Multivariable (ϕ, Γ)-modules and products of Galois groups

Multivariable (ϕ, Γ)-modules and products of Galois groups

Gergely Zábrádi

14th March 2016

Abstract

We show that the category of continuous representations of the dth direct power of the absolute Galois group of Qp on finite dimensional Fp-vector spaces (resp. finitely generated Zp-modules, resp. finite dimensional Qp-vector spaces) is equivalent to the category of étale(ϕ,Γ)-modules over a d-variable Laurent-series ring overFp (resp. over Zp, resp. over Qp).

1 Introduction

This note serves as a complement to the work [11] where we relate multivariable (ϕ,Γ)- modules to smooth modulopⁿ representations of a split reductive groupGover Qp. The goal here is to show that the category ofd-variable(ϕ,Γ)-modules is equivalent to the category of representations of thedth direct power of the absolute Galois group of Qp.

LetK be a finite extension of Qp with ring of integers O_K, prime element $, and residue field κ. For a finite set ∆ let G_Qp,∆:=Q

α∈∆Gal(Q^p/Q^p) denote the direct power of the ab- solute Galois group ofQp indexed by∆. We denote byRep_κ(G_Qp,∆)(resp. byRep_O

K(G_Qp,∆), resp. byRep_K(G_Qp,∆)) the category of continuous representations of the profinite groupG_Qp,∆

on finite dimensionalκ-vector spaces (resp. finitely generatedO_K-modules, resp. finite dimen- sionalK-vector spaces). On the other hand, for independent commuting variablesX_α(α∈∆) we put

E_∆,κ := κ[[X_α|α ∈∆]][X_α⁻¹ |α∈∆] , OE_∆,K := lim←−

h

O_K/$^h[[X_α |α ∈∆]][X_α⁻¹ |α∈∆]

, E_∆,K := OE_∆,K[p⁻¹] .

Moreover, for each element α ∈ ∆ we have the partial Frobenius ϕ_α, and group Γ_α ∼= Gal(Q^p(µp^∞)/Q^p)acting on the variable Xα in the usual way and commuting with the other

∗This research was supported by a Hungarian OTKA Research grant K-100291 and by the János Bolyai Scholarship of the Hungarian Academy of Sciences. I would like to thank the Arithmetic Geometry and Number Theory group of the University of Duisburg–Essen, campus Essen, for its hospitality and for financial support from SFB TR45 where parts of this paper was written.

variablesX_β (β ∈∆\{α}) in the above rings. A(ϕ_∆,Γ_∆)-module overE_∆,κ(resp. overOE_∆,K, resp. overE_∆,K) is a finitely generated E_∆,κ-module (resp. OE_∆,K-module, resp. E_∆,K-module) Dtogether with commuting semilinear actions of the operators ϕ_α and groupsΓ_α (α∈∆). In case the coefficient ring isE_∆,κ orOE_∆,K, we say thatDis étale if the mapid⊗ϕ_α:ϕ^∗_αD→D is an isomorphism for all α ∈ ∆. For the coefficient ring E_∆,K we require the stronger as- sumption for the étale property that D comes from an étale (ϕ_∆,Γ_∆)-module over O_E∆,K by inverting p. The main result of the paper is that Rep_κ(G_Qp,∆) (resp. Rep_O

K(G_Qp,∆), resp.

Rep_K(G_Qp,∆)) is equivalent to the category of étale (ϕ_∆,Γ_∆)-modules over E_∆,κ (resp. over O_E∆,K, resp. over E_∆,K).

Passing from the Galois side to(ϕ_∆,Γ_∆)-modules is rather straightforward. One constructs a big ringE_∆^sepas an inductive limit of completed tensor products of finite separable extensions E_α⁰ of Eα = F^p((Xα)) (α ∈ ∆) over which the action of H_Qp,∆ = Ker(G_Qp,∆ Q

α∈∆Γ∆) trivializes. The other direction is more involved. In order to trivialize the action of the partial Frobeniiϕ_α (α∈∆) using induction, the main step is to find a latticeD_α^+∗ integral in the variableXα for some fixed α∈∆which is an étale (ϕ∆\{α},Γ∆\{α})-module over the ring Fp[[X_β |β ∈ ∆]][X_β⁻¹ | β ∈ ∆\ {α}]. This uses the ideas of Colmez [3] constructing lattices D⁺ and D⁺⁺ in usual(ϕ,Γ)-modules.

We remark here that Scholze [7] recently realizedG_Qp,∆ (using Drinfeld’s Lemma for dia- monds) as a geometric fundamental groupπ₁((Spd Qp)^|∆|/p.Fr.) of the diamond (Spd Qp)^|∆|

modulo the partial Frobenii ϕ_β (β ∈ ∆ \ {α}) for some fixed α ∈ ∆: one can endow E_∆⁺ = F^p[[Xα | α ∈ ∆]] with its natural compact topology, and look at the subset of its adic spectrum SpaE_∆⁺ where all X_α (α ∈ ∆) are invertible. This defines an analytic adic space overFp, whose perfection modulo the action of allΓ_α’s is a model for (SpdQp)^d. Thus, after taking the action modulo partial Frobenii ϕβ (β ∈∆\ {α} for some fixed α ∈ ∆), the fundamental group will be G_Qp,∆. Now, quite generally étale local systems on diamonds are equivalent to ϕ-modules. This introduces the last missing Frobenius, and one ends up with an equivalence between representations of G_Qp,∆, and some sheaf of modules with Γ∆-action and commuting actions ofϕ_α for all α∈∆. However, this will not produce an actual module over a ring, but a sheaf of modules over a sheaf of rings. One can perhaps deduce the result of this paper along these lines, but that would require some further nontrivial input (replacing the above method of finding a latticeD^+∗_α ).

1.1 Acknowledgements

I would like to thank Christophe Breuil, Elmar Große-Klönne, Kiran Kedlaya, and Vytas Pašk¯unas for useful discussions on the topic. I would like to thank Peter Scholze for clarifying the relation of this work to his theory of realizing G_Qp,∆ as the étale fundamental group of a diamond.

2 Algebraic properties of multivariable (ϕ, Γ)-modules

2.1 Definition and projectivity

For a finite set∆(which is the set of simple roots of Gin [11]) consider the Laurent series ring E_∆ := E_∆⁺[X_∆⁻¹] where E_∆⁺ := Fp[[X_α | α ∈ ∆]] and X_∆ := Q

α∈∆X_α ∈ E_∆⁺. E_∆⁺ is a

regular noehterian local ring of global dimension|∆|, thereforeE_∆is a regular noetherian ring of global dimension |∆| −1. For each index α we define the action of the partial Frobenius ϕ_α and of the group Γ_α with χ_α: Γ_α →^∼ Z^×p onE_∆ as

ϕ_α(X_β) :=

(X_β if β∈∆\ {α}

(X_α+ 1)^p−1 =X_α^p if β=α γ_α(X_β) :=

(X_β if β ∈∆\ {α}

(X_α+ 1)^χα(γα)−1 if β =α (1) for allγ_α ∈Γ_α extending the above formulas to continuous ring endomorphisms ofE_∆ in the obvious way. By an étale (ϕ_∆,Γ_∆)-module over E_∆ we mean a (unless otherwise mentioned) finitely generated moduleD overE_∆ together with a semilinear action of the (commutative) monoidT_+,∆:=Q

α∈∆ϕ^N_αΓ_α (also denote by ϕ_t the action of ϕ_t∈T_+,∆) such that the maps id⊗ϕ_t: ϕ^∗_tD:=E_∆⊗_E∆,ϕt D→D

are isomorphisms for all elements ϕ_t ∈ T_+,∆. Here we put Γ_∆ := Q

α∈∆Γ_α. We denote by D^et(ϕ∆,Γ∆, E∆) the category of étale (ϕ∆,Γ∆)-modules over E∆.

The categoryD^et(ϕ_∆,Γ_∆, E_∆)has the structure of a neutral Tannakian category: For two objects D₁ and D₂ the tensor product D₁ ⊗_E∆ D₂ is an étale T_+,∆-module with the action ϕt(d1⊗d2) :=ϕt(d1)⊗ϕt(d2) for ϕt ∈T+,∆,, di ∈ Di (i= 1,2). Moreover, since E∆ is a free module over itself via ϕ_t, putting (·)^∗ := Hom_E∆(·, E_∆) we have an identification (ϕ^∗_tD)^∗ ∼= ϕ^∗_t(D^∗). So the isomorphism id⊗ϕ_t: ϕ^∗_tD → D dualizes to an isomorphism D^∗ → ϕ^∗_t(D^∗).

The inverse of this isomorphism (for all ϕt ∈T+,∆) equips D^∗ with the structure of an étale T_+,∆-module.

Lemma 2.1. There exists aΓ_∆-equivariant injective resolution of E_∆⁺ as a module over itself.

Proof. Consider the Cousin complex (see IV.2 in [6]) 0→E_∆→E_∆,(0) → · · · → M

p∈Spec(E_∆),codimp=r

J(p)→. . .

where J(p) is the injective envelope of the residue field κ(p) as a module over the local ring E_∆,p. This is aΓ_∆-equivariant injective resolution since the action ofΓ_∆onSpec(E_∆)respects the codimension.

Proposition 2.2. Any object D in D^et(ϕ_∆,Γ_∆, E_∆) is a projective module over E_∆.

Proof. Since E_∆has finite global dimension, letn be the projective dimension of D. Then by Lemma 4.1.6 in [9] we have Extⁱ(D, M) = 0 for all i > n and E_∆-module M and there exists anR-module M₀ with Extⁿ(D, M₀)6= 0. By the long exact sequence ofExt and choosing an onto module homomorphism F M₀ from a free module F we find that Extⁿ(D, F) 6= 0 whence Extⁿ(D, E_∆) 6= 0. However, Extⁿ(D, E_∆) is a finitely generated torsion E_∆-module forn >0admitting a semilinear action ofΓ_∆. Therefore the global annihilator ofExtⁿ(D, E_∆) inE_∆is a nonzero Γ_∆-invariant ideal in E_∆hence equalsE_∆ by Lemma 2.1 in [11]. So n= 0 and D is projective.

Lemma 2.3. We have K₀(E_∆) ∼= Z, ie. any finitely generated projective module over E_∆ is stably free.

Proof. E_∆⁺ ∼= Fp[[X_α | α ∈ ∆]] is a regular local ring, so it has finite global dimension and K0(E_∆⁺) ∼= G0(E_∆⁺) ∼= Z (Thm. II.7.8 in [10]). Therefore the localization E∆ = E_∆⁺[X_∆⁻¹] also has finite global dimension whence we have K₀(E_∆) ∼= G₀(E_∆). The statement follows noting that the mapG₀(E_∆⁺)→G₀(E_∆)is onto by the localization exact sequence of algebraic K-theory (Thm. II.6.4 in [10]).

Remark. I am not aware of the analogue of the Theorem of Quillen and Suslin on the freeness of projective modules over E_∆. However, using the equivalence of categories of D^et(ϕ_∆,Γ_∆, E_∆) with Rep_Fp(G_Qp,∆) we shall see later on (Cor. 3.16) that any object D in D^et(ϕ∆,Γ∆, E∆) is in fact free over E∆.

We equip E_∆⁺ with the X_∆-adic topology. Then (E_∆, E_∆⁺) is a Huber pair (in the sense of [7]) if we equip E_∆ with the inductive limit topology E_∆ = S

nX_∆⁻ⁿE_∆⁺. In fact, E_∆ is a complete noetherian Tate ring (op. cit.). Note that this is not the natural compact topology on E_∆⁺ as in the compact topology E_∆⁺ would not be open in E_∆ since the index of E_∆⁺ in X_∆⁻ⁿE_∆⁺ is not finite. On the other hand, the inclusion Fp((X_α)) ,→ E_∆ is not continuous in the X_∆-adic topology therefore we cannot apply Drinfeld’s Lemma (Thm. 17.2.4 in [7]) directly in this situation.

LetDbe an object inD^et(ϕ_∆,Γ_∆, E_∆). By Banach’s Theorem for Tate rings (Prop. 6.18 in [8]), there is a uniqueE_∆-module topology onDthat we call theX_∆-adic topology. Moreover, any E_∆-module homomorphism is continuous in the X_∆-adic topology.

2.2 Integrality properties

Put ϕ_s := Q

α∈∆ϕ_α ∈ T_+,∆ and define D⁺⁺ := {x ∈ D | limk→∞ϕ^k_s(x) = 0} where the limit is considered in theX∆-adic topology (cf. II.2.1 in [3] in case |∆|= 1). Note that ϕs is the absolute Frobenius onE_∆, it takes any element to its pth power.

Lemma 2.4. Let M be a finitely generated E_∆⁺-submodule in D. Then E_∆⁺ϕ_s(M) is also finitely generated.

Proof. If M is generated by m₁, . . . , m_n then ϕ_s(m₁), . . . , ϕ_s(m_n)generate E_∆⁺ϕ_s(M).

Proposition 2.5. D⁺⁺ is a finitely generated E_∆⁺-submodule in D that is stable under the action ofT_+,∆ and we have D=D⁺⁺[X_∆⁻¹].

Proof. Choose an arbitrary finitely generated E_∆⁺-submoduleM of DwithM[X_∆⁻¹] =D (e.g.

take M = E_∆⁺e₁ +· · ·+E_∆⁺e_n for some E_∆-generating system e₁, . . . , e_n of D). By Lemma 2.4 we have an integer r ≥0 such that ϕ(M)⊆ X_∆^−rM, since E_∆⁺ is noetherian and we have D=S

rX_∆^−rM. Then we have

ϕs(X_∆^kM) = X_∆^pkϕs(M)⊆X_∆^pk−rM ⊆X_∆^k+1M for any integer k ≥ _p−1^r+1. Therefore we have X[^r+1p−1]⁺¹

∆ M ⊆ D⁺⁺ whence D⁺⁺[X_∆⁻¹] = M[X_∆⁻¹] =D.

Since T_+∆ is commutative and the action of each ϕ_t (t ∈ T_+,∆) is continuous, D⁺⁺ is stable under the action ofT_+,∆. There is a system of neighbourhoods of 0in D consisting of E_∆⁺-submodules therefore D⁺⁺ is an E_∆⁺-submodule.

To prove thatD⁺⁺is finitely generated overE_∆⁺ suppose first thatDis a free module over E_∆ generated by e₁, . . . , e_n and put M :=E_∆⁺e₁+· · ·+E_∆⁺e_n. We may assumeM ⊆D⁺⁺ by replacing M with X[^r+1p−1]⁺¹

∆ M. Moreover, further multiplying M =E_∆⁺e₁ +· · ·+E_∆⁺e_n by a power ofX_∆, we may assume that the matrixA:= [ϕ_s]_e1,...,en ofϕ_s in the basis e₁, . . . , e_n lies inE_∆^+n×n as we have[ϕ_s]_X^r

∆e1,...,X_∆^ren =X_∆^(p−1)r[ϕ_s]_e1,...,en. Now we choose the integer r >0so that it is bigger thanval_Xα(detA)for all α∈∆and claim thatD⁺⁺⊆X_∆^−rM whenceD⁺⁺is finitely generated overE_∆⁺ as E_∆⁺ is noetherian. Assume for contradiction that d=Pn

i=1d_ie_i lies in D⁺⁺ for some d_i ∈ E_∆ (i = 1, . . . , n) such that at least one d_i, say d₁, does not lie in X_∆^−rE_∆⁺. In particular, there exists an α in∆ such that val_Xα(d₁)<−r. Since M is open in Dand d ∈D⁺⁺, there exists an integer k > 0such that ϕ^k_s(d)is in M which is equivalent to saying that the column vector

Aϕ_s(A). . . ϕ^k−1_s (A)





 ϕ^k_s(d1)

... ϕ^k_s(d_n)







lies in E_∆⁺ⁿ. Multiplying this by the matrix built from the (n − 1)× (n −1) minors of Aϕ_s(A). . . ϕ^k−1_s (A) we deduce that det(Aϕ_s(A). . . ϕ^k−1_s (A))ϕ^k_s(d₁) = det(A)^pk−1p−1 d^p₁^k lies in E_∆⁺. We compute

0≤val_Xα(det(A)^pk−1p−1 d^p₁^k) = p^k−1

p−1 val_Xα(det(A)) +p^kval_Xα(d₁)<

< p^k−1

p−1 valXα(det(A))−p^kr <0 by our assumption thatr >val_Xα(det(A)), yielding a contradiction.

In the general case note that D is always stably free by Prop. 2.2 and Lemma 2.3. So D1 := D ⊕E_∆^k is a free module over E∆ for k large enough. We make D1 into an étale T_+,∆-module by the trivial action ofT_+,∆onE_∆^k to deduce thatD₁⁺⁺is finitely generated over E_∆⁺. The result follows noting that D⁺⁺ ⊆D₁⁺⁺ and E_∆⁺ is noetherian.

For an object D inD^et(ϕ_∆,Γ_∆, E_∆)we define

D⁺ :={x∈D| {ϕ^k_s(x) : k ≥0} ⊂D is bounded}.

Sinceϕ^k_s(X_∆)tends to0in theX_∆-adic topology, we haveX_∆D⁺ ⊆D⁺⁺, ie.D⁺ ⊆X_∆⁻¹D⁺⁺. In particular,D⁺ is finitely generated overE_∆⁺. On the other hand, we also have D⁺⁺⊆D⁺ by construction whence we deduceD=D⁺[X_∆⁻¹].

Lemma 2.6. We have ϕ_t(D⁺)⊂D⁺ (resp. ϕ_t(D⁺⁺)⊂D⁺⁺) for all ϕ_t∈T_+,∆.

Proof. For any generating system e₁, . . . , e_n of D and any ϕ_t ∈ T_+,∆ there exists an integer k = k(ϕt, M) > 0 such that we have ϕt(X_∆^kM) ⊆ X_∆^kE_∆⁺ϕt(M) ⊆ M where we put M :=

E_∆⁺e₁ +· · ·+E_∆⁺e_n by Lemma 2.4. Indeed, X_∆ divides ϕ_t(X_∆) in E_∆⁺, and we have D = M[1/X_∆] by construction. The statement on D⁺⁺ follows from the commutativity of the monoid T_+,∆ noting that there exists a basis of neighbouhoods of 0 in D consisting of E_∆⁺- submodules of the form M. To see that ϕ_t(D⁺) ⊆D⁺ note that ϕ_t(D⁺) is bounded and we haveϕ^k_s(ϕ_t(D⁺)) = ϕ_t(ϕ^k_s(D⁺))⊂ϕ_t(D⁺).

Now fix an α ∈ ∆ and define D_α⁺ := D⁺[X_∆\{α}⁻¹ ] where for any subset S ⊆ ∆ we put X_S :=Q

β∈SX_β. Then D_α⁺ is a finitely generated module overE_α⁺ :=E_∆⁺[X_∆\{α}⁻¹ ]. We denote byT_+,α⊂T_+,∆ the monoid generated by ϕ_β (β ∈∆\ {α}) andΓ_∆.

Lemma 2.7. D⁺_α/D⁺ is X_α-torsion free: If both X_αⁿ¹d and X_∆\{α}ⁿ² d lie in D⁺ for some element d∈ D⁺, α ∈∆, and integers n₁, n₂ ≥0 then we have d ∈D⁺. The same statement holds if we replace D⁺ by D⁺⁺.

Proof. At first assume that D is free as a module over E∆ with basis e1, . . . , en. Then the denominators of ϕ^k_s(X_αⁿ¹d) = X_α^n1pkϕ^k_s(d) in the basis e₁, . . . , e_n are bounded for k ≥ 0 by assumption. Therefore the X_β-valuations of the denominators of ϕ^k_s(d) are bounded for all β∈∆\ {α}since E_∆⁺is a unique factorization domain. On the other hand, theXα-valuations of these denominators are also bounded since the denominators ofϕ^k_s(X_∆\{α}ⁿ² d) = X_∆\{α}^n2pk ϕ^k_s(d) are bounded. To prove the statement we have the same argument but ‘being bounded’ replaced by ‘tends to0’.

Finally, by Prop. 2.2 and Lemma 2.3 D ⊕ E_∆^k is free over E_∆ and we equip it with the structure of an étale (ϕ,Γ)-module (trivially on E_∆^k). The statement follows from the additivity of the constructions D7→D⁺ and D7→D⁺_α in direct sums.

Lemma 2.8. Assume thatD is generated by a single element e₁ ∈D over E_∆. Then for any ϕt in T+,α we have ϕt(e1) = ate1 for some unit at in (E_α⁺)^×.

Proof. Define a_t ∈ E_∆ and a_α ∈ E_∆ so that ϕ_t(e₁) = a_te₁ and ϕ_α(e₁) =a_αe₁. By the étale property both a_t and a_α are units in E_∆, so it remains to show that val_Xα(a_t) = 0. We compute

ϕ_α(a_t)a_αe₁ =ϕ_α(a_t)ϕ_α(e₁) =ϕ_α(a_te₁) = ϕ_α(ϕ_t(e₁)) =

=ϕ_t(ϕ_α(e₁)) =ϕ_t(a_αe₁) = ϕ_t(a_α)ϕ_t(e₁) = ϕ_t(a_α)a_te₁ whence we deduce

pval_Xα(a_t) + val_Xα(a_α) = val_Xα(ϕ_α(a_t)a_α) = val_Xα(ϕ_t(a_α)a_t) = val_Xα(a_α) + val_Xα(a_t) . This yieldsval_Xα(a_t) = 0 as required.

Lemma 2.9. There exists an integer k = k(D) > 0 such that for any ϕt ∈ T+,α we have X_α^kD⁺_α ⊆E_∆⁺ϕ_t(D⁺_α).

Proof. At first assume that D is free, choose a basis e₁, . . . , e_n contained in D⁺, and put M := E_∆⁺e1 +. . . E_∆⁺en, Mα := E_α⁺e1 +· · ·+E_α⁺en. There exists an integer k0 > 0 such that D⁺ ⊆ X_∆^−k0M. In particular, we have D⁺_α ⊆ X_α^−k0M_α. Now for a fixed ϕ_t ∈ T_+,α let A_t ∈ E_∆^n×n be the matrix of ϕ_t in the basis e₁, . . . , e_n. Since ϕ_t(e_i) lies in D⁺ ⊆ X_α^−k0M_α,

all the entries of the matrixA_t are in X_α^−k0E_α⁺. Applying Lemma 2.8 to the single generator e₁∧ · · · ∧e_n of Vn

D we obtain val_Xα(detA_t) = 0. In particular, all the entries ofA⁻¹_t lie in Xα^−(n−1)k0E_α⁺by the formula for the inverse matrix using the(n−1)×(n−1)minors inA_t. Now note that the elementse1, . . . , en can be written as a linear combination of ϕt(e1), . . . , ϕt(en) with coefficients from A⁻¹_t . Using Lemma 2.6 this shows

X_α^k0D⁺_α ⊆Mα ⊆X_α^−(n−1)k0ϕt(Mα)⊆X_α^−(n−1)k0D⁺_α . So we may choosek :=nk₀ independent of ϕ_t.

The general case follows from Prop. 2.2 and Lemma 2.3 noting that the functorD7→D⁺_α commutes with direct sums.

In view of the above Lemma we define D_α^+∗ := \

ϕt∈T_+,α

E_α⁺ϕ_t(D_α⁺) .

D_α^+∗ is finitely generated over E_α⁺ as it is contained in D_α⁺ and E_α⁺ is noetherian. On the other hand, by Lemma 2.9 we have X_α^kD⁺_α ⊆D^+∗_α for some integer k =k(D)>0 whence, in particular,D=D^+∗_α [X_α⁻¹].

Proposition 2.10. D^+∗_α is an étale T_+,α-module over E_α⁺, ie. the maps id⊗ϕ_t: ϕ^∗_tD_α^+∗ =E_α⁺⊗_E⁺

α,ϕt D_α^+∗ →D_α^+∗ (2)

are bijective for all ϕt∈T+,α.

Proof. At first note that we have ϕ_t(D^+∗_α ) ⊆ D_α^+∗ for all ϕ_t ∈ T_+,α by Lemma 2.6 and the commutativity ofT_+,α, so the map (2) exists. Now letϕ_t0 ∈T_+,αbe arbitrary. SinceE_α⁺(resp.

E_∆) is a finite free module over ϕ_t0(E_α⁺) (resp. over ϕ_t0(E_∆)) with generators contained in E_∆⁺, we have a natural identification ϕ^∗_t0D_α^+∗ ∼=E_∆⁺⊗_E⁺

∆,ϕt0D^+∗_∆ (resp.ϕ^∗_t0D∼=E_∆⁺⊗_E⁺

∆,ϕt0 D).

Since E_∆⁺ is finite free (hence flat) over ϕ_t0(E_∆⁺), the inclusion D_α⁺ ⊂ D induces an inclusion ϕ^∗_t0D_α⁺ ⊂ϕ^∗_t0D. It follows that (2) is injective since D is étale. Similarly, for each ϕ_t ∈T_+,α, the map

id⊗ϕ_t0: ϕ^∗_t

0(E_α⁺ϕ_t(D_α⁺))→E_α⁺ϕ_t(D⁺_α)

is injective with imageE_α⁺ϕt0ϕt(D_α⁺). On the other hand, since E_∆⁺is finite free over ϕt0(E_∆⁺), we haveϕ^∗_t

0D^+∗_α =T

t∈T_+,αϕ^∗_t

0(E_α⁺ϕ_t(D_α⁺))where the intersection is taken insideϕ^∗_t

0D. There- fore (2) is bijective as we have D_α^+∗ =T

ϕt∈T_+,αE_α⁺ϕ_t0ϕ_t(D⁺_α).

Lemma 2.11. There exists a finitely generated E_∆⁺-submodule D₀ ⊂ D_α^+∗ such that D₀ ⊆ E_∆⁺ϕ_α(D₀) and D^+∗_α = D₀[X_∆\{α}⁻¹ ] where ϕ_α := Q

β∈∆\{α}ϕ_β. Moreover, we have D_α^+∗ = S

r≥0E_∆⁺ϕ^r_α(X_∆\{α}⁻¹ D₀).

Proof. Put D₁ :=D⁺∩D^+∗_α . By Prop. 2.10 and the fact that D_α^+∗ =D₁[X_∆\{α}⁻¹ ] we find an integer k₀ >0 such that X_∆\{α}^k0 D₁ ⊆E_∆⁺ϕ_α(D₁). So for k > _p−1^k0 we have

X_∆\{α}^−k D₁ ⊆X_∆\{α}^−k−k0E_∆⁺ϕ_α(D₁)⊆X_∆\{α}^−pk E_∆⁺ϕ_α(D₁) = E_∆⁺ϕ_α(X_∆\{α}^−k D₁) .

So we put D₀ := X_∆\{α}^−k D₁ so that the first part of the statement is satisfied. Iterating the inclusion D₀ ⊆E_∆⁺ϕ_α(D₀) we obtainD₀ ⊆E_∆⁺ϕ^r_α(D₀) for all r ≥1. Finally, we compute

X_∆\{α}^−pr D₀ ⊆X_∆\{α}^−pr E_∆⁺ϕ^r_α(D₀) =E_∆⁺ϕ^r_α(X_∆\{α}⁻¹ D₀) . The statement follows noting that we have D_α^+∗ =D₀[X_∆\{α}⁻¹ ] =S

rX_∆\{α}^−pr D₀.

3 The equivalence of categories for F

-representations

3.1 The functor D

Take a copy G_Qp,α ∼= Gal(Qp/Qp) of the absolute Galois group of Qp for each element α ∈ ∆ and let G_Qp,∆ := Q

α∈∆G_Qp,α. Let Rep

F^p(G_Qp,∆) be the category of continuous representations of the group G_Qp,∆ on finite dimensional Fp vectorspaces. We identify Γ_α with the Galois group Gal(Qp(µ_p^∞)/Qp) as a quotient of G_Qp,α via the cyclotomic character χ_α: Gal(Q^p(µ_p^∞)/Q^p)→Z^×p. Further, we denote byH_Qp,α the kernel of the natural quotient map G_Qp,α → Γ_α and put H_Qp,∆ := Q

α∈∆H_Qp,αCG_Qp,∆. Putting E_α := Fp((X_α)) we have the following fundamental result of Fontaine and Wintenberger (Thm. 4.16 [5]).

Theorem 3.1. The absolute Galois group Gal(E_α^sep/E_α) is isomorphic to H_Qp,α. Moreover, G_Qp,α acts on the separable closure E_α^sep via automorphisms such that the action of Γ_α ∼= G_Qp,α/H_Qp,α on E_α= (E_α^sep)^HQp,α coincides with the one given in (1).

For eachα∈∆consider a finite separable extensionE_α⁰ ofE_α together with the Frobenius ϕ_α: E_α⁰ → E_α⁰ acting by raising to the power p. We denote by E_α⁰⁺ the integral closure of E_α⁺ = F^p[[Xα]] in E_α⁰. Note that E_α⁰ is isomorphic to F^qα((X_α⁰)) for some power qα of p and uniformizerX_α⁰ such that we haveE_α⁰⁺ ∼=Fqα[[X_α⁰]]. We normalize theX_α-adic (multiplicative) valuation onE_α so that we have |X_α|_Xα =p⁻¹. This extends uniquely to the finite extension E_α⁰. Moreover, we equip the tensor product E_∆,◦⁰ := N

α∈∆,FpE_α⁰ with a norm | · |prod by the formula

|c|_prod:= inf max

i (Y

α∈∆

|c_α,i|_α) | c=

n

X

i=1

O

α∈∆

c_α,i

!

. (3)

Note that the restriction of | · |prod to the subring E_∆,◦⁰⁺ := N

α∈∆,F^pE_α⁰⁺ induces the valu- ation with respect to the augmentation ideal Ker(E_∆,◦⁰⁺ N

α∈∆,FpFqα). The norm | · |_prod is not multiplicative in general, as the ring N

α∈∆,F^pF^qα) is not a domain. However, it is submultiplicative. We define E_∆⁰⁺ as the completion of E_∆,◦⁰⁺ with respect to | · |_prod and put E_∆⁰ :=E_∆⁰⁺[1/X_∆]. Note thatE_∆⁰ isnot complete with respect to| · |_prod (unless|∆|= 1) even thoughE_∆,◦⁰ =E_∆,◦⁰⁺ [1/X_∆] is a dense subring in E_∆⁰ . Since we have a containment

( O

α∈∆,F^p

F^qα)[X_α⁰, α∈∆] = O

α∈∆,F^p

F^qα[Xα]≤denseE_∆,◦⁰⁺

we may identify E_∆⁰⁺ with the power series ring (N

α∈∆,FpF^qα)[[X_α⁰, α ∈ ∆]] which is the completion of the polynomial ring above. In particular, the special case E_α⁰ = E_α for all α∈∆yields a ring E_∆⁰ isomorphic to E_∆. Therefore E_∆ is a subring of E_∆⁰ for all collection

of finite separable extensions E_α⁰ of E_α (α ∈∆). Further, ϕ_α acts on E_∆,◦⁰⁺ (and on E_∆,◦⁰ ) by the Frobenius on the component inE_α⁰ and by the identity on all the other components inE_β⁰, β∈∆\ {α}. This action is continuous in the norm| · |_prod therefore extends to the completion E_∆⁰⁺ and the localization E_∆⁰ . We have the following alternative characterization of the ring E_∆⁰ .

Lemma 3.2. Put ∆ ={α₁, . . . , α_n}. We have E_∆⁰ ∼=E_α⁰₁ ⊗_Eα

1 E_α⁰₂ ⊗_Eα

2 · · ·(E_α⁰_n ⊗_Eαn E_∆) . Proof. By rearranging the order of tensor products we have an identification

E_∆,◦⁰⁺ = O

α∈∆,Fp

(E_α⁰⁺⊗_E⁺

α E_α⁺)∼=E_α⁰⁺₁ ⊗_E⁺

α1

E_α⁰⁺₂ ⊗_E⁺

α2

. . .(E_α⁰⁺_n⊗_E⁺

αn E_∆,◦⁺ ) .

The statement follows by completing this with respect to the maximal ideal of E_∆⁺ and in- vertingX_∆.

We define the multivariable analogue ofE^sep as E_∆^sep := lim−→

Eα≤E_α⁰≤E_α^sep,∀α∈∆

E_∆⁰ .

For any subset S ⊆ ∆ we define the similar notions E_S⁰⁺, E_S⁰, and E_S^sep with ∆ replaced byS. We equip E_∆^sep with the relative Frobeniiϕ_α for each α∈∆defined above on each E_∆⁰ . Further,E_∆^sep admits an action ofG_Qp,∆ satisfying

Proposition 3.3. Assume that the extensions E_α⁰/E_α are Galois for all α ∈ ∆ and let H⁰ := Q

α∈∆H_α⁰ where H_α⁰ := Gal(E_α^sep/E_α⁰). Then we have (E_∆^sep)^H∆0 = E_∆⁰ . In particular, the subring (E_∆^sep)^HQp,∆ of H_Qp,∆-invariants in E_∆^sep equals E_∆ with the previously defined action ofΓ_∆ ∼=G_Qp,∆/H_Qp,∆.

Proof. Since X_∆ is H_∆⁰ -invariant and lim−→ can be interchanged with taking H_∆⁰ -invariants, it suffices to show that whenever

Eα =F^p((Xα))≤E_α⁰ =F^q0α((X_α⁰))≤E_α⁰⁰ =F^qα⁰⁰((X_α⁰⁰))

is a a sequence of finite Galois extensions for eachα∈∆ then we have (E_∆⁰⁰⁺)^H∆0 =E_∆⁰⁺. The containment(E_∆⁰⁰⁺)^H∆0 ⊇ E_∆⁰⁺ is clear. We prove the converse by induction on|∆|. Note that the idealMαCE_∆⁰⁰⁺ generated byX_α⁰⁰ is invariant under the action of H_∆⁰ for any fixed α in

∆. Moreover, for any integer k≥1the ringE_α⁰⁰⁺/M^k_α is finite dimensional overFp. Therefore the image of(E_∆⁰⁰⁺)^H∆0 under the quotient map E_∆⁰⁰⁺E_∆⁰⁰⁺/M^k_α is contained in

E_∆⁰⁰⁺/M^k_αH_∆⁰

⊆ E_∆⁰⁰⁺/M^k_αH_∆\{α}⁰

=

E_∆\{α}⁰⁰⁺ ⊗_Fp E_α⁰⁰⁺/M^k_αH⁰_∆\{α}

=

=

E_∆\{α}⁰⁰⁺ H_∆\{α}⁰

⊗_Fp E_α⁰⁰⁺/M^k_α

=E_∆\{α}⁰⁺ ⊗_Fp E_α⁰⁰⁺/M^k_α

by induction. Taking the projective limit with respect to k ≥ 1 we deduce that (E_∆⁰⁰⁺)^H∆0 is contained in the power series ring



F^q00α⊗_Fp O

β∈∆\{α},Fp

Fq⁰_β



[[X_α⁰⁰, X_β⁰ |β ∈∆\ {α}]]⊆E_∆⁰⁰⁺ .

Now using the action of H_α⁰ in a similar argument as above (reducing modulo the kth power of the ideal generated by all theX_β⁰, β ∈∆\ {α}for all k ≥1) we deduce the statement.

The subring E_∆,◦^sep ∼= N

α∈∆,FpE_α^sep in E_∆^sep is the inductive limit of E_∆,◦⁰ ⊆ E_∆⁰ where E_α⁰ runs through the finite separable extensions of Eα for each α∈∆.

LetV be a finite dimensional representation of the groupG_Qp,∆ over Fp. The basechange E_∆^sep⊗_Fp V is equipped with the diagonal semilinear action of G_Qp,∆ and with the Frobenii ϕα for α ∈∆. These all commute with each other. We define the value of the functor D at V by putting

D(V) := (E_∆^sep⊗_Fp V)^HQp,∆ .

By Lemma 3.3D(V) is a module over E_∆ inheriting the action of the monoid T_+,∆ from the action of ϕ_α (α ∈ ∆) and the Galois group G_Qp,∆ on E_∆^sep ⊗_Fp V. Our key Lemma is the following.

Lemma 3.4. TheE_∆^sep-moduleE_∆^sep⊗_FpV admits a basis consisting of elements fixed byH_Qp,∆. Proof. At first consider the E_∆,◦^sep-module E_∆,◦^sep ⊗_Fp V. We show by induction on |∆| that E_∆,◦^sep ⊗_FpV admits a basis consisting of H_Qp,∆-invariant vectors. The statement follows from this noting thatE_∆,◦^sep is a subring inE_∆^septherefore the required basis exists also inE_∆^sep⊗_FpV ∼= E_∆^sep⊗_E^sep

∆,◦(E_∆,◦^sep ⊗_FpV).

By Hilbert’s Thm. 90 theH_Qp,α-module E_α^sep⊗_Fp V is trivial for each α∈∆. So we have anE_α^sep-basise^(α)₁ , . . . , e^(α)_d ofE_α^sep⊗_FpV consisting of H_Qp,α-invariant elements. Since we have an action of the direct productH_Qp,∆ on V, the E_α-vector space

V_α :=E_αe^(α)₁ +· · ·+E_αe^(α)_d = (E_α^sep⊗_FpV)^HQp,α

admits a linear action of the groupH_Qp,∆\{α}. Now note that the representationsV andVα of the groupH_Qp,∆\{α} become isomorphic over the field E_α^sep by construction. Since H_Qp,∆\{α}

acts through a finite quotient onV, there is a finite extensionE_α⁰ ofE_αcontained inE_α^sep such that we have an isomorphism E_α⁰ ⊗_Fp V ∼= E_α⁰ ⊗Eα Vα of H_Qp,∆\{α}-representations. Making this identification and writinge_i := 1⊗e_i ∈E_α⁰ ⊗_FpV (resp.e^(α)_i := 1⊗e^(α)_i ), i= 1, . . . , d, for a basise₁, . . . , e_d in V (resp. for the basis e^(α)₁ , . . . e^(α)_d inV_α) by an abuse of notation, we find a matrix B ∈ GL_d(E_α⁰) with Bρ(h) = ρ_α(h)B for all h ∈ H_Qp,∆\{α} where ρ(h) ∈ GL_d(F^p) (resp. ρ_α(h) ∈ GL_d(E_α)) is the matrix of the action of h on V (resp. on V_α) in the basis e₁, . . . , e_d (resp. e^(α)₁ , . . . e^(α)_d ). Now E_α⁰/E_α is a finite separable extension, so there exists a primitive element u ∈ E_α⁰ with E_α⁰ = E_α(u). Hence we may write B is a sum B = B(u) = B0+B1u+· · ·+Bn−1uⁿ⁻¹ for some matrices B0, B1, . . . , Bn−1 ∈ E_α^d×d with n := |E_α⁰ : Eα|.

Since detB 6= 0, the polynomial det(B(x)) := det(B₀ +B₁x+· · ·+Bn−1xⁿ⁻¹) ∈ E_α[x] is not identically0. As E_α is an infinite field, there exists a u₀ ∈ E_α with detB(u₀)6= 0. Now