Galois Representations and Lubin-Tate Groups

Using Lubin-Tate groups, we develop a variant of Fontaine's theory of (ϕ,)-modules, and we use it to give a description of the Galois stable lattices inside certain crystalline representations. K of K. Fontaine's theory starts with an infinite extension K∞/K which is required to have certain ramification properties. Miraculously, these properties ensure that GK1 = Gal( ¯ K/K∞) can be identified with the absolute Galois group of a local field of equal characteristic p, X(K). It is well known that representations of such a Galois group on finite dimensional Fp-vector spaces can be classified rather concretely in terms of finite dimensional vector spaces over X(K) equipped with anFrobenius. If K∞/K is Galois, then = Gal(K∞/K) acts naturally on X(K), and one obtains a classification of GK- representations on finite dimensional Fp-vector spaces by adding a semi-linear action of to the ´ ϕ-modules over X(K). To obtain a classification of GK-representations on finite Zp-modules, one needs to lift the action of ϕ and on X(K) to commuting operators on a Cohen ring for X(K). This is probably not always possible, but can be done when K∞ is the p-cyclotomic extension of K. Much of the work on Fontaine's theory by Berger, Colmez, Wach and others has focused on this case. In this paper we focus on the case when K∞ is generated by the p-power torsion points of a Lubin-Tate group for a finite extension L/Qp contained in K. As an application we obtain a description of the GK-stable lattices in a certain class of crystalline GK-representations. This is possible using the p-cyclotomic theory only when K is an unramified extension of some Qp(� pn).


Introduction
In his Grothendieck Festschrift paper [Fo 1], Fontaine introduced a new way to classify local Galois representation, using the theory of so called (ϕ, Γ)modules.To recall this, let k be a perfect field of characteristic p, K 0 = Fr W (k) and K/K 0 a finite, totally ramified extension.Fix an algebraic closure K of K. Fontaine's theory starts with an infinite extension K ∞ /K which is required to have certain ramification properties.Miraculously, these properties ensure that G K∞ = Gal( K/K ∞ ) can be identified with the absolute Galois group of a local field of equal characteristic p, X(K).It is well known that representations of such a Galois group on finite dimensional F p -vector spaces can be classified rather concretely in terms of finite dimensional vector spaces over X(K) equipped with an étale Frobenius.If K ∞ /K is Galois, then Γ = Gal(K ∞ /K) acts naturally on X(K), and one obtains a classification of G Krepresentations on finite dimensional F p -vector spaces by adding a semi-linear action of Γ to the étale ϕ-modules over X(K).
To obtain a classification of G K -representations on finite Z p -modules, one needs to lift the action of ϕ and Γ on X(K) to commuting operators on a Cohen ring for X(K).This is probably not always possible, but can be done when K ∞ is the p-cyclotomic extension of K.Much of the work on Fontaine's theory by Berger, Colmez, Wach and others has focused on this case.In this paper we focus on the case when K ∞ is generated by the p-power torsion points of a Lubin-Tate group for a finite extension L/Q p contained in K.As an application we obtain a description of the G K -stable lattices in a certain class of crystalline G K -representations.This is possible using the p-cyclotomic theory only when K is an unramified extension of some Q p (µ p n ).
More precisely, let G be a Lubin-Tate group over O L , write k L for the residue field of L, fix a uniformizer π L of L, and write R = lim ← − O K /p, where the transition maps in the inverse limit are given by Frobenius.The action of G K on the Tate module T G of G gives rise to a character χ : Γ → O × L .It turns out that, using the periods of T G one can construct a subring O E ⊂ W (Fr R) ⊗ W (kL) L which is naturally a Cohen ring for X(K).The action of O × L ⊂ O L on G gives rise to a natural lifting of the action of Γ to O E (via χ), while the action of π L on G allows one to lift the q-Frobenius ϕ q = ϕ r to O E , where q = |k L |.This allows one to classify G K -representations on finite O L -modules in terms of étale (ϕ q , Γ)-modules (see Theorem 1.6 below), and is explained in §1 of the paper.At least some part of this construction was certainly known to experts.The construction of the periods involved is in Colmez's paper [Col 1], and some of the ideas go back to Coleman [Co].This material is also closely related to the subject of Fourquaux's thesis [Fou,§1.4].
In §2,3 we use this classification to give a classification of Galois stable lattices in certain crystalline G K -representations, assuming that K ⊂ K 0 • L ∞ where L ∞ /L is the field generated by the torsion points of G. To explain the classification, assume for simplicity that K = K 0 • L, and let the category of finite free S L -modules equipped with a continuous semi-linear action of Γ which induces the trivial action of M/uM, and an isomorphism with the action of Γ. Inside this category is a subcategory Mod To describe the crystalline representations we allow, consider any crystalline G K -representation on an L-vector space V. Then where m runs over the maximal ideals of K ⊗ Qp L. We say that V is L-crystalline if the filtration on D dR (V ) m is trivial, unless m is the kernel of the natural map K ⊗ Qp L ֒→ K corresponding to the inclusion L → K.One of our main results is then the following Theorem (0.1).There is an exact equivalence of ⊗-categories between Mod ϕq,Γ,an /SL and the category of The theorem is a generalization of the classification of G K -stable lattices in crystalline representations in terms of Wach lattices due to Wach [Wa], Colmez [Col 2] and Berger [Be 3], when K ∞ is the p-cyclotomic extension and K is unramified.
It is also analogous to the classification G K∞ -stable lattices, obtained in [Ki] in the case when K ∞ is obtained from K by adjoining the p-power roots of a uniformizer.The advantage of Theorem (0.1) is that it applies without restriction on the ramification of K, and gives a precise description of G K -stable lattices.Unfortunately, it applies only to a rather special kind of crystalline G K -representation.It seems likely that in order to obtain a classification valid for any crystalline G K -representation one needs to consider higher dimensional subrings of W (Fr R), constructed using the periods of all the conjugates of G.

Acknowledgment:
The results presented here were to be the subject of the Ph.D thesis of Wei Ren.They were written up by the first author after Ren's premature and tragic death.We would like to thank Barry Mazur and Jean-Marc Fontaine for useful conversations on some of the material presented here.Finally we thank the referee for a careful reading of the paper and some useful remarks.§1 Étale (ϕ q , Γ)-modules (1.1) Throughout the paper we fix a perfect field k, of characteristic p > 0. Let W = W (k), K 0 = W [1/p] and K/K 0 a finite totally ramified extension with ring of integers O K , and uniformizer π.We also fix an algebraic closure K of K with ring of integers O K , and set For n ≥ 1, let K n ⊂ K denote the subfield generated by the π n L -torsion points of G.We set K ∞ = ∪ n K n and we write Γ = Gal(K ∞ /K) and G K∞ = Gal( K/K ∞ ).Let T G denote the p-adic Tate module of G. Then T G is a free O L -module of rank 1, and the action of Γ induces a faithful character χ : Γ → O × L .We let R = lim ← − O K /p with the transition maps being given by Frobenius.We may also identify R with lim ← − O K /π L with the transition map being given by the q-Frobenius ϕ r .Evaluation of X at π L -torsion points then induces a map ι :

Lemma (1.2).
There is a unique map { } : R → W (R) L such that {x} is a lifting of x, and ϕ r ({x}) = [π L ](x).Moreover { } respects the action of G K , and for v ∈ T G we have Documenta Mathematica 14 (2009) 441-461 (2) The action of G K on {ι(T G)} factors through Γ and for Proof.The existence and uniqueness of { } is [Col 1, Lem. 9.3].The map {x} is given by where x ∈ W (R) L is any lifting of x.That { } respects the action of G K follows by functoriality.In particular, the action of G K on {ι(T G)} factors through Γ.
For (1) note that Since Proof.This is a consequence of the theory of norm fields [Wi].Since Γ is a p-adic Lie group the theory of loc.cit applies [Wi,1.2.2].For any finite where the maps in the inverse limit are given by the norm.We set X K ( K) = ∪ F X K (F ) where the limit runs over finite extensions F/K in K. Then X K (F ) has the structure of a local field of characteristic p, which is a finite separable extension of X K (K), X K ( K) is a separable closure of X K (K), and the functor X K induces an isomorphism [Wi,3.2 On the other hand, there is a natural embedding To see this explicitly note that one has well defined maps of rings where the transition maps in the first two inverse limits are given by the norm, and the final inverse limit by x → x q .The image of (1.4.1) is easily seen to be k The lemma follows.
(1.5) Note that the above proof shows that the map ι induces a map where the transition maps are given by the norm.This is Coleman's map [Co, Thm.A].
We will write ϕ q for the q-Frobenius ϕ r (for example on the ring W (Fr R)). ) we set

Now denote by Mod
Theorem and taking G K∞ invariants of both sides induces a map M OE (V (M )) → M. To show that this map is an isomorphism one reduces to the case of objects killed by p, using the exactness proved above.In this case, (1.6.1) is an isomorphism, because étale locally M is spanned by its ϕ q -invariants.Similarly, one obtains an isomorphism V (M OE (V )) → M for V in Rep tor GK ∞ , using dévissage, Hilbert theorem 90, and (1.4).§2 (ϕ q , Γ)-modules and weakly admissible modules (2.1)We keep the notation of the previous section, so in particular we write K 0,L for the field K 0 ⊗ L0 L = Fr W L ⊂ K.In order not to overload notation we will write v n for (v n ) * (X) ∈ K.We now also assume that As in [Ki,1.1.1],denote by D[0, 1) the rigid analytic disk of radius 1, over K 0,L , and denote by u the co-ordinate on D[0, 1).For I ⊂ [0, 1) an interval, denote by D(I) ⊂ D[0, 1) the open subspace whose K points consist of x ∈ K with |x| ∈ I.We denote by O I the ring of rigid analytic functions on D(I), and we write O = O [0,1) .We will often use the fact that D[0, 1) is a p-adic Stein space, so that a coherent sheaf on D[0, 1) can be recovered from its global sections.In particular, we may regard a finite free O-module as a coherent sheaf on D[0, 1).We regard S L ⊂ O by u → u.The action of ϕ q and Γ on S L have a unique continuous extension to O, regarded with its canonical Frechet topology.equipped with a continuous semi-linear action of Γ such that Γ acts trivially on M/uM and the ϕ q -semi-linear map 1 ⊗ ϕ q : M → M[1/Q] commutes with Γ.
We now explain how to differentiate the action of Γ on an object in Mod Lemma (2.1.1).The action of Γ on O, defined above, is continuous.In particular, O with its action of Γ and ϕ q is an object of Mod We have to show that, for any r, The lemma follows.
Lemma (2.1.2).Let M be in Mod ϕq,Γ /O .For each r ∈ (0, 1) and γ ∈ Γ sufficiently close to 1 (depending on r) the series induces a well defined operator on M| D[0,r) .This induces a well defined Z plinear map of Lie algebras This shows that logγ is well defined.It follows that the map is well defined for β sufficiently small, and we extend it to all of Lie Γ by Z plinearity.That dΓ O (β) is a derivation and dΓ M (β) is a differential operator over dΓ O (β) follows from a simple computation, as does the fact that dΓ M is a map of Lie algebras.Note that the latter statement just means that the differential operators dΓ M (β) for β ∈ Lie Γ commute.
(2.1.3)We say that M in Mod (2) follows by viewing M as a coherent module on D[0, 1).The fact that N ∇ commutes with ϕ q follows from the fact that ϕ q commutes with the action of Γ.
To see (3), we first compute the derivation  .The construction is analogous to that in [Be 2] and [Ki,1.2].Since many of the proofs from [Ki] go over verbatim, we often only sketch the argument. 3or n ≥ 0, denote by S n the complete local ring at the point x n of D[0, 1), corresponding to u = v m+n .That is, S n is the completion of the localization of O at the maximal ideal generated by and write ϕ q,WL : O → O for the O L -linear automorphism given by applying ϕ r to the coefficients of a series in O.

Given D in Mod
F,ϕq /K0,L and n ≥ 0, we denote by ι n the composite We set

Lemma (2.2.1). For D in Mod
The same argument shows that γ induces an automorphism of S n for n ≥ 0. As ϕ q,WL [χ(γ)] = [χ(γ)], one sees that M(D) is stable by the action of Γ.Finally, this action is O L -analytic, as the action of Γ on O is O L -analytic.
Since N ∇ commutes with ϕ q , the section ϕ q • ξ r • ϕ −1 q is also ∇-parallel, and hence equal to ξ r .Hence ξ r is ϕ q -invariant.Similarly γ • ξ r • γ −1 is ∇-parallel for γ ∈ Γ, so ξ r is Γ-invariant.Now ξ may be constructed from ξ r exactly as in [Ki,1.2.6], by repeatedly pulling ξ r back by ϕ * q and using the isomorphism 2) and (3) also follow exactly as in loc.cit.
(2.2.3) Suppose that M is in Mod ϕq,Γ,an /O .For i an integer denote by Fil i ϕ * q M the preimage of Give the right hand side of (2.2.4) the filtration induced by that on ϕ * q M, and pull this filtration back to D(M) ⊗ K0,L K(v m ).This gives rise to a Γ-stable filtration on D(M) ⊗ K0,L K(v m ), which necessarily descends to a filtration on D(M) K .This gives D(M) the structure of an object in Mod Proof.This is the analogue of [Ki,1.2.12(4)] in our situation, and the proof is identical, so we only sketch it here.Since D(M) K ⊗ S 0 and S 0 ⊗ O ϕ * q (M) induce the same filtration on their common quotient D(M) K , one sees easily that it suffices to check that for all i ∈ Z, We will identify ϕ * q M with its image (1 Thus it suffices to show that for all i ∈ Z, N ∇ induces a bijection on M i , for then The general case follows by descending induction on i and an application of the snake lemma.

Proposition (2.2.6). The functors M and D between Mod
/O , to show these submodules are equal it suffices to check that these two submodules coincide at u = v m .This follows from (2.2.5).(cf.[Ki,1.2.13]).Hence we have a natural isomorphism That M and D are exact follows from (2.2.5).One checks easily that M and D respect ⊗-products (cf.[Ki,1.2.15]).
(2.3) We now apply Kedlaya's slope filtration as in [Be 2] and [Ki,§1.3] to show that an object M of Mod ϕq,Γ,an /O can be descended to S L if and only if D(M) is weakly admissible.4Again, as many of the arguments are identical to those of [Ki] we sometimes only sketch the proofs.Let R = lim r→1 − O (r,1) denote the Robba ring, and and uniformizer π L .The endomorphism ϕ q and the derivation N ∇ of O induce an automorphism and a derivation respectively of R and R b , which we will again denote by ϕ q and N ∇ .

Denote by Mod
We write s i for the slope of the pure slope quotient M i /M i−1 , which is finite free over R. The filtration is functorial for maps in Mod .The slope filtration on M R is induced by a unique filtration on M by saturated, finite free O-submodules.This filtration on M is stable by ϕ q and the action of Γ.
Proof.It is clear that a such filtration on M, if it exists is unique and stable by ϕ q .The functoriality of the slope filtration on M R implies that it is stable by Γ, and hence so is the filtration on M. It remains to show the existence of such a filtration. As L ]λ, for any integer s, the slope filtration on λ −s M R is given by λ −s M R,i , and the slopes of ]π L has a unique, simple zero on D[0, 1) at x 0 , we may replace M by λ −s M for s sufficiently large, and assume that ϕ q induces a map ϕ * q M → M. We first show that the slope filtration is induced by a filtration on M| D(0,1) by saturated O (0,1) -submodules.For some r 0 sufficiently close to 1, the slope filtration on M R is induced by a filtration on M| D(r0,1) by saturated O (r0,1)submodules.Let r 0 > r 1 > . . .be a sequence approaching 0, and such that ϕ −1 q (D(r i , 1)) ⊂ D(r i−1 , 1) for i ≥ 1.The same argument as in [Ki,1.3.4]shows that for j ≥ 0, M R,i is induced by a filtration on M| D(rj ,1) by closed, saturated O (rj ,1) -submodules, and hence by a filtration on M| D(0,1) by closed, saturated O (0,1) -submodules.The filtration on M D(0,1) is stable by Γ, by uniqueness, and hence it is stable by N ∇ .Consider the operator ∂ = ∇, −u d du on M.This is well defined in a neighbourhood of 0, and preserves the filtration on M| D(0,1) over this neighbourhood as N ∇ does.Hence the filtration on M| D(0,1) is induced by a filtration on M by closed, saturated O-submodules by [Ki,1.3.5].
(2.3.2) D in Mod F,ϕq /K0,L be 1-dimensional over K 0,L .Choose a basis vector v for D, and set t N,L (D) = v πL (α) where α ∈ K 0,L satisfies ϕ q (v) = αv.We write t H,L (D) for the unique integer i such that gr i D K is non-zero.For D of arbitrary dimension d, we set t N,L (D) = t N,L (∧ d D) and t H,L (D) = t H,L (∧ d D).We will say that D is weakly admissible if the usual conditions of weak admissibility are satisfied with these invariants in place of the usual ones.That is if t H,L (D) = t N,L (D) and t H,L (D ′ ) t N,L (D ′ ) for all ϕ q -stable submodules D ′ ⊂ D. Proof.Suppose first that dim K0,L D = 1.Let v be a basis vector for D, and write ϕ(v) = αv for some α ∈ K × 0,L .From the definition of M(D) one finds that M(D) = λ −tH,L(D) (D ⊗ K0,L O), so

Proposition (2.3.3). Let D be in Mod
This proves the proposition when D has dimension 1.The general case follows from exactly the same argument as in [Ki,1.3.8], using the equivalence (2.2.6) and (2.3.1).
(2.4) Denote by Mod ϕq /SL the category consisting of finite free S L -modules M equipped with an isomorphisms 1 ⊗ ϕ We denote by Mod ϕq,Γ /SL the category whose objects consist of an object of Mod ϕq /SL equipped a semi-linear action of Γ on M which commutes with the action of ϕ q , and such that Γ acts trivially on M/uM.We denote by Mod  Moreover,(3.1.3)implies that there is a perfect pairing ) is an isomorphism, and both sides are free O L -modules of rank rk SL M.Moreover, if ϕ q on M induces a map ϕ * q M → M then the left hand side of (3.2.2) is equal to Hom SL,ϕq (M, S ur L ).Proof.Suppose first that ϕ q induces a map ϕ * q M → M. In this case the proof of the lemma is identical to the proof of [Ki,2.1.4],using [Fo 1,§A.1.2].Next let t L = log G u ∈ O as in the proof of (3.1.3).Then ϕ q (λ −1 t L ) = Q(u)λ −1 t L , and the zeroes of λ −1 t L on D[0, 1) coincide with those of P. Hence λ −1 t L P −1 ∈ S L [1/p] × , and there exists w ∈ S L [1/P ] × such that ϕ q (w) = Q(u)w.Let M(G) = S L equipped with a semi-linear action of ϕ q which takes 1 to Q(u).Then multiplication by w i induces a bijection Hence the lemma for general M in Mod ϕq /SL follows from the case considered above.where the right hand side means maps compatible with filtrations in the sense explained in (3.1).In particular, both sides of the above isomorphism have dimension dim K0,L D over L.

Proposition (3.2.3). Let M be in Mod
Proof.The argument is similar to [Ki,2.1.5] so that D is a finite free K 0 ⊗ Qp L-module.Here we have denoted by The essential image of this functor consists of those objects such that the filtration on ⊕ m =m0 DK is trivial.

Proof. Given D in Mod
one sees that there is a canonical isomorphism D′ ∼ −→ D. In particular this makes D ′ into an object of Mod F,ϕq /K0,L .One checks immediately that these two functors are quasi-inverse.Proof.Since the functor of (3.3.1)respects ⊗-products, it suffices to prove (3.3.3) when D is 1-dimensional over K 0,L .Moreover, as the essential image of the functor in (3.3.1) is stable under subobjects, (3.3.3)implies the claim regarding weak admissibility.

Lemma (3.3.2). Let D be in Mod
Let D be 1-dimensional over K 0,L with basis vector v, and ϕ q (v) = αv for some α ∈ K 0,L .Then for i = 0, . . ., r − 1 the K 0 -vector space ∧ This is a ϕ q -compatible map, such that the composite is obtained from f K by localizing at m 0 .In particular, θ(f ) K is compatible with filtrations.Note also that f can be recovered from θ(f ) : The decomposition allows us to view B cris,L as a direct summand in B cris ⊗ Qp L. Then f is the unique ϕ-linear extension of the ϕ q -linear map on which the Γ-action is O L -analytic.this means that there is O L -linear map of Lie algebras dΓ : Lie Γ → End K (M ⊗ SL K[[u]]), such that the action of an open subgroup of Γ is obtained by exponentiating dΓ.
the ring of integers of L, and k L ⊂ k its residue field.Write O L0 = W (k L ), L 0 = O L0 [1/p], and q = p r = |k L |.For an O L0 -algebra A, it will be convenient to write A L = A ⊗ OL 0 O L .Let G be a Lubin-Tate group over L corresponding to a uniformizer π L ∈ L. Fix a local co-ordinate X on G so that the formal Hopf algebra O G may be identified with O L [[X]].For a ∈ O L we denote by [a] ∈ O L [[X]] = O G the power series giving the endomorphism a of G.
the category of finite free (resp.finite torsion) O E -modules M, equipped with an isomorphism (ϕ q ) of consisting of a module M in Mod ϕq /OE (resp.Mod ϕq,tor /OE ) equipped with a continuous semi-linear action of Γ which commutes with the action of ϕ q .We denote by Rep GK ∞ (resp.Rep tor GK ∞ ) the category of finite free (resp.finite torsion) O L -modules V, equipped with a linear action of G K∞ Similarly, we denote by Rep GK (resp.Rep tor GK ) the category of finite free (resp.finite torsion) O K -modules V, equipped with a linear action of G K .
Proof.Let M 0 ⊂ M be finite free W L -submodule of rank equal to d = rk O M, which spans M. Choosing a basis for M 0 , we may identify M with O d .As in (2.1.1),choose r ∈ (0, 1) and denote by | | r the norm on M D[0,r] = O d D[0,r] induced by the sup norm on O D[0,r] .For any ǫ > 0, and γ sufficiently small we have |γ(m) − m| r ǫ|m| r for m ∈ M 0 and |γ(f

.( 1 )
of O L -analytic objects.One checks easily that this is a ⊗-subcategory, which is stable under taking subobjects and quotients.Lemma (2.1.4).Let M be in Mod ϕq,Γ,an /O For each r ∈ (0, 1) the operator N ∇ =: logγ/logχ(γ) is well defined for γ = 1 sufficiently close to 1, and is independent of γ.(2)The operators in (1) induce a K 0,L -linear map N ∇ : M → M, which is a differential operator over the derivation N ∇ : O → O, and which commutes with ϕ q on M.(3) There is a singular connection ∇ on M with simple poles at the zeroes of [π n L ]/u for n ≥ 1 (that is at the non-trivial π L -power torsion points of G) such that N ∇ = ∇, ∂F ∂Y (u, 0)log G u • d/du , where F (X, Y ) denotes the formal group law of G with respect to X, and ∂F ∂Y (u, 0) ∈ O L [[u]] × .Proof.For γ sufficiently close to 1, we may write γ = exp β with β ∈ Lie Γ.Since β → log(exp β) is O L -linear by assumption.and β → log(χ(exp(β))) is obviously O L -linear, logγ/logχ(γ) is independent of γ.This proves (1) and 0) has constant term 1, and coefficients in O L [[u]], it is a unit O. Now for any M, define ∇(m) for m ∈ M by ∇(m) = ( ∂F ∂Y (u, 0)log G u) −1 N ∇ (m).Since N ∇ on M is a differential operator over the derivation ∂F ∂Y (u, 0)(log G u) df du on O, ∇ is a (singular) connection.A priori ∇(m) has a simple pole at each [π L ]-torsion point of G, however since the action of Γ on M is trivial mod u, the operator N ∇ vanishes mod u, and ∇(m) has no pole at u = 0.This proves (3).
L the category of finite dimensional K 0,L -vector spaces D equipped with an isomorphism ϕ * q D ∼ −→ D and a decreasing, separated filtration on D K = D ⊗ K0,L K, indexed by Z, by K-subspaces.Our next task to to show that there is an exact ⊗-equivalence between Mod F,ϕq /K0,L and Mod ϕq,Γ,an /O That M(D) is a finite free O-module, and the fact that Documenta Mathematica 14 (2009) 441-461 Lemma (2.2.5).Let M be in Mod ϕq,Γ,an /O and D = D(M).Then for all i ∈ Z the map ξ induces an isomorphism j≥0

.
inverse, exact, ⊗-equivalences.Proof.Let D be in Mod F,ϕq /K0,L .From the definition of M(D), there is a natural Γ-equivariant inclusion D ⊂ M(D)[1/Q], which induces an isomorphism of D with D(M(D)) = M(D)/uM(D).Hence the image of this inclusion coincides with ξ(D(M(D))), and one sees from the definitions that the filtration on D K coincides with the one on D(M(D)) K .This produces a natural isomorphism D ∼ −→ D(M(D)).Conversely, let M be in Mod ϕq,Γ,an /O Then both M and M(D(M)) may be identified with O-submodules of D ϕq /R .One of Kedlaya's results about the filtration says that a module of pure slope s has a canonical descent to a module M b in Mod ϕq /R b which has slope s in the sense of Dieudonné-Manin theory (and the valuation on R b normalized so that v(π L ) = 1).For M in Mod ϕ,Γ,an /O we write M R = M ⊗ O R. The operators ϕ q and N ∇ on M induce operators ϕ q and N ∇ on M R .Lemma (2.3.1).Let M be in Mod ϕq,Γ,an O L and M = M(D).Then D is weakly admissible if and only if M R is pure of slope 0.
consisting of objects M such that M R = M ⊗ O R is pure of slope 0. Documenta Mathematica 14 (2009) 441-461 for some b ∈ C cris,L .As 0 the weakly admissible module associated to M by (2.4.4).Then there is a a canonical G Kequivariant bijection Hom SL,ϕq (M, S ur L [1/P ]) ∼ −→ Hom Fil,ϕq (D, B cris,L ) L0 L. We put DK = D ⊗ K0 K. We denote by Mod F,ϕ /K0⊗ Qp L the category of finite free K ⊗ Qp L-modules D equipped with an isomorphism ϕ * ( D) ∼ −→ D and a decreasing separated filtration on DK = D ⊗ K0 K, indexed by Z, by K ⊗ Qp L-submodules.For any D in Mod F,ϕ KO⊗ Qp L we denote by t N ( D) and t H ( D) the usual invariants when D is considered as a filtered ϕ-module over K 0 .For any D in Mod F,ϕ /K0⊗ Qp L we may write DK = ⊕ m ( DK ) m where m runs over the maximal ideals of K ⊗ Qp L. Denote by m 0 the kernel of the natural map K ⊗ Qp L → K. Given D in Mod F,ϕq /K0,L we give D the structure of an object in Mod F,ϕ /K0⊗ Qp L by noting that ( DK ) m0 may be identified with D K , and giving DK the direct sum of the filtration on D K and the trivial filtration on ⊕ m =m0 ( DK ) m .Lemma (3.3.1).The functor D → D induces an fully faithful ⊗-functor Mod there is a natural isomorphismϕ * ( D) = ⊕ r i=1 ϕ i * (D) ∼ −→ ⊕ r−1 i=0 ϕ i * (D) = D,which sends ϕ i * (D) identically to ϕ i * (D) for i = r and maps ϕ r * (D) = ϕ * q (D) to D using the map ϕ q on D. This defines the functor of the lemma.To define a quasi-inverse on the essential image of the functor, let D be in Mod F,ϕ /K0⊗ Qp L be such that the filtration on ⊕ m =m0 DK is trivial, and setD ′ = D ⊗ K0⊗ Qp L (K 0 ⊗ L0 L).There is an isomorphism ϕ * r (D ′ ) ∼ −→ D ′ inducedby ϕ * r ( D) ∼ −→ D. Using the decomposition t N,L (D) = t N ( D) and t H,L (D) = t H ( D) Documenta Mathematica 14 (2009) 441-461 and D is weakly admissible if and only if D is weakly admissible.

→
B cris,L ֒→ B cris ⊗ Qp L. Documenta Mathematica 14 (2009) 441-461 L .Let O E ur ⊂ W (Fr R) L denote the maximal integral, unramified extension of O E .We denote by O b E ur the p-adic completion of O E ur , which is again naturally a subring of W (Fr R) L .We write E, E ur and E ur for the fields of fractions of O E , O E ur and O b E ur respectively.These rings are all stable by ϕ r , and by the action of G (1.6).V and M OE are quasi-inverse equivalences between the exact tensor categories Mod Rep GK ∞ (resp.Rep tor GK ∞ , resp.Rep GK , resp.Rep tor GK )Proof.The argument for this is identical to that in[Fo 1, 1.2.6, 3.4.3].For the convenience of the reader we sketch it: It suffices to prove that V and M induce quasi-inverse, exact tensor equivalences between Mod We first remark that both functors are exact.It suffices to prove this for objects killed by p.For M OE this follows from the fact that for M in Mod extends uniquely to a continuous embedding O ֒→ B + cris,L , where B + cris,L = B + cris ⊗ OL 0 O L , as usual.Lemma (3.2.1).Let M be in Mod ϕq /SL .The natural map