , V . . i .
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JONATHAN DEE
Proof. For V of finite length morphisms, Hi Ž V . ª H i Ž K , V . are constructed in wHerrx using the fact that H i Ž K, V . are universal ␦-functors on the category of discrete representations of GK . One then proves this map to be an isomorphism, and it remains for us to show how to pass to the limit. First recall that since Vn is finite and K is a finite extension of ⺡ p it is known that H i Ž K, Vn . is finite. It follows from wTatex that 6
H i Ž K , V . s lim H i Ž K , Vn . , where we recall that the left-hand side is computed using continuous cochains. On the other hand, we may show 6
Hi Ž V . s lim Hi Ž Vn . . Indeed, the inverse limit functor is exact on the category of torsion ´ etale ⌽-⌫-modules with coefficients in R, since these modules are finite over the Artin ring ORrᒊ Rn . Also, recall that we showed in Lemma 2.1.8 that 6
⺔R Ž V . s lim ⺔R Ž Vn . . The result follows formally from these two facts. 3.2. The Operator One may define a one-sided inverse to , denoted by . Recall that since the residue field of O is not perfect, will not be an automorphism but will be injective. The field Eˆ, which we recall is the fraction field of Oˆn r , is an extension of degree p of Ž Eˆ.. Define
: Eˆ ª Eˆ by
Ž x. s
1 p
y1 Ž Tr Eˆr Ž Eˆ. Ž x . . .
Note that maps Oˆn r to itself and O to itself. Indeed, this follows from the fact that the residue extensions E sepr Ž E . sep and Er Ž E . are totally inseparable. The trace map defined by Trace Eˆr Ž Eˆ. Ž x . s Trace Ž Eˆ. Ž y ¬ xy .
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is then trivial for these extensions. Hence if x g Oˆn r then Trace Eˆr Ž Eˆ.Ž x . g p Oˆn r. It is easy to verify that commutes with the action of GK . Furthermore, Tr Eˆr Ž Eˆ. Ž Ž x . . s p Ž x . implies that
Ž Ž x . . s x. We may extend to Oˆn r m⺪ p R by making it act trivially on R, and then it will map ᒊ Oˆn r m R q Oˆn r m R to itself. We may thus extend to an R-linear map
: OˆRn r ª OˆRn r . Making act on OˆRn r m V via its action on OˆRn r we obtain an operator on DŽ V . Ž acts Galois equivariantly.. It is clear that if V has finite length then the on ⺔R Ž V . agrees with the on ⺔ ⺪ pŽ V . under the identification ⺔R Ž V . s ⺔ ⺪ pŽ V . . Now suppose
Ž x, y . g Z 1 Ž CR, ␥ Ž K , V . . is a 1-cocycle. We claim that
Ž y Ž x . , y . g Z 1 Ž CR, ␥ Ž K , V . . . Indeed, y Ž ␥ y 1 . Ž x . y Ž y 1 . Ž y . s y Ž y 1 . Ž y . y Ž y 1 . Ž y . s yy q Ž y . y y 1 Ž y . s 0, since Ž␥ y 1.Ž x . s Ž y 1.Ž y .. Further,
Ž y Ž y 1 . Ž x . , Ž ␥ y 1 . Ž x . . s Ž Ž y 1 . x, Ž ␥ y 1 . y . , so we have a well-defined induced homomorphism H 1 Ž CR, ␥ Ž K , V . . ª H 1 Ž CR, ␥ Ž K , V . . . The following extends Proposition 4.1 in wHerrx. PROPOSITION 3.2.1. The map Ž x, y . ¬ Žy Ž x ., y . induces an isomorphism H 1 Ž CR, ␥ Ž K , V . . ª H 1 Ž CR, ␥ Ž K , V . . .
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Proof. If V has finite length then this follows from Proposition 4.1 in wHerrx and the compatibility of the operator with the isomorphism ⺔ ⺪ pŽ V . ( ⺔R Ž V .. If V is arbitrary, then the result holds for Vn for arbitrary n. We may pass to the limit as in the proof of Proposition 3.1.1. The following exact sequence follows formally from the definition of the complex C , ␥ Ž K, V .. 0 ª ⺔R Ž V .
s1
⌫
r Ž ␥ y 1 . ª H1 Ž V . ª ⺔R Ž V . r Ž y 1 . ª 0. Ž 6 .
where the maps are y ¬ Ž0, y . and Ž x, y . ¬ x. 3.3. Iwasawa Theory We still assume for the moment that ⌫ is cyclic. We work in this paragraph with ⌽-⌫-modules in the sense of Cherbonnier and Colmez wCCx Žnote that there is no difference between the two approaches while we still assume ⌫ to be cyclic, a condition we shall drop presently.. First, if V is an R-representation of GK we write 6
HIiw Ž K , V . s lim H i Ž K n , V . , where K n s K ⬁⌫n and ⌫n s ⌫ p and the transition maps are corestriction. We have immediately: n
LEMMA 3.3.1.
For any V, HI0w Ž K, V . s 0.
Proof. Since V is finitely generated as an R-module it is in particular Noetherian. The sequence V K n is an ascending chain of sub R-modules, so must be constant for large n. But then the transition maps for suitably large n are just multiplication by the degree. On the other hand, p⬁ N w K ⬁ : K x, and p is in the maximal ideal of R. Hence F p n V s 0, and the proof is complete. It is the HI1w that we shall be particularly interested in. First note that it follows immediately from the definitions that if V is an R-representation of GK , then ⺔R Ž V . is canonically isomorphic to ⺔R Ž V⬘. as ⌽-⌫n-modules, where V ⬘ is the restriction of V to GK n for any n. Combining the exact sequence Ž6. and the isomorphisms of cohomology groups in Proposition 3.2.1 and Proposition 3.1.1 we deduce a diagram 6
H 1Ž K n , V . cores
id
H K ny1 , V ..
6
6
⺔R Ž V .
1Ž
6
⺔R Ž V .
Ž7.
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Here the complexes over K n should be constructed using the generator n n ␥ p of ⌫ p . This diagram is commutative when R s ⺪ p by unpublished work of Fontaine Žbut see the proof given in wCCx.. It follows that the diagram is commutative when R is arbitrary but V has finite length. It is easy to see from this that we have a commutative diagram for all m and n, s1
ª
H 1 Ž K n , Vm . cores x
s1
ª
H 1 Ž K ny1 , Vm . .
⺔R Ž V . id x ⺔R Ž V .
The diagram Ž7., being the inverse limit over all m of these diagrams, is therefore also commutative. We deduce from this a natural homomorphism of R-modules ⺔R Ž V .
s1
ª HI1w Ž K , V . .
Observe that this will still exist even if K does not contain p , and possibly p s 2. Indeed, ⺔R Ž V . is the same when computed over K or K n as an R-module for any n. Similarly, HIiw Ž K , V . s HIiw Ž K n , V . . We may thus work over K 2 , and ⌫K 2 is cyclic. For R s ⺪ p this homomorphism is known, after Fontaine, to be an isomorphism Žsee wCCx.. By using the result for Vm and passing to the limit we deduce the following. PROPOSITION 3.3.2. There is a canonical isomorphism of R-modules ⺔R Ž V .
s1 ;
ª HI1w Ž K , V . .
Remark 3.3.3. If we work with ⌽-⌫-modules in the sense of Fontaine then the same methods yield an isomorphism ⺔R Ž V .
s1 ;
ª HI1w Ž K , V . ,
where now the right-hand side is defined using the cyclotomic ⺪ p-extension of K. We have already computed HI0w Ž K, V . s 0 for arbitrary V. An argument as above, using the method of wCCx for the case of finite length V, shows that ;
⺔R Ž V . r Ž y 1 . ª HI2w Ž K , V . .
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Putting these results together we deduce the following. THEOREM 3.3.4. The groups HIiw Ž K, V . are computed by the complex y1
⺔R Ž V . ª 0
6
0 ª ⺔R Ž V . for all i.
REFERENCES wCCx
F. Cherbonnier and P. Colmez, Theorie d’Iwasawa des representations p-adiques ´ ´ d’un corps local, J. Amer. Math. Soc. 12 Ž1999., 241᎐268. wEGAx A. Grothendieck and J. Dieudonne, de geometrie algebrique 0 IV , Publ. ´ Elements ´ ´ ´ ´ Math. Inst. Hautes Etudes Sci. 20 Ž1964.. wFonx J.-M. Fontaine, Representations p-adiques des corps locaux, I, in ‘‘The Grothendieck ´ Festschrift,’’ pp. 249᎐310, Birkhauser, Basel, 1990. ¨ wFon2x J.-M. Fontaine, Groupes p-divisible sur les corps locaux, Asterisque 47–48 Ž1977.. ´ wFTx A. Frohlich and M. J. Taylor, ‘‘Algebraic Number Theory,’’ Cambridge Univ. Press, ¨ Cambridge, UK, 1991. wHerrx L. Herr, Sur la cohomologie galoisienne des corps p-adique, Bull. Soc. Math. France 126 Ž1998., 563᎐600. wMatx H. Matsumura, ‘‘Commutative Ring Theory,’’ Cambridge Univ. Press, Cambridge, UK, 1989. wMazx B. Mazur, Deformation theory of Galois representations, in ‘‘Modular Forms and Fermat’s Last Theorem,’’ pp. 243᎐313, Springer, New York, 1997. wTatex J. Tate, Relations between K 2 and Galois cohomology, In¨ ent. Math. 36 Ž1976., 257᎐274. wWeix C. Weibel, ‘‘An Introduction to Homological Algebra,’’ Cambridge Univ. Press, Cambridge, UK, 1994. wWinx J.-P. Wintenberger, Le corps des normes de certaines extensions infinies des corps locaux; applications, Ann. Sci. Ecole Norm. Super. 16 Ž1983., 59᎐89.