EXTENSIONS OF CRYSTALLINE REPRESENTATIONS

Matthew Emerton and Mark Kisin

Department of Mathematics, University of Michigan, Ann Arbor (M.E.)

Department of Mathematics, University of Sydney, Sydney, and SFB 478, Westfälische Wilhelms Universität, Münster (M.K.)

Contents

Introduction. 1

1. Tannakian Formalism and Comparison of Ext’s.

1.1 Injectives in Tannakian Categories. 4 1.2 Ext’s For Topological Groups. 6 1.3 Ext’s for Local Galois Groups. 8

2. Rings of Periods and Admissible Modules.

2.1 Some Categories of Modules 10 2.2 Review of Period Rings. 11 2.3 Algebraic Period Rings. 12 2.4 Ext’s for Admissible Modules. 16 2.5 Analysis of Complexes. 18

3. Weakly Admissible Modules.

3.1 Ext’s for Weakly Admissible Modules. 21 3.2 Analysis of Complexes. 24 3.3 Strongly Admissible Modules. 27

4. Extensions of Crystalline Representations.

4.1 Continuous Galois Cohomology. 33 4.2 Topologies on Period Rings. 35 4.3 The Basic Long Exact Sequences. 36 4.4 Galois Cohomology of Period Rings. 41 4.5 Hyodo’s Theorem. 47

References. 49

Introduction

The theme of this paper is extensions of Galois representations, and especially extensions of crystalline and semi-stable representations in the sense of Fontaine. Here “extensions” includes higher Ext groups. Crystalline and semi-stable representations appear in comparison theorems for crystalline and p-adic étale cohomology. This paper was motivated by the authors’ desire to understand the derived categories of the categories of crystalline and semi-stable representations, with an eye to applications to comparison theorems with coefficients.

Before giving a detailed description of our results, let us point out the highlights. Fix a complete, discretely valued field K whose ring of integers is of mixed characteristic with perfect residue field. Denote by K_{0} its maximal absolutely unramified subfield, and by G_{K} its absolute Galois group. Recall that to a crystalline representation V is attached a finite dimensional K_{0} vector space M equipped with a Frobenius semi-linear operator ϕ and a filtration on M ⊗_{K_{0}} K. Conversely, given a finite dimensional K_{0} vector space M with these structures, one can ask when it comes from a crystalline representation. M’s which arise in this way are called admissible. Fontaine [Fo 2] has defined another category of finite dimensional vector spaces equipped with a ϕ and a filtration on M ⊗_{K_{0}} K, subject to certain explicit restrictions. He calls such modules weakly admissible, and conjectures that “weakly admissible ⟹ admissible.”

We construct explicit finite length complexes which compute Rhom’s in the category of admissible modules, and we show that Rhom’s in the category of weakly admissible modules are computed by the same complexes. This may be regarded as evidence for Fontaine’s conjecture. It also has the added consequence that extensions of admissible modules which are weakly admissible are admissible. Since the category of weakly admissible modules is in fact closed under extensions, so is the category of admissible modules. This reduces Fontaine’s conjecture to the case of irreducible weakly admissible modules.

We obtain analogous results in the semi-stable situation. In this case one works with K_{0} vector spaces M as above, except that in addition, M is equipped with a nilpotent operator N, which satisfies pϕN = Nϕ.

The complexes we use have previously been considered by Nekovář in [Ne]. He proved the stability of admissibility under extensions, at least for K/ℚ_{p} finite. In the case N = 0, these complexes go back to Bloch and Kato [BK]. They showed that they compute Ext’s in the category of crystalline representations whose Dieudonné modules satisfied certain strong restrictions on filtrations [BK, 4.4] (see also [Ni]). The key point is that under these restrictions, one has the results of Fontaine-Laffaille [FL], so that one knows Fontaine’s conjecture in this case. Perhaps one of the more surprising points in this paper is that one can compute Ext’s for crystalline representations without knowing Fontaine’s conjecture.^{1}

Given a weakly admissible module M such that F il^{0}M = M, Fontaine calls M strongly admissible if every extension of M by a certain period ring B_{st}^{+} (in the category of vector spaces with ϕ, N and filtration after applying ⊗_{K_{0}}K ) is trivial. He has conjectured (unpublished) that “admissible ⟹ strongly admissible.” We prove his conjecture.

Finally, we give a proof of a theorem of Hyodo, which says that an extension of semi-stable representations which is de Rham is semi-stable. This theorem is mentioned in [Fo 2, 6.2.2], and was proved in [Ne] by a different method.

We now give a more detailed overview of the paper. §1 begins with a general result on Tannakian categories, which may well be useful in other situations. Recall that to a (neutral) Tannakian category C equipped with a fibre functor there is attached a pro-algebraic group. Let R denote the corresponding (Ind-) Hopf algebra. We show that R is an injective object for C. This has the rather pleasant consequence that objects in C acquire canonical injective resolutions.

1Since the writing of this paper, Colmez and Fontaine have proved Fontaine’s conjecture [FC].

In this preprint they independently show that any extension of admissible modules is again admissible (without explicitly computing the Ext groups), and the consequent reduction of Fontaine’s conjecture to the case of simple admissible modules is an important step in their argument.

We use this abstract theory to study the following rather natural question, which however seems not to have been addressed anywhere in the literature. Consider a pro-finite group G and the category of continuous representations of G on finite dimensional ℚ_{p} vector spaces. For V, W in this category there are two obvious ways to try to define Ext(V, W); firstly via Yoneda Ext’s and secondly as the continuous group cohomology of Hom_{ℚ_{p}}(V, W). It seems unlikely that these always give the same answer. However we show that these two alternatives always agree in degrees 0, 1 and that they agree completely when G is the absolute Galois group of a local field with either finite or algebraically closed residue field. We do not know, for example, if the two definitions agree for the Galois group of the maximal extension of a number field unramified outside a finite number of places.

The connection of §1 with the rest of the paper is largely thematic. The idea that the affine ring of a pro-algebraic group (or sometimes a homogeneous space over this group) attached to a Tannakian category should be an injective object for this Tannakian category pervades much of the paper. In particular it play a critical role in the results of §2. Note though that the result on Ext’s for Galois representations is used in §4 to know that certain spaces computed via Galois cohomology really correspond to extensions of Galois representations.

In §2 we study Ext groups for admissible modules. After reviewing the definitions of various categories of K_{0} vector spaces with additional structure in §2.1 and some constructions involving period rings in §2.2, we study in §2.3 the subrings of algebraic periods B_{cris}^{alg} and B_{st}^{alg} of B_{cris} and B_{st} respectively, and among other things show that they are injective objects for suitable categories of admissible modules. In §2.4 we construct explicit finite length complexes which compute the Ext groups. The key point, proved in §2.5, is that that when these constructions are applied to B_{cris}^{alg} and B_{st}^{alg} they yield complexes which are acyclic in degree > 0. Combined with the injectivity of B_{cris}^{alg} and B_{st}^{alg}, this shows that our complexes really compute Rhom’s in these categories. It should be stressed that the acyclicity result mentioned above is far from formal. It is a consequence of certain delicate properties of the rings B_{cris} and B_{st}.

In §3 we show that the complexes of §2 also compute Rhom’s in suitable categories of weakly admissible modules. As mentioned above, this implies that the category of admissible modules is stable under extension. Again, the key point is an acyclicity result: injectives for the category of weakly admissible modules yield complexes which are acyclic in degree > 0. The proof of this result is given in §3.2. It involves showing that a failure of acyclicity always yields a non-trivial extension of a weakly admissible module by an injective, which is impossible.

In §3.3, we prove Fontaine’s conjecture that “strongly admissible ⟺ admissible.” In fact the implication ⟹ was already known (Fontaine – unpublished). For completeness, we give a proof of both implications. Our proof of ⟹ relies largely on results of Fontaine. In fact it turns out that it is this direction, rather than the (until now) conjectural one, which takes us a little more effort to prove. The ideas of the proof have much in common with the calculations in §2.

In §4 we apply the theory of §2 to study the difference between Ext’s computed in the category of Galois representations and Ext’s computed in the category of crystalline (respectively semi-stable) representations. In particular this bears on the question of when an extension of crystalline (respectively semi-stable) representations is again crystalline (respectively semi-stable). It turns out that the difference between these two Ext’s is controlled by the Galois cohomology of the period ring B_{cris} (respectively B_{st}). Motivated by this we study in §4.4 the Galois cohomology of these rings. By an argument in the style of Tate/Faltings, we reduce the calculation of this Galois cohomology to the calculation of the cohomology of a ℤ_{p} extension with coefficients in certain smaller rings. In particular this means that the Galois cohomology of B_{cris} and B_{st} vanishes in degree bigger than one. This fact is used in §4.3 to construct two exact sequences which give the precise relationship between extensions of crystalline representations, and extensions of the corresponding admissible modules.

In §4.5, we apply our results to give a proof of Hyodo’s theorem, mentioned above. A key ingredient in the proof is a certain calculation due to Bloch and Kato [BK]. We do not know how our proof compares with Hyodo’s original one. To close, we give a brief discussion of how one might try to use our results to prove the analogue of Hyodo’s theorem when the residue field is not finite.

§1 Tannakian Formalism and Comparison of Ext’s

Consider a pro-finite group G, and denote by Rep_{G} the category of finite dimensional ℚ_{p} vector spaces equipped with a continuous action by G. If V, W are in Rep_{G}, there are two possible definitions of Ext^{i}(V, W) which seem reasonable:

(1) Ext^{i}(W, V) = H^{i}(G, W^{∗}⊗_{ℚ_{p}} V), where the second term denotes cohomology computed using continuous cochains;

(2) Yoneda Ext.

We do not know if anyone has studied the general question of when these two definitions coincide. Here we show that they always coincide when i = 0, 1. It seems unlikely that the two definitions coincide in general. However, we prove that they do when G is the absolute Galois group of a local field whose residue field is either finite or algebraically closed. We begin with a general discussion of injective objects in Tannakian categories.

1.1 Injectives in Tannakian Categories. Let C be an F linear Tannakian category, for some field F, equipped with a fibre functor ω : C → Vect_{F} , where as usual Vect_{F} denotes the category of finite dimensional F -vector spaces. (Thus C is neutral.) The fundamental theorem of Tannakian categories asserts that C is equivalent to the category of finite dimensional representations of a pro-algebraic group over F. We fix the data given above, and we denote by R the affine ring of the afore mentioned pro-algebraic group. (1.1.1) Recall that given an artinian abelian category A, there is a canonical way of embedding A into an category Ind(A) with enough injectives [Oor]. This embedding induces a fully faithful functor D^{+}(A) → D^{+}(Ind(A)) on the corresponding derived categories. Indeed this amounts to showing that Yoneda Ext groups computed in A coincide with those computed in Ind(A), and this is the main result of [Oor].

The objects of Ind(A) are filtered direct limits \underrightarrow{lim}_{i∈I}V_{i}, with V_{i} in A and I some index set. Given two such objects V = \underrightarrow{lim}_{i∈I}V_{i}, W = \underrightarrow{lim}_{j∈J}W_{j}, one defines

Hom_{Ind(A)}(W, V) = \underleftarrow{lim}_{j∈J} \underrightarrow{lim}_{i∈I}Hom_{A}(W_{j}, V_{i}).

We have Lemma (1.1.2). C is Artinian. If C is identified with the category of finite dimensional representations of Spec (R), then R has a natural structure as an object of Ind(C).

Proof. As C is equivalent to the category of finite dimensional representations of a pro-algebraic group, the first claim is clear. For the second claim, note that R is a direct limit of finitely generated Hopf algebras R_{i}, and it is a standard fact from the theory of affine groups schemes that every h ∈ R_{i} is contained in an R_{i} comodule N of finite F - dimension. Such an N corresponds to a finite dimensional representation of Spec (R), whence to an object of C. Thus each R_{i} corresponds to an object of Ind(A), and so does R. □

(1.1.3) From now on we identify C with the category of finite dimensional representations of Spec (R), in which case ω becomes the functor “forget the R comodule structure.”We extend ω to a functor from Ind(C) to Ind(V ect_{F}), which we again denote ω. Note that ω remains faithful, and exact.

If N is in Ind(V ect_{F}), then N ⊗_{F} R has the structure of an object of Ind(C), as the Hopf algebra structure on R makes N ⊗_{F} R into an R comodule. We then have the key

Proposition (1.1.4). If V is in Ind(C) and N is in Ind(V ect_{F}), then there is a canonical isomorphism

Hom_{Ind(C)}(V, N ⊗_{F} R) ˜ ⟶Hom_{Ind(V ect_{F})}(ω(V), N).

Proof. To construct the required morphism, we use the unit section ϵ : R → F. Given V → N ⊗_{F} R, we form the composite

ω(V) = V ⟶ N ⊗_{F} R ⟶^{1⊗ϵ} N.

To show that this is an isomorphism, we reduce immediately to the case where V is in C, and N is finite dimensional. In that case, let us construct a map in the other direction. Let us denote by μ : V → V ⊗_{F} R the comodule map, and by m the comultiplication on R. If g : V → N is a morphism in V ect_{F} , then we have a commutative diagram in V ect_{F} .

V  ⟶^{μ} V ⊗_{F} R ⟶^{g⊗1} N ⊗_{F} R
μ↓  ^{1⊗m}↓  ^{1⊗m}↓
V ⊗_{F} R ⟶^{μ⊗1} V ⊗_{F} R ⊗_{F} R ⟶^{g⊗1⊗1} N ⊗_{F} R ⊗_{F} R

Indeed, the left square commutes by the definition of a comodule, and the right square obviously commutes. Thus the outer square commutes, which shows that (g ⊗ 1) ∘ μ is actually a map of R comodules, giving the required morphism in C. One checks easily that the two maps we have constructed are inverses. □

Corollary (1.1.5). R is an injective object in Ind(C), and every object in Ind(C) has a canonical (i.e. functorial) resolution by injectives.

Proof. For the first claim we have to show that Hom_{Ind(C)}( , R) is an exact functor, and this follows immediately from the formula in (1.1.4).

Likewise the second claim follows from the fact that ⊗_{F} R is right adjoint to ω, using the “triple” formalism (see [Wei, 8.7], where the dual case of projective resolutions is discussed).

Specifically, for V in Ind(C), write T (V) = ω(V)⊗_{F} R. Then there is a resolution T(V) = T (V) → T^{2}(V) ⋯ → of V. As an object of the form T (V) is injective (as R is), this is an injective resolution. The functoriality follows from the construction of the complex (see the next section for a more detailed description). Note that we also have

(1.1.6)  T(V) ˜ ⟶V ⊗_{F} T(F)

where F denotes the trivial representation of Spec (R). □

1.2 Ext’s for Pro-finite Groups. Let G be a pro-finite group. We want to apply the theory of the previous section to the category C = Rep_{G} of continuous representations of G on finite dimensional ℚ_{p} vector spaces. (1.2.1) Denote by R̂ the ring of continuous functions from G to ℚ_{p}, which is a ℚ_{p} Banach algebra with respect to its topology of uniform convergence, equipped with a continuous comultiplication m : R̂ → R̂ ˆ⊗R̂. We will recall some standard facts about the calculation of continuous group cohomology of G [Se, I 2.2], [Wei, 6.5].

We can form a complex T̂which provides a right resolution of the trivial representation ℚ_{p} of G, in analogy to the complex T(F) constructed in the previous section, as follows: the degree n term of T̂ is R̂^{ˆ⊗n+1}, with g ∈ G acting by 1 ⊗ 1 ⊗ ⋯ ⊗ g (in other words, we regard each R̂^{ˆ⊗n+1} as a right R̂ comodule, via the right R̂ comodule structure on the right-most factor in the tensor product), and the differential is given by defining, for any r_{0} ⊗ ⋯ ⊗ r_{n} ∈ R̂^{ˆ⊗n+1},

d(r_{0} ⊗ ⋯ ⊗ r_{n}) = 1 ⊗ r_{0} ⊗ ⋯ ⊗ r_{n} + ∑_{i=0}^{n}(−1)^{i+1}r_{0} ⊗ ⋯ ⊗ m(r_{i}) ⊗ ⋯ ⊗ r_{n}.

Concretely, one may think of an element r of R̂^{ˆ⊗n+1} as a function of n + 1-tuples (g_{0}, …, g_{n}) of elements of G, in which case the above differential can be rewritten via the formula

dr(g_{0}, …, g_{n+1}) = r(g_{1}, ⋯, g_{n+1}) + ∑_{i=0}^{n}(−1)^{i+1}r(g_{0}, ⋯, g_{i}g_{i+1}, ⋯ g_{n}).

From this point of view the G action on functions r is described by

(g · f)(g_{0}, ⋯, g_{n}) = f(g_{0}, ⋯, g_{n}g).

Let ℚ_{p}⟨G^{n}⟩ denote the ring obtained by completing the group ring ℤ_{p}[G^{n}] with respect to the topology given by ideals of the form I_{H} = {h − 1 : h ∈ H} for an open subgroup H ⊂ G^{n}, and then inverting p. Regard ℚ_{p}⟨G^{n}⟩ as a left G module by having G act via the formula

g · (g_{0}, …, g_{n−1}) = (g_{0}, ⋯, g_{n−1}g^{−1}).

Then R̂^{ˆ⊗n} is naturally isomorphic to Hom_{ℚ_{p},cts}(ℚ_{p}⟨G^{n}⟩, ℚ_{p}), so that for any continuous representation of G on a finite dimensional ℚ_{p} vector space V there is a natural isomorphism of G modules

V ⊗ R̂ ˜ ⟶Hom_{ℚ_{p}}(ℚ_{p}⟨G^{n}⟩, V),

and consequently a natural isomorphism

(V ⊗ R^{ˆ⊗n})^{G} ˜ ⟶Hom_{ℚ_{p}⟨G⟩}(ℚ_{p}⟨G^{n}⟩, V).

(In both the previous displayed isomorphisms, Hom denotes continuous homomorphisms.) Let 𝒢^{•} denote the complex whose degree n term is equal to ℚ_{p}⟨G^{n+1}⟩ which is dual to the complex T̂. Then we have a natural isomorphism

(V ⊗ T̂)^{G} ˜ ⟶Hom_{ℚ_{p}⟨G⟩}(𝒢^{•}, V).

If f ∈ Hom_{ℚ_{p}⟨G⟩}(ℚ_{p}⟨G^{n+1}⟩, V) then f(g_{0}, ⋯, g_{n−1}, g_{n}) = g_{n}^{−1}f(g_{0}, ⋯, g_{n−1}), so that f is determined by its restriction to the first n variables, and one has the isomorphism

Hom_{ℚ_{p}⟨G⟩}(ℚ_{p}⟨G^{n+1}⟩, V) ˜ ⟶Hom_{ℚ_{p}}(ℚ_{p}⟨G^{n}⟩, V).

This yields an isomorphism between the complex Hom_{ℚ_{p}⟨G⟩}(𝒢^{•}, V) and the complex of continuous V valued cochains on G. Thus

H^{i}(G, V) = H^{i}(Hom_{ℚ_{p}⟨G⟩}(𝒢^{•}, V)) = H^{i}((V ⊗ T̂)^{G})

for each i. (1.2.2) Note that for any V in Rep_{G}, (R̂ ⊗ V)^{G} has ℚ_{p} dimension equal to that of V ([Fo 2, 2.2]). Thus every V is R̂ admissible. It follows that the subring of periods R of R̂ consists of those h ∈ R̂ whose images under G span a finite dimensional ℚ_{p} vector subspace of R̂. Indeed, this is clear, as a period of R̂ is simply an element of R̂ which is in the image of a G equivariant map from an R̂ admissible representation of G.

Spec (R) is a principal homogeneous space for the pro-algebraic group associated to the Tannakian category Rep_{G} and the fibre functor “forget the G action” [Fo 2, 1.7.3], and in our case it admits a section R → ℚ_{p}, given by evaluation on the identity of G. Thus R is the affine ring of the afore mentioned pro-algebraic group. (1.2.3) We may apply the results of §1.1. Using the notation of (1.1.5), write T = T(ℚ_{p}). If V is in Rep_{G}, then V ⊗_{ℚ_{p}} T is a resolution of V by injectives. Thus if W is in Rep_{G}, then using the main result of [Oor], we have

(1.2.4) Ext_{Rep_{G}}^{i}(W, V) = H^{i}(Hom_{ℚ_{p}⟨G⟩}(W, V ⊗_{ℚ_{p}} T)) ˜ ⟶H^{i}((W^{∗}⊗_{ℚ_{p}} V ⊗_{ℚ_{p}} T)^{G}),

where the left hand side denotes Yoneda Ext in Rep_{G}.

It follows immediately from the constructions of T and T̂ that there is a natural map of complexes T → T̂, induced by the inclusion R ↪ R̂. In fact, T is defined by replacing R̂ with R, and completed tensor products with usual tensor products, in all of the discussion of 1.2.1. Hence for any W, V there is a natural map of complexes

(1.2.5.)  (W^{∗} ⊗_{ℚ_{p}} V ⊗_{ℚ_{p}} T)^{G} ⟶ (W^{∗} ⊗_{ℚ_{p}} V ⊗_{ℚ_{p}} T̂)^{G}

By (1.2.1) and (1.2.4) we see that the two possible definitions of Ext^{i}(W, V) discussed in the introduction to this section agree if and only if the map (1.2.5) is a quasi-isomorphism.

For V in Rep_{G} write H_{Y}^{i} (G, V) := (V ⊗_{ℚ_{p}} T)^{G}, and H^{i}(G, V) for the usual cohomology of V, computed using continuous cochains. (The subscript ‘Y’ is for Yoneda.)

Proposition (1.2.6). The following conditions are equivalent.

(1) The map of (1.2.5) is a quasi-isomorphism.

(2) For every V in Rep_{G}, the canonical map

H_{Y}^{i} (G, V) ⟶ H^{i}(G, V)

is an isomorphism for all i.

(3) The morphism in (2) is surjective for all V and i.

(4) For i > 0, we have H^{i}((R ⊗_{ℚ_{p}} T̂)^{G}) = 0. When i = 0, 1 the morphism of (2) is an isomorphism.

Proof. The equivalence of (1) and (2) follows by replacing W^{∗} ⊗_{ℚ_{p}} V with V in (1.2.5). That (2) is an isomorphism when i = 0 is immediate, since in fact both sides are then equal to V^{G}.

Now H_{Y}^{i} (G, V) = Ext_{Rep_{G}}^{i}(ℚ_{p}, V) extends to a functor on Ind(Rep_{G}), and as R is an injective object by (1.1.5) this functor erases R for i > 0. Similarly H^{i}(G, V) extends to a functor on Ind(Rep_{G}) given by V ↦ H^{i}((V ⊗_{ℚ_{p}} T̂)^{G}).

The morphism in (2) induces a morphism of ∂ functors which is an isomorphism when i = 0. Thus it is an isomorphism if this ∂ functor erases enough injectives. By the explicit description of the resolution in (1.1.5), it is enough that it erases R. Thus (4) ⟹ (2). If (3) holds, then evaluating the left hand side of the morphism in (2) on R, for i > 0, we see that the right hand side is 0, whence (4). Thus we have (2) ⟹ (3) ⟹ (4) ⟹ (2).

It remains to check that the morphism of (2) is always an isomorphism when i = 1. But this is well-known and easily proved, since any continuous 1-cocyle of G with coefficients in some V can be used to construct an explicit extension of ℚ_{p} by V in Rep_{G}, which is split if and only if the continuous 1-cocycle is a continuous coboundary. □

1.3 Ext’s for Local Galois Groups. In this section we consider the situation where G = G_{K} is the absolute Galois group of a mixed characteristic (0, p) discretely valued local field K. In particular, we show that the two possible definitions of the Ext groups coincide if K is a finite extension of ℚ_{p}, or when K has algebraically closed residue field. In fact this second case is easy, using the theory of the previous section.

We denote by Rep_{K} the category of continuous representations of G_{K} on finite dimensional ℚ_{p} vector spaces.

Theorem (1.3.1). Suppose K has algebraically closed residue field. If V, W are in Rep_{K}, then the two definitions of Ext^{i}(V, W) given in the introduction coincide.

Proof. We use the criterion 1.2.6 (3). We already know the map of 1.2.6 (2) is surjective when i = 0, 1, and as K has algebraically closed residue field, the target of this map vanishes when i > 1, by [Se, II 4.3]. □

Theorem (1.3.2). Suppose that K is a finite extension of ℚ_{p}. If V, W are in Rep_{K}, then the two definitions of Ext^{i}(V, W) given in the introduction coincide.

We begin with two simple Lemmas.

Lemma (1.3.3). If K is as in (1.3.2) and j is an integer, then there exists a non-trivial extension E of ℚ_{p} by ℚ_{p}(j) (as G_{K} representations).

Proof. As K is finite over ℚ_{p}, we have

∑_{i=0}^{2}(−1)^{i}dim_{ℚ_{p}}H^{i}(G_{K}, ℚ_{p}(j)) = −[K : ℚ_{p}].

This shows that dim_{ℚ_{p}}H^{1}(G_{K}, ℚ_{p}(j)) ≥ 1, and by (1.2.6) a non-zero element in H^{1}(G_{K}, ℚ_{p}(j)) corresponds to the required extension.

We remark that if j > 0 then we can even choose E to be crystalline, and we need no longer assume that K is a finite extension of ℚ_{p}. (See (3.1.5).) □

Lemma (1.3.4). If V is in Rep_{K}, there are only finitely many j with

H^{2}(G_{K}, V (j)) ≠ 0.

Proof. By Tate duality, it is enough to show that

Hom(ℚ_{p}(j − 1), V^{∗}) = H^{0}(G_{K}, V^{∗}(−j + 1)) = 0

for all but finitely many j. This follows by induction on the dimension of V. If V is irreducible the result is clear. Otherwise write V as an extension V′ → V → V′′. A map from ℚ_{p}(j − 1) to V either factors through V′, or gives a non-zero map into V′′, so the result follows by induction. □

Proposition (1.3.5). If V is in Rep then

(1) There exists W in Rep_{K}, and an injective map V ↪ W, such that the induced map H^{2}(G_{K}, V) → H^{2}(G_{K}, W) is zero.

(2) There exists W in Rep_{K}, and a surjective map W ↠ V, such that the induced map H^{0}(G_{K}, W) → H^{0}(G_{K}, V) is zero.

Proof. By Tate duality the two statements are equivalent. We prove (2). Consider the exact sequence

0 → V^{G_{K}} → V → V_{G_{K}} → 0,

and choose an integer j > 0, such that H^{2}(G_{K}, (V_{G_{K}})^{∗}(j)) = 0. This is possible by (1.3.4). Choose a non-trivial extension E of ℚ_{p} by ℚ_{p}(j). I claim that there is a W making the following diagram commute:

[figure: V^{G_{K}} (j); 0; E ⊗_{ℚ_{p}} V^{G_{K}}; W; V_{G_{K}}; 0; 0; V^{G_{K}}; V; V_{G_{K}}; 0]

Indeed, since the group of extensions can be computed using Galois cohomology by (1.2.6), the cokernel of the map

Ext_{Rep_{K}}^{1} (V_{G_{K}} , E ⊗_{ℚ_{p}} V^{G_{K}}) → Ext_{Rep_{K}}^{1} (V_{G_{K}} , V^{G_{K}})

is contained in

H^{2}(G_{K}, V_{G_{K}}^{∗} ⊗_{ℚ_{p}} V^{G_{K}} (j)) = H^{2}(G_{K}, V_{G_{K}}^{∗} (j)) ⊗_{ℚ_{p}} V^{G_{K}} = 0.

Now

W^{G_{K}} ⊂ (E ⊗_{ℚ_{p}} V^{G_{K}})^{G_{K}} = E^{G_{K}} ⊗_{ℚ_{p}} V^{G_{K}} = 0,

as E is not a split extension. □

Finally we give the

Proof of 1.3.2. We check the condition (1.2.6) (4). By [Se, II 5.6 Prop 20] G_{K} has cohomological dimension 2, so it is enough to check (1.2.6) (4) when i = 1, 2. When i = 1, it follows from the fact that the map (1.2.6) (2) is an isomorphism when i = 1, while H_{Y}^{i} (G, R) = 0 for i > 0.

To see the result when i = 2, we use (1.3.5) (1). As R is a direct limit of objects of Rep_{K}, we have

H^{2}(R ⊗_{ℚ_{p}} T̂) = \underrightarrow{lim}H^{2}(V_{i} ⊗_{ℚ_{p}} T̂) = \underrightarrow{lim}H^{2}(G_{K}, V_{i}),

where the V_{i} ⊂ R are in Rep_{K}. For any such V_{i}, choose a W which satisfies (1.3.5)

(1) with V = V_{i}. As R is an injective for Rep_{K}, the inclusion V_{i} ↪ R lifts to a map W → R. Hence the natural map

H^{2}(G_{K}, V_{i}) → H^{2}((R ⊗_{ℚ_{p}} T̂))

factors through 0, and the result follows. □

§2 Rings of Periods and Admissible Modules

In this section we consider admissible modules, in the sense of Fontaine, and we study extensions of such modules. The theme which emerged in the previous section, that algebraic rings of periods should act like injective objects in a suitable category, is also present in this new situation. The main results consist of giving explicit complexes which compute the total derived functor Rhom in the category of admissible modules.

Throughout we fix a discretely valued field K of characteristic 0 whose residue field is perfect of characteristic p > 0. We denote by K_{0} the quotient field of the ring of Witt vectors of the residue field of K. We denote by G_{K} the absolute Galois group of K. We denote by ℂ_{p} the completion of the algebraic closure of K, and by O_{ℂ_{p}} its ring of integers.

2.1 Some Categories of Modules. We briefly recall the definitions of various categories of modules that will be used in what is to come.

Let F be the Frobenius on K_{0}. We denote by K_{0}[ϕ] the twisted polynomial ring defined by the relations ϕa = F (a)ϕ, for all a ∈ K_{0}. We denote by K_{0}[ϕ, N] = K_{0}[ϕ][N] the twisted polynomial ring satisfying pϕN = Nϕ, aN = Na and ϕa = F (a)ϕ for a ∈ K_{0}.

A ϕ module (respectively (ϕ, N) module) is a left K_{0}[ϕ] module (respectively left K_{0}[ϕ, N]) module. We say that such a module is finite provided that the underlying K_{0} vector space is finite dimensional and that ϕ acts injectively, and hence bijectively (it follows from this that N acts nilpotently). A finite K_{0}[ϕ] or finite K_{0}[ϕ, N] module M is called effective if the characteristic polynomial of ϕ is in 𝒪_{K_{0}}[X] for some (and hence any) choice of basis for M as a K_{0} vector space.

A filtered ϕ module (respectively filtered (ϕ, N) module) is a ϕ module (respectively (ϕ, N) module) equipped with an exhaustive, decreasing, filtration on M_{K} = M ⊗_{K_{0}} K (indexed by ℤ.) The filtered ϕ modules (respectively filtered (ϕ, N) modules) form a category in the obvious way: a morphism between two such modules should be a morphism of left K_{0}[ϕ] modules (respectively left K_{0}[ϕ, N] modules) which, after applying ⊗_{K_{0}}K, is compatible with the filtrations on its source and target. We say that a filtered (ϕ, N) module is finite if the underlying (ϕ, N) module is finite.

2.2 Review of Period Rings. (2.2.1) We are going to review the construction, due to Fontaine, of certain rings of p-adic periods attached to the field K (see [Fo]). If we define R = \underleftarrow{lim}O_{ℂ_{p}}/p, where the terms in the inverse limit are indexed by the positive integers and the transition maps are given by the absolute Frobenius, then R is a perfect ring of characteristic p. We denote by W (R) its ring of Witt vectors.

Consider the map W (R) → R → O_{ℂ_{p}}/p, where the second map is given by projection onto the first term in the inverse limit defining R. As O_{ℂ_{p}} is p-adically complete, the universal mapping property of Witt vectors implies that this map lifts uniquely to a map W (R) → O_{ℂ_{p}}, which we denote by θ. (2.2.2) The map θ is surjective, and its kernel is a principal ideal. We can extend θ to a map θ : W (R)[1/p] → ℂ_{p}. The completion of W (R)[1/p] with respect to the ideal ker(θ) is denoted by B_{dR}^{+}. B_{dR}^{+} is a discrete valuation ring with residue field ℂ_{p}, and we denote by B_{dR} its fraction field. We again denote by θ the induced map θ : B_{dR}^{+} → ℂ_{p}.

Consider a sequence {ϵ_{i}}_{i∈ℕ} of elements of O_{ℂ_{p}}, with ϵ_{i+1}^{p} = ϵ_{i} for each i, and ϵ_{1} a primitive p^{th} root of unity. This sequence corresponds to an element ϵ ∈ R. The Teichmüller representative [ϵ] ∈ W (R) satisfies θ([ϵ]) = 1, so that [ϵ] − 1 ∈ ker(θ). We thus have

log([ϵ]) = ∑_{i=1}^{∞}([ϵ] − 1)^{i}/i(−1)^{i+1} ∈ B_{dR}^{+}.

The element log([ϵ]) is usually denoted by t. It is a uniformiser for B_{dR}^{+}. We equip B_{dR} with the filtration defined by F il^{i}B_{dR} = t^{i}B_{dR}^{+}, for i ∈ ℤ.

¿From the construction there is a natural inclusion K_{0} ↪ W (R)[1/p] ↪ B_{dR}^{+}. This map extends to an inclusion of the algebraic closure K̄ ↪ B_{dR}^{+}. (2.2.3) Consider a sequence {p_{i}}_{i∈ℕ} of elements of O_{ℂ_{p}} with p_{i+1}^{p} = p_{i}, and p_{0} = p. This sequence corresponds to an element \underline{p} ∈ R. The element ζ = [\underline{p}] − p ∈ W (R) is a generator for ker(θ) ⊂ W (R).

We denote by A_{cris} the p-adic completion of the divided power envelope of W (R) with respect to the ideal (ζ). This is the same as the p-adic completion of the divided power envelope of W (R) with respect to the ideal (ζ, p), as (p) is already equipped with divided powers. Since the ideal (ζ, p) ⊂ W (R) is stable under the Frobenius map (whereas (ζ) is not!), the Frobenius on W (R) extends continuously to A_{cris}.

As [ϵ] − 1 ∈ (ζ), we have t ∈ A_{cris}. We write B_{cris}^{+} = A_{cris}[1/p], and

B_{cris} = B_{cris}^{+}[1/t] = A_{cris}[1/t].

The Frobenius ϕ on A_{cris} extends to B_{cris} by ϕ(1/t) = p^{−1}t.

From its construction we see that B_{cris} is a K_{0} algebra, and that there is a natural map B_{cris} → B_{dR}. The induced map B_{cris} ⊗_{K_{0}} K → B_{dR} turns out to be an injection, and we equip B_{cris} ⊗_{K_{0}} K with the induced filtration. Of course B_{cris} ⊂ B_{dR} is also equipped with a filtration (the restriction of the filtration on B_{dR}), but the filtration on B_{cris} ⊗_{K_{0}} K is not induced by that of B_{cris}; that is, it is not K_{0} rational. (2.2.4) Write

u = log(1 + ζ/p) = ∑_{i=1}^{∞}(−1)^{i+1} \frac{(ζ/p)^{i}}{i} ∈ B_{dR}^{+}.

One can show that u is a transcendental element over B_{cris}. We set B_{st}^{+} = B_{cris}^{+}[u] ⊂ B_{dR}, and B_{st} = B_{cris}[u] ⊂ B_{dR}.

The natural map B_{st} ⊗_{K_{0}} K → B_{dR} turns out to be an injection, and we equip B_{st} ⊗_{K_{0}} K → B_{dR} with the induced filtration. We may of course also regard B_{st} as being filtered via the inclusion B_{st} ⊂ B_{dR}, but just as for B_{cris} the filtration on B_{st} ⊗_{K_{0}} K is not induced from B_{st}.

The Frobenius extends to B_{st} using the formula ϕ(u) = pu. Moreover, there is a monodromy operator N on B_{st}. If B_{st} is regarded as a polynomial ring over B_{cris} in one variable, then N acts as d/du. (2.2.5) Finally, let us note that as all the constructions involved are natural, the rings R, W (R), A_{cris}, B_{cris}^{+}, B_{cris}, B_{cris} ⊗_{K_{0}} K, B_{dR}^{+} and B_{dR} are all equipped with an action of G_{K}. This action commutes with the action of Frobenius on rings when the latter is defined, and respects the filtrations on all these rings.

Observe that B_{st} ⊂ B_{dR} is Galois stable. In fact an easy calculation shows that if g ∈ G_{K}, then g(u) = u + a_{g}t for some a_{g} ∈ ℤ_{p}. From this one sees that G_{K} commutes with the operators N and ϕ on B_{st}. As usual the G_{K} action on B_{st} ⊗_{K_{0}} K respects the filtrations.

We have B_{st}^{G_{K}} = B_{cris}^{G_{K}} = K_{0}.

The question of the continuity of the G_{K} action will play no role until §4. Thus, we also postpone till §4.2, the, slightly subtle discussion of the correct topologies on these period rings. (2.2.6) An important and non-trivial property of B_{cris} is that the map

F il^{0}B_{cris} →^{ϕ−1} B_{cris}

is surjective with kernel equal to ℚ_{p} [Fo, 5.3.7]. The surjectivity of this map follows in turn from the surjectivity (loc. cit.) for each positive integer j of

F il^{j}B_{cris}^{+} →^{p^{−j}ϕ−1} B_{cris}^{+}.

Much easier is that fact that N is surjective on B_{st}, and has kernel B_{cris}.

2.3 Algebraic Period Rings. (2.3.1) Recall that a continuous representation of G_{K} on a finite dimensional ℚ_{p} vector space V is called crystalline if the natural map

(V ⊗_{ℚ_{p}} B_{cris})^{G_{K}} ⊗_{K_{0}} B_{cris} ⟶ V ⊗_{ℚ_{p}} B_{cris}

is an isomorphism.

For any such representation V, one knows that M = (V ⊗_{ℚ_{p}} B_{cris})^{G_{K}} is a finite filtered ϕ module. If V is crystalline, then we have dim_{K_{0}}M = dim_{ℚ_{p}}V, and M is called admissible. More generally, any finite filtered ϕ module isomorphic to such a finite filtered ϕ module is called admissible. Sometimes we will say that M is admissible with N = 0 (in order to contrast with the semi-stable case considered below).

We denote by M_{K}^{ad} the full subcategory of the category of filtered ϕ modules consisting of the admissible modules (with N = 0). It turns out that all morphisms in M_{K}^{ad} are strict for filtrations, and that M_{K}^{ad} is a Tannakian category which is equivalent to the category of crystalline representations of G_{K} ([Fo 2, 5.3.5]).

For M an object of M_{K}^{ad} we write F il^{i}M for the filtration on M induced by the inclusion M ⊂ M ⊗_{K_{0}} K. However, when we speak of maps between admissible modules being compatible with filtrations, we will always mean the filtrations on M ⊗_{K_{0}} K. (2.3.2) A continuous representation of G_{K} on a finite dimensional ℚ_{p} vector space V is called semi-stable if the natural map

(V ⊗_{ℚ_{p}} B_{st})^{G_{K}} ⊗_{K_{0}} B_{st} ⟶ V ⊗_{ℚ_{p}} B_{st}

is an isomorphism.

For any such representation V, one knows that M = (V ⊗_{ℚ_{p}} B_{cris})^{G_{K}} is a finite filtered (ϕ, N) module. If V is semi-stable, then we have dim_{K_{0}}M = dim_{ℚ_{p}}V, and M is called admissible. More generally, any finite filtered (ϕ, N) module isomorphic to such a module is called admissible. Sometimes we will say that M is admissible with N ≠ 0. (Note that this is a slight abuse of language, because the operator N will be identically zero on those admissible modules which arise from crystalline, rather than just semi-stable, representations.)

We denote by M_{st,K}^{ad} the category of admissible modules (with N ≠ 0). Morphisms are K_{0} semi-linear maps compatible with ϕ’s and N’s and respecting filtrations after applying ⊗_{K_{0}}K.

It turns out that all morphisms in M_{st,K}^{ad} are strict for filtrations. M_{st,K}^{ad} is a Tannakian category, which is equivalent to the category of semi-stable representations of G_{K}.

For M an object of M_{st,K}^{ad} we write F il^{i}M for the filtration on M induced by the inclusion M ⊂ M ⊗_{K_{0}} K, however, when we speak of maps between admissible modules being compatible with filtrations we will always mean filtrations on M ⊗_{K_{0}} K. (2.3.3) An element b ∈ B_{cris} (respectively B_{st}) is called a period if it is in the image of a G_{K} equivariant map V → B_{cris} for some crystalline (respectively semi-stable) representation V. The set of periods forms a subring B_{cris}^{alg} ⊂ B_{cris} (respectively B_{st}^{alg} ⊂ B_{st}) ([Fo 2, 1.7.6]). It is clear from the definition that B_{cris}^{alg} (respectively B_{st}^{alg}) is stable by the action of G_{K}, and is the union of its finite dimensional G_{K} stable subspaces.

Proposition (2.3.4). B_{cris}^{alg} ⊂ B_{cris} is stable under ϕ and B_{st}^{alg} ⊂ B_{st} is stable under ϕ and N.

Proof. If b ∈ B_{cris} is a period, denote by V a crystalline representation such that b is in the image of a G_{K} equivariant map V → B_{cris}. Replacing V by its image (also crystalline), we may assume that V ⊂ B_{cris}. As the ϕ action on B_{cris} commutes with the G_{K} action ϕ(V) is also G_{K} stable, and it is isomorphic to V as a G_{K} module, hence also crystalline. This shows that B_{cris}^{alg} is stable under ϕ.

A similar argument applies for B_{st}^{alg} as N also commutes with the Galois action. □

(2.3.5) We equip B_{cris}^{alg} ⊗_{K_{0}} K ⊂ B_{cris} ⊗_{K_{0}} K and B_{st}^{alg} ⊗_{K_{0}} K ⊂ B_{st} ⊗_{K_{0}} K with the induced filtrations. Sometimes we will also consider the filtrations on B_{cris}^{alg} and B_{st}^{alg} by restricting the above filtrations to them.

We call a K_{0}-subspace M ⊂ B_{cris}^{alg} (respectively M ⊂ B_{st}^{alg}) admissible if it is finite dimensional, stable under ϕ (respectively ϕ and N) and induces an object in M_{K}^{ad} (respectively M_{st,K}^{ad}) when equipped with the filtration induced by the inclusion M ⊗_{K_{0}} K ⊂ B_{cris}^{alg} ⊗_{K_{0}} K (respectively M ⊗_{K_{0}} K ⊂ B_{st}^{alg} ⊗_{K_{0}} K )

Proposition 2.3.6. B_{cris}^{alg} is the union of its admissible K_{0}-subspaces. If M is in M_{K}^{ad}, then any K_{0} linear map M → B_{cris}^{alg} which respects ϕ and filtrations (after applying ⊗_{K_{0}}K) is strict, and has admissible image.

Similarly, B_{st}^{alg} is the union of its admissible K_{0}-subspaces. If M is in M_{st,K}^{ad}, then any K_{0} linear map M → B_{st}^{alg} which respects ϕ, N and filtrations (after applying ⊗_{K_{0}}K) is strict, and has admissible image.

Proof. First consider the case of B_{cris}^{alg}. Note that as M_{K}^{ad} is an abelian category, and all morphisms in M_{K}^{ad} are strict for filtrations, the proposition will follow once we have proved the first claim.

Let b ∈ B_{cris}^{alg} be the image of a G_{K} equivariant map V →^{f} B_{cris}^{alg}, with V crystalline. We may replace V by its image, and so assume V ⊂ B_{cris}^{alg}, and that f is the inclusion map. Moreover, we may replace V by the smallest G_{K} stable ℚ_{p} subspace of B_{cris}^{alg} containing b, and so assume that V is generated by b as a ℚ_{p}-vector space with G_{K} action. Write

M = Hom_{ℚ_{p}⟨G_{K}⟩}(V, B_{cris}) = Hom_{ℚ_{p}⟨G_{K}⟩}(V, B_{cris}^{alg});

then M is an admissible module, and we have an isomorphism

V ˜ ⟶Hom_{K_{0}[ϕ],F il}(M, B_{cris}^{alg})

(here and below “F il” means that the maps respect filtrations after applying ⊗_{K_{0}}K), given by the map v ↦ (m ↦ m(v)), as well as an isomorphism

M ˜ ⟶Hom_{ℚ_{p}⟨G_{K}⟩}(Hom_{K_{0}[ϕ],F il}(M, B_{cris}^{alg}), B_{cris}^{alg}),

given by the map m ↦ (g ↦ g(m)). Thus the inclusion map f corresponds to an element m_{f} ∈ M, and b ∈ V corresponds to a K_{0}-linear map g_{b} : M → B_{cris}^{alg} respecting Frobenius and filtrations. We have b = f(b) = g_{b}(m_{f}).

We claim that g_{b} is an injection, and identifies M with its image, compatible with the filtrations. As M is admissible the proof of the proposition in the case of B_{cris}^{alg} will be complete once we prove this claim.

To see that g_{b} is an injection, note that V ˜ ⟶Hom_{K_{0}[ϕ],F il}(M, B_{cris}^{alg}) is generated by g_{b} as a ℚ_{p}-vector space with G_{K} action, so every map in Hom_{K_{0}[ϕ],F il}(M, B_{cris}^{alg}) kills the kernel of g_{b}. However, if V^{∗} denotes the dual of V, then we have

M = (V^{∗} ⊗_{ℚ_{p}} B_{cris}^{alg})^{G_{K}} ⊂ V^{∗} ⊗_{ℚ_{p}} B_{cris}^{alg} ˜ ⟶(B_{cris}^{alg})^{d},

where d = dim_{ℚ_{p}}V and the last isomorphism is a map of finite free B_{cris}^{alg} modules with semi-linear ϕ and filtration. Thus M ⊂ (B_{cris}^{alg})^{d}. It follows that g_{b} is an injection.

Denote by M′ the image of M under g_{b}. We have to show that the filtrations on M ⊗_{K_{0}} K and M′ ⊗_{K_{0}} K coincide. As the G_{K} action on B_{cris}^{alg} commutes with the ϕ action and respects filtrations, and Hom_{K_{0}[ϕ],F il}(M, B_{cris}^{alg}) is generated by g_{b}, we see that every map M → B_{cris}^{alg} factors through the map M → M′. Thus we obtain maps respecting ϕ′s and filtrations

M′ ⟶ Hom_{ℚ_{p}⟨G_{K}⟩}(Hom_{K_{0}[ϕ],F il}(M′, B_{cris}^{alg}), B_{cris}^{alg})

˜ ⟶Hom_{ℚ_{p}⟨G_{K}⟩}(Hom_{K_{0}[ϕ],F il}(M, B_{cris}^{alg}), B_{cris}^{alg}) ˜ ⟶M,

and applying ⊗_{K_{0}}K to the composite provides an inverse to the map M ⊗_{K_{0}} K → M′ ⊗_{K_{0}} K in the category of filtered K-vector spaces.

The proof of the claim in the case of B_{st}^{alg} is identical, except that we consider everywhere morphisms which also respect N’s. □

By virtue of this result, we may regard B_{cris}^{alg} (respectively B_{st}^{alg}) as an object of Ind(M_{K}^{ad}) (respectively Ind(M_{st,K}^{ad})). We have

Theorem (2.3.7). B_{cris}^{alg} is an injective object in Ind(M_{K}^{ad}). B_{st}^{alg} is an injective object in Ind(M_{st,K}^{ad}).

We first need the following criterion for injectives.

Lemma (2.3.8). Let C be either M_{K}^{ad} or M_{st,K}^{ad}. Let I be an object of Ind(C) such that

(1) Hom_{Ind(C)}( , I) is exact on C.

(2) For every object M in C, Hom_{C}(M, I) is a finite dimensional ℚ_{p} vector space.

Then I is an injective in Ind(C).

Proof. If M^{•} is a short exact sequence in Ind(C), then one checks easily that M^{•} = \underrightarrow{lim}_{j∈J}M_{j}^{•}, with each M_{j}^{•} a short exact sequence in C. Then, from the definition of morphisms in Ind(C), we see that

Hom_{Ind(C)}^{•}(M^{•}, I) = \underleftarrow{lim}_{j∈J}Hom_{C}^{•}(M_{j}^{•}, I).

The right hand side is an exact sequence, as Hom_{Ind(C)}( , I) is exact on M_{K}^{ad}, and the inverse limit does not destroy exactness, because condition (2) guarantees that the relevant Mittag-Leffler condition is satisfied. □

Proof of (2.3.7). We will prove the theorem for B_{cris}^{alg}. The proof for B_{st}^{alg} is formally identical.

If M is an admissible module, corresponding to the crystalline Galois representation V, then Hom_{Ind(M_{K}^{ad})}(M, B_{cris}^{alg}) = V^{∗}, and so is finite dimensional. This

shows that B_{cris}^{alg} satisfies criterion (2) of Lemma (2.3.8).

To show that it satisfies criterion (1), it will suffice to show that if M_{1} → M_{2} is an injection of admissible modules, then

(2.3.9)  Hom_{Ind(M_{K}^{ad})}(M_{1}, B_{cris}^{alg}) → Hom_{Ind(M_{K}^{ad})}(M_{2}, B_{cris}^{alg})

is a surjection. But if M_{1} (respectively M_{2}) corresponds to the crystalline representation V_{1} (respectively V_{2}) then there is a canonical isomorphism Hom_{Ind(M_{K}^{ad})}(M_{1}, B_{cris}^{alg}) ˜ ⟶V_{1}^{∗}

(respectively Hom_{Ind(M_{K}^{ad})}(M_{2}, B_{cris}^{alg}) ˜ ⟶V_{2}^{∗}). Also the injection M_{1} → M_{2} corresponds to an injection V_{1} → V_{2}. We now see that (2.3.9) is surjective, because under the identifications made the morphism (2.3.9) is just the morphism V_{2}^{∗} → V_{1}^{∗} dual to injection V_{1} → V_{2}. □

2.4 Ext’s for Admissible Modules. In this section we describe explicit finite length complexes which compute Rhom’s in the category of admissible modules in both the cases N = 0 and N ≠ 0. (2.4.1) Let M be a filtered ϕ module. We denote by 𝒦_{cris}^{•}(M) the total complex of the following double complex

\xymatrix{M \ar[r] & 0 \\ M \ar[r] \ar[u]^{ϕ−1} & M_{K}/F il^{0}M_{K}, \ar[u]}

in which the lower horizontal map is the natural one, and the lower left hand term is concentrated in bidegree (0, 0). The complex 𝒦_{cris}^{•} is functorial in M. Its restriction to M_{K}^{ad} takes short exact sequences to exact triangles of complexes, and so induces a functor on D^{+}(M_{K}^{ad}), the derived category of bounded below complexes of objects of M_{K}^{ad}. Indeed, this follows immediately from the fact that morphisms in M_{K}^{ad} are strict for filtrations, so that the functor M ↦ F il^{0}M_{K} is exact. Similarly 𝒦_{cris}^{•} induces a functor on D^{+}(Ind(M_{K}^{ad})). (2.4.2) We consider the analogue of (2.4.1) when N ≠ 0. Let M be a filtered (ϕ, N) module. We denote by 𝒦_{st}^{•}(M) the total complex of the following double complex

\xymatrix{M \ar[r] & 0 \\ M ⊕ M \ar[u]^{(1−pϕ)+N} \ar[r] & 0 \ar[u] \\ M \ar[u]^{N⊕ϕ−1} \ar[r] & M_{K}/F il^{0}M_{K}, \ar[u]}

in which the lower horizontal map is the natural one, and the lower left hand term is in bidegree (0, 0). To make a little more sense of this double complex, note that, for example, taking kernels under the horizontal differentials gives a complex which is itself the total complex of the following double complex

\xymatrix{M \ar[r]^{N} & M \\ F il^{0}M \ar[r]^{N} \ar[u]^{ϕ−1} & M. \ar[u]^{1−pϕ}}

The complex 𝒦_{st}^{•} is functorial in M. As above, since morphisms in M_{st,K}^{ad} are strict for filtrations, the restriction of 𝒦_{st}^{•} to M_{st,K}^{ad} takes short exact sequences to exact triangles of complexes, and so induces a functor on D^{+}(M_{st,K}^{ad}), the derived category of bounded below complexes of objects of M_{st,K}^{ad}. Similarly 𝒦_{st}^{•} induces a functor on D^{+}(Ind(M_{st,K}^{ad})).

The key technical result for the complexes 𝒦_{cris}^{•} and 𝒦_{st}^{•} is the following

Proposition (2.4.3).

(1) 𝒦_{cris}^{•}(B_{cris}^{alg}) is quasi-isomorphic to ℚ_{p} concentrated in degree 0.

(2) 𝒦_{st}^{•}(B_{st}^{alg}) is quasi-isomorphic to ℚ_{p} concentrated in degree 0.

The proof of (2.4.3) will occupy §2.5. For the moment we state the following corollary:

Corollary (2.4.4). Let M^{•} be a complex in D^{b}(M_{K}^{ad}). There is an isomorphism of functors on D^{+}(Ind(M_{K}^{ad}))

𝒦_{cris}^{•}(Hom_{K_{0}}^{•}(M^{•}, )) ˜ ⟶Rhom_{Ind(M_{K}^{ad})}(M^{•}, )

This isomorphism is functorial in M^{•}.

Similarly, let M^{•} be a complex in D^{b}(M_{st,K}^{ad}). There is an isomorphism of functors on D^{+}(Ind(M_{st,K}^{ad}))

𝒦_{st}^{•}(Hom_{K_{0}}^{•}(M^{•}, )) ˜ ⟶Rhom_{Ind(M_{st,K}^{ad})}(M^{•}, )

This isomorphism is functorial in M^{•}.

Proof. Consider first the case of M_{K}^{ad}. If P is in Ind(M_{K}^{ad}) then we have P ⊂ P ⊗_{K_{0}} B_{cris}^{alg}, and the latter is a (possibly infinite) direct sum of copies of B_{cris}^{alg}. As this embedding is canonical (i.e. functorial in P), it is enough to construct the above isomorphism when both sides are evaluated on complexes whose terms are abstractly isomorphic to a direct sum of copies of B_{cris}^{alg}. Let P^{•} be such a complex. Then we have

𝒦_{st}^{•}(Hom_{K_{0}}^{•}(M^{•}, P^{•})) ˜ ⟶𝒦_{st}^{•}((M^{•})^{∗} ⊗_{K_{0}}^{•} P^{•})

˜ ⟶F il^{0}((M^{•})^{∗} ⊗_{K_{0}}^{•} P^{•})^{ϕ=1} ˜ ⟶Hom_{Ind(M_{K}^{ad})}^{•}(M^{•}, P^{•}).

Here we have used the fact that the terms of (M^{•})^{∗} ⊗_{K_{0}}^{•} P^{•} are again isomorphic to a direct sum of copies of B_{cris}^{alg}, so that the second isomorphism follows from (2.4.3). On the other hand (2.3.7) shows that P^{•} is a complex of injectives, and so the natural map

Hom_{Ind(M_{K}^{ad})}^{•}(M^{•}, P^{•}) ⟶ Rhom_{Ind(M_{K}^{ad})}(M^{•}, P^{•})

is an isomorphism.

This completes the proof in the case of M_{K}^{ad}, and the proof for M_{st,K}^{ad} is formally identical. □

2.5 Analysis of Complexes.

Proposition (2.5.1). The map

F il^{0}B_{cris}^{alg} ⟶^{ϕ−1} B_{cris}^{alg}

is surjective and its kernel is equal to ℚ_{p}.

Proof. The claim regarding the kernel follows from (2.2.6), since ℚ_{p} ⊂ B_{cris}^{alg} (the trivial representation is crystalline). Let us show that ϕ − 1 is surjective on B_{cris}^{alg}.

We know this is true for B_{cris} (2.2.6). If b ∈ B_{cris} is a period, denote by V a crystalline representation such that b is in the image of a G_{K} equivariant map V → B_{cris}. Replacing V by its image (also crystalline), we may assume that V ⊂ B_{cris}. Set W = (ϕ − 1)^{−1}(V) ∩ F il^{0}B_{cris}. As ϕ commutes with the G_{K} action, W is G_{K} stable, and it has finite ℚ_{p} dimension equal to dim_{ℚ_{p}}V + 1. We have to show W ⊂ B_{cris}^{alg}, so it suffices to show that W is crystalline. For this note that

dim_{K}Hom_{ℚ_{p}⟨G_{K}⟩}(W, B_{cris}) ≥ dim_{K}Hom_{ℚ_{p}⟨G_{K}⟩}(V, B_{cris}) + 1 = dim_{ℚ_{p}}V + 1 = dim_{ℚ_{p}}W,

as any map map from V into B_{cris} induces a map of W into B_{cris}, but the inclusion W ⊂ B_{cris} is not induced by one of these, as it does not have ℚ_{p} in its kernel. It follows that W is crystalline (and that the above inequality is necessarily an equality). □

Proposition (2.5.2). We have

F il^{0}(B_{cris}^{alg} ⊗_{K_{0}} K) + B_{cris}^{alg} = B_{cris}^{alg} ⊗_{K_{0}} K,

so that the map B_{cris}^{alg} → B_{cris}^{alg} ⊗_{K_{0}} K/F il^{0}(B_{cris}^{alg} ⊗_{K_{0}} K) is surjective.

Proof. First we claim that

(2.5.3)  F il^{1}(B_{cris}^{alg} ⊗_{K_{0}} K) + F il^{0}B_{cris}^{alg} = F il^{0}(B_{cris}^{alg} ⊗_{K_{0}} K).

This is equivalent to the statement that under the map θ : B_{dR}^{+} → ℂ_{p}, F il^{0}B_{cris}^{alg} and F il^{0}(B_{cris}^{alg} ⊗_{K_{0}} K) have the same image, so it is enough to show that K ⊂ θ(F il^{0}(B_{cris}^{alg})).

We already know K_{0} ⊂ θ(F il^{0}(B_{cris}^{alg})). Let π be a uniformiser for K, and choose a large integer n prime to [K : K_{0}]. Using the notation of §2.2, consider a sequence {v_{i}}_{i∈N} of elements of O_{ℂ_{p}}, with v_{i+1}^{p} = v_{i}, and v_{0} = 1 + π^{n}. This sequence corresponds to an element v ∈ R. We have log([v]) ∈ F il^{0}B_{cris}. If g ∈ G_{K} then g(log([v])) = c_{g}t + log([v]) with c_{g} ∈ ℤ_{p}. Thus V = ℚ_{p} · t + ℚ_{p} · log([v]) ⊂ B_{cris} is G_{K} stable, and it is crystalline, as it admits two linearly independent maps into B_{cris} : the natural embedding V ⊂ B_{cris}, and the map given by log([v]) ↦ 1 and t ↦ 0 (in fact V is the Galois representation associated to an ordinary elliptic curve with good ordinary reduction, and Serre-Tate parameter q = 1 + π^{n}).

It follows that log[v] is a period, and its image under θ is log(1 + π^{n}). As v_{K}(log(1 + π^{n})) = n, which is prime to [K : K_{0}], log(1 + π^{n}) generates K over K_{0}, proving (2.5.3).

Multiplying (2.5.3) by t^{−i} for i ∈ ℕ gives

F il^{−i+1}(B_{cris}^{alg} ⊗_{K_{0}} K) + F il^{−i}B_{cris}^{alg} = F il^{−i}(B_{cris}^{alg} ⊗_{K_{0}} K),

whence

F il^{0}(B_{cris}^{alg} ⊗_{K_{0}} K) + F il^{−i}B_{cris}^{alg} = F il^{−i}(B_{cris}^{alg} ⊗_{K_{0}} K)

by induction on i. The proposition follows. □

(2.5.4) We can now give the proof of (2.4.3)(1). By definition 𝒦_{cris}^{•}(B_{cris}^{alg}) is the total complex of

\xymatrix{B_{cris}^{alg} \ar[r] & 0 \\ B_{cris}^{alg} \ar[r] \ar[u]^{ϕ−1} & B_{cris}^{alg} ⊗_{K_{0}} K/F il^{0}(B_{cris}^{alg} ⊗_{K_{0}} K) \ar[u]}

Taking cohomology of this double complex with respect to the horizontal differentials, and using (2.5.2), we see that 𝒦_{cris}^{•}(B_{cris}^{alg}) is quasi-isomorphic to

F il^{0}B_{cris}^{alg} ⟶^{ϕ−1} B_{cris}^{alg},

which is quasi-isomorphic to ℚ_{p} by (2.5.1). □

We proceed to the proof of (2.4.3)(2).

Proposition (2.5.5). The map

F il^{0}B_{st} ⟶^{N} B_{st}

is surjective, with kernel F il^{0}B_{cris}.

Proof. The statement regarding the kernel is clear, as N acts as differentiation on B_{st} ˜ ⟶B_{cris}[X]. After replacing X by Xt^{i} for a suitable integer i, we may assume that X ∈ F il^{0}B_{st}\F il^{1}B_{st}.

It is enough to show that for n ∈ ℕ, B_{cris}X^{n} ⊂ N(F il^{0}B_{st}). We proceed by induction, starting with n = 0. In this case consider a ∈ F il^{−i}B_{cris} with i ≥ 0. We have to find b ∈ B_{cris} such that b+aX ∈ F il^{0}B_{st}. If i = 0 then we have N(aX) = a, and we may take b = 0. Suppose i > 0. By induction, we may assume that there exists b′ ∈ B_{cris} with b′ + atX ∈ F il^{0}B_{st}. As θ maps F il^{0}B_{cris} onto ℂ_{p}, we may choose b′′ ∈ F il^{0}B_{cris} with θ(b′′ + b′ + atX) = 0. That is, b′′ + b′ + atX ∈ F il^{1}B_{st}. Taking b = (b′′ + b′)/t, we see that b + aX ∈ F il^{0}B_{st}, as required, which completes the proof if n = 0.

Suppose n > 0, and consider aX^{n} ∈ B_{st} with a ∈ B_{cris}. Using the case n = 0, we may choose b ∈ B_{cris} with b+aX ∈ F il^{0}B_{st}, so that also bX^{n}/(n+1)+aX^{n+1}/(n+ 1) ∈ F il^{0}B_{st}. Now N(bX^{n}/(n + 1) + aX^{n+1}/(n + 1)) = (bn/n + 1)X^{n−1} + aX^{n}. By induction on n, we already know that (bn/n + 1)X^{n−1} ∈ N(F il^{0}B_{st}), and thus aX^{n} ∈ N(F il^{0}B_{st}), which is what had to be proved. □

Corollary (2.5.6). The complex

(2.5.7)  F il^{0}B_{st} →^{N⊕ϕ−1} B_{st} ⊕ B_{st} →^{(1−pϕ)+N} B_{st}

with the left most term concentrated in degree 0, is quasi-isomorphic to ℚ_{p} concentrated in degree 0.

Proof. The complex (2.5.7) is the total complex attached to the double complex

\xymatrix{B_{st} \ar[r]^{N} & B_{st} \\ F il^{0}B_{st} \ar[r]^{N} \ar[u]^{ϕ−1} & B_{st} \ar[u]^{1−pϕ}}

Taking cohomology of this double complex with respect to to the horizontal differentials, and using (2.5.5) we see that it is quasi-isomorphic to F il^{0}B_{cris} ⟶^{ϕ−1} B_{cris}, which we have already seen is quasi-isomorphic to ℚ_{p}. □

Proposition (2.5.8). We have

F il^{0}(B_{st}^{alg} ⊗_{K_{0}} K) + B_{st}^{alg} = B_{st}^{alg} ⊗_{K_{0}} K,

so that the map B_{st}^{alg} → B_{st}^{alg} ⊗_{K_{0}} K/F il^{0}(B_{st}^{alg} ⊗_{K_{0}} K) is surjective.

Proof. The same argument as in (2.5.2) shows that it is enough to show that θ(F il^{0}(B_{st}^{alg})) contains K. However we saw in the proof of (2.5.2) that we even have θ(F il^{0}(B_{cris}^{alg})) contains K, so the result follows. □

(2.5.9) Note that the element u ∈ B_{st} defined in (2.2.4) is a period in B_{st}. Indeed the calculation in (2.2.5) shows that V = ℚ_{p} · t ⊕ ℚ_{p} · u is G_{K} stable, and it is semistable, as it admits two linearly independent maps to B_{st} : the natural inclusion V ⊂ B_{st}, and the map given by u ↦ 1 and t ↦ 0 (in fact V is the representation attached to an elliptic curve with bad multiplicative reduction).

Write A = B_{st}^{alg} ∩ B_{cris} ⊂ B_{st}. Clearly B_{cris}^{alg} ⊂ A, and one might be tempted to conjecture that this is an equality, however that is not the case. Indeed, there exist irreducible semi-stable representations W which are not crystalline, and for any such W, the operator N is nilpotent but non-zero on (W^{∗} ⊗_{ℚ_{p}} B_{st})^{G_{K}} . Hence there exists a non-zero Galois equivariant map W → B_{cris}, which is necessarily injective.

Lemma (2.5.10). With the notation above, consider an element a ∈ B_{st} and write a as a sum a = ∑_{i=0}^{k} a_{i}u^{i} with a_{i} ∈ B_{cris}, for some k ∈ ℕ. Then a ∈ B_{st}^{alg} if and only if a_{i} ∈ A for each i.

Proof. If all the a_{i} are in A then clearly a ∈ B_{st}^{alg}. Suppose that a ∈ B_{st}^{alg}. We proceed by induction on k. If k = 0, there is nothing to prove.

As B_{st}^{alg} is stable under N, and u is a period, we have a − Na(u/k) ∈ B_{st}^{alg}. However

a − Na(u/k) = ∑_{i=0}^{k−1} a_{i}(1 − i/k)u^{i},

so we see by induction that a_{i} ∈ A for i < k. It follows also that a_{k}u^{k} ∈ B_{st}^{alg}, whence so is N^{k}(a_{k}u^{k}) = k!a_{k}. Hence a_{k} ∈ B_{st}^{alg}, and therefore in A. □

Corollary (2.5.11). The map B_{st}^{alg} →^{N} B_{st}^{alg} is surjective.

Proof. This follows from the formula B_{st}^{alg} = A[u], which follows from (2.5.10). □

Proposition (2.5.12). The complex

(2.5.13)  F il^{0}B_{st}^{alg} →^{N⊕ϕ−1} B_{st}^{alg} ⊕ B_{st}^{alg} →^{(1−pϕ)+N} B_{st}^{alg}

with the left most term concentrated in degree 0, is quasi-isomorphic to ℚ_{p} concentrated in degree 0.

Proof. That the last map of (2.5.13) is surjective follows from (2.5.11), and it is clear (from (2.2.6)) that the kernel of the first map in (2.5.13) has kernel ℚ_{p}. Thus, we just have to check that (2.5.13) is exact at the middle term. This is shown by an argument analogous to that used in the proof of (2.5.1):

Let b ∈ B_{st}^{alg} ⊕ B_{st}^{alg} be in the kernel of the second map of (2.5.13). By (2.5.6), we know that b = (N ⊕ ϕ − 1)(b′) for some b′ ∈ F il^{0}B_{st}, and we have to show that b′ is a period. As the components of b are periods, the arguments of (2.5.1) show that the images of b under G_{K} generate a finite dimensional ℚ_{p} vector space V, and that V is a semi-stable representation. Setting W = (N ⊕ ϕ − 1)^{−1}(V) we have dim_{ℚ_{p}}W = dim_{ℚ_{p}}V + 1, and we deduce that W is semi-stable as in the proof of (2.5.1). □

(2.5.14) We can now show that 𝒦_{st}^{•}(B_{st}^{alg}) is quasi-isomorphic to ℚ_{p} concentrated in degree 0. In (2.4.2) 𝒦_{st}^{•}(B_{st}^{alg}) is defined as the total complex associated to a certain double complex. Using (2.5.8) we see that the cohomology of this double complex with respect to the horizontal differentials is quasi-isomorphic to the complex (2.5.13), so that the result follows from (2.5.12). □ (2.5.15) Let us note for future use that the complexes 𝒦_{cris}^{•}(B_{cris}) and 𝒦_{st}^{•}(B_{st}) are also both quasi-isomorphic to ℚ_{p} concentrated in degree 0. In fact the proof in this case is easier than in the algebraic case: As K ⊂ θ(F il^{0}B_{cris}) = ℂ_{p} we see as in (2.5.2) that F il^{0}(B_{cris}⊗_{K_{0}} K) surjects onto B_{cris}⊗_{K_{0}} K/B_{cris}, and a similarly with B_{st} in place of B_{cris}. That 𝒦_{cris}^{•}(B_{cris}) (respectively 𝒦_{st}^{•}(B_{st})) is quasi-isomorphic to ℚ_{p} concentrated in degree 0 then follows from (2.2.6) (respectively (2.5.6)).

In §4 we will need the slightly more refined fact that for each positive integer j, 𝒦_{cris}^{•}(t^{−j}B_{cris}^{+}) is quasi-isomorphic to ℚ_{p}. Using (2.2.6), this is equivalent to the surjectivity of the map

t^{−j}B_{cris}^{+} → t^{−j}B_{cris}^{+} ⊗_{K_{0}} K/F il^{0}(t^{−j}B_{cris}^{+} ⊗_{K_{0}} K),

which follows from the surjectivity of

F il^{j}(B_{cris}^{+} ⊗_{K_{0}} K) + B_{cris}^{+} → B_{cris}^{+} ⊗_{K_{0}} K.

That this last map is surjective follows by exactly the same argument as in (2.5.2), using K ⊂ θ(B_{cris}^{+}).

§3 Weakly Admissible Modules

3.1 Ext’s for weakly admissible modules. In this section we show that the complexes which compute extensions in the category of weakly admissible modules are the same as those which are used for admissible modules. As a result we deduce that admissible modules are stable under extensions, and that Fontaine’s conjecture “weakly admissible ⟹ admissible” is true if and only if it is true for irreducible modules.

In the last part of this section we apply our methods to prove Fontaine’s conjecture that “admissible ⟹ strongly admissible.”

A general reference for facts regarding weakly admissible modules is [Fo 2]. (3.1.1) Suppose that M is a finite filtered ϕ module (respectively a filtered (ϕ, N) module). If dim_{K_{0}}M = d then ⋀^{d} M is also naturally a filtered ϕ module (respectively a filtered (ϕ, N) module).

Suppose d = 1. In this case ϕ is given by multiplication by a non-zero element a of K_{0}, which depends on a choice basis element for M. The p-adic valuation of a is however independent of this choice, and we set t_{N}(M) = v_{p}(a). We set t_{H}(M) equal to the integer j such that F il^{j}M_{K} = M_{K}, F il^{j+1}M = 0.

In general if d ≥ 1 then we set t_{N}(M) = t_{N}(⋀^{d} M), and t_{H}(M) = t_{H}(⋀^{d} M).

Definition (3.1.2). With the notation of (3.1.1) a finite filtered ϕ module (respectively a finite filtered (ϕ, N) module) M is called “weakly admissible” t_{H}(M) = t_{N}(M), and for each K_{0} subspace M′ ⊂ M which is stable under ϕ (respectively ϕ and N), regarded as finite filtered ϕ submodule (respectively finite filtered (ϕ, N) submodule) of M, we have t_{H}(M′) ⩽ t_{N}(M′).

(3.1.3) In the situation of (3.1.1) (respectively 3.1.2) we denote by M_{K}^{wa} (respectively M_{st,K}^{wa}) the full subcategory of the category of filtered ϕ modules (respectively filtered (ϕ, N) modules) consisting of weakly admissible modules.

An important property of M_{K}^{wa} and M_{st,K}^{wa} is that morphisms in these categories are strict for filtrations. From this one deduces that M_{K}^{wa} (respectively M_{st,K}^{wa}) is an abelian category. Furthermore, as a subcategory of the category of all filtered ϕ modules (respectively filtered (ϕ, N) modules), M_{st,K}^{wa} (respectively M_{st,K}^{wa}) is stable under taking duals, taking direct summands, taking extensions [Fo 2, 4.4.4], and also under taking tensor products (due to Faltings [Fa 2] in the case of M_{K}^{wa} and Totaro [To] in the case of M_{st,K}^{wa}). Thus M_{K}^{wa} and M_{st,K}^{wa} are Tannakian categories.

It turns out that every object of M_{K}^{ad} (respectively of M_{st,K}^{ad}) belongs to M_{K}^{wa} (respectively M_{st,K}^{wa}) ([Fo 2, 5.4.2]). Since both are full subcategories of the categories of the category of filtered ϕ modules (respectively the category of filtered (ϕ, N) modules) we see in particular that M_{K}^{ad} is a full subcategory of M_{K}^{wa} (respectively M_{st,K}^{ad} is a full subcategory of M_{st,K}^{wa}). Fontaine has conjectured that conversely “weakly admissible ⟹ admissible,” and thus that M_{K}^{ad} is equal to M_{K}^{wa} (respectively M_{st,K}^{ad} is equal to M_{st,K}^{wa}).

As usual we denote by Ind(M_{K}^{wa}) (respectively Ind(M_{st,K}^{wa})) the Ind categories of M_{K}^{wa} (respectively M_{st,K}^{wa}).

If M is in Ind(M_{K}^{wa}) (respectively Ind(M_{st,K}^{wa})) then we can consider the complex 𝒦_{cris}^{•}(M) (respectively 𝒦_{st}^{•}(M)) defined in §2.4. Since morphisms in M_{K}^{wa} (respectively M_{st,K}^{wa}) are strict for filtrations, 𝒦_{cris}^{•} (respectively 𝒦_{st}^{•}) induces a functor on D^{+}(Ind(M_{K}^{wa})) (respectively D^{+}(Ind(M_{st,K}^{wa})), as in (2.4.1), (2.4.2). The key technical result for these complexes is the following

Proposition (3.1.4). Let I be an injective in Ind(M_{K}^{wa}) (respectively Ind(M_{st,K}^{wa})). Then 𝒦_{cris}^{•}(I) (respectively 𝒦_{st}^{•}(I)) is acyclic in degree > 0.

The proof of the proposition will be given in §3.2. For the moment we deduce some consequences.

Theorem (3.1.5). Let M^{•} be a complex in D^{b}(M_{K}^{wa}). There is an isomorphism of functors on D^{+}(Ind(M_{K}^{wa}))

𝒦_{cris}^{•}(Hom_{K_{0}}^{•}(M^{•}, )) ˜ ⟶Rhom_{Ind(M_{K}^{wa})}(M^{•}, )

This isomorphism is functorial in M^{•}.

Similarly, let M^{•} be a complex on D^{b}(M_{st,K}^{wa}). There is an isomorphism of functors on D^{+}(Ind(M_{st,K}^{wa}))

𝒦_{st}^{•}(Hom_{K_{0}}^{•}(M^{•}, )) ˜ ⟶Rhom_{Ind(M_{st,K}^{ad})}(M^{•}, )

This isomorphism is functorial in M^{•}.

Proof. We prove the first claim. The proof is formally similar to that of (2.4.9). It is enough to construct the isomorphisms when both sides are evaluated on bounded below complexes of injectives I^{•} in Ind(M_{K}^{wa}).

Now if I is an injective, L are in Ind(M_{K}^{wa}), and M is in M_{K}^{wa}, then we have a functorial isomorphism

Hom_{Ind(M_{K}^{wa})}(L, Hom_{K_{0}}(M, I)) ˜ ⟶Hom_{Ind(M_{K}^{wa})}(L ⊗_{K_{0}} M, I).

This shows that Hom_{K_{0}}(M, I) is also an injective. By (3.1.4), and the definition of 𝒦_{cris}^{•}, we get

𝒦_{cris}^{•}(Hom_{K_{0}}^{•}(M^{•}, I^{•})) ˜ ⟶H^{0}(𝒦_{cris}^{•})(Hom_{K_{0}}^{•}(M^{•}, I^{•})) ˜ ⟶Hom_{Ind(M_{K}^{wa})}^{•}(M^{•}, I^{•}) ˜ ⟶Rhom_{Ind(M_{K}^{wa})}(M^{•}, I^{•}).

The proof of the second claim is formally identical. □

Corollary (3.1.6). Let M, N be in M_{K}^{ad} (respectively in M_{st,K}^{ad}). Every extension of M by N is in M_{K}^{ad} (respectively M_{st,K}^{ad}). That is, M_{K}^{ad} (respectively M_{st,K}^{ad}) is closed under extensions as a subcategory of M_{K}^{wa} (respectively M_{st,K}^{wa}).

Proof. Consider the case where M and N are in M_{K}^{ad}. Let E be an extension of M by N. As M_{K}^{wa} is closed under extensions E is an object of M_{K}^{wa}. On the other hand, by (2.4.9) and (3.1.5) we have

Ext_{M_{K}^{ad}}^{1}(M, N) = H^{1}(𝒦_{cris}^{•}(Hom_{K_{0}}(M, N))) = Ext_{M_{K}^{wa}}^{1}(M, N),

so E is an object of M_{K}^{ad} as required.

The proof in case M and N are in M_{st,K}^{ad} is formally similar. □

More generally we have Corollary (3.1.7). The natural functors

D^{b}(Ind(M_{K}^{ad})) ⟶ D^{b}(Ind(M_{K}^{wa}))

and

D^{b}(Ind(M_{st,K}^{ad})) ⟶ D^{b}(Ind(M_{st,K}^{wa}))

are fully faithful

Proof. Immediate from (2.4.9) and (3.1.5). □

Corollary (3.1.8). Fontaine’s conjecture “weakly admissible ⟹ admissible” is true if and only if it is true for irreducible objects of M_{K}^{wa} (respectively M_{st,K}^{wa}.)

More precisely an object M of M_{K}^{wa} (respectively M_{st,K}^{wa}) is in M_{K}^{ad} (respectively M_{st,K}^{ad}) if and only if it is a successive extension of objects of M_{K}^{ad} (respectively M_{st,K}^{ad}).

Proof. Immediate from (3.1.6) □

As a final application of the theory, we give a reformulation of Fontaine’s conjecture.

Proposition (3.1.9). Fontaine’s conjecture is true if and only if B_{cris}^{alg} (respectively B_{st}^{alg}) is an injective object in Ind(M_{K}^{wa}) (respectively Ind(M_{st,K}^{wa})).

Proof. The implication ⟹ follows from (2.3.7). Let us prove the reverse implication.

Suppose that B_{cris}^{alg} is an injective object in Ind(M_{K}^{wa}). We show that each object M of M_{K}^{wa} is in M_{K}^{ad} by induction on the dimension of M. If dim_{K_{0}}M = 1 this is known [Fo 2].

For each n ∈ ℤ, we denote by K_{0}(n) the object of M_{K}^{ad} corresponding (covariantly) to the one dimensional crystalline representation ℚ_{p}(n). As a vector space K_{0}(n) is simply K_{0}, which we may regard as a one dimensional K_{0} vector space with a fixed basis vector e. The associated graded of K_{0}(n) is concentrated in degree −n, and ϕ acts by sending e to p^{−n}e.

If M is any object of M_{K}^{wa}, then one sees easily from (3.1.5) that for sufficiently large n, there exist non-trivial extensions of M by K_{0}(n) in M_{K}^{wa}. (For example take n so large that M ⊗_{K_{0}} K_{0}(−n) has no slope zero part, so that 1−ϕ is injective on M ⊗_{K_{0}} K_{0}(−n).) Let

0 → K_{0}(n) → E → M → 0

be such an extension.

As M is in M_{K}^{wa}, it is in M_{K}^{ad} provided that E is ([Fo 2. 5.4.2]). Since K_{0}(n) is in M_{K}^{ad}, there is a non-zero map K_{0}(n) → B_{cris}^{alg} (for example the one given by e ↦ t^{−n}). By hypothesis, B_{cris}^{alg} is injective in M_{K}^{wa}, so this lifts to a map f : E → B_{cris}^{alg}. We have an exact sequence

0 → ker(f) → E → Im(f) → 0.

Now as K_{0}(n) is isomorphic to its image in B_{cris}^{alg}, Im(f) cannot be one dimensional, as it would then give a splitting of E. As Im(f) ⊂ B_{cris}^{alg} it is in M_{K}^{ad}, hence in M_{K}^{wa}. Thus ker(f) is also in M_{K}^{wa}, as M_{K}^{wa} is abelian. By induction ker(f) is in M_{K}^{ad}, whence so is E, by (3.1.6).

Now assume that B_{st}^{alg} is an injective in Ind(M_{st,K}^{wa}). The proof that M_{st,K}^{ad} is equivalent to M_{st,K}^{wa} is formally identical to the argument above. The only difference is that in order to to check that there exist non-trivial extensions of E by K_{0}(n), we use the complex 𝒦_{st}^{•}(Hom_{K_{0}}(E, K_{0}(n))) rather than 𝒦_{cris}^{•}(Hom_{K_{0}}(E, K_{0}(n))). □

3.2 Analysis of Complexes. The purpose of this section is to prove Proposition (3.1.4). The basic idea is to show that if I is an object in Ind(M_{K}^{wa}) (respectively Ind(M_{st,K}^{wa})), then obstructions to 𝒦_{cris}^{•}(I) (respectively 𝒦_{st}^{•}(I)) being acyclic in degree > 0 give rise to non-trivial extensions of weakly admissible modules by I. Thus, if I is injective all these obstructions must vanish.

Proposition (3.2.1). Let I be an injective object in Ind(M_{K}^{wa}). The map

F il^{0}I →^{ϕ−1} I

is surjective.

Proof. Let a ∈ I. We define an extension E_{a} of K_{0} by I as follows. E_{a} = I ⊕ K_{0} on the underlying K_{0} vector spaces, and the filtration E_{a} ⊗_{K_{0}} K is just the direct sum filtration (as usual K_{0} denotes the unit object in M_{K}^{wa}). Let e be a basis vector for K_{0} with ϕ(e) = e. We define ϕ on E_{a} be the formula ϕ(b ⊕ αe) = (ϕ(b) − ϕ(α)a) ⊕ ϕ(α)e, for b ∈ I, and α ∈ K_{0}. It is clear that this gives E_{a} the structure of an extension of K_{0} by I in Ind(M_{K}^{wa}).

As I is injective E_{a} splits. If s : K_{0} → E_{a} is a splitting. then we have s(e) = b⊕e for some b ∈ F il^{0}I, and we must have ϕ(s(e)) = s(ϕ(e)) = s(e). That is ϕ(b)−a = b, or ϕ(b) − b = a. □

Proposition (3.2.2). Let I be an injective object in Ind(M_{K}^{wa}). Then the map F il^{0}I_{K} → I_{K}/I is surjective. (Here I_{K} = I ⊗_{K_{0}} K, as usual).

Proof. Let a ∈ I_{K}. We define an extension E_{a} of K_{0} by I as follows. E_{a} = I ⊕ K_{0} as left K_{0}[ϕ] modules. We define a filtration on E_{aK} as follows.

F il^{i}E_{aK} = \{ \begin{matrix}F il^{i}I_{K} i > 0 \\ F il^{i}I_{K} + K · (a ⊕ e) i ⩽ 0 \end{matrix}

This gives E_{a} the structure of an extension of I by K_{0}, which splits as I is injective. Let s : K_{0} → E_{a} be a splitting. Then we have s(e) = b ⊕ e ∈ F il^{0}E_{a} ⊂ F il^{0}E_{aK}. Hence b = b′ + a with b′ ∈ F il^{0}I_{K}. As b ∈ I we see that −b′ maps onto the class of a in I_{K}/I. □

(3.2.3) We can now prove the part of Proposition (3.1.4) pertaining to M_{K}^{wa}. Namely, suppose that I is injective object in Ind(M_{K}^{wa}). By definition (2.3.1), 𝒦_{cris}^{•}(I) is the total complex associated to a certain double complex. By (3.2.2) the cohomology of this double complex with respect to the horizontal differentials is simply F il^{0}I →^{ϕ−1} I, with the first term in degree 0, and this complex is acyclic in degree > 0 by (3.2.1). □

We now consider the case of an injective object I of Ind(M_{st,K}^{wa}).

Proposition (3.2.4). The map I →^{N} I is surjective.

Proof. Recall the weakly admissible modules K_{0}(n), which were discussed in the proof of (3.1.9). In particular the underlying vector spaces of all the K_{0}(n) are identified with a one dimensional K_{0} vector space, with fixed basis vector e. Slightly abusing this notation, we again denote this last space by K_{0}.

For each non-negative integer k, we define an object L(k) of M_{st,K}^{wa} as follows. As a left K_{0}[ϕ] module with filtration (which in this case is K_{0}-rational)

L(k) = K_{0}(0) ⊕ K_{0}(1) ⊕ ⋯ ⊕ K_{0}(k).

The operator N on L(k) is given by

N(a_{0} ⊕ … a_{k}) = a_{1} ⊕ a_{2} ⋯ ⊕ a_{k} ⊕ 0.

It is easy to check that the equation pϕN = Nϕ holds, and L(k) is weakly admissible, as it is a successive extension of weakly admissible modules.

Now let M be any object of M_{st,K}^{wa}. Then

M = M ⊗_{K_{0}} K_{0}(0) ⊂ M ⊗_{K_{0}} L(k).

Consider m ∈ M, with N^{k}m = 0. I claim that there is an n ∈ M ⊗_{K_{0}} L(k) of the form n = 0 ⊕ n_{1} ⊕ ⋯ ⊕ n_{k−1}, with

Nn = m ∈ M ⊂ M ⊗_{K_{0}} L(k).

To see the claim we proceed by induction on k. For k = 1, we take n = 0 ⊕ m, so that Nn = m ⊕ Nm = m ⊕ 0. If the result is known for k − 1, then there exists an n′ = 0 ⊕ n_{2} ⊕ ⋯ ⊕ n_{k−1} ∈ M ⊗_{K_{0}} L(k − 1) with

Nn′ = −Nm ∈ M ⊂ M ⊗_{K_{0}} L(k − 1).

Write n = 0 ⊕ m ⊕ n_{2} ⊕ ⋯ ⊕ n_{k−1} ∈ M ⊗_{K_{0}} L(k). Then we compute

Nn = m ⊕ Nm ⊕ 0 ⊕ ⋯ ⊕ 0 − 0 ⊕ −Nm ⊕ 0 ⋯ ⊕ 0 = m ⊕ 0 ⊕ ⋯ ⊕ 0,

as required.

Now take any a ∈ I. As I is a direct limit of object in M_{st,K}^{wa}, a is in the image of a map f : M → I, with M in M_{st,K}^{wa}. Suppose that m ∈ M maps to a, and choose k such that N^{k}m = 0. As I is an injective, and M ⊂ M ⊗_{K_{0}} L(k), f lifts to a map f̃: M ⊗_{K_{0}} L(k) → I. Choose n ∈ M ⊗_{K_{0}} L(k) with Nn = m. Then we have

N(f̃(n)) = f̃(Nn) = f̃(m) = f(m) = a,

so that a ∈ N(I), as required. □

Proposition 3.2.5. The map F il^{0}I_{K} → I_{K}/I is surjective.

Proof. The proof is very similar to that of (3.2.2). Given an element a ∈ I, we construct an extension E_{a}. E_{a} = I ⊕ K_{0} as K_{0} vector spaces with ϕ and N, and the filtration is given by the same formulas as in (3.2.2). The argument given there then shows that the class of a in I_{K}/I is in the image of F il^{0}I_{K}. □

Proposition 3.2.6. The complex

(3.2.7)  F il^{0}I →^{N⊕ϕ−1} I ⊕ I →^{(1−pϕ)+N} I

with the first term in degree 0, is acyclic in degree > 0.

Proof. That (3.2.7) is acyclic in degree 2 follows from (3.2.4), so it remains to show it is acyclic in degree 1.

The argument is formally similar to that of (3.2.1). Take a ⊕ b ∈ I ⊕ I, with a − pϕ(a) + Nb = 0. We define an extension E_{(a,b)} of K_{0} by I as follows. On the underlying K_{0} vector spaces E_{(a,b)} = I ⊕ K_{0}, and the filtration on E_{(a,b)K} is the direct sum filtration. N and ϕ on E_{(a,b)} are given by the formulas. N(x ⊕ e) = (Nx + a) ⊕ 0, and ϕ(x ⊕ e) = (ϕ(x) + b) ⊕ e. One checks easily that pϕN = Nϕ.

Since I is an injective, E_{(a,b)} admits a splitting s : K_{0} → E_{(a,b)}, and we have s(e) = h ⊕ e for some h ∈ F il^{0}I. Moreover we must have N(s(e)) = 0, and ϕ(s(e)) = s(e), from which we deduce that N(h) = −a, and ϕ(h) − h = −b. Thus −h ∈ F il^{0}I is mapped onto a ⊕ b. □

(3.2.8) Finally we can show the remaining case of Proposition (3.1.4), which says that 𝒦_{st}^{•}(I) is acyclic in degree > 0. By definition 𝒦_{st}^{•}(I) is the total complex of a certain double complex. Proposition (3.2.5) shows that the cohomology of this double complex with respect to the horizontal differentials is simply the complex (3.2.7), which, according to (3.2.6) is acyclic in degree > 0. □

3.3 Strongly Admissible Modules. In this section we prove Fontaine’s conjecture that “strongly admissible ⟺ admissible. ” The implication ⟹ is a result of Fontaine (unpublished). We prove both directions here for the convenience of the reader. Ironically, with our methods it is the previously known implication which is a little bit more difficult to prove. (3.3.1) Let M be in M_{st,K}^{wa}. We say that M is positive if F il^{0}M_{K} = M_{K}. Note that the weak admissibility condition implies that any such module is effective as a filtered (ϕ, N) module.

Given a positive weakly admissible module M, we are going to consider the set of extensions E of M by B_{st}^{+}.

(3.3.2)  0 → B_{st}^{+} → E → M → 0

This means that E is a left K_{0}[ϕ, N] module with a filtration on E_{K}. The maps in 3.3.2 should be compatible with ϕ and N, and strictly compatible with filtrations.

Abusing notation, we write Ext_{M_{st,K}^{wa}}^{1} (M, B_{st}^{+}) for the set of such extensions E, up

to isomorphism, the idea being that B_{st}^{+} is an honorary member of M_{st,K}^{wa}. Similarly we write Hom_{M_{st,K}^{wa}} (M, B_{st}^{+}) for the group of K_{0} linear maps M → B_{st}^{+} which respect ϕ, N and filtrations, and likewise with B_{st} in place of B_{st}^{+}.

M is called strongly admissible if Ext_{M_{st,K}^{wa}}^{1} (M, B_{st}^{+}) = 0. To prove Fontaine’s conjecture we need some preparation. We begin with several results due to Fontaine. These were presented by him in his course at IHP, Paris in the Spring of 1997. They will be the main ingredients in proving the implication “strongly admissible ⟹ admissible”.

Proposition (3.3.3). (Fontaine) Suppose M is positive. The natural map

Hom_{M_{st,K}^{wa}} (M, B_{st}^{+}) → Hom_{M_{st,K}^{wa}} (M, B_{st})

is an isomorphism.

Proof. With the notation of (2.2.4), we have B_{st}^{+} = B_{cris}^{+}[u], and B_{st} = B_{cris}[u]. We have to show that for any m ∈ M, and any map f : M → B_{st}, f(m) = ∑_{i=0}^{∞} b_{i}u^{i} with b_{i} ∈ B_{cris}^{+}.

First we claim that for every non-negative integer n and each i, we have ϕ^{n}(b_{i}) ∈ F il^{0}B_{cris}. We prove this by descending induction on i. For i >> 0, b_{i} = 0, and there is nothing to prove. Suppose the result is known for i > r. Then by the induction hypothesis we have f(N^{r}m) = N^{r}f(m) = r!b_{r} + y, where y is a sum of terms of the form b′_{i}u^{i−r} with ϕ^{n}(b′_{i}) ∈ F il^{0}B_{cris}, for each n ≥ 0, and i > r. As M is positive, we also have

ϕ^{n}N^{r}(f(m)) = f(ϕ^{n}N^{r}m) ∈ F il^{0}B_{st} ⊂ B_{dR}^{+}.

It follows that ϕ^{n}(b_{r}) ∈ F il^{0}B_{cris} for all non-negative integers n, which proves the claim.

By [Fo, 5.3.7] the above shows that f(ϕ^{2}(m)) = ϕ^{2}(f(m)) ∈ B_{st}^{+}, and as ϕ is bijective on M, we are done. □

Proposition (3.3.4). (Fontaine) Let M be any object of M_{st,K}^{wa}. Let M′ be the smallest quotient of M (as a K_{0} vector space) such that every map M → B_{st} respecting ϕ, N and filtrations factors through M′.

Then V = Hom_{M_{st,K}^{wa}} (M, B_{st}) is a finite dimensional semi-stable representation of G_{K}, there is a canonical isomorphism M′ ˜ ⟶Hom_{G_{K}} (V, B_{st}), and M′ is the largest quotient of M which is in M_{st,K}^{ad}.

Proof. Write d′ = dim_{K_{0}}M′. We first show that dim_{ℚ_{p}}V ⩽ d′.

Let C_{st} be the quotient field of B_{st} ⊂ B_{dR}, and denote by r the dimension of the C_{st} vector space generated by V in Hom_{K_{0}}(M, C_{st}). We have r ⩽ d′. Set

W = ⋀_{ℚ_{p}}^{r} V ⊂ Hom_{M_{st,K}^{wa}} (⋀_{K_{0}}^{r} M, B_{st}) ⊂ Hom_{K_{0}}(⋀_{K_{0}}^{r} M, C_{st}).

Note that W generates a one dimensional C_{st} subspace of Hom_{K_{0}}(⋀_{K_{0}}^{r} M, C_{st}).

Take f ∈ W, f ≠ 0, let L ⊂ B_{st} be the image of f. Let l_{1}, … l_{m} be a K_{0} basis for L. If g ∈ G_{K}, then the composite

⋀_{K_{0}}^{r} M →^{f} L →^{g} L

is also in W. Hence we must have g(l_{i}) = c_{g}l_{i} for some c_{g} ∈ C_{st}, for each i = 1, … m. This shows that for each i, j, g(l_{i}/l_{j}) = (l_{i}/l_{j}). Since C_{st} ⊗_{K_{0}} K ⊂ B_{dR}, we have C_{st}^{G_{K}} ⊗_{K_{0}} K ⊂ B_{dR}^{G_{K}} = K, so that C_{st}^{G_{K}} = K_{0}. As the l_{i} were assumed to be linearly independent over K_{0} we must have m = 1, and so L is one dimensional.

If we set L′ = ⋀_{K_{0}}^{r} M/ ker(f) (L′ differs from L only in the filtration on L′_{K}), then L′ is one dimensional over K_{0}, and as ⋀_{K_{0}}^{r} M is weakly admissible (because M is), we have that t_{N}(L′) ⩽ t_{H}(L′). However as L′ admits a non-zero map to B_{st}, one checks easily that we must have that t_{N}(L′) = t_{H}(L′), and Hom_{M_{st,K}^{wa}} (L′, B_{st}) has ℚ_{p} dimension 1. As every map in W differs from f by a C_{st} scalar, each such map factors through L′, so we must have dim_{ℚ_{p}}W = 1, and dim_{ℚ_{p}}V = r, which shows that V is finite dimensional.

Now set M_{2} = Hom_{ℚ_{p}⟨G_{K}⟩}(V, B_{st}). There is an obvious natural map M′ → M_{2}, and it is injective by definition of M′. Thus we get

d′ = dim_{K_{0}}M′ ⩽ dim_{K_{0}}M_{2} ⩽ dim_{ℚ_{p}}V = r ⩽ d′,

whence all the above inequalities are equalities, and we get that V is semi-stable, that d′ = r, and that M′ ˜ ⟶M_{2} has the structure of an object of M_{st,K}^{ad}.

Finally, if M′′ is any quotient of M which is in M_{st,K}^{ad} then M′′ admits an embedding M′′ ⊂ (B_{st})^{q} for some integer q, so that every element in the kernel of M → M′ maps to 0 in M′′, and M′′ is a quotient of M′. □

Corollary (3.3.5). Let M be in M_{st,K}^{wa}. Then any map h : M → B_{st} respecting ϕ, N and filtrations factors through the ring of algebraic periods B_{st}^{alg} ⊂ B_{st}. In particular h is strict for filtrations, and its image is admissible.

Proof. With the notation of the previous proposition, h factors through M′, which is in M_{st,K}^{ad}. But every map M′ → B_{st} factors through B_{st}^{alg}, so we are done. □

Lemma (3.3.6). (Fontaine) Let P (x) ∈ 𝒪_{K_{0}}[x] be a polynomial with non-zero reduction modulo p. With the notation of §2.2 we have

(1) The map P (ϕ) : R → R is surjective.

(2) The map P (ϕ) : W (R) → W (R) is surjective.

(3) The map P (ϕ) : B_{cris}^{+} → B_{cris}^{+} is surjective.

Proof. (1) Note that the truth of (1) only depends on the reduction of P (x) modulo p, since R is of characteristic p. Thus we may replace P (x) by the sum of those of its terms whose coefficients are units in 𝒪_{K_{0}}; multiplying P (x) by a unit in 𝒪_{K_{0}}, we may and do furthermore assume that P (x) is monic.

As ℂ_{p} is algebraically closed the map P (ϕ) : O_{ℂ_{p}}/p → O_{ℂ_{p}}/p is surjective with kernel Q (say). Thus surjectivity will follow for the map in (1) if we show that the inverse system

… Q →^{x↦x^{p}} Q

satisfies the Mittag-Leffler condition.

Let s be the degree of P. If x̂ ∈ O_{ℂ_{p}} with |P (x̂^{p})| ⩽ |p|, then there exists a root r of P (X^{p}) with |x̂^{p} − r| ⩽ |p|^{1/sp}. If n is an integer with n ≥ log_{p}s + 1, then a simple calculation shows that |x̂^{p^{n+1}} − r^{p^{n}}| ⩽ |p| so that ϕ^{n+1}(Q) is actually a finite set, whence (1).

(2) As P (ϕ)(p) = pP (ϕ)(1) and W (R) is p-adically complete and separated, (2) follows from (1).

(3) Recall that B_{cris}^{+} = A_{cris}[1/p]. Since multiplication by P (ϕ) commutes with multiplication by 1/p, it suffices to show that for any a ∈ A_{cris}, we can find b ∈ B_{cris}^{+} such that P (ϕ)(b) = a. By [Fo , 5.2] each element a ∈ A_{cris} has an (non unique) expansion a = ∑_{i=0} a_{i}t^{{i}} where a_{i} ∈ W (R), a_{i} → 0 p-adically, and t^{{i}} denotes a certain fixed rational multiple of t^{i}. We are going to find b = ∑ b_{i}t^{{i}}, with b_{i} ∈ W (R)[1/p] and the b_{i} tending p-adically to zero, such that P (ϕ)b = a.

Write P (x) = c_{k}x^{k} + ⋯ + c_{0}. Let P_{i}(x) = P (p^{i}x). Then we may write P_{i}(x) = p^{r_{i}}Q_{i}(x), where r_{i} = min_{j}(v_{p}(p^{ij}c_{j})) and Q_{i}(x) is polynomial in O_{K_{0}}(x) with nonzero reduction modulo p. Note that for large i, r_{i} = v_{p}(c_{0}) is constant.

For any b_{i} ∈ W (R)[1/p] we see that

P (ϕ)(b_{i}t^{{i}}) = P_{i}(ϕ)(b_{i})t^{{i}} = p^{r_{i}}Q_{i}(ϕ)(b_{i})t^{{i}}.

By (2) we may choose b′_{i} ∈ W (R) such that Q_{i}(ϕ)b′_{i} = p^{−v_{p}(a_{i})}a_{i}. If we write b_{i} = p^{v_{p}(a_{i})−r_{i}}b′_{i}, then since r_{i} is constant for large i, the fact that the a_{i} tend to zero implies that the b_{i} also tend to zero. Thus b = ∑ b_{i}t^{{i}} is an element of B_{cris}^{+}, and by construction P (ϕ)b = a. □

Proposition (3.3.7). (Fontaine) Let M be an effective filtered (ϕ, N) module. Then any extension of D by B_{cris}^{+} (respectively by B_{st}^{+}) in the category of left K_{0}[ϕ] (respectively K_{0}[ϕ, N]) modules is split.

Proof. First consider the case where D is equipped with a left K_{0}[ϕ] module structure. By dévissage, we reduce to the case where D = K_{0}[X]/(P (ϕ)), for some monic P [X] ∈ 𝒪_{K_{0}}[X] (as D is effective). So D = ⊕_{i=0}^{s−1}ϕ^{i}e, where s is the degree of P.

If E is an extension

0 → B_{cris}^{+} → E → D → 0.

let ê ∈ E be a lift of e. We have P (ϕ)(ê) ∈ B_{cris}^{+}. By (3.3.6) there exists x ∈ B_{cris}^{+} with P (ϕ)(x) = P (ϕ)(ê), so that e ↦ ê − x is a splitting of E.

Now suppose that D is equipped with a ϕ and nilpotent N. By dévissage, we may assume that N = 0. Let E be an extension

0 → B_{st}^{+} → E → D → 0.

Consider any map s : γ : D → E which splits E as an extension of K_{0} vector spaces. As Ns(D) ⊂ B_{st}^{+}, and N is surjective on B_{st}^{+}, after modifying s, we may assume that Ns(D) = 0. We may regard E as B_{st}^{+} ⊕ D, via s, though this decomposition does not respect ϕ. However, as Ns(D) = 0, we have Nϕs(D) = pϕNs(D) = 0, so ϕ leaves B_{cris}^{+} ⊕ D ⊂ E stable. As the extension

0 → B_{cris}^{+} → B_{cris}^{+} ⊕ D → D → 0

is split, we see that after modifying s by a K_{0} linear map D → B_{cris}^{+} (after which we still have Ns(D) = 0) s becomes a splitting of E. □

Proposition (3.3.8). Let M′, M be positive objects of M_{st,K}^{wa}, with M′ ⊂ M. Then the natural map

Ext_{M_{st,K}^{wa}}^{1} (M, B_{st}^{+}) → Ext_{M_{st,K}^{wa}}^{1} (M′, B_{st}^{+})

is surjective. In particular if M is strongly admissible, then so is M′.

Proof. If E is an extension of M′ by B_{st}^{+} then by (3.3.7) we know that E admits a splitting which respects ϕ and N, but not necessarily the filtrations. Write E = M′ ⊕ B_{st}^{+} (respecting only ϕ and N). To show that E is induced by an extension of M by B_{st}^{+}, we simply have to show that there is a filtration on (M ⊕ B_{st}^{+}) ⊗_{K_{0}} K which extends the filtration on E_{K}, and makes

0 → B_{st}^{+} ⊗_{K_{0}} K → (M ⊕ B_{st}^{+}) ⊗_{K_{0}} K → M → 0

into a short exact sequence of filtered vector spaces. This is clear, for example, by choosing a K linear splitting M_{K} → M′_{K} which respects the filtrations. □

(3.3.9) Now let M be a positive object of M_{st,K}^{ad}. To show Ext_{M_{st,K}^{wa}}^{1} (M, B_{st}^{+}) = 0,

we want to reduce to the case M = K_{0}. If we replace B_{st}^{+} by B_{st}^{+alg} = B_{st}^{+} ∩ B_{st}^{alg}, then the Tannakian formalism gives

Ext_{M_{K}^{wa}}^{1}(M, B_{st}^{+alg}) = Ext_{M_{K}^{wa}}^{1}(K_{0}, M^{∗} ⊗_{K_{0}} B_{st}^{+alg}) = Ext_{M_{K}^{wa}}^{1}(K_{0}, (B_{st}^{+alg})^{d}),

where d = dim_{K_{0}}M, and the second equality follows from (3.3.3) and the fact that M is positive and admissible. Since we want to work with B_{st}^{+} rather than B_{st}^{+alg} we cannot use a Tannakian argument. Instead we construct the isomorphism

Ext_{M_{K}^{wa}}^{1}(M, B_{st}^{+}) = Ext_{M_{K}^{wa}}^{1}(K_{0}, M^{∗} ⊗_{K_{0}} B_{st}^{+})

“by hand.”

For this we do not need to assume that M is positive, but just that M is in M_{st,K}^{wa}. Consider an extension

0 → B_{st}^{+} → E → M → 0.

Applying ⊗_{K_{0}}M^{∗} and pulling back by the canonical map K_{0} → M ⊗_{K_{0}} M^{∗} (the dual of the pairing M^{∗} ⊗_{K_{0}} M → K_{0}) gives an extension

0 → B_{st}^{+} ⊗_{K_{0}} M^{∗} → E′ → K_{0} → 0.

On the other hand given an extension E′ as above we can apply ⊗_{K_{0}}M and form the pushout under the map B_{st}^{+} ⊗_{K_{0}} M^{∗} ⊗_{K_{0}} M → B_{st}^{+} to get an extension

0 → B_{st}^{+} → E′′ → M → 0.

We have to check these two constructions are inverses. We check the one direction we will need, and leave the other to the reader. Let us check that E ˜ ⟶E′′ as extensions. If H denotes the kernel of the map B_{st}^{+} ⊗_{K_{0}} M^{∗} ⊗ M → B_{st}^{+}, and e ∈ M^{∗} ⊗_{K_{0}} M denotes the image 1 under the canonical map K_{0} → M ⊗_{K_{0}} M^{∗}, then we have a commutative diagram

[figure: B_{st}^{+}; E; M; B_{st}^{+} ⊗_{K_{0}} M^{∗} ⊗_{K_{0}} M/H; E ⊗_{K_{0}} M^{∗} ⊗_{K_{0}} M/H; M ⊗_{K_{0}} M^{∗} ⊗_{K_{0}} M; m⊗a⊗b↦m⟨a,b⟩; B_{st}^{+}; E′′; M; m↦e⊗m]

where the bottom row is obtained from the middle one by the pull back M → M ⊗_{K_{0}} M^{∗} ⊗_{K_{0}} M, m ↦ e ⊗ m.

It remains to be check that composite of the maps in the right vertical column is the identity on M. This is a little exercise in linear algebra, which is left to the reader.

Proposition (3.3.10). The complex

F il^{0}B_{st}^{+} →^{N⊕ϕ−1} B_{st}^{+} ⊕ B_{st}^{+} →^{(1−pϕ)+N} B_{st}^{+}

is acyclic in degree > 0.

Proof. As usual this complex is the total complex of

\xymatrix{B_{st}^{+} \ar[r]^{N} & B_{st}^{+} \\ F il^{0}B_{st}^{+} \ar[r]^{N} \ar[u]^{ϕ−1} & B_{st}^{+} \ar[u]^{1−pϕ}}

But in fact we have F il^{0}B_{st}^{+} = B_{st}^{+}, and the map N is surjective on B_{st}^{+} = B_{cris}^{+}[u]. It follows that the cohomology of this double complex with respect to the horizontal differentials is just B_{cris}^{+} →^{ϕ−1} B_{cris}^{+}, and this map is surjective by [Fo, 5.3.7]. □

Theorem (3.3.11). Let M be positive. Then M is admissible if and only if it is strongly admissible.

Proof. First suppose that M is admissible. By the discussion in (3.3.9), we have maps

Ext_{M_{st,K}^{wa}}^{1} (M, B_{st}^{+}) → Ext_{M_{st,K}^{wa}}^{1} (K_{0}, M^{∗} ⊗_{K_{0}} B_{st}^{+}) → Ext_{M_{st,K}^{wa}}^{1} (M, B_{st}^{+}),

with the composite being the identity, so it is enough to show that the middle term is 0. As M is admissible, (3.3.3) shows that M^{∗} ⊗_{K_{0}} B_{st}^{+} ˜ ⟶(B_{st}^{+})^{d}, with d = dim_{K_{0}}M. So consider an extension

0 → (B_{st}^{+})^{d} → E → K_{0} → 0.

As usual, fix a basis vector e ∈ K_{0}, with ϕ(e) = e. Since E = F il^{0}E we can lift e to ẽ ∈ F il^{0}E. As 1 ∈ K_{0} is invariant under ϕ, and is killed by N, we see that

f = (N ⊕ ϕ − 1)(e) ∈ B_{st}^{+} ⊕ B_{st}^{+} ⊂ E ⊕ E.

By (3.3.10) we have an exact sequence

(3.3.12)  0 → ℚ_{p} → F il^{0}B_{st}^{+} →^{N⊕ϕ−1} B_{st}^{+} ⊕ B_{st}^{+} →^{(1−pϕ)+N} B_{st}^{+} → 0

Now ((1 − pϕ) + N)(f) = 0, and so the exactness of (3.3.12) implies that there exists e′ ∈ F il^{0}B_{st}^{+} with (N ⊕ ϕ − 1)(e′) = (N ⊕ ϕ − 1)(e). Then the map 1 ↦ e − e′ gives a splitting of E, and we are done.

Suppose now that M is strongly admissible. We show that M is admissible by induction on d = dim_{K_{0}}M. The case d = 1 follows from the fact that any one-dimensional weakly admissible module is in fact admissible.

If F il^{0}M_{K}^{∗} = M_{K}^{∗}, then we have t_{H}(M) = t_{N}(M) = 0, and there exists a finite unramified extension K′_{0} of K_{0} such that M′ = M ⊗_{K_{0}} K′_{0} ˜ ⟶(K′_{0})^{d}. As M′ is spanned by elements m with ϕ(m) = m, we see immediately that (the linear extension of) N maps M′ into a subspace spanned by elements m ∈ M′ with ϕ(m) = p^{−1}m, hence is 0. Thus M′ is obviously admissible, whence so is M by [Fo 2, 5.3.4].

So assume that F il^{0}M_{K}^{∗} ≠ M_{K}^{∗}. We will show that there is a non-trivial extension E of M by K_{0}. We show the dual statement that there is a non-trivial extension E^{∗} of K_{0} by M^{∗}.

First, suppose that K_{0} is finite dimensional over ℚ_{p}. Consider the double complex of (2.4.2), whose total complex equals Rhom_{M_{st,K}^{wa}} (by (3.1.5)). This is a complex of ℚ_{p}-linear maps of finite dimensional ℚ_{p} vector spaces. Since we are assuming that F il^{0}M_{K}^{∗} ≠ M_{K}^{∗}, a dimension count over ℚ_{p} shows that it has nontrivial first cohomology, so there is a non-trivial extension of K_{0} by M^{∗}.

If K_{0} is not finite dimensional over ℚ_{p}, then let L be the ℚ_{p} subvector space of M^{∗} which is fixed by ϕ. Note that L is finite dimensional over ℚ_{p}, because L ⊗_{ℚ_{p}} K_{0} → M is injective. Thus the map L → M^{∗}/F il^{0}M^{∗} cannot be surjective, because the right hand side is a non-zero K_{0} vector space, hence infinite dimensional over ℚ_{p}. Choose m in M^{∗} whose class in M^{∗}/F il^{0}M^{∗} is not in the image of L. We define an extension E^{∗} of K_{0} by M^{∗} as follows: E^{∗} = M^{∗} ⊕ K_{0}, as (ϕ, N) modules, and the filtration on E_{K} is given by

F il^{i}E_{K}^{∗} = \{ \begin{matrix}F il^{i}M_{K}^{∗} + K · (m ⊕ e), i ⩽ 0 \\ 0, i > 0 \end{matrix}

We claim that E^{∗} is nontrivial. Indeed, if s : K_{0} → E^{∗} is a section to the natural projection E^{∗} → K_{0}, then s(e) = b ⊕ e, with ϕ(b) = b. Then b is an element of L congruent to m modulo F il^{0}M_{K}, contradicting our assumptions. Thus E^{∗} must be non-trivial.

Now choose a non-zero map K_{0} → B_{st}^{+}. As Ext_{M_{st,K}^{wa}}^{1} (M, B_{st}^{+}) = 0, standard

manipulations show that this lifts to a map f : E → B_{st}^{+}. As E is not split, Im(f) must have dimension at least 2. Im(f) is admissible by (3.3.5), and ker(f) is strongly admissible by (3.3.8), hence admissible by induction. It follows from (3.1.6) that E is also admissible. Alternatively one can use the fact that as Im(f) is strongly admissible, we have

dim_{ℚ_{p}}Hom_{M_{st,K}^{wa}} (E, B_{st}^{+})

= dim_{ℚ_{p}}Hom_{M_{st,K}^{wa}} (Im(f), B_{st}^{+}) + dim_{ℚ_{p}}Hom_{M_{st,K}^{wa}} (ker(f), B_{st}^{+}) = dim_{K_{0}}E,

so E is admissible by (3.3.4). It follows that M is also admissible. □

§4 Extensions of Crystalline Representations

In this section we study the obstruction to an extension of crystalline (respectively semi-stable) Galois representations being crystalline (respectively semistable). We show that this obstruction is controlled by the Galois cohomology of the corresponding period ring B_{cris} (respectively B_{st}). We give a partial calculation of these Galois cohomology groups in the style of Tate/Faltings, using an “almost étale” approach. Finally we recover a result of Hyodo, which says that when the residue field is finite, an extension of semi-stable representations which is de Rham is in fact semi-stable.

We begin with some preliminaries on continuous Galois cohomology.

4.1 Continuous Galois Cohomology Throughout this section we denote by G a pro-finite group. The main point of this section is to provide some general criteria under which one has some of the the usual machinery of Galois cohomology available - for example long exact cohomology sequences. (4.1.1) Recall that a continuous G module is a topological abelian group M, equipped with a G action such that the map G × M → M is continuous. A map of continuous G modules is a continuous map of abelian groups, respecting the G action.

We define the continuous cohomology H^{i}(G, M) of a continuous G module M, using the complex of continuous cochains RG^{•}(M).

A map M → N of topological G modules is called pro-finitely surjective if for any pro-finite group H, every continuous map H → N can be lifted to a map of sets H → M. The importance of such maps is explained by the following proposition.

Proposition (4.1.2). Let

0 → A →^{f} B →^{g} C → 0

be a short exact sequence of continuous G modules. Assume that f is an isomorphism onto its image, and that g is pro-finitely surjective. Then there is a short exact sequence

0 → RG^{•}(A) → RG^{•}(B) → RG^{•}(C) → 0,

and so consequently there is also a long exact cohomology sequence

⋯ → H^{i}(RG^{•}(A)) → H^{i}(RG^{•}(B)) → H^{i}(RG^{•}(C)) → H^{i+1}(RG^{bullet}(A)) → · · · .

Proof. In the notation of (1.2.1), we have an exact sequence of complexes

0 → RG(A) → RG(B) → RG(C) → 0,

where the surjectivity of the final map follows because g is pro-finitely surjective, so that any continuous cochain map G^{n} → C on G can be lifted to B. Taking cohomology yields the proposition. □

(4.1.3) We now give a criterion for when a map of continuous G modules is pro-finitely surjective.

We call a continuous G module M pro-discrete if it has a countable basis of neighbourhoods of the identity consisting of open G-stable subgroups, and if the natural map

M → \underleftarrow{lim}M/M_{n},

where M_{n} runs through open G-stable subgroups of M, is an isomorphism.

Proposition (4.1.4). Let f : M → N be a surjective open map of continuous G modules, and suppose that M and N are pro-discrete. Then f is pro-finitely surjective.

Proof. The conditions imply that there exists a basis of neighbourhoods {M_{n}}_{n∈ℕ} of M, consisting of continuous G submodules, and satisfying M_{1} ⊃ M_{2} ⊃ M_{3} …, n = 1, 2 …. Set N_{n} = f(M_{n}). Then {N_{n}}_{n∈ℕ} is a basis of open neighbourhoods of N consisting of open sub G modules, and f is the inverse limit of the induced maps f_{n} : M/M_{n} → N/N_{n}.

If H is a pro-finite group, and h : H → N is a continuous map, denote by h_{n}, the composite H → N → N/N_{n}. Since the map

M/M_{n+1} → N/N_{n+1} ×_{N/N_{n}} M/M_{n}

is a surjection we can construct inductively a sequence of maps g_{n} : H → M_{n}, such that f_{n} ∘ g_{n} = h_{n}, and g_{n+1} = g_{n} modulo M_{n}. Then we may take for g the inverse limit of the maps g_{n}. □

Proposition (4.1.5). Suppose that G ˜ ⟶ℤ_{p}, with topological generator γ, and let M be a pro-discrete, continuous G module. There is a canonical quasi-isomorphism

RG(M) ˜ ⟶(M →^{1−γ} M).

Proof. As in the previous proposition let M_{1} ⊃ M_{2} … be a basis of continuous G submodules. Recall the construction of the quasi-isomorphism

RG(M/M_{n}) ˜ ⟶(M/M_{n} ⟶^{1−γ} M/M_{n}).

In degree 0 this is the identity map, in degree 1 it is the map which sends a cochain s to its value on γ ∈ G, and in higher degrees it is zero. This is a surjective map of complexes; let K_{n}^{•} denote its kernel.

The natural map RG(M/M_{n+1}) → RG(M/M_{n}) induces maps K_{n+1} → K_{n} which are surjective. Indeed, by (4.1.4) this is obvious except possibly in degree 1, and there it follows because any continuous cochain h_{n} : G → M/M_{n} with h_{n}(γ) = 0 can be lifted to a cochain h_{n+1} : G → M/M_{n+1} with h_{n+1}(γ) = 0. Indeed, take any lifting h′_{n+1} of h_{n}, and define h_{n+1} by h_{n+1}(s) = h′_{n+1}(s)−h′_{n+1}(γ).

It follows that we may pass to the inverse limit to obtain an exact sequence

0 → \underleftarrow{lim}K_{n}^{•} → RG(M) → (M →^{1−γ} M) → 0.

Since the K_{n}^{•} are all acyclic, with surjective transition maps, we see that \underleftarrow{lim}K_{n}^{•} is acyclic. □

4.2 Topologies on Period Rings (4.2.1) We want to define topologies on the rings introduced in (2.2), which make them into continuous G_{K} modules. The slight subtlety is that the obvious topologies are not always the right ones. For example, if R is given its discrete topology, then it is not a continuous G_{K} module, and it follows easily that W (R) is not a continuous G_{K} module when given its p-adic topology, (4.2.2) We begin by giving the ring R = \underleftarrow{lim}OK̄/p its inverse limit topology. If p_{n} : R → OK̄/p denotes the projection onto the n^{th} term of the inverse limit, then this is the topology induced by the collection of semi-norms given by r ↦ |p_{n}(r)|, n = 1, 2, …, where | · | denotes the natural semi-norm on OK̄/p. This topology gives R the structure of a continuous, pro-discrete G_{K} module. (4.2.3) Let p̃_{n} : W (R) → O_{ℂ_{p}} be the lifting of the map p_{n} introduced above. We give W (R) the topology induced by the collection of semi-norms given by w ↦ |p̃_{n}(w)|, n = 1, 2 …, where | · | denotes the norm on O_{ℂ_{p}}.

It is easy to check that a sequence {w_{n}}_{n∈ℕ} of elements of W (R) goes to 0 in this topology if and only if the Witt co-ordinates w_{n}^{(i)} of the w_{n} go to 0 in R co-ordinatewise. That is, if and only if for each i we have

lim_{n} w_{n}^{(i)} = 0 ∈ R.

This topology gives W (R) the structure of a continuous G_{K} module. It is easy to check that W (R) is complete for this topology, and in fact it is pro-discrete. To see this let ζ ∈ W (R) be any generator of the kernel of θ (which is just p̃_{1} in the above notation). Then we have ker p̃_{n} = ϕ^{n−1}(ζ), and

W (R) = \underleftarrow{lim}_{n,m,r}W (R)/(p^{n} + ϕ^{m}(ζ)).

Finally, using the congruence ϕ(ζ) = ζ^{p} modulo p, we see that the above topology is simply the one generated by the ideal (p, ζ). (4.2.4) We equip W (R)[x], the polynomial ring in one variable over W (R), with the direct limit topology that results by viewing it as a direct sum W (R)[x] = ⊕_{i}W (R)x^{i}. We give the ring

E_{n,r} = W (R)/p^{n}W (R)[ζ^{p^{r}}/(p^{r})!]

the quotient topology induced by the surjection

\xymatrix{W (R)[x] \ar[r]^{x↦ζ^{p^{r}} /(p^{r})!} & W (R)/p^{n}W (R)[ζ^{p^{r}}/(p^{r})!].}

It is not hard to check that this makes E_{n,r} into a continuous G_{K} module.

In fact the topology on E_{n,r} is the discrete one (!), because the open set (p^{n} + ζ^{p^{r}s})E_{n,r} ⊂ E_{n,r} is 0 if s is a sufficiently large integer. Thus

A_{cris}/p^{n}A_{cris} = \underrightarrow{lim}_{r}W (R)/p^{n}W (R)[ζ^{p^{r}}/(p^{r})!]

becomes a continuous G_{K} module, when given the discrete topology, and A_{cris} becomes a continuous G_{K} module when given its p-adic topology. (4.2.5) We give B_{cris}^{+} = \underrightarrow{lim}p^{−i}A_{cris} and B_{cris} = \underrightarrow{lim}t^{−i}B_{cris} their direct limit topologies. Note that this makes B_{cris}^{+} into a p-adic Banach space. Similarly, we give

B_{st} = ⊕_{i}B_{cris}u^{i}

the direct limit topology.

Note that for each j, F il^{j}B_{cris}^{+} with the topology induced from B_{cris}^{+} is prodiscrete. Indeed, this just amounts to the fact that it is p-adically complete, which is clear from the definition of the filtration.

4.3 The Basic Long Exact Sequences. We are going to construct a long exact sequence which measures the difference between Ext’s in the category of Galois representations and Ext’s in the category of crystalline (respectively semistable) Galois representations.

(4.3.1) If M is a continuous G_{K} module, we follow the notation of §4.1 and denote by RG_{K}(M) the complex of continuous cochains of G_{K} with values in M. (4.3.2) Suppose that M is a continuous G_{K} module which is equipped with a compatible structure of filtered ϕ module, such that each F il^{j}M is an open submodule of M.

For each i the Frobenius endomorphism ϕ of M induces a σ-linear endomorphism of RG^{i}(M). Also, since K has finite degree over K_{0} there is an isomorphism

RG^{i}(M) ⊗_{K_{0}} M ˜ ⟶RG^{i}(M_{K}).

Since G_{K} is pro-finite, and so compact, and since the F il^{j}M_{K} provide an exhaustive filtration of M_{K}, we see that any M_{K}-valued continuous cochain on G_{K} lies in one of the F il^{j}M. Thus the images of the RG_{K}^{i}(F il^{j}M_{K}) in RG_{K}^{i}(M_{K}) form an exhaustive filtration of RG_{K}^{i}(M_{K}) ˜ ⟶RG_{K}^{i}(M) ⊗_{K_{0}} K for each i.

Putting these structures together endows each RG_{K}^{i}(M) with the structure of a filtered ϕ module, so that RG_{K}^{•}(M) becomes a complex of filtered ϕ modules.

Now the complex 𝒦_{cris}^{•}(M) is a complex of continuous Galois modules, and so we may form the double complex RG_{K}^{•}(𝒦_{cris}^{•}(M)) of ℚ_{p}-vector spaces. On the other hand, we may apply the 𝒦_{cris}^{•} construction to the complex RG_{K}^{•}(M) of filtered ϕ-modules to obtain the double complex of ℚ_{p}-vector spaces 𝒦_{cris}^{•}(RG_{K}^{•}(M)). By virtue of the definition of the filtration and Frobenius on RG_{K}^{•}(M), there is a natural morphism of double complexes

RG_{K}^{•}(𝒦_{cris}^{•}(M)) → 𝒦_{cris}^{•}(RG_{K}^{•}(M)),

which is an isomorphism of double complexes.

A similar discussion holds with 𝒦_{cris}^{•} replaced by 𝒦_{st}^{•} in the case when M is a filtered (ϕ, N) module. (4.3.3) Since B_{cris} = \underrightarrow{lim}t^{−j}B_{cris}^{+} is the union of the open G_{K} submodules t^{−j}B_{cris}, and since G_{K} is compact, we see that

RG_{K}^{•}(B_{cris}) = \underrightarrow{lim}RG_{K}^{•}(t^{−j}B_{cris}).

Similarly, if we write B_{st}^{i,j} = ⊕_{r=0}^{i}t^{−j}B_{cris}^{+}u^{r}, then B_{st} = \underrightarrow{lim}B_{st}^{i,j}, and so

RG_{K}^{•}(B_{st}) = \underrightarrow{lim}RG_{K}^{•}(B_{st}^{i,j}).

Proposition (4.3.4). Let V be a crystalline representation of G_{K}, and let M = (V ⊗_{ℚ_{p}} B_{cris})^{G_{K}} . Then there is a canonical quasi-isomorphism

RG_{K}^{•}(V) ˜ ⟶(RG_{K}^{•}(𝒦_{cris}^{•}(B_{cris} ⊗_{K_{0}} M)))^{T ot} ˜ ⟶(𝒦_{cris}^{•}(RG_{K}^{•}(B_{cris} ⊗_{K_{0}} M)))^{T ot}.

Similarly if V is a semi-stable representation of G_{K}, and M = (V ⊗_{ℚ_{p}} B_{st})^{G_{K}} , then there is a canonical quasi-isomorphism

RG_{K}^{•}(V) ˜ ⟶(RG_{K}^{•}(𝒦_{st}^{•}(B_{st} ⊗_{K_{0}} M)))^{T ot} ˜ ⟶(𝒦_{st}^{•}(RG_{K}^{•}(B_{st} ⊗_{K_{0}} M)))^{T ot}.

Proof. First consider the case where V is crystalline. By (2.2.6), we have for each sufficiently large positive integer j an exact sequence

0 → V → F il^{0}(t^{−j}B_{cris}^{+} ⊗_{K_{0}} M) →^{ϕ−1} t^{−j}B_{cris}^{+} ⊗_{K_{0}} M → 0.

The last two terms are p-adic Banach spaces, so the map between them is open. Since they are pro-discrete (4.2.5), (4.1.2) yields an exact sequence

0 → RG_{K}^{•}(V) → RG_{K}^{•}(F il^{0}(t^{−j}B_{cris}^{+} ⊗_{K_{0}} M)) →^{ϕ−1} RG_{K}^{•}(t^{−j}B_{cris}^{+} ⊗_{K_{0}} M) → 0.

Now the remark at the end of (2.5.15), and another application of (4.1.2), shows that there is a quasi-isomorphism

(RG_{K}^{•}(F il^{0}(t^{−j}B_{cris}^{+} ⊗_{K_{0}} M) →^{ϕ−1} t^{−j}B_{cris}^{+} ⊗_{K_{0}} M))^{T ot}

˜ ⟶(RG_{K}^{•}(𝒦_{cris}^{•}(t^{−j}B_{cris}^{+} ⊗_{K_{0}} M)))^{T ot}.

Combining this with the previous exact sequence and letting j go to infinity proves the proposition when V is crystalline. (Here we have also used remark 4.3.3.)

The proof of the second quasi-isomorphism is similar, but slightly more involved: For positive integers i, j, recall the definition of B_{st}^{i,j} from remark 4.3.3, and denote by 𝒦_{st}^{•}(i, j) the total complex of

\xymatrix{B_{st}^{i,j} \ar[r] & 0 \\ B_{st}^{i,j} ⊗_{K_{0}} M ⊕ B_{st}^{i+1,j} ⊗_{K_{0}} M \ar[u]^{(1−pϕ)+N} \ar[r] & 0 \ar[u] \\ B_{st}^{i+1,j} ⊗_{K_{0}} M \ar[u]^{N⊕ϕ−1} \ar[r] & B_{st}^{i+1,j} ⊗_{K_{0}} M_{K}/F il^{0}(B_{st}^{i+1,j} ⊗_{K_{0}} M_{K}) \ar[u]}

Using (4.1.2), and the open mapping theorem as above, we see that for sufficiently large i, j, (RG_{K}^{•}(𝒦_{st}^{•}(i, j)))^{T ot} is quasi-isomorphic to (RG_{K}^{•}(𝒦_{st}^{•}(i, j)′))^{T ot}, where we denote by 𝒦_{st}^{•}(i, j)′ the total complex of

\xymatrix{B_{st}^{i+1,j} ⊗_{K_{0}} M \ar[r]^{N} & B_{st}^{i,j} ⊗_{K_{0}} M \\ F il^{0}B_{st}^{i+1,j} ⊗_{K_{0}} M \ar[r]^{N} \ar[u]^{ϕ−1} & B_{st}^{i,j} ⊗_{K_{0}} M. \ar[u]^{1−pϕ}}

Similarly, (RG_{K}^{•}(𝒦_{st}^{•}(i, j)′))^{T ot} is quasi-isomorphic to

(RG_{K}^{•}(F il^{0}t^{−j}B_{cris} ⊗_{ℚ_{p}} V) →^{ϕ−1} RG_{K}^{•}(t^{−j}B_{cris}^{+} ⊗_{ℚ_{p}} V))^{T ot},

which is quasi-isomorphic to RG_{K}^{•}(V) for all sufficiently large j. Taking a limit over i and j (and applying remark 4.3.3) establishes the required quasi-isomorphism. □

Lemma (4.3.5). Let V be a crystalline representation, and M = (V ⊗_{ℚ_{p}} B_{cris})^{G_{K}} . Then boundary maps in the complex RG_{K}^{•}(B_{cris} ⊗_{K_{0}} M) are strict for the filtration.

Similarly, if V is a semi-stable representation, and M = (V ⊗_{ℚ_{p}} B_{st})^{G_{K}} , then the boundary maps in the of complex RG_{K}^{•}(B_{st} ⊗_{K_{0}} M) are strict for the filtration.

Proof. We will give the proof in the case that V is crystalline. The case when V is semi-stable is dealt with in the same way.

The statement of the lemma is equivalent to proving, for each i, that the morphism H^{i}(G_{K}, F il^{0}(B_{cris} ⊗_{K_{0}} M)) → H^{i}(G_{K}, B_{cris} ⊗_{K_{0}} M) is an injection. (Here H^{i} denotes continuous Galois cohomology, of course.)

Choose a direct sum decomposition

M_{K} ˜ ⟶ ⊕_{j∈ℤ} M_{K}^{j}

where M_{K}^{j} is a subspace of M_{K} mapping isomorphically to F il^{j}M_{K}/F il^{j+1}M_{K}. Since

F il^{0}(B_{cris} ⊗_{K_{0}} M_{K}) = ⊕_{j∈ℤ}F il^{−j}(B_{cris} ⊗_{K_{0}} K) ⊗_{K} M_{K}^{j}

it is enough to show that for each j ∈ ℤ the map

H^{i}(G_{K}, F il^{j}B_{cris} ⊗_{K_{0}} K) → H^{i}(G_{K}, B_{cris} ⊗_{K_{0}} K)

is an injection or, by remark 4.3.3, that for each positive integer r,

H^{i}(G_{K}, F il^{j}(t^{−r}B_{cris}^{+} ⊗_{K_{0}} K)) → H^{i}(G_{K}, t^{−r}B_{cris}^{+} ⊗_{K_{0}} K)

is an injection. For this, it is enough to show that

H^{i}(G_{K}, F il^{j+1}(t^{−r}B_{cris}^{+} ⊗_{K_{0}} K)) → H^{i}(G_{K}, F il^{j}(t^{−r}B_{cris}^{+} ⊗_{K_{0}} K))

is an injection for each j. The quotient F il^{j}(t^{−r}B_{cris}^{+} ⊗_{K_{0}} K)/F il^{j+1}(t^{−r}B_{cris}^{+} ⊗_{K_{0}} K) is equal to ℂ_{p}(j) if j ≥ −r and 0 otherwise. By [Ta, Thms. 1,2] the cohomology H^{i}(G_{K}, ℂ_{p}(j)) vanishes unless j = 0 and i = 0, 1. Thus by (4.1.2), if j ≠ 0, we see that the above map is actually an isomorphism. If j = 0, then the results of [Ta] show that the map H^{i}(G_{K}, K) → H^{i}(G_{K}, ℂ_{p}) is a surjection. More precisely, if H_{K} is the absolute Galois group of the cyclotomic ℤ_{p} extension of K, and Γ_{K} = G_{K}/H_{K}, then the composite map

H^{i}(Γ_{K}, K) → H^{i}(G_{K}, K) → H^{i}(G_{K}, ℂ_{p})

is an isomorphism for i = 0, 1. In particular, it follows that the natural map

H^{i}(G_{K}, F il^{0}(t^{−r}B_{cris}^{+} ⊗_{K_{0}} K)) → H^{i}(G_{K}, ℂ_{p})

is a surjection, since K ⊂ B_{cris}^{+} ⊗_{K_{0}} K. The required injectivity now follows from (4.1.2). □

Corollary (4.3.6). If V is a crystalline representation of G_{K} then there is a convergent spectral sequence

E_{2}^{pq} = H^{p}(𝒦_{cris}^{•}(H^{q}(G_{K}, B_{cris}) ⊗_{K_{0}} M)) ⇒ H^{p+q}(G_{K}, V)

Similarly, if V is semi-stable there is a convergent spectral sequence

E_{2}^{pq} = H^{p}(𝒦_{st}^{•}(H^{q}(G_{K}, B_{st}) ⊗_{K_{0}} M)) ⇒ H^{p+q}(G_{K}, V)

The first spectral sequences degenerates at E_{2}^{pq}, and the second at E_{3}^{pq}.

Proof. Let us first remark that if V is crystalline (respectively semi-stable) then RG_{K}^{•}(B_{cris} ⊗_{K_{0}} M) is a complex of filtered ϕ modules (respectively RG_{K}^{•}(B_{st} ⊗_{K_{0}} M) is a complex of filtered (ϕ, N) modules), and so for each i the continuous cohomology module H^{i}(G_{K}, B_{cris} ⊗_{K_{0}} M) is a filtered ϕ-module (respectively for each i the continuous cohomology module H^{i}(B_{st} ⊗_{K_{0}} M) is a filtered (ϕ, N) module).

The required spectral sequences are constructed from the quasi-isomorphisms of (4.3.4), using one of the spectral sequences associated to the double complex RG_{K}^{•}(𝒦_{cris}^{•}(B_{cris} ⊗_{K_{0}} M)) (respectively the double complex RG_{K}^{•}(𝒦_{st}^{•}(B_{st} ⊗_{K_{0}} M))) together with the fact that the natural map 𝒦_{cris}^{•}(H^{i}(G_{K}, B_{cris} ⊗_{K_{0}} M)) → H^{i}(G_{K}, 𝒦_{cris}^{•}(B_{cris} ⊗_{K_{0}} M)) (respectively the natural map 𝒦_{st}^{•}(H^{i}(G_{K}, B_{st} ⊗_{K_{0}} M)) → H^{i}(G_{K}, 𝒦_{cris}^{•}(B_{st} ⊗_{K_{0}} M))) is an isomorphism of complexes for each i, as follows from (4.1.2) and the previous lemma.

That these spectral sequences degenerate at E_{2}^{pq} and E_{3}^{pq} respectively follows from the fact that 𝒦_{cris}^{•} (respectively 𝒦_{st}^{•}) gives rise to complexes which are acyclic in degree > 1 (respectively > 2) when evaluated on filtered ϕ modules (respectively filtered (ϕ, N) modules). □

Theorem (4.3.7). The (continuous) Galois cohomology groups H^{i}(G_{K}, B_{cris}) and H^{i}(G_{K}, B_{st}) vanish if i is greater than one.

This result is included in corollary (4.4.23) of the following section. Here we will use it to deduce the following corollaries of (4.3.4):

Corollary (4.3.8). Let V, W be crystalline representations of G_{K}, set M = (V ⊗_{ℚ_{p}} B_{cris})^{G_{K}} , N = (W ⊗_{ℚ_{p}} B_{cris})^{G_{K}} , and denote

Ext_{ℚ_{p}⟨G_{K}⟩}^{i}(V, W) = H^{i}(RG_{K}(V^{∗} ⊗_{ℚ_{p}} W)).

(1) There is an exact sequence

0 → Ext_{M_{K}^{ad}}^{1}(M, N) → Ext_{ℚ_{p}⟨G_{K}⟩}^{1}(V, W)

→ Hom_{K_{0}[ϕ],F il}(M, N ⊗_{K_{0}} H^{1}(G_{K}, B_{cris})) → 0

(2) There is an isomorphism

Ext_{ℚ_{p}⟨G_{K}⟩}^{2}(V, W) ˜ ⟶H^{1}(𝒦_{cris}^{•}(Hom_{K_{0}}(M, N ⊗_{K_{0}} H^{1}(G_{K}, B_{cris}))))

Proof. First note that replacing W by V^{∗} ⊗_{ℚ_{p}} W, we may assume that V = ℚ_{p}. The spectral sequence in (4.3.6), together with (2.3.9) and the fact that B_{cris}^{G_{K}} = K_{0}, gives an exact sequence

0 → Ext_{M_{K}^{ad}}^{1}(M, N) → Ext_{ℚ_{p}⟨G_{K}⟩}^{1}(V, W)

→ Hom_{K_{0}[ϕ],F il}(M, N ⊗ H^{1}(G_{K}, B_{cris})) → Ext_{M_{K}^{ad}}^{2}(M, N) …,

However the last term is 0, as H^{2}(𝒦_{cris}^{•}(M^{∗} ⊗_{K_{0}} N)) = 0. This proves (1).

The fact that H^{i}(𝒦_{cris}^{•}( )) = 0, i ≥ 2, together with (4.3.7), shows that E_{2}^{pq} = 0 in the first spectral sequence of (4.3.6) for p > 1 or q > 1. Thus Ext_{ℚ_{p}⟨G_{K}⟩}^{2}(V, W) = E_{2}^{1,1}, which is (2). □

Corollary (4.3.9). Let V, W be semi-stable representations of G_{K}, and set M = (V ⊗_{ℚ_{p}} B_{st})^{G_{K}} , N = (W ⊗_{ℚ_{p}} B_{st})^{G_{K}} . Then

(1) There is an exact sequence

0 → Ext_{M_{st,K}^{ad}}^{1} (M, N) → Ext_{ℚ_{p}⟨G_{K}⟩}^{1}(V, W)

→ Hom_{K_{0}[ϕ,N],F il}(M, N ⊗_{K_{0}} H^{1}(G_{K}, B_{st})) → Ext_{M_{st,K}^{ad}}^{2} (M, N) →

Ext_{ℚ_{p}⟨G_{K}⟩}^{2}(V, W) → H^{1}(𝒦_{st}^{•}(M^{∗} ⊗_{K_{0}} N ⊗_{K_{0}} H^{1}(G_{K}, B_{st}))) → 0

(2) We have

H^{2}(𝒦_{st}^{•}(M^{∗} ⊗_{K_{0}} N ⊗_{K_{0}} H^{1}(G_{K}, B_{st}))) = 0

Proof. The proof is similar to that of (4.3.8). The fact that H^{i}(𝒦_{st}^{•}( )) = 0 for i > 2, together with (4.3.7), shows that E_{2}^{p,q} = 0 in the second spectral sequence of (4.3.6), provided p > 2 or q > 1. Using this observation, we see that this spectral sequence, together with (2.3.9) and the fact that B_{st}^{G_{K}} = K_{0}, yields (1). The same observation, combining the spectral sequence with the fact that G_{K} has cohomological dimension at most 2 [Se, 4.3 Prop 12], proves (2). □

(4.3.10) Remarks: (1) By (1.3.1), if the residue field of K is either finite or algebraically closed, then the groups Ext_{ℚ_{p}⟨G_{K}⟩}^{i}(V, W) which appear in (4.3.8), (4.3.9) are actually Ext groups computed in the category of finite dimensional, continuous ℚ_{p} representations of G_{K}, so the above results can be regarded as studying the difference between extensions in this category and extensions in the category of crystalline (respectively semi-stable) representations.

In particular, when i = 1, this shows that the obstruction to an extension of crystalline (respectively semi-stable) representations being crystalline (respectively semi-stable) is controlled by the Galois cohomology of B_{cris} (respectively B_{st}). (Here we do not need to assume the residue field is finite – see (1.2.6).)

(2) Groups like H^{1}(𝒦_{cris}^{•}(Hom_{K_{0}}(M, N ⊗_{K_{0}} H^{1}(G_{K}, B_{cris})))) which appear in (4.3.8) and (4.3.9) should be the extensions of M by N ⊗_{K_{0}} H^{1}(G_{K}, B_{cris}) in some suitable category. We do not know how to formulate this rigorously, however. On the other hand, it is easy to show that the expression above is in bijective correspondence with the set of extensions of M by N ⊗_{K_{0}} H^{1}(G_{K}, B_{cris}), when these are considered in the exact (but not abelian) category of K_{0} vector spaces equipped with ϕ, N and filtration after ⊗_{K_{0}}K.

4.4 Galois Cohomology of Period Rings. We saw in the previous section that the obstruction to an extension of crystalline (respectively semi-stable) representations being crystalline (respectively semi-stable) is, in some sense, controlled by H^{1}(G_{K}, B_{cris}) (respectively H^{1}(G_{K}, B_{st})). The purpose of this section is to try to compute this group. We are only partly successful, in the sense that we reduce this calculation to a calculation of the Galois cohomology of a ℤ_{p} extension, with a coefficients in a certain subring of B_{cris} (respectively B_{st}). In particular, we show that the Galois cohomology of B_{cris} (respectively B_{st}) vanishes in degree ≥ 2, proving (4.3.7). (4.4.1) We will denote by K′_{∞} the extension of K obtained by adjoining all p-power roots of unity, and by K_{∞} ⊂ K′_{∞} the sub-extension with Gal(K_{∞}/K) ˜ ⟶ℤ_{p}. We denote by O_{K′_{∞}} and O_{K_{∞}} the rings of integers of K′_{∞} and K_{∞} respectively. Finally we denote by K̂_{∞} and OK̂_{∞} the completions of K_{∞} and O_{K_{∞}} respectively.

If p ≠ 2 and μ_{p} ⊂ K, or p = 2 and μ_{4} ⊂ K, then we have K′_{∞} = K_{∞}. Otherwise, we have Gal(K′_{∞}/K) ˜ ⟶ℤ_{p}^{∗}, and Gal(K′_{∞}/K_{∞}) corresponds to the subgroup μ_{p−1} of Teichmüller lifts if p ≠ 2, and to {±1} if p = 2.

We write Γ_{K} = Gal(K_{∞}/K), and H_{K} = Gal(K̄/K_{∞}). (4.4.2) Recall that given any ring V, such that the Frobenius map on V/pV is a surjection one can apply Fontaine’s “B_{cris}” construction to it. We will describe this more explicitly in the case we need, which is V = OK̂_{∞}. To know the construction applies we need

Lemma (4.4.3). The Frobenius map

O_{K_{∞}}/p ⟶^{x↦x^{p}} O_{K_{∞}}/p

is a surjection.

Proof. Recall the following fact, due to Faltings [Fa 1.2, 2.4]. If M is a p-adically complete and separated 𝒪K̄ module with a continuous, semi-linear G_{K} action, then for each i, H^{i}(H_{K}, M) is an 𝒪_{K_{∞}} module, and if i > 0, it is annihilated by the maximal ideal 𝔪_{∞} of 𝒪_{K_{∞}}.

Now consider the Frobenius on 𝒪K̄/p. It is surjective, and its kernel satisfies the hypotheses above. Thus by taking H_{K} invariants, we see that the cokernel of the Frobenius on (𝒪K̄/p)^{H_{K}} is killed by 𝔪_{∞}.

Next, applying the theory to the exact sequence

0 → OK̄ →^{p} OK̄ → OK̄/p → 0,

we see that the cokernel of the natural map 𝒪_{K_{∞}}/p → (𝒪_{K}/p)^{H_{K}} is killed by 𝔪_{∞}.

It follows that the cokernel of the Frobenius on (𝒪_{K_{∞}}/p) is killed by 𝔪_{∞}. To see that this cokernel is actually zero, take x ∈ 𝒪_{K_{∞}}/p, and choose x_{0} ∈ O_{K_{0}}, such that x − x_{0} ∈ 𝔪_{∞}. Then x − x_{0} is in the image of Frobenius by the above. Since K_{0} has a perfect residue the same is true of x_{0}, and so we see that it is also true of x. □

(4.4.4) We set R_{∞} = \underleftarrow{lim}O_{K_{∞}}/p, where the transition maps are given by Frobenius. R_{∞} is a perfect ring, and we set W_{∞} = W (R_{∞}).

Recall that R = \underleftarrow{lim}OK̄ is equipped with a valuation v_{R}, defined as follows: if x ∈ R, x ≠ 0, then x corresponds to a sequence {x_{i}}_{i∈ℕ}, with x_{i} ∈ OK̄/p. We set v_{R}(x) = p^{−i}v_{OK̄/p}(x_{i}) for i >> 0. This is clearly independent of the choice of i >> 0. There is an obvious inclusion R_{∞} ⊂ R, and we also denote by v_{R} the induced valuation on R_{∞}.

As OK̂_{∞} is a complete ring, the universal mapping property of the Witt vectors shows that the projection W_{∞} → R_{∞} → O_{K_{∞}}/p onto the first factor lifts to a map θ_{∞} : W_{∞} → OK̂_{∞}, which extends to the usual map θ : W (R) → O_{ℂ_{p}}. We have

Proposition (4.4.5). The kernel of θ_{∞} is a principal ideal

Proof. One sees, as in [Fo,5.1.2], that an element x ∈ ker(θ_{∞}), is a generator of the kernel if and only if its image x̄ in R_{∞} satisfies v_{R}(x̄) = 1. Thus we have to show that such elements exist.

Assume first that p is odd. Following [Fo, 5.1.2] if x ∈ W (R), we set x′ = ϕ^{−1}(x). Let ζ_{p−1} ∈ ℤ_{p}^{∗} be a primitive p − 1^{st} roots of unity, and set η = 1 + [ϵ′] + … [ϵ′]^{p−1}, and ξ = 1 + ∑_{i=1}^{p−1}[ϵ′]^{ζ_{p−1}^{i}}, where ϵ ∈ R is the element constructed in (2.2.2). According to [Fo 5.1.2] the element η (which is denoted ξ there) satisfies θ(η) = 0, and v_{R}(η̄) = 1, so it is a generator of ker(θ). We also have v_{R}(ϵ − 1) = p/(p − 1) (loc cit.).

Now if i is an integer not divisible by p denote by [i] ∈ ℤ_{p}^{∗} the Teichmüller lift of its image in 𝔽_{p}^{∗}. Then [i] = i + pr_{i} for some r_{i} ∈ ℤ_{p}. We compute

ξ̄− η̄ = ∑_{i=1}^{p−1}(ϵ^{′[i]} − ϵ^{′i}) = ∑_{i=1}^{p−1} ϵ^{′i}(ϵ^{′pr_{i}} − 1) = ∑_{i=1}^{p−1} ϵ^{′i}(ϵ^{r_{i}} − 1).

Thus v_{R}(ξ̄− η̄) ≥ p/(p − 1), whence we have v_{R}(ξ̄) = v_{R}(η̄) = 1.

Next we compute

θ_{∞}(ξ) = 1 + ∑_{i=1}^{p−1}(ϵ^{(1)})^{[i]} = ∑_{i=0}^{p−1}(ϵ^{(1)})^{i} = 0.

Finally, we must show that show that ξ ∈ W_{∞}. Let us first see that ξ is invariant under the action of H_{K}. It is clear that ξ is invariant under the group Gal(K̄/K′_{∞}), so if μ_{p} ⊂ K, then we are done. Otherwise, the group Gal(K′_{∞}/K_{∞}) ˜ ⟶μ_{p−1} has a generator which acts on [ϵ′] by [ϵ′] ↦ [ϵ′]^{ζ_{p−1}}, so we see directly from the formula for ξ that H_{K} fixes ξ. The proposition now follows from the equality W (R)^{H_{K}} = W_{∞}, whose proof we postpone.

If p = 2, a similar analysis applies to ξ = [ϵ′] + [ϵ′]^{−1}, where ϵ ∈ R corresponds to a coherent system of p-power roots of unity {ϵ^{(i)}}_{i∈ℕ^{+}}, with ϵ^{(0)} = −1. We leave the details to the reader.

Note that ξ is also a generator of ker(θ) ⊂ W (R).

(4.4.6) We denote by A_{∞} the p-adic completion of the divided power envelope E_{∞} of W_{∞} relative to the ideal ker(θ_{∞}). Similarly, let A_{cris} denote the p-adic completion of the divided power envelope E_{cris} of W (R) relative to the ideal ker(θ).

By (4.4.5) we have E_{cris} = W (R)[{ξ^{n}/n!}_{n≥1}], and E_{∞} = W_{∞}[{ξ^{n}/n!}_{n≥1}]. Note also that, just as for A_{cris}, A_{∞} is equipped with a Frobenius, which continuously extends the Frobenius on W_{∞}.

The we postpone the proof of the following fact: A_{cris}^{H_{K}} = A_{∞}. Assuming this, we can prove that some power of t lies in A_{∞}. To see this, first suppose that p ≠ 2. In this case the image of H_{K} under the cyclotomic character consists of the Teichmüller representatives in ℤ_{p}^{∗}. As these are p − 1^{st} roots of unity, we see that t^{p−1} is fixed by H_{K}, and so we conclude that t^{p−1} ∈ A_{∞}. Similarly, if p = 2, then we conclude that t^{2} ∈ A_{∞}.

Granted this result, we set B_{∞}^{+} = A_{∞}[1/p], and define B_{∞} = B_{∞}^{+}[1/t^{p−1}] for p odd and B_{∞} = B_{∞}^{+}[1/t^{2}] for p = 2.

We equip the rings R_{∞}, W (R_{∞}), A_{∞}, B_{∞}^{+}, and B_{∞} with topologies in a manner which is completely analogous to the constructions of §4.2. By the functoriality of all our constructions these rings are all equipped with a Γ_{K} action making them into continuous Γ_{K} modules. Furthermore, from the construction we see that there is an inclusion B_{∞} ↪ B_{cris}, compatible with the Galois and Frobenius action. We may give B_{∞} ⊗_{K_{0}} K ⊂ B_{cris} ⊗_{K_{0}} K the filtration induced by this inclusion. (The filtration may also be defined by defining an “∞” analogue of B_{dR} and embedding B_{∞} ⊗_{K_{0}} K into it.) (4.4.7) We are going to reduce the calculation of the Galois cohomology of B_{cris} (respectively B_{st}) to the calculation of the cohomology of Γ_{K} with coefficients in B_{∞} (respectively B_{st}^{H_{K}}). (Unfortunately, as B_{st} = B_{cris}[u], and u is not fixed by H_{K}, one cannot expect a simple description of B_{st}^{H_{K}} , although we do prove later that if B_{n} = ⊕_{i=0}^{n}B_{cris}u^{i}, then B_{n}^{H_{K}} is a successive extension of copies of B_{∞}, so that B_{st}^{H_{K}} is the union of these.)

The key technical result is the following

Proposition (4.4.8). We have

H^{i}(H_{K}, B_{cris}) = \{ \begin{matrix}B_{∞} & i = 0 \\ 0 & i > 0. \end{matrix}

H^{i}(H_{K}, B_{st}) = 0 i > 0.

(4.4.9) The proof of (4.4.8) proceeds by tracing through the construction of B_{cris} and B_{st} and proving an analogous result for each of the intermediate rings that arise. (H_{K} acts on each of these.)

To begin with recall that an O_{K_{∞}} module is called almost zero if it is killed by the maximal ideal of O_{K_{∞}}. (This idea is due to Faltings and was already used in the proof of (4.4.3)). A map M → N of O_{K_{∞}} modules is called an almost isomorphism if its kernel and cokernel are almost zero. In this case we write M ≈ N. We saw in the proof of (4.4.3) that

(4.4.10)  H^{i}(H_{K}, OK̄/p) ≈ \{ \begin{matrix}O_{K_{∞}}/p & i = 0 \\ 0 & i > 0 \end{matrix}

(4.4.11) We call an R_{∞} module almost zero if it is killed by the ideal of R_{∞} consisting of elements of positive valuation (under v_{R}). A map M → N of R_{∞} modules is called an “almost isomorphism” if its kernel and cokernel are almost zero. In this case we write M ≈ N or N ≈ M (in practice the direction of the underlying map of modules will be obvious). (4.4.12) With this notation we have

(4.4.13)  H^{i}(H_{K}, R) ≈ \{ \begin{matrix}R_{∞} & i = 0 \\ 0 & i > 0. \end{matrix}

To see this note that since OK̄/p carries the discrete topology, the cohomology of H_{K} with coefficients in R is computed by

(4.4.14)  RH_{K}(R) = \underleftarrow{lim}RH_{K}(OK̄/p)

Using (4.4.3) and (4.4.10), we see that for the inverse system in (4.4.14), the transition maps induce almost surjections on cohomology (that is, their cokernels are almost zero). Moreover, they evidently induce surjections on the terms of the complexes in the inverse system. Repeating the usual proof that the Mittag-Leffler condition implies that inverse limits are exact in this “almost” context, one finds that

H^{i}(\underleftarrow{lim}RG_{K}(OK̄/p)) ≈ \underleftarrow{lim}H^{i}(RG_{K}(OK̄/p))

(almost isomorphisms of R_{∞} modules), and by (4.4.10) the last term is almost isomorphic to R_{∞} or 0, depending on whether i = 0 or i > 0. (4.4.15) We say that a W_{∞} module M is “almost zero” if it is killed by the ideal generated by the x ∈ W_{∞} such that the image x̄ ∈ R_{∞} of x satisfies v_{R}(x̄) > 0. We define almost isomorphisms, and adopt the notation ≈, in the same way as above. Note that, in particular, such maps become isomorphisms after inverting p.

With this notation we have

(4.4.16.)  H^{i}(H_{K}, W (R)) ≈ \{ \begin{matrix}W_{∞} & i = 0 \\ 0 & i > 0 \end{matrix}

Moreover we claim that if i = 0 the above is actually an equality. To prove (4.4.16) we prove the analogous statement for each W_{n}(R) : For each non-negative integer n, we have

(4.4.17.)  H^{i}(H_{K}, W_{n}(R)) ≈ \{ \begin{matrix}W_{n}(R_{∞}) & i = 0 \\ 0 & i > 0 \end{matrix}

Indeed, when n = 0 this is just (4.4.13), and in general it follows immediately by a dévissage argument, using (4.1.2) and (4.1.4). (4.2.16) now follows in the same way as we proved (4.4.13), using the fact that the transition maps in W (R) = \underleftarrow{lim}W_{n}(R) are open, and pro-finitely surjective by (4.1.4).

Let us see that W (R)^{H_{K}} = W_{∞}. Let x ∈ W (R) be fixed by H_{K}. Then (4.4.16) shows that px ∈ W_{∞}. Thus, the Witt vector expansion of x must have all terms in R_{∞}, whence x ∈ W_{∞}.

This provides the last fact needed in the proof of (4.4.5). Note that we have not yet used the fact that ξ ∈ W_{∞}. Thus the proof of (4.4.5) is now also complete. (4.4.18) We have

(4.4.19)  H^{i}(H_{K}, A_{cris}) ≈ \{ \begin{matrix}A_{∞} & i = 0 \\ 0 & i > 0 \end{matrix}

Moreover, if i = 0, then the above is actually an equality. Here ≈ means as W_{∞} modules. To prove (4.4.19), we consider the analogous statement for the ring E_{n,r} introduce in (4.2.4). Fix n, and assume that r ≥ n. As (p, ξ) form a regular sequence in W (R), we have an exact sequence

0 → W_{n}(R)[x] ⟶^{−ξ^{p^{r}}} W_{n}(R)[x] ⟶^{x↦ξ^{p^{r}} /(p^{r})!} E_{n,r} → 0

Taking H_{K} cohomology, applying 4.2.17, yields

(4.4.20)  H^{i}(H_{K}, E_{n,r}) ≈ \{ \begin{matrix}E_{n,r,∞} & i = 0 \\ 0 & i > 0 \end{matrix}

where E_{n,r,∞} = W_{n}(R_{∞})[ξ^{p^{r}}/(p^{r})!]. Passing to the direct limit over r, we obtain

(4.4.21)  H^{i}(H_{K}, A_{cris}/p^{n}A_{cris}) ≈ \{ \begin{matrix}A_{∞}/p^{n}A_{∞} & i = 0 \\ 0 & i > 0 \end{matrix} .

Now (4.4.19) follows by passing to the inverse limit in 4.4.21.

As taking Galois invariants commutes with direct and inverse limits, the above constructions, together with the fact that W (R)^{H_{K}} = W_{∞}, show that A_{cris}^{H_{K}} = A_{∞}. (Use the fact that p · H^{1}(H_{K}, E_{cris}) = 0.)

Note that we have not yet used the result that t^{p−1} ∈ A_{∞}, and so we have also completed the proof of this fact. (4.4.22) Finally we come to the proof of Proposition 4.4.8. Since almost isomorphism of W (R_{∞}) modules become isomorphisms after inverting p, the formula for H^{i}(G_{K}, B_{cris}) in (4.4.8) follows immediately from (4.4.19) by inverting p and t^{p−1} ∈ A_{∞} (or t^{2} if p = 2) and using the fact that cohomology commutes with direct limits.

Now recall that B_{st} = B_{cris}[u], and that g ∈ G_{K} acts on u by g(u) = c_{g}t + u, for some c_{g} ∈ ℤ_{p}. It follows that if we set B_{n} = ⊕_{i=0}^{n}u^{n}B_{cris}, for each non-negative integer n, then as G_{K} modules, each B_{n} is a successive extension of copies of B_{cris}. By dévissage, we obtain for i > 0,

H^{i}(H_{K}, B_{st}) = \underrightarrow{lim}_{n}H^{i}(H_{K}, B_{n}) = 0.

Corollary 4.4.23. We have isomorphisms

H^{i}(B_{∞} →^{γ−1} B_{∞}) ˜ ⟶H^{i}(Γ_{K}, B_{∞}) ˜ ⟶H^{i}(G_{K}, B_{cris})

and

H^{i}(B_{st}^{H_{K}} →^{γ−1} B_{∞}^{H_{K}}) ˜ ⟶H^{i}(Γ_{K}, (B_{st})^{H_{K}}) ˜ ⟶H^{i}(G_{K}, B_{st}),

In particular we have H^{i}(G_{K}, B_{cris}) = H^{i}(G_{K}, B_{st}) = 0 if i > 1.

Proof. Fix a positive integer j, divisible by p − 1 of p is odd, and by 2, if p = 2. Using the Hochschild-Serre spectral sequence for discrete G_{K} modules, (4.4.8) and (4.4.19) we see that for each positive integer n, the cohomology of the cone of the composite

RΓ_{K}(t^{−j}A_{∞}[1/p]/t^{−j}p^{n}A_{∞}) → RΓ_{K}(RH_{K}(t^{−j}B_{cris}^{+}/t^{−j}p^{n}A_{cris}))^{T ot}

˜ ⟶RG_{K}(t^{−j}B_{cris}^{+}/t^{−j}p^{n}A_{cris})

is killed by p. Denote by C_{n}^{•} the cokernel of the above, injective map of complexes. Passing to the inverse limit with n, and using (4.1.4), we get an exact triangle

RΓ_{K}(t^{−j}A_{∞}[1/p]) → RG_{K}(t^{−j}B_{cris}^{+}) → \underleftarrow{lim}C_{n}^{•}.

Now the transition maps in the inverse system {C_{n}^{•}}_{n∈ℕ} are surjective on the level of complexes, and the cohomology of all the terms is killed by p. On the other hand \underleftarrow{lim}C_{n}^{•} is a complex of ℚ_{p} vector spaces, hence it must by acyclic. We conclude that there are isomorphisms

H^{i}(t^{−j}A_{∞}[1/p] →^{1−γ} t^{−j}A_{∞}[1/p]) ˜ ⟶H^{i}(Γ_{K}, t^{−j}A_{∞}[1/p]) ˜ ⟶H^{i}(G_{K}, t^{−j}B_{cris}^{+}),

where the first isomorphism follows from (4.1.5). Passing to the limit over j, gives the first two isomorphisms of the corollary. The proof of the second two is similar. □

Corollary (4.4.24). With the notation of (4.4.22), each B_{n}^{H_{K}} is a successive extension of copies of B_{∞}, so that B_{st}^{H_{K}} is the union of these.

Proof. We saw in (4.4.22) that B_{n} is a successive extension of copies of B_{cris}. Thus the result follows from (4.4.8) by taking H_{K} invariants. □

4.5 Hyodo’s Theorem. In this section we prove an unpublished theorem of Hyodo, which says that if K has finite residue field, an extension of semi-stable representations which is de Rham is semi-stable. In particular, it follows that an extension of potentially semi-stable representations which is de Rham is potentially semi-stable.

The key ingredients are the exact sequences discussed in §4.3, the results of §2, and some calculations due to Bloch and Kato [BK].

Throughout this section we assume that K has finite residue field unless otherwise stated. At the end we discuss how one might try to prove the theorem without making this assumption. (4.5.1) We recall some definitions from [BK]. Let V be a de Rham representation. Then Bloch-Kato define three subgroups of H^{1}(G_{K}, V).

H_{e}^{1}(G_{K}, V) = ker(H^{1}(G_{K}, V) ⟶ H^{1}(G_{K}, V ⊗_{ℚ_{p}} B_{cris}^{ϕ=1}))

H_{f}^{1}(G_{K}, V) = ker(H^{1}(G_{K}, V) ⟶ H^{1}(G_{K}, V ⊗_{ℚ_{p}} B_{cris}))
H_{g}^{1}(G_{K}, V) = ker(H^{1}(G_{K}, V) ⟶ H^{1}(G_{K}, V ⊗_{ℚ_{p}} B_{dR}))

To this we add

H_{st}^{1}(G_{K}, V) = ker(H^{1}(G_{K}, V) ⟶ H^{1}(G_{K}, V ⊗_{ℚ_{p}} B_{st}))

The significance of these subspaces is given by

Proposition (4.5.2). If V is crystalline (respectively semi-stable, respectively de Rham) then H_{f}^{1}(G_{K}, V) (respectively H_{st}^{1}(G_{K}, V), respectively H_{g}^{1}(G_{K}, V)) classifies extensions of ℚ_{p} by V which are crystalline (respectively semi-stable, respectively de Rham).

Proof. The results of §4.3 imply the proposition in the crystalline and the semistable cases. For B_{dR} we have the following argument (which in fact also works for the other cases.) Suppose V is de Rham, and let E be an extension of ℚ_{p} by V. Then

(4.5.3)  0 → V ⊗_{ℚ_{p}} B_{dR} → E ⊗_{ℚ_{p}} B_{dR} → B_{dR} → 0

is an extension of B_{dR} modules equipped with a semi-linear G_{K} action. It splits if and only if 1 ∈ B_{dR} lifts to a Galois invariant element of E ⊗_{ℚ_{p}} B_{dR}; that is, if and only if the sequence obtained by pulling back (4.5.3) by the inclusion ℚ_{p} → B_{dR} splits. However this latter sequence is the image of [E] in H^{1}(G_{K}, V ⊗_{ℚ_{p}} B_{dR}). □

Theorem (4.5.4). (Hyodo) Let V, W be semi-stable representations of G_{K}. If an extension of W by V is de Rham, then it is semi-stable.

Proof. As de Rham and semi-stable representations form Tannakian categories, replacing V by V ⊗_{ℚ_{p}} W^{∗} allows us to assume that W = ℚ_{p}. In this case (4.5.2) shows that we have to prove that the natural inclusion H_{st}^{1}(G_{K}, V) ⊂ H_{g}^{1}(G_{K}, V) is an equality. We will show that both sides have the same dimension.

Recall that H^{1}(G_{K}, V) and H^{1}(G_{K}, V^{∗}(1)) are dual (Tate duality). By [BK, 3.8] H_{g}^{1}(G_{K}, V) and H_{e}^{1}(G_{K}, V^{∗}(1)) are exact annihilators under the duality pairing. Thus it is enough to show that

dim_{ℚ_{p}}H_{st}^{1}(G_{K}, V) + dim_{ℚ_{p}}H_{e}^{1}(G_{K}, V^{∗}(1)) = dim_{ℚ_{p}}H^{1}(G_{K}, V).

Write M = (V ⊗_{ℚ_{p}} B_{st})^{G_{K}} . Using [BK, 3.8.4] we compute, for any semi-stable representation V,

dim_{ℚ_{p}}H_{e}^{1}(G_{K}, V) = dim_{ℚ_{p}}H_{f}^{1}(G_{K}, V) − dim_{ℚ_{p}}M^{N=0}/(ϕ − 1)(M^{N=0}).

Computing dim_{ℚ_{p}}H_{f}^{1}(G_{K}, V) = dim_{ℚ_{p}}H^{1}𝒦_{cris}^{•}(M)) via Euler characteristics, we find:

(4.5.5) dim_{ℚ_{p}}H^{1}(𝒦_{cris}^{•}(M)) = −χ(𝒦_{cris}^{•}(M)) + dim_{ℚ_{p}}H^{0}(𝒦_{cris}^{•}(M))

= dim_{ℚ_{p}}M_{K}/F il^{0}M_{K} + dim_{ℚ_{p}}H^{0}(G_{K}, V) − dim_{ℚ_{p}}M^{N=0}/(ϕ − 1)(M^{N=0})

= dim_{ℚ_{p}}M_{K}/F il^{0}M_{K} + dim_{ℚ_{p}}H^{0}(G_{K}, V) − dim_{ℚ_{p}}M^{ϕ=1,N=0}.

We can also compute the dimension of H_{st}^{1}(G_{K}, V) = H^{1}(𝒦_{st}^{•}(M)) using Euler characteristics:

(4.5.6) dim_{ℚ_{p}}H_{st}^{1}(G_{K}, M) = −χ(𝒦_{st}^{•}(M)) + dim_{ℚ_{p}}H^{0}(𝒦_{st}^{•}(M)) + dim_{ℚ_{p}}H^{2}(𝒦_{st}^{•}(M))

= dim_{ℚ_{p}}M_{K}/F il^{0}M_{K} + dim_{ℚ_{p}}H^{0}(G_{K}, V) + dim_{ℚ_{p}}M/((1 − pϕ)M + NM).

Note that

(4.5.7) M^{∗}(1)^{ϕ=1,N=0} ˜ ⟶Hom_{K_{0}[ϕ,N]}(K_{0}, M^{∗}(1))

˜ ⟶Hom_{K_{0}[ϕ,N]}(M(−1), K_{0}) ˜ ⟶Hom_{K_{0}}(M/((1 − pϕ)M + NM), K_{0}),

and

(4.5.8)  dim_{ℚ_{p}}M_{K}/F il^{0}M_{K} + dim_{ℚ_{p}}M^{∗}(1)_{K}/F il^{0}M^{∗}(1)_{K} = dim_{ℚ_{p}}M_{K}.

Now in (4.5.5) replace V by V^{∗}(1) and M by M^{∗}(1), and add the resulting expression to (4.5.6). Using (4.5.7), (4.5.8) we get

dim_{ℚ_{p}}H_{st}^{1}(G_{K}, V) + dim_{ℚ_{p}}H_{e}^{1}(G_{K}, V^{∗}(1))

= dim_{ℚ_{p}}M_{K} + dim_{ℚ_{p}}H^{0}(G_{K}, V) + dim_{ℚ_{p}}H^{0}(G_{K}, V^{∗}(1))
= [K : ℚ_{p}]dim_{ℚ_{p}}V + dim_{ℚ_{p}}H^{0}(G_{K}, V) + dim_{ℚ_{p}}H^{0}(G_{K}, V^{∗}(1))
= dim_{ℚ_{p}}H^{1}(G_{K}, V),

where the last equality follows from Tate duality. □

Corollary (4.5.9). (Hyodo) An extension of potentially semi-stable representations which is de Rham is potentially semi-stable.

(4.5.10) Let us make some final remarks about how one might try to prove the analogue of Theorem (4.5.4) when we do not assume that K has finite residue field.

Take V semi-stable, and set M = (V ⊗_{ℚ_{p}} B_{st})^{G_{K}} . A simple diagram chase, using the exact sequence (4.3.9)(1) shows that there is a natural map

(4.5.11) ker(F il^{0}(M ⊗_{K_{0}} H^{1}(G_{K}, B_{st}))^{ϕ=1,N=0}→Ext_{M_{st,K}^{ad}}^{2} (K_{0}, M))

→H^{1}(G_{K}, V ⊗_{ℚ_{p}} B_{dR})),

and that an arbitrary extension of ℚ_{p} by V which is de Rham is semi-stable if and only if this map is an injection.

Using the arguments of §4.3 it is easy to see that H^{1}(G_{K}, V ⊗_{ℚ_{p}} B_{dR}) = H^{1}(G_{K}, B_{dR}) ⊗_{K} M_{K}. The group H^{1}(G_{K}, B_{dR}) is much simpler than the analogues with B_{cris} and B_{st}. Indeed a result of Tate shows that H^{i}(G_{K}, ℂ_{p}(j)) = 0 unless j = 0 and i = 0 or 1, while H^{1}(G_{K}, ℂ_{p}) = K. A simple dévissage argument, together with the fact that B_{dR}^{+} is complete with respect to the topology given by the kernel of the canonical map B_{dR}^{+} → ℂ_{p}, shows that H^{1}(G_{K}, B_{dR}) = K, with the trivial filtration. Thus, the target of the map (4.5.11) is very simple.

So far we have not been able to gain a sufficiently good understanding of the source of the map (4.5.11). One can try to analyze it by using the results of §4.4:

H^{1}(G_{K}, B_{st}) ˜ ⟶H^{1}(Γ_{K}, B_{st}^{H_{K}}) = coker(B_{st}^{H_{K}} →^{γ−1} B_{st}^{H_{K}}).

Denote by J the source of the map (4.5.11). By considering the complex

𝒦_{st}^{•}(B_{st}^{H_{K}} ⊗_{K_{0}} M →^{γ−1} B_{st}^{H_{K}} ⊗_{K_{0}} M → H^{1}(Γ_{K}, B_{st}^{H_{K}})) ⊗_{K_{0}} M

one obtains an exact sequence

0 → V^{H_{K}} /(1 − γ)V^{H_{K}} → J → H^{1}(H_{K}, V)^{Γ_{K}} /H^{1}(𝒦_{st}^{•}(M)).

Unfortunately the last term seems somewhat mysterious at the moment.

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