Higher Fields of Norms and (ϕ, Γ)-Modules

Dedicated to John Coates

on the occasion of his 60th birthday

Anthony J. Scholl

Received: December 1, 2005 Revised: March 3, 2006

Abstract. We describe a generalisation of the Fontaine-Wintenberger theory of the “field of norms” functor to local fields with imperfect residue field, generalising work of Abrashkin for higher dimensional local fields. We also compute the cohomology of associated p-adic Galois representations using (ϕ, Γ)-modules.

2000 Mathematics Subject Classification: 11S15, 11S23, 11S25, 12G05 Keywords and Phrases: local fields, ramification theory, Galois representations

Introduction

Abrashkin [3] has found an analogue of the field of norms functor for higherdimensional local fields. His construction uses the theory of ramification groups [24] for such fields. As an application of his results (include the transfer of the ramification group structure from characteristic zero to characteristic p) he obtains the analogue of Grothendieck’s anabelian conjecture for higherdimensional local fields. In the first part of this paper we construct an analogue of the field of norms for fairly general^{1} local fields with imperfect residue field. Like Abrashkin’s, as a starting point it uses the alternative characterisation of the ring of integers of the (classical) field of norms as a subring of Fontaine’s ring R = Ẽ^{+} (the perfection of 𝔬\overline{K} ⊗ 𝔽_{p}). However we differ from him, and the original construction by Fontaine and Wintenberger [12], [13], by making no appeal to higher ramification theory. We instead restrict to extensions which are “strictly deeply ramified” (see §1.3 and Remark 1.3.8 below) and appeal instead to the differential characterisation of deeply ramified extensions which forms the basis for Faltings’s approach to p-adic Hodge theory [10] (although we only use the most elementary parts of Faltings’s work). These extensions are (in the classical case) closely related to strictly APF extensions; one may hope that by using Abbès and Saito’s higher ramification theory ([1], [2]) a theory for all APF extensions could be developed. We hope to clarify this relation in a subsequent paper. In any case, the theory presented here includes those extensions which arise in the theory of (ϕ, Γ)-modules. It is also perhaps worth noting that in the classical case (perfect residue field), the 2 key propositions on which the theory depends (1.2.1 and 1.2.8) are rather elementary. In the second part of the paper we begin the study of (ϕ, Γ)-modules in this setting, and prove the natural generalisation of Herr’s formula [15] for the cohomology of a p-adic Galois representation. We also describe a natural family of (non-abelian) extensions to which this theory applies. We hope to develop this further in a subsequent paper. This work grew out of the preparation of talks given during a study group at Cambridge in winter 2004, and the author is grateful to the members of the study group, particularly John Coates and Sarah Zerbes, for their comments and encouragement, to Victor Abrashkin, Ivan Fesenko and Jan Nekovář for useful discussions, to Pierre Colmez for letting me have some of his unpublished work, and to the referee for his careful reading of the paper. He also wishes to thank Bilkent University, Ankara, for their hospitality while parts of this paper were being written. As the referee has pointed out, the possibility of such constructions has been known to the experts for some time (see for example the remarks on page 251 of [11]). After this paper was written the author received a copy of Andreatta and Iovita’s preprints [4, 5], which construct rings of norms and compute the cohomology of (ϕ, Γ)-modules for Kummer-like extensions of more general padic base rings.

1The only requirement is that the residue field has a finite p-basis.

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Notation

Throughout this paper p denotes a fixed prime number. If A is an abelian group and ξ an endomorphism of A, or more generally an ideal in a ring of endomorphisms of A, we write A/ξ for A/ξA, and A[ξ] for the ξ-torsion subgroup of A. If R is a ring of characteristic p, we denote by f = f_{R}: x ↦ x^{p} the Frobenius endomorphism of R. If K is any p-adically valued field and λ ∈ ℚ belongs to the value group of K, we will by abuse of notation write p^{λ} for the fractional ideal comprised of all x ∈ K with v_{p}(x) ≥ λ. We use the sign = to denote equality or canonical isomorphism, and A := B to indicate that A is by definition B.

Higher Fields of Norms and (ϕ, Γ)-Modules 687

1 Fields of norms

1.1 Big local fields

By a big local field we mean a complete discretely-valued field, whose residue field k has characteristic p and satisfies [k : k^{p}] = p^{d} for some d ≥ 0 (we then talk of a “d-big local field”). If K is such a field we use the usual notations: 𝔬_{K} for its valuation ring, ϖ_{K} for a uniformiser (not always fixed), k_{K} or (if no confusion is likely) simply k for its residue field, and v_{K} for the normalised valuation on K with v_{K}(ϖ_{K}) = 1. When char K = 0, we write e_{K} for its absolute ramification degree, and v_{p} for the p-adic valuation with v_{p}(p) = 1. Of course, d = 0 if and only if K is a local field in the usual sense (i.e., with perfect residue field). We recall for convenience some facts about big local fields and their extensions, and fix some notation. If L/K is a finite separable extension of d-big local fields, then [L : K] = ef_{0}p^{s} where e = e(L/K) = v_{L}(ϖ_{K}) is the (reduced) ramification degree, and f_{0} and p^{s} are the separable and inseparable degrees of the extension k_{L}/k_{K}, respectively, so that f = f_{0}p^{s} = [k_{L} : k_{K}]. If L/K is a finite separable extension of big local fields, the valuation ring 𝔬_{L} is not necessarily of the form 𝔬_{K}[x]. There are two particular cases when this is true:

(i) when the residue class extension k_{L}/k_{K} is separable [21, III, §6 Lemme 4]. Then there exists x ∈ 𝔬_{L} with 𝔬_{L} = 𝔬_{K}[x]; and if k_{L} = k_{K} then x = ϖ_{L} for any uniformiser ϖ_{L} will do, and its minimal polynomial is an Eisenstein polynomial. (ii) when ϖ_{K} = ϖ_{L} and the residue class extension is purely inseparable and simple^{2}. Let k_{L} = k_{K}(b) for some b with b^{q} = a ∈ k_{K} \ k_{K}^{p}, and let u ∈ 𝔬_{L} be any lift of b. Then 𝔬_{L} = 𝔬_{K}[u] where the minimal polynomial of u has the form g(T) = T^{q} + ∑_{i=1}^{q−1} c_{i}T^{i} − v, with ϖ_{K}|c_{i} and a = v mod ϖ_{K}. Conversely, let g = T^{q} +∑_{i=0}^{q−1} c_{i}T^{i} ∈ 𝔬_{K}[T] be any polynomial. Let us say that g is a fake Eisenstein polynomial if (a) its degree q is a power of p; (b) for every i ≥ 1, c_{i} ≡ 0 (mod ϖ_{K}); and (c) c_{0} is a unit whose reduction mod ϖ_{K} is not a p^{th} power. Then g is irreducible (since it is irreducible mod ϖ_{K}) and 𝔬_{K}[T]/(g) is a discrete valuation ring. It is the valuation ring of a totally fiercely ramified extension of K of degree q. In particular, if L/K is Galois of prime degree then one of (i), (ii) applies, so 𝔬_{L} = 𝔬_{K}[x]. For any big local field K of characteristic zero there exists a complete subfield K_{u} ⊂ K which is absolutely unramified (that is, p is a uniformiser) having the same residue field as K. (This holds by the existence of Cohen subrings; see for example [EGA4, 19.8.6] or [18, pp. 211–212]). If d = 0 then K_{u} is unique; otherwise (except when e_{K} = 1) it is non-unique [EGA4, 19.8.7]. If L/K is a finite extension it is not in general possible to find such subfields K_{u} ⊂ K, L_{u} ⊂ L satisfying K_{u} ⊂ L_{u} (even when K itself is absolutely unramified).

2In the terminology of [24], L/K is totally fiercely ramified.

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Let K be a big local field with residue field k, and choose K_{u} ⊂ K as above. Then for any m with 0 < m ≤ e_{K}, the quotient 𝔬_{K}/(ϖ_{K}^{m}) contains 𝔬_{K_{u}}/(p) = k and therefore 𝔬_{K}/(ϖ_{K}^{m}) ≃ k[ϖ_{K}]/(ϖ_{K}^{m}). When k is perfect (but not in general) this isomorphism is canonical, since the projection 𝔬_{K}/(ϖ_{K}^{m}) → ⟶ k has a unique section, whose image is the maximal perfect subring of 𝔬_{K}/(ϖ_{K}^{m}). If K is a big local field of characteristic p then it contains a coefficient field (non-unique if d > 0), so that K ≃ k_{K}((ϖ_{K})). If L/K is a finite separable extension then one cannot in general find a coefficient field of L containing one of K. From now on, unless stated explicitly to the contrary, all big local fields will be assumed to have characteristic zero. For a finite extension L/K we then write

δ(L/K) = ∑ δ_{i}(L/K) = v_{p}(𝔇_{L/K})

where the δ_{i}(L/K) are the p-adic valuations of the primary factors of Ω(L/K).

1.2 Differentials and ramification

If L/K is an extension of big local fields, we usually write Ω(L/K) := Ω_{𝔬_{L}/𝔬_{K}} for the module of relative Kähler differentials, which is an 𝔬_{L}-module of finite length. Then Ω(L/K) can be generated by ≤ (d + 1) generators (for example, by equation (1.2.2) below). The Fitting ideal of Ω(L/K) (the product of its primary factors) equals the relative different 𝔇_{L/K}, defined in the usual way as the inverse of the 𝔬_{K}-dual of 𝔬_{L} with respect to the trace form; see for example [10, Lemma 1.1].

Proposition 1.2.1. Let L/K be a finite extension of d-big local fields with [L : K] = p^{d+1}. Assume that there exists a surjection

Ω(L/K) ⟶ ⟶ (𝔬_{L}/ξ)^{d+1}

for some ideal ξ ⊂ 𝔬_{K} with 0 < v_{p}(ξ) ≤ 1. Then e(L/K) = p and k_{L} = k_{K}^{1/p}, and the Frobenius endomorphism of 𝔬_{L}/ξ has a unique factorisation

\xymatrix{𝔬_{L}/ξ \ar@{->>}[d]_{mod ξ′} \ar[rr]^{f} &  & 𝔬_{L}/ξ \\ 𝔬_{L}/ξ′ & ∼ \ar@{-->}[r] & 𝔬_{K}/ξ \ar@{^{(}->}[u]_{inclusion}}

where ξ′ ⊂ 𝔬_{L} is the ideal with valuation p^{−1}v_{p}(ξ). In particular, Frobenius induces a surjection f: 𝔬_{L}/ξ ⟶ ⟶ 𝔬_{K}/ξ.

Proof. Let ϖ_{L} be a uniformiser. We have [L : K] = p^{d+1} = ef_{0}p^{s}, and if p^{r} = [k_{L} : k_{L}^{p}k] then dim_{k_{L}} Ω_{k_{L}/k} = r ≤ s. We have the exact sequence of differentials

(ϖ_{L})/(ϖ_{L}^{2}) ⟶ Ω(L/K) ⊗_{𝔬_{L}} k_{L} ⟶ Ω_{k_{L}/k} ⟶ 0  (1.2.2)

and if e = 1 the first map is zero (taking ϖ_{L} = ϖ_{k}). It follows that

dim_{k_{L}}(Ω(L/K) ⊗_{𝔬_{L}} k_{L}) \begin{cases}≤ 1 + r & in general \\ = r & if e = 1. \end{cases}

By definition, d = [k_{L} : k_{L}^{p}] ≥ r and by hypothesis dim_{k_{L}}(Ω(L/K) ⊗_{𝔬_{L}} k_{L}) ≥ d + 1, so we must have r = s = d, f_{0} = 1, e = p and k_{L} = k^{1/p}. Let {t_{α} | 1 ≤ α ≤ d} ⊂ 𝔬_{L}^{∗} be a lift of a p-basis for k_{L}. Then dϖ_{L}, {dt_{α}} is a basis for Ω(L/K) ⊗ k_{L}. Introduce a multi-index notation I = (i_{1}, …, i_{d}), t^{I} = ∏ t_{α}^{i_{α}}. Then the k-vector space 𝔬_{L}/(ϖ_{K}) has as a basis the reduction mod ϖ_{K} of the p^{d+1} monomials {t^{I}ϖ_{L}^{j} | 0 ≤ j < p, 0 ≤ i_{α} < p}. So by Nakayama’s lemma,

𝔬_{L} = 𝔬_{K}[ϖ_{L}, \{t_{α}\}] = ⨁_{\begin{matrix}0≤j<p \\ 0≤i_{α}<p \end{matrix}} t^{I}ϖ_{L}^{j}𝔬_{K}.  (1.2.3)

Lemma 1.2.4. If x = ∑_{0≤j<p, 0≤i_{α}<p} x_{I,j}t^{I}ϖ_{L}^{j} with x_{I,j} ∈ 𝔬_{K}, then

v_{p}(x) = min_{I,j}(v_{p}(x_{I,j}) + \frac{j}{e_{L}} ).

Proof. If y_{I} ∈ 𝔬_{K} for 0 ≤ i_{α} < p, then since the elements t^{I} are linearly independent mod (ϖ_{L}), we have

ϖ_{L}|∑_{I} y_{I}t^{I} ⟺ for all I, y_{I} ≡ 0 (mod ϖ_{K}) ⟺ ϖ_{K} | ∑_{I} y_{I}t^{I}

from which we see that

v_{K}(∑_{I} y_{I}t^{I}) = min_{I} v_{K}(y_{I})  (1.2.5)

and that this is an integer. Therefore

v_{K}(ϖ_{L}^{j} ∑_{I} x_{I,j}t^{I}) ≡ \frac{j}{p} (mod ℤ)

and so

v_{p}(x) = v_{p}(∑_{j=0}^{p−1} ϖ_{L}^{j} ∑_{I} x_{I,j}t^{I}) = min_{j}{v_{p}(ϖ_{L}^{j} ∑_{I} x_{I,j}t^{I})}.

Then the lemma follows from (1.2.5).

From (1.2.3) we obtain (d + 1) relations in 𝔬_{L} of the shape:

ϖ_{L}^{p} = ∑_{j=0}^{p−1} A_{j}(t)ϖ_{L}^{j},  t_{α}^{p} = ∑_{j=0}^{p−1} B_{α,j}(t)ϖ_{L}^{j} (1 ≤ α ≤ d)  (1.2.6)

where A_{j}, B_{α,j} ∈ 𝔬_{K}[X_{1}, …, X_{d}] are polynomials of degree < p in each variable. Write D_{γ} for the derivative with respect to X_{γ}, and δ_{αγ} for Kronecker delta. Therefore in Ω(L/K) the following relations hold:

(−pϖ_{L}^{p−1} + ∑_{j=1}^{p−1} jA_{j}(t)ϖ_{L}^{j−1})dϖ_{L} + ∑_{γ}(∑_{j=0}^{p−1} D_{γ}A_{j}(t)ϖ_{L}^{j})dt_{γ} = 0

(∑_{j=1}^{p−1} jB_{α,j}(t)ϖ_{L}^{j−1})dϖ_{L} − pt_{α}^{p−1}dt_{α} + ∑_{γ}(∑_{j=0}^{p−1} D_{γ}B_{α,j}(t)ϖ_{L}^{j})dt_{γ} = 0

The condition on Ω(L/K) forces all the coefficients in these identities to be divisible by ξ. From (1.2.4) this implies that for all j > 0, A_{j}(t)ϖ_{L}^{j−1} ≡ 0 ≡ B_{α,j}(t)ϖ_{L}^{j−1} (mod ξ). Therefore

ϖ_{L}^{p} ≡ A_{0}(t) and t_{α}^{p} ≡ B_{α,0}(t) (mod ϖ_{L}ξ).

Similarly, for every γ and every j ≥ 0,

D_{γ}A_{j}(t) ≡ D_{γ}B_{α,j}(t) ≡ 0 (mod ϖ_{L}^{−j}ξ).

This last congruence implies that the nonconstant coefficients of A_{j} and B_{α,j} are divisible by ϖ_{L}^{−j}ξ, so especially

A_{0}(t) ≡ A_{0}(0), B_{α,0}(t) ≡ B_{α,0}(0) (mod ξ).

The first of these congruences, together with 1.2.4 and the first equation of (1.2.6), implies that v_{L}(A_{0}(0)) = p. We will therefore choose ϖ_{K} = A_{0}(0) as the uniformiser of K. Then

ϖ_{L}^{p} ≡ ϖ_{K}, t_{α}^{p} ≡ b_{α} (mod ξ)

where b_{α} = B_{α,0}(0) ∈ 𝔬_{K}^{∗}. If m = v_{K}(ξ) then, as noted just before the statement of this Proposition, 𝔬_{K}/ξ \underrightarrow{∼} k[ϖ_{K}]/(ϖ_{K}^{m}). We fix such an isomorphism. If b̄_{α} ∈ k denotes the reduction of b_{α} mod ϖ_{K}, then by (1.2.3) there are compatible isomorphisms

𝔬_{L}/ξ \underrightarrow{∼} k[ϖ_{L}, \{t_{α}\}]/(ϖ_{L}^{mp}, \{t_{α}^{p} − b̄_{α}\})
𝔬_{L}/ξ′ \underrightarrow{∼} k[ϖ_{L}, \{t_{α}\}]/(ϖ_{L}^{m}, \{t_{α}^{p} − b̄_{α}\})

such that the inclusion 𝔬_{K}/ξ ⟶ 𝔬_{L}/ξ induces the identity on k and maps ϖ_{K} to ϖ_{L}^{p}. Therefore

𝔬_{L}/ξ′ \underrightarrow{∼}_{f} (𝔬_{L}/ξ)^{p} = 𝔬_{K}/ξ ⊂ 𝔬_{L}/ξ

as required.

Remark 1.2.7. It is perhaps worth noting that in the case d = 0 the proof just given simplifies greatly; in this case L/K is totally ramified by hypothesis, so ϖ_{L} satisfies an Eisenstein polynomial over K, whose constant term we may take to be −ϖ_{K}. We then have canonical isomorphisms 𝔬_{K}/ξ = k[ϖ_{K}]/(ϖ_{K}^{m}), 𝔬_{L}/ξ = k[ϖ_{L}]/(ϖ_{L}^{mp}), and the minimal polynomial of ϖ_{L} gives at once the congruence ϖ_{L}^{p} ≡ ϖ_{K} (mod ξ) — cf. [21], Remark 1 after Proposition 13 of §III.6.

Recall now the key lemma in the theory ([9], [10], [22]) of deep ramification of local fields:

Proposition 1.2.8. (Faltings) Let L and K′ be linearly disjoint finite extensions of a d-big local field K, and set L′ = LK′ ≃ L ⊗_{K} K′. Assume there exists a surjection Ω(K′/K) ⟶ ⟶ (𝔬_{K′}/p^{λ})^{d+1} for some λ ≥ 0. Then

δ(L′/K′) ≤ δ(L/K) − \frac{1}{d + 2} min(λ, δ(L/K)).

Proof. (expanded from the proof of [10, Theorem 1.2]). For simplicity of notation write:

R = 𝔬_{K}, S = 𝔬_{L}, R′ = 𝔬_{K′}, S′ = 𝔬_{L′}
δ = δ(L/K), δ_{i} = δ_{i}(L/K), δ′ = δ(L′/K′), δ′_{i} = δ_{i}(L′/K′).

If M is an S′-module of finite length, write ℓ_{p}(M) for 1/e_{L′} times the length of M (so ℓ_{p}(M) also equals the p-adic valuation of the Fitting ideal of M). Consider the homomorphism γ = βα, which links the two exact^{3} sequences of differentials in the commutative diagram:

\xymatrix{ &  & 0 \ar[d] \\  &  & S′ ⊗_{S} Ω_{S/R} \ar[d]_{α} \ar@{-->}[dr]^{γ} \\ 0 \ar[r] & S′ ⊗_{R′} Ω_{R′/R} \ar[r] & Ω_{S′/R} \ar[r]_{β} & Ω_{S′/R′} \ar[r] & 0}

In this diagram, all entries are torsion S′-modules which can be generated by ≤ (d + 1) elements. We then have the following inequalities:

(a) ℓ_{p}(ker γ) ≥ min(λ, δ)

(b) ℓ_{p}(im γ) ≥ (d + 2)δ′ − (d + 1)δ Since ℓ_{p}(im γ) + ℓ_{p}(ker γ) = ℓ_{p}(S′ ⊗ Ω_{S/R}) = δ, combining (a) and (b) gives the desired inequality. Proof of (a): We have α: ker γ \underrightarrow{∼} im α ∩ ker β. Therefore as there is a surjection Ω_{R′/R} → (R′/p^{λ})^{d+1}, and as Ω_{S′/R} can be generated by (d + 1) elements, we have

3See [20, p.420, footnote] or [10, Lemma 1.1]

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ker β ⊃ Ω_{S′/R}[p^{λ}] ≃ (S′/p^{λ})^{d+1}

and so

ker γ ⊃ S′ ⊗_{S} Ω_{S/R}[p^{λ}] ≃ ⨁_{i=0}^{d} S′/p^{min(λ,δ_{i})}.

Therefore

ℓ_{p}(ker γ) ≥ ∑ min(λ, δ_{i}) ≥ min(λ, ∑ δ_{i}) = min(λ, δ).

Proof of (b): Evidently im γ = S′d(S) = S′d(R′S). Now since under the trace form we have 𝔇_{L/K}^{−1} = Hom_{R}(S, R), it follows that

R′ 𝔇_{L/K}^{−1} = Hom_{R′}(R′ ⊗ S, R′) ⊃ Hom_{R′}(S′, R′) = 𝔇_{L′/K′}^{−1}

and so S′ ⊃ R′S ⊃ 𝔇_{L/K}𝔇_{L′/K′}^{−1} = ϖ^{j}S′ say, where ϖ = ϖ_{L′} is a uniformiser and j = e_{L′}(δ − δ′). Therefor we have inclusions

im γ ⊃ S′d(ϖ^{j}S′) ⊃ ϖ^{j}Ω_{S′/R′} = p^{δ−δ′}Ω_{S′/R′} ≃ ⨁_{i=0}^{d} S′/(p^{max(0,δ′_{i}−δ+δ′)})

and therefore

ℓ_{p}(im γ) ≥ ∑_{i=0}^{d}(δ′_{i} − δ + δ′) = (d + 2)δ′ − (d + 1)δ.

1.3 Deep ramification and norm fields

In this section we will work with towers K_{0} ⊂ K_{1} ⊂ … of finite extensions of d-big local fields. If K_{•} = {K_{n}} is such a tower, write K_{∞} = ⋃ K_{n}. We abbreviate 𝔬_{n} = 𝔬_{K_{n}}, ϖ_{n} = ϖ_{K_{n}} and k_{n} = k_{K_{n}}. Define an equivalence relation on towers by setting K_{•} ∼ K′_{•} if there exists r ∈ ℤ such that for every n sufficiently large, K′_{n} = K_{n+r}. We shall say that a tower K_{•} is strictly deeply ramified if there exists n_{0} ≥ 0 and an ideal ξ ⊂ 𝔬_{n_{0}} with 0 < v_{p}(ξ) ≤ 1, such that the following condition holds:

For every n ≥ n_{0}, the extension K_{n+1}/K_{n} has degree p^{d+1}, and there exists a surjection Ω(K_{n+1}/K_{n}) ⟶ ⟶ (𝔬_{n+1}/ξ)^{d+1}.

(1.3.1)

If K_{•} is strictly deeply ramified then so is any equivalent tower (with the same ξ and possible different n_{0}). See 1.3.8 below for some comments on this definition. Let K_{•} be a strictly deeply ramified tower, and (n_{0}, ξ) a pair for which (1.3.1) holds. Then by 1.2.1, for every n ≥ n_{0} we have e(K_{n+1}/K_{n}) = p, and Frobenius induces a surjection f: 𝔬_{n+1}/ξ ⟶ ⟶ 𝔬_{n}/ξ. We can then choose uniformisers ϖ_{n} of K_{n} such that ϖ_{n+1}^{p} ≡ ϖ_{n} (mod ξ) for every n ≥ n_{0}. Define

X^{+} = X^{+}(K_{•}, ξ, n_{0}) := \underleftarrow{lim}_{n≥n_{0}} (𝔬_{n}/ξ, f)

and wite pr_{n}: X^{+} ⟶ ⟶ 𝔬_{n}/ξ for the n^{th} projection in the inverse limit. Set Π = (ϖ_{n} mod ξ) ∈ X^{+}. Let k′ = \underleftarrow{lim}_{n≥n_{0}}(k_{n}, f); since k_{n+1} = k_{n}^{1/p}, the projections pr_{n}: k′ → k_{n} for any

n ≥ n_{0} are isomorphisms. (Note that the residue field k_{∞} of K_{∞} is then the perfect closure (k′)^{1/p^{∞}} of k′.)

Theorem 1.3.2. X(K_{•}, ξ, n_{0}) is a complete discrete valuation ring of characteristic p, with uniformiser Π, and residue field k′. Up to canonical isomorphism (described in the proof below) X^{+}(K_{•}, ξ, n_{0}) depends only on the equivalence class of the tower K_{•}, and not on the choices of ξ and n_{0} satisfying (1.3.1).

Proof. Define a partial order on triples (K_{•}, ξ, n_{0}) satisfying (1.3.1) by setting (K′_{•}, ξ′, n′_{0}) ≥ (K_{•}, ξ, n_{0}) if and only if v_{p}(ξ′) ≤ v_{p}(ξ) and for some r ≥ 0 one has n′_{0} + r ≥ n_{0} and K′_{n} = K_{n+r} for every n ≥ 0. It is obvious that under this order any two triples have an upper bound if and only if the associated towers of extensions are equivalent. If (K′_{•}, ξ′, n′_{0}) ≥ (K_{•}, ξ, n_{0}) and r is as above then there is a canonical map

X^{+}(K_{•}, ξ, n_{0}) → X^{+}(K′_{•}, ξ′, n′_{0})
g: (x_{n})_{n≥n_{0}} ↦ (x_{n+r} mod ξ′)_{n≥n′_{0}}.

If ξ = ξ′, g is obviously an isomorphism. In general we can define a map h in the other direction by

h: (y_{n})_{n≥n′_{0}} ↦ (y_{n+s−r}^{p^{s}})_{n≥n_{0}}

which is well-defined and independent of s for s sufficiently large. Then g and h are mutual inverses. For three triples (K′′_{•}, ξ′′, n′′_{0}) ≥ (K′_{•}, ξ′, n′_{0}) ≥ (K_{•}, ξ, n_{0}) the isomorphisms just described are obviously transitive, so we obtain the desired independence on choices. Truncating K_{•} if necessary we may therefore assume that n_{0} = 0 and ξ = ϖ_{0}. We then have by 1.2.1

X^{+}/(Π^{p^{m}}) = \underleftarrow{lim} 𝔬_{n}/(ϖ_{0}, ϖ_{n}^{p^{m}}) ∼ ; ⟶ ; pr_{m} 𝔬_{m}/(ϖ_{0}).

Therefore \underleftarrow{lim} X^{+}/(Π^{p^{m}}) = \underleftarrow{lim} 𝔬_{m}/(ϖ_{0}) = X^{+}, so X^{+} is Π-adically complete

and separated, and Π is not nilpotent. Since X^{+}/(Π) is a field, X^{+} is therefore a discrete valuation ring with uniformiser Π.

To make the definition of X^{+} truly functorial, we define for an equivalence class K of towers

X_{K}^{+} := \underrightarrow{lim} X^{+}(K_{•}, ξ, n_{0})

where the limit is taken over triples (K_{•}, ξ, n_{0}) with K_{•} ∈ K and (ξ, n_{0}) satisfying (1.3.1), and the transition maps are the isomorphisms g in the preceding proof. We let Π_{K} denote any uniformiser of X_{K}^{+}, and define k_{K} = X_{K}^{+}/(Π_{K}) to be its residue field.

Definition. The field of fractions X_{K} of X_{K}^{+} is the norm field of K.

Of course this is illogical terminology, because when d > 0 this has nothing to do with norms. But when d = 0 it is just the field of norms X_{K}(K_{∞}) for the extension K_{∞}/K in the sense of Fontaine and Wintenberger ([12], [13], and [23] — especially 2.2.3.3), and for d > 0 see also remark 1.3.9 below. Let K_{•} be a tower of d-big local fields, K its equivalence class, and L_{∞}/K_{∞} a finite extension. Then there exists a finite extension L_{0}/K_{0} contained in L_{∞} such that L_{∞} = K_{∞}L_{0}; write L_{n} = K_{n}L_{0}. The equivalence class L of L_{•} depends only on L_{∞}.

Theorem 1.3.3. Let K and L be as above. Then if K is strictly deeply ramified so is L.

Proof. The condition on the extension degrees is clear. By Proposition 1.2.8 with (K, K′, L, L′) = (K_{n}, K_{n+1}, L_{n}, L_{n+1}) we have

δ(L_{n+1}/K_{n+1}) ≤ δ(L_{n}/K_{n}) − \frac{1}{d + 2} min(v_{p}(ξ), δ(L_{n}/K_{n}))

and so δ(L_{n}/K_{n}) → 0 as n → ∞. Using the exact sequences of differentials for the extensions L_{n+1}/L_{n}/K_{n} and L_{n+1}/K_{n+1}/K_{n}, it follows that the annihilators of the kernel and cokernel of the canonical map

𝔬_{L_{n+1}} ⊗_{𝔬_{K_{n+1}}} Ω(K_{n+1}/K_{n}) → Ω(L_{n+1}/L_{n})

have p-adic valuation tending to zero as n → ∞. Therefore L_{•} satisfies (1.3.1) for any ξ′ with 0 < v_{p}(ξ′) < v_{p}(ξ) (and suitable n_{0}).

Theorem 1.3.4. Let K_{•} be strictly deeply ramified, K its equivalence class and L_{∞}/K_{∞} a finite extension.

(i) X_{L} is a finite separable extension of X_{K}. More generally, if L′_{∞}/K_{∞} is another finite extension and τ: L_{∞} → L′_{∞} is a K_{∞}-homomorphism, the maps τ: 𝔬_{L_{n}}/ξ ↪ 𝔬_{L′_{n}}/ξ, for n sufficiently large and v_{p}(ξ) sufficiently small, induce an injection X_{K}(τ): X_{L}^{+} ↪ X_{L′}^{+} which makes X_{L′}/X_{L} a separable extension of degree [L′_{∞} : τL_{∞}].

(ii) The sequences (e(L_{n}/K_{n})), (s(L_{n}/K_{n})) and (f_{0}(L_{n}/K_{n})) are stationary for n sufficiently large. Their limits equal e(X_{L}/X_{K}), s(X_{L}/X_{K}) and f_{0}(X_{L}/X_{K}) respectively.

(iii) There exists a constant c ≥ 0 such that δ(L_{n}/K_{n}) = cp^{−n} for n sufficiently large.

Proof. It suffices in (i) to consider the case of a single extension L_{∞}/K_{∞}. Let m = [L_{∞} : K_{∞}]. Changing ξ and n_{0} if necessary, we can assume that (1.3.1) holds for both K_{•} and L_{•} with the same ξ and n_{0}, and that [L_{n} : K_{n}] = [L_{∞} : K_{∞}] = m for n ≥ n_{0}. Then for every n ≥ n_{0}, 𝔬_{L_{n}}/ξ is a finite flat 𝔬_{n}/ξ-algebra of rank m. Therefore by Nakayama’s lemma X_{L}^{+} is a finite flat X_{K}^{+}-algebra of rank m, so X_{L}/X_{K} is a finite extension of degree m. Consider the discriminant 𝔡 = 𝔡_{X_{L}/X_{K}} ⊂ X_{K}^{+} of X_{L}^{+}/X_{K}^{+}. The projection of 𝔡 to 𝔬_{n}/ξ equals the discriminant of 𝔬_{L_{n}}/ξ over 𝔬_{n}/ξ. Since δ(L_{n}/K_{n}) → 0 the latter is nonzero for n sufficiently large. So X_{L}/X_{K} is separable. Its residue field extension is isomorphic to k_{L_{n}}/k_{n} for n sufficiently large. So the sequences (f_{0}(L_{n}/K_{n})) and (s(L_{n}/K_{n})) are ultimately stationary, hence the same holds for e(L_{n}/K_{n}) = [L_{n} : K_{n}]/f(L_{n}/K_{n}). Let v_{X_{K}}(𝔡) = r; then for n ≥ n_{0}, (ϖ_{n}^{r}) equals the discriminant of 𝔬_{L_{n}}/ξ over 𝔬_{n}/ξ. So for n sufficiently large, v_{p}(ϖ_{n}^{r}) = mδ(L_{n}/K_{n}). Therefore δ(L_{n}/K_{n}) = p^{−n}c where c equals rp^{n}/me_{K_{n}}, which is constant for n sufficiently large.

So if K is strictly deeply ramified, for any finite L_{∞}/K_{∞} we may define

X_{K}^{+}(L_{∞}) := X_{L}^{+}, X_{K}(L_{∞}) := X_{L}

which by the above is a functor from the category of finite extensions of K_{∞} to that of X_{K}.

Theorem 1.3.5. The functor X_{K}(−) defines an equivalence between the category of finite extensions of K_{∞} and the category of finite separable extensions of X_{K}.

Proof. The functor is fully faithful. It is enough to show that if L_{∞}/K_{∞} is a finite Galois extension then any non-trivial σ ∈ Gal(L_{∞}/K_{∞}) induces a nontrivial automorphism X_{K}(σ) of X_{K}(L_{∞}) = X_{L}. In that case since [X_{L} : X_{K}] = [L_{∞} : K_{∞}] it follows that X_{L}/X_{K} is a Galois extension, and that X_{K}(−): Gal(L_{∞}/K_{∞}) \underrightarrow{∼} Gal(X_{L}/X_{K}), from which the fully faithfulness is formal by Galois theory. Assume that X_{K}(σ) = 1. Then replacing σ by a suitable power, we may assume it has prime order. Replacing K_{∞} by the fixed field of σ, and truncating the tower if necessary we may then assume that L_{∞}/K_{∞} is cyclic of prime degree ℓ, with Galois group G say. In this case for n sufficiently large, L_{n}/K_{n} is cyclic of degree ℓ and so 𝔬_{L_{n}} = 𝔬_{K_{n}}[x_{n}] for some x_{n} ∈ 𝔬_{L_{n}}. If g_{n} ∈ 𝔬_{K_{n}}[T] is the minimal polynomial of x_{n} then

𝔇_{L_{n}/K_{n}} = (g′_{n}(x_{n})) = ∏_{1≠σ∈G}(x_{n} − σx_{n}).

So since δ(L_{n}/K_{n}) → 0, it follows that if 1 ≠ σ ∈ G and n is sufficiently large, then σx_{n} ≢ x_{n} (mod ξ). So σ acts nontrivially on 𝔬_{L_{n}}/ξ hence also on X_{L}. The functor is essentially surjective. Using fully faithfulness, it is enough to show that if Y/X_{K} is a finite Galois extension then there exists L_{∞}/K_{∞} and a X_{K}-isomorphism X_{K}(L_{∞}) \underrightarrow{∼} Y. Let Y^{+} ⊂ Y be the valuation ring of Y. Building the extension step-by-step we are reduced to the cases:

(a) Y/X_{K} is unramified. The categories of finite unramified extensions of X_{K} and K_{∞} are equivalent to the categories of finite separable extensions of their respective residue fields k_{K} and k_{∞}. But as k_{∞} is the perfect closure of k_{K} these categories are equivalent.

(b) Y/X_{K} is ramified and of prime degree ℓ. There are two subcases: (b1) e(Y/X_{K}) = ℓ. Then Y^{+} = X_{K}^{+}[Π_{Y}] where the uniformiser Π_{Y} satisfies an Eisenstein polynomial G(T) ∈ X_{K}^{+}[T]. Choose n_{0} such that (1.3.1) holds and v_{p}(ξ) > v_{p}(ϖ_{n_{0}}). For every n ≥ n_{0}, let g_{n} ∈ 𝔬_{n}[T] be any monic polynomial such that ḡ_{n} = pr_{n}(G) ∈ (𝔬_{n}/ξ)[T]. Then g_{n} is an Eisenstein polynomial, and g_{n}(T^{p}) ≡ g_{n+1}(T)^{p} (mod ξ). Fix an algebraic closure \overline{K} of K_{∞} and let 𝔬̄ be its valuation ring.

Claim: There exist n_{1} ≥ n_{0}, ξ′ ∈ 𝔬_{n_{1}} with v_{p}(ξ′) ≤ v_{p}(ξ), and roots x_{n} ∈ 𝔬̄ of g_{n}, such

(i) For every n ≥ n_{1}, x_{n+1}^{p} ≡ x_{n} (mod ξ′)

(ii) If L_{n} := K_{n}(x_{n}) ⊂ \overline{K} then L_{n+1} = K_{n+1}L_{n} for all n ≥ n_{1}.

(iii) If n ≥ n_{1} then (𝔬_{L_{n+1}}/ξ′)^{p} = 𝔬_{L_{n}}/ξ′, and there is an isomorphism of

X_{K}^{+}-algebras

Y^{+} \underrightarrow{∼} \underleftarrow{lim}_{n≥n_{1}}(𝔬_{L_{n}}/ξ′, f)

mapping Π_{Y} to (x_{n} mod ξ′)_{n}.

Granted this claim, L_{∞} := ⋃ L_{n} is an extension with X_{K}(L_{∞}) ≃ Y.

Proof of claim. (i) Let S_{n} = {x_{n,i} | 1 ≤ i ≤ ℓ} ⊂ 𝔬̄ be the set of roots of g_{n}. Then for all n ≥ 0 and all i we have

∏_{j=1}^{ℓ}(x_{n+1,i}^{p} − x_{n,j}) = g_{n}(x_{n+1,i}^{p}) ≡ g_{n+1}(x_{n+1,i})^{p} ≡ 0 (mod ξ).

Choose n_{1} ≥ n_{0} and ξ′ ⊂ 𝔬_{n_{1}} such that 0 < v_{p}(ξ′) ≤ ℓ^{−1}v_{p}(ξ). Then for each i there exists j with x_{n+1,i}^{p} ≡ x_{n,j} (mod ξ′). Choosing such a j for each i then determines a map S_{n+1} → S_{n}, and by compactness \underleftarrow{lim} S_{n} is nonempty. Let (x_{n}) be any element of the inverse limit; then (i) is satisfied. If L_{n} = K_{n}(x_{n}), then [L_{n} : K_{n}] = e(L_{n}/K_{n}) = ℓ. Since it satisfies an Eisenstein polynomial, x_{n} is a uniformiser of L_{n}, and 𝔬_{L_{n}}/ξ′ = (𝔬_{n}/ξ′)[x_{n}] = (𝔬_{n}/ξ′)[T]/ḡ_{n}(T). Therefore for each n there is a unique surjection

f: 𝔬_{L_{n+1}}/ξ′ ⟶ ⟶ 𝔬_{L_{n}}/ξ′  (1.3.6)

which is Frobenius on 𝔬_{n+1}/ξ and maps x_{n+1} to x_{n} (mod ξ′). Let μ_{n}: Y^{+} ⟶ ⟶ 𝔬_{L_{n}}/ξ′ be the map taking Π_{Y} to x_{n}, and whose restriction to X_{K}^{+} is pr_{n}. The different of Y/X_{K} is (G′(Π_{Y})), and it is nonzero since Y/X_{K} is separable. Let r = v_{Y} (G′(Π_{Y})). Then ḡ′_{n}(x_{n}) = μ_{n}(G′(Π_{Y})) equals x_{n}^{r} times a unit. Therefore if n is large enough so that v_{L_{n}}(ξ) > r, we have v_{L_{n}}(g′_{n}(x_{n})) = r. Therefore δ(L_{n}/K_{n}) = v_{p}(g′_{n}(x_{n})) → 0. Order the roots of g_{n} so that x_{n} = x_{n,1}. Since

∏_{i≠1}(x_{n+1}^{p} − x_{n,i}) ≡ ∏_{i≠1}(x_{n} − x_{n,i}) ≡ g′_{n}(x_{n}) (mod ξ)′

it follows that for n sufficiently large, x_{n+1}^{p} is closer to x_{n} than to any of the other roots {x_{n,i} | i ≠ 1} of g_{n}. By Krasner’s lemma, x_{n} ∈ K_{n}(x_{n+1}^{p}), so L_{n} ⊂ L_{n+1} and the map (1.3.6) is induced by the Frobenius endomorphism of 𝔬_{L_{n+1}}/ξ′ (by its uniqueness). We have to check that L_{n+1} = K_{n+1}L_{n} for n sufficiently large. Since [L_{n+1} : K_{n+1}] = ℓ = [L_{n} : K_{n}] it is enough to show that the extensions L_{n}/K_{n} and K_{n+1}/K_{n} are linearly disjoint. If not, since [L_{n} : K_{n}] is prime, there exists a K_{n}-homomorphism τ: L_{n} → K_{n+1}, and so ℓ = p. But as δ(L_{n}/K_{n}) → 0 and Ω(K_{n+1}/K_{n}) surjects onto (𝔬_{n+1}/ξ)^{d+1} this implies that for n sufficiently large, Ω(K_{n+1}/τL_{n}) surjects onto k_{n+1}^{d+1}, which is impossible as [K_{n+1} : τL_{n}] = p^{d}. Finally, making n_{1} sufficiently large, we have a commutative diagram

\xymatrix{X_{K}^{+} \ar@{->>}[d]^{pr_{n+1}} \ar@{^{(}->}[r] & Y^{+} \ar@{->>}[d]_{μ_{n+1}} \ar@{->>}[ddr]^{μ_{n}} \\ 𝔬_{n+1}/ξ′ \ar@{^{(}->}[r] \ar@{->>}[dr]_{f} & 𝔬_{L_{n+1}}/ξ′ \ar@{->>}[dr]_{f} \\  & 𝔬_{n}/ξ′ \ar@{^{(}->}[r] & 𝔬_{L_{n}}/ξ′}  (1.3.7)

where L_{n+1} = K_{n+1}L_{n} for n ≥ n_{1}, inducing a X_{K}^{+}-homorphism

Y^{+} → X_{K}^{+}(L_{∞}) = \underleftarrow{lim}_{n≥n_{1}}(𝔬_{L_{n}}/ξ′, f).

Since Y^{+} and X_{K}^{+}(L_{∞}) are both valuation rings of extensions of X_{K} of the same degree, this is an isomorphism.

(b2) e = 1 and s = 1. Then Y^{+} = X_{K}^{+}[U] for some U ∈ (Y^{+})^{∗}, whose reduction mod Π_{K} generates k_{Y} /k_{K}. As in (b1), let G be the minimal polynomial of U, and get ḡ_{n} ∈ (𝔬_{n}/ξ)[T] be its image, and g_{n} ∈ 𝔬_{n}[T] any monic lift. Then g_{n} is a fake Eisenstein polynomial (cf. §1.1) hence is irreducible; just as above we find roots u_{n} ∈ 𝔬̄ of g_{n} such that u_{n+1}^{p} ≡ u_{n} (mod ξ′) for n sufficiently large and suitable ξ′. The remainder of the argument proceeds exactly as for (b1).

Remark 1.3.8. The condition 1.3.1 is closely related, in the case d = 0, to that of strictly arithmetically profinite extension [23, §1.2.1]. It is possible to weaken the condition without affecting the results: one could instead just require that there exist surjections Ω(K_{n+1}/K_{n}) ⟶ ⟶ (𝔬_{n+1}/ξ_{n+1})^{d+1} where ξ_{n} ⊂ 𝔬_{n} is a sequence of ideals whose p-adic valuations do not tend too rapidly to zero.

Remark 1.3.9. Suppose that K (and therefore also X_{K}) is a (d+1)-dimensional local field. Then, as Fesenko and Zerbes have remarked to the author, local class field theory for higher dimensional local fields [17] gives a reciprocity homomorphism K_{d+1}^{M}(K) → Gal(\overline{K}/K)^{ab}, where K_{∗}^{M}() is Milnor K-theory,

which becomes an isomorphism after passing to a suitable completion { \atop K_{d+1}^{M}(K)}. Therefore there is a commutative diagram

\underleftarrow{lim}_{norms} { \atop K_{d+1}^{M}(K_{n})}  \underrightarrow{∼} { \atop K_{d+1}^{M}(X_{K})}
∥  ∥
\underleftarrow{lim} Gal(\overline{K}/K_{n})^{ab} = Gal(\overline{K}/K_{∞})^{ab} \underrightarrow{∼} Gal(\overline{X_{K}}/X_{K})^{ab}

which may be viewed as the generalisation of the Fontaine-Wintenberger definition (for d = 0) of X_{K} as the inverse limit of the K_{n} with respect to the norm maps.

2 (ϕ, Γ)-modules

2.1 Definitions

We review Fontaine’s definition [11] of the (ϕ, Γ)-module associated to a p-adic representation, in an appropriately axiomatic setting. The key assumptions making the theory possible are (2.1.1) and (2.1.2) below. We begin with a strictly deeply ramified tower K_{•} of d-big local fields (always of characteristic zero) such that K_{n}/K_{0} is Galois for each n, and set K = K_{0}, Γ_{K} = Gal(K_{∞}/K). Fixing an algebraic closure \overline{K} of K containing K_{∞}, write G_{K} = Gal(\overline{K}/K) ⊃ H_{K} = Gal(\overline{K}/K_{∞}). All algebraic extensions of K will be tacitly assumed to be subfields of \overline{K}. Let E_{K} = X_{K} be the norm field of the tower K_{•}, and E_{K}^{+} its valuation ring. To be consistent with the notation established in [8], we write π̄, or when there is no confusion simply π, for a uniformiser of E_{K}. Then E_{K}^{+} is (noncanonically) isomorphic to k_{K}[[π̄]]. For a finite extension L/K, one writes E_{L} for the norm field of the tower LK_{•}, and E for \underrightarrow{lim} E_{L} (the limit over all finite extensions L/K). The group G_{K} then acts continuously (for the valuation topology) on E = E_{K}^{sep}, and this action identifies the subgroup H_{K} with Gal(E/E_{K}). If E is any of these rings of characteristic p, write E^{rad} for the perfect closure \sqrt[p^{∞}]{E} of E, and Ẽ for the completion of E^{rad}. In particular, Ẽ^{+} is the valuation ring of the algebraic closure of E_{K}, and can be alternatively described as \underleftarrow{lim}(𝔬\overline{K}/p, f), also known as R. By continuity the action of G_{K} on E ex-

tends uniquely to a continuous action on E^{rad} and Ẽ, and for any L on has Ẽ_{L} = Ẽ^{H_{L}}. In the theory of (ϕ, Γ)-modules there are two kinds of rings of characteristic zero which appear. The first are those with perfect residue ring, which are completely canonical. These are:

• Ã^{+} = W (Ẽ^{+}) ⊂ Ã = W (Ẽ);

• Ã_{L} = W (Ẽ_{L}), for any finite L/K;

• Ã_{L}^{+} = W (Ẽ_{L}^{+}) = Ã^{+} ∩ Ã_{L}

They carry a unique lifting of Frobenius (namely the Witt vector endomorphism F), and the action of G_{K} on Ẽ defines an action on Ã. The ring Ã has a canonical topology (also called the weak topology) which is the weakest structure of topological ring for which Ã → Ẽ is continuous (for the valuation topology on Ẽ). Equivalently, in terms of the definition of W (Ẽ) as Ẽ^{ℕ} with Witt vector multiplication and addition, it is the product of the valuation topologies on the factors. The G_{K}-action is evidently continuous with respect to the canonical topology. The other natural topology to put on Ã is the p-adic (or strong) topology. The other rings of characteristic zero have imperfect residue rings, and depend on certain choices. Let A_{K}^{+} be a complete regular local ring of dimension 2, together with an isomorphism A_{K}^{+}/(p) ≃ E_{K}^{+}. Such a lift of E_{K}^{+} exists and is unique up to nonunique isomorphism. If C is a p-Cohen ring with residue field k, then any A_{K}^{+} is (non-canonically) isomorphic to C[[π]]. Define A_{K} to be the p-adic completion of (A_{K}^{+})_{(p)}; it is a p-Cohen ring with residue field E_{K}. Fix a principal ideal I = (π) of A_{K}^{+} lifting (π̄) ⊂ E_{K}^{+}. Then A_{K} is the p-adic completion of A_{K}^{+}[1/π]. The essential choice to be made is a lifting ϕ: A_{K}^{+} → A_{K}^{+} of the absolute Frobenius endomorphism of E_{K}^{+}, which is required to satisfy two conditions. The first is simply

ϕ(I) ⊂ I.  (2.1.1)

It is clear that ϕ extends to an endomorphism of A_{K}, whose reduction mod p is the absolute Frobenius of E_{K}. For any finite extension L/K there exists a finite étale extension A_{L}/A_{K}, unique up to unique isomorphism, with residue field E_{L}. Let A\overline{K} = \underrightarrow{lim} A_{L}, the direct limit taken over finite extensions L/K, and let A be the p-adic completion of A\overline{K}. Then A\overline{K} is the maximal unramified extension of A_{K}, and the isomorphism H_{K} ≃ Gal(E/E_{K}) extends to an isomorphism with Aut(A\overline{K}/A_{K}). This in turn extends to a unique action of H_{K} on A, continuous for both the canonical and p-adic topologies, and for any finite L/K one has A^{H_{L}} = A_{L} by the Ax-Sen-Tate theorem [6]. Since A_{L}/A_{K} is étale there is a unique extension of ϕ to an endomorphism of A_{L} whose reduction mod p is Frobenius; by passage to the limit and completion it extends to an endomorphism of A. We use ϕ to denote any of these endomorphisms. The lifting ϕ of Frobenius determines (see [7, Ch,IX, §1, ex.14] and [11, 1.3.2]) a unique embedding

μ_{K}: A_{K} ⟶ W (E_{K})

such that μ∘ϕ = F ∘μ, which maps A_{K}^{+} into W (E_{K}^{+}). We identify A_{K} with its image under this map. An alternative description of μ_{K} is as follows: consider the direct limit

ϕ^{−∞}A_{K} = \underrightarrow{lim}(A_{K}, ϕ)

on which ϕ is an automorphism. Its p-adic completion is a complete unramified DVR of characteristic zero, with perfect residue field E_{K}^{rad}, hence is canonically isomorphic to W (E_{k}^{rad}). Likewise the action of ϕ on A determines an embedding μ: A ↪ W (E), which is uniquely characterised by the same properties as μ_{K}. The embeddings A_{K} ↪ A ↪ W (Ẽ) induce topologies on A_{K} and A. One writes A^{+} = A ∩ Ã^{+}. Then A^{+}/pA^{+} ≃ E^{+} by [11, 1.8.3], and a basis of neighbourhoods of 0 for the canonical topology on A is the collection of A^{+}-submodules

p^{m}A + π^{n}A^{+}, m, n ≥ 0.

The reduction map A → E is H_{K}-equivariant by construction, and so μ is H_{K}-equivariant. The second, and much more serious, condition to be satisfied by ϕ is:

A ⊂ Ã is stable under the action of G_{K}.  (2.1.2)

In particular, A inherits an action of G_{K}, and A_{K} and A_{K}^{+} inherit an action of Γ_{K}, continuous for the canonical topology. A ℤ_{p}-representation of G_{K} is by definition a ℤ_{p}-module of finite type with a continuous action of G_{K}. Assuming (2.1.1) and (2.1.2) above are satisfied, Fontaine’s theory associates to a ℤ_{p}-representation of G_{K} the A_{K}-module of finite type

D(V) = D_{K}(V) := (A ⊗_{ℤ_{p}} V)^{H_{K}} .

The functor D is faithful and exact. The A_{K}-module D(V) has commuting semilinear actions of ϕ and Γ_{K}. Being a finitely-generated A_{K}-module, D(V) has a natural topology (which is the quotient topology for any surjection A_{K}^{d} → D(V)), for which the action of Γ_{K} is continuous. Therefore D(V) has the structure of an étale (ϕ, Γ_{K})-module, and just as in [11] we have:

Theorem 2.1.3. Assume conditions (2.1.1) and (2.1.2) are satisfied. The functor D is an equivalence of categories

(ℤ_{p}-representations of G_{K}) ⟶ (étale (ϕ, Γ_{K})-modules over A_{K})

and an essential inverse is given by D ↦ (A ⊗_{A_{K}} D)^{ϕ=1}.

Lemma 2.1.4. (i) The sequences

0 → ℤ_{p} → A ⟶^{ϕ−1} A → 0  (2.1.5)
0 → ℤ_{p} → A^{+} ⟶^{ϕ−1} A^{+} → 0  (2.1.6)

are exact, and for every n > 0, the map

ϕ − 1: π^{n}A^{+} → π^{n}A^{+}  (2.1.7)

is an isomorphism. (ii) For any n > 0 and for any L/K, the map ϕ − 1: E_{L}^{+} → E_{L}^{+} is an isomorphism.

Proof. It suffices (by passage to the limit) to prove the corresponding statements mod p^{m}. By dévissage it is enough to check them mod p. Therefore (2.1.5), (2.1.6) follow from the Artin–Schreier sequences for E and E^{+}, and (2.1.7) follows from (ii), since A^{+}/pA^{+} = E^{+}. Rewriting the map as π^{n(p−1)}ϕ − 1: E_{L}^{+} → E_{L}^{+}, by Hensel’s lemma it is an isomorphism.

2.2 Cohomology

We assume that we are in the situation of the previous subsection. In particular, we assume that conditions (2.1.1) and (2.1.2) are satisfied. If G is a profinite group and M a topological abelian group with a continuous G-action, by H^{∗}(G, M) we shall always mean continuous group cohomology. Write C^{•}(G, M) for the continuous cochain complex of G with coefficients in M, so that H^{∗}(G, M) = H^{∗}(C^{•}(G, M)). If ϕ ∈ End_{G}(M) write C_{ϕ}^{•}(G, M) for the sim-

ple complex associated to the double complex [C^{•}(G, M) ⟶^{ϕ−1} C^{•}(G, M)]. Write H_{ϕ}^{∗}(G, M) for the cohomology of C_{ϕ}^{•}(G, M), and H_{ϕ}^{∗}(M) for the cohomology of

the complex M ⟶^{ϕ−1} M (in degrees 0 and 1). If H ⊂ G is a closed normal subgroup and M is discrete then there are two Hochschild–Serre spectral sequences converging to H_{ϕ}^{∗}(G, M), whose E_{2}-terms are respectively

H^{a}(G/H, H_{ϕ}^{b}(H, M)) and H_{ϕ}^{a}(G/H, H^{b}(H, M)),

and which reduce when H = {1} and H = G respectively to the long exact sequence

H^{a}(G, M^{ϕ=1}) → H_{ϕ}^{a}(G, M) → H^{a−1}(G, M/(ϕ − 1)) → H^{a+1}(G, M^{ϕ=1}) and the short exact sequences

0 → H^{b−1}(G, M)/(ϕ − 1) → H_{ϕ}^{b}(G, M) → H^{b}(G, M)^{ϕ=1} → 0.

Theorem 2.2.1. Let V be a ℤ_{p}-representation of G_{K}, and set D = D_{K}(V). There are isomorphisms

H^{∗}(G_{K}, V) \underrightarrow{∼} H_{ϕ}^{∗}(Γ_{K}, D)  (2.2.2)
H^{∗}(H_{K}, V) \underrightarrow{∼} H_{ϕ}^{∗}(D)  (2.2.3)

which are functorial in V, and compatible with restriction and corestriction.

Remarks. (i) In the case when K has perfect residue field, and K_{∞} is the cyclotomic ℤ_{p}-extension, we recover Théorème 2.1 of [15], since taking γ to be a topological generator of Γ_{K} ≃ ℤ_{p}, the complex

D \underrightarrow{\binom{ϕ−1}{γ−1}} D ⊕ D ⟶^{(γ−1,1−ϕ)} D

computes H_{ϕ}^{∗}(Γ_{K}, D). (ii) An oversimplified version of the proof runs as follows: from the short exact sequence (2.1.5) we have, tensoring with V and applying the functor RΓ(H_{K}, −), an isomorphism (in an unspecified derived category)

RΓ(H_{K}, V) \underrightarrow{∼} RΓ(H_{K}, A ⊗ V ⟶^{ϕ−1} A ⊗ V).  (2.2.4)

But for i > 0, H^{i}(H_{K}, A⊗V) = 0, and H^{0}(H_{K}, A⊗V) = D, so the right-hand side of (2.2.4) is isomorphic to [D ⟶^{ϕ−1} D]. Applying RΓ(Γ_{K}, −) then would give

RΓ(G_{K}, V) \underrightarrow{∼} RΓ(Γ_{K}, D ⟶^{ϕ−1} D).

Since the formalism of derived categories in continuous cohomology requires extra hypotheses (see for example [16] or [19, Ch.4]) which do not hold in the present situation, we fill in this skeleton by explicit reduction to discrete modules. (Note that in general these Galois cohomology groups will not be of finite type over ℤ_{p}, hence need not commute with inverse limits.)

Proof. We construct a functorial isomorphism (2.2.2); once one knows that it is compatible with restriction, one may obtain (2.2.3) by passage to the limit over finite extensions L/K; alternatively it can be proved directly (and more simply) by the same method as (2.2.2). The compatibility of the constructed isomorphisms with restriction and corestriction is an elementary verification which we leave to the interested reader. Write V_{m} = V/p^{m}V and D_{m} = D/p^{m}D; we have D_{m} = D_{K}(V_{m}) since D_{K} is exact. A basis of neighbourhoods of 0 in D_{m} is given by the open subgroups

D_{m} ∩ (π^{n}A^{+} ⊗ V_{m}) = (π^{n}A^{+} ⊗ V_{m})^{H_{K}}

which are stable under Γ_{K} and ϕ. Write also

D_{m,n} = D_{m}/(π^{n}A^{+} ⊗ V_{m})^{H_{K}}

which is a discrete Γ_{K}-module; we have topological isomorphisms

D_{m} = \underleftarrow{lim}_{n}(D_{m,n}), D = \underleftarrow{lim}_{m}(D_{m})

and H_{ϕ}^{∗}(Γ_{K}, D) is the cohomology of \underleftarrow{lim}_{m,n} C_{ϕ}^{•}(Γ_{K}, D_{m,n}).

From 2.1.4 we obtain for every m, n ≥ 1 a short exact sequence

0 → V_{m} → (A/π^{n}A^{+}) ⊗ V_{m} ⟶^{ϕ−1} (A/π^{n}A^{+}) ⊗ V_{m} → 0

and so the canonical map

C^{•}(G_{K}, V_{m}) → C_{ϕ}^{•}(G_{K}, (A/π^{n}A^{+}) ⊗ V_{m})  (2.2.5)

is a quasi-isomorphism, for every m, n ≥ 1. The inclusion D_{m,n} ⟶ (A/π^{n}A^{+}) ⊗ V_{m} induces a morphism of complexes

α_{m,n}: C_{ϕ}^{•}(Γ_{K}, D_{m,n}) → C_{ϕ}^{•}(G_{K}, (A/π^{n}A^{+}) ⊗ V_{m}).

Passing to the inverse limit and taking cohomology, this together with (2.2.5) defines a functorial map

H_{ϕ}^{∗}(Γ_{K}, D) → H^{∗}(G_{K}, V)  (2.2.6)

whose inverse will be (2.2.2). To prove it is an isomorphism, it is enough to show:

Proposition 2.2.7. For every m ≥ 1, \underleftarrow{lim}_{n}(α_{m,n}) is a quasi-isomorphism.

Proof. First note that the exactness of D implies that there is a short exact sequence

0 → D_{m} → D_{m+1} → D_{1} → 0

which clearly has a continuous set-theoretical splitting (it is enough to give a continuous section of the surjection A_{K} → E_{K} which is easy), so gives rise to a long exact sequence of continuous cohomology. Suppose the result is shown for m = 1. Then (2.2.6) is an isomorphism for every V with pV = 0, and so by the 5-lemma it is an isomorphism for every V of finite length, whence the result holds for all m ≥ 1. So we may assume for the rest of the proof that pV = 0 and m = 1, and therefore replace A by E. Fix a finite Galois extension L/K such that H_{L} acts trivially on V. We then have a natural map

D_{1,n} = \frac{(E ⊗ V)^{H_{K}}}{(π^{n}E^{+} ⊗ V)^{H_{K}}} → \frac{(E ⊗ V)^{H_{L}}}{(π^{n}E^{+} ⊗ V)^{H_{L}}} = E_{L}/π^{n}E_{L}^{+} ⊗ V.

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The map α_{1,n} therefore factors as the composite of two maps

C_{ϕ}^{•}(Γ_{K}, D_{1,n}) ⟶^{β_{n}} C_{ϕ}^{•}(Gal(L_{∞}/K), E_{L}/π^{n}E_{L}^{+} ⊗ V)

⟶^{γ_{n}} C_{ϕ}^{•}(G_{K}, E/π^{n}E^{+} ⊗ V)

which we treat in turn:

(a) γ_{n} is a quasi-isomorphism. We may compute the induced map H^{∗}(γ_{n}) on cohomology using the morphism of associated spectral sequences, which on E_{2}-terms is the map

H^{a}(Gal(L_{∞}/K), H_{ϕ}^{b}(E_{L}/π^{n}E_{L}^{+}) ⊗ V)

→ H^{a}(Gal(L_{∞}/K), H_{ϕ}^{b}(H_{L}, E_{L}/π^{n}E_{L}^{+}) ⊗ V) (2.2.8)

We then have a commutative square (where E is regarded as a discrete H_{L}- module)

H_{ϕ}^{b}(E_{L}) ⟶ H_{ϕ}^{b}(E_{L}/π^{n}E_{L}^{+})
↓  ↓
H_{ϕ}^{b}(H_{L}, E) ⟶ H_{ϕ}^{b}(H_{L}, E/π^{n}E^{+})

in which all the arrows are isomorphisms; in fact by 2.1.4(ii), the horizontal arrows are isomorphisms, and since H^{b}(H_{L}, E) = 0 for b > 0 the same is true of the left vertical arrow. Therefore the maps (2.2.8) are isomorphisms, and hence γ_{n} is a quasi-isomorphism, for every n ≥ 1.

(b) \underleftarrow{lim}(β_{n}) is a quasi-isomorphism. We consider the cohomology of the finite group Δ = Gal(L_{∞}/K_{∞}) acting on the short exact sequence

0 → π^{n}E_{L}^{+} ⊗ V → E_{L} ⊗ V → E_{L}/π^{n}E_{L}^{+} ⊗ V → 0.  (2.2.9)

Lemma 2.2.10. (i) H^{j}(Δ, E_{L} ⊗ V) = 0 for j > 0. (ii) There exists r ≥ 0 such that for all j > 0 and n ∈ ℤ, the group H^{j}(Δ, π^{n}E_{L}^{+} ⊗ V) is killed by π^{r}.

Proof. It is enough to prove (ii) for n = 0 (since π is fixed by Δ) and since E_{L} = \underrightarrow{lim} π^{−n}E_{L}^{+}, (ii) implies (i). It is therefore enough to know that if M

is any E_{L}^{+}-module with a semilinear action of Δ, then there exists r ≥ 0 such that π^{r}H^{j}(Δ, M) = 0 for any j > 0, which is standard.^{4}

To complete the computation of β_{n}, we next recall [16, 1.9] that an inverse system (X_{n}) of abelian groups is ML-zero if for every n there exists r = r(n) ≥ 0 such that X_{n+r} → X_{n} is the zero map. The class of ML-zero inverse systems is a Serre subcategory [16, 1.12]. A morphism (X_{n}) → (Y_{n}) is said to be an ML-isomorphism if its kernel and cokernel are ML-zero, and if this is so, the induced maps

4Let M → N^{•} be the standard resolution. Choose y ∈ E_{L}^{+} such that x = tr_{E_{L}/E_{K}} (y) ≠ 0, and let λ(m) = ∑_{g∈Δ} g(ym). Then the composite (N^{•})^{Δ} ⟶ N^{•} ⟶^{λ} (N^{•})^{Δ} is multiplication by x, hence by passing to cohomology, multiplication by x kills H^{j}(Δ, M) for j > 0.

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\underleftarrow{lim} X_{n} → \underleftarrow{lim} Y_{n}, R^{1}\underleftarrow{lim} X_{n} → R^{1}\underleftarrow{lim} Y_{n}  (2.2.11)

are isomorphisms. This implies that if (f_{n}): (X_{n}^{•}) → (Y_{n}^{•}) is a morphism of inverse systems of complexes with surjective transition maps X_{n+1}^{i} → X_{n}^{i}, Y_{n+1}^{i} → Y_{n}^{i}, then if (H^{∗}(f_{n})): (H^{∗}(X_{n}^{•})) → (H^{∗}(Y_{n}^{•})) is an ML-isomorphism, the map \underleftarrow{lim}(f_{n}): \underleftarrow{lim} X_{n}^{•} → \underleftarrow{lim} Y_{n}^{•} is a quasi-isomorphism. (Consider the induced map between the exact sequences [16, (2.1)] for X_{n}^{•} and Y_{n}^{•}.) From the exact sequence of cohomology of (2.2.9) and the lemma, we deduce that:

• for all j > 0, the inverse system (H^{j}(Δ, E_{L}/π^{n}E_{L}^{+} ⊗ V))_{n} is ML-zero;

• the map of inverse systems

(D_{1,n})_{n} → (H^{0}(Δ, E_{L}/π^{n}E_{L}^{+} ⊗ V))_{n}

is an ML-isomorphism.

We now have a spectral sequence of inverse systems of abelian groups (_{n}E_{2}^{ij})_{n} ⇒ (_{n}E_{∞}^{i+j})_{n} with

nE_{2}^{ij} = H_{ϕ}^{i}(Γ_{K}, H^{j}(Δ, E_{L}/π^{n}E_{L}^{+} ⊗ V))
nE_{∞}^{k} = H_{ϕ}^{k}(Gal(L_{∞}/K), E_{L}/π^{n}E_{L}^{+} ⊗ V).

such that, for all i ≥ 0 and j > 0, the inverse system (_{n}E_{2}^{ij})_{n} are ML-zero. Therefore the edge homomorphism

(_{n}E_{2}^{i0})_{n} → (_{n}E_{∞}^{i})_{n}

is an ML-isomorphism. Moreover for all i ≥ 0 the map of inverse systems

(H_{ϕ}^{i}(Γ_{K}, D_{m,n}))_{n} → (_{n}E_{2}^{i0})_{n}

is an ML-isomorphism, so composing with the edge homomorphism gives an ML-isomorphism

(H_{ϕ}^{i}(Γ_{K}, D_{1,n}))_{n} → (H_{ϕ}^{i}(Gal(L_{∞}/K), E_{L}/π^{n}E_{L}^{+} ⊗ V))_{n}.

Hence \underleftarrow{lim}(β_{n})_{n} is a quasi-isomorphism.

2.3 Kummer towers

Let F be any local field of characteristic 0, with perfect residue field. Set ϖ = ϖ_{F}, k = k_{F}, 𝔬 = 𝔬_{F}. (Later in this section we will require further that F is absolutely unramified.) Let K ⊃ F be any d-big local field such that 𝔬_{K}/𝔬_{F} is formally smooth (i.e., ϖ is a uniformiser of K). Let {t_{α} | 1 ≤ α ≤ d} ⊂ 𝔬_{K}^{∗} be a set of units whose reductions {t̄_{α}} ⊂ k_{K} form a p-basis for k_{K}. Fix an algebraic closure \overline{K} of K. Let (ε_{n})_{n≥0} be a compatible system of primitive p^{n}-th roots of unity in \overline{K}, and for each α let (t_{α,n})_{n≥0} be a compatible system of p^{n}-th roots of t_{α}. Set F_{n} = F (ε_{n}), 𝔬_{n} = 𝔬_{F_{n}}, k_{n} = k_{F_{n}}, K′_{n} = K(t_{1,n}, …, t_{d,n}) and K_{n} = K′_{n}(ε_{n}). The tower {F_{n}} is strictly deeply ramified; choose n_{0} ≥ 0, ξ ∈ K_{n_{0}} with 0 < v_{p}(ξ) ≤ 1, and uniformisers ϖ_{n} ∈ 𝔬_{n} such that ϖ_{n+1}^{p} ≡ ϖ_{n}^{p} (mod ξ) for all n ≥ n_{0}. Let X_{F} be the field of norms of \{F_{n}\} and k_{F} = \underleftarrow{lim}(k_{n}, f) its residue field. Put π̄ = Π_{F}, so that X_{F} ≃ k_{F}[[π̄]], and the isomorphism is canonical once the uniformisers ϖ_{n} are fixed (since k_{F} is perfect). Write for reduction mod ξ. We have 𝔬_{K′_{n}} = 𝔬_{K}[t_{1,n}, …, t_{d,n}] since this ring is a DVR, and so ϖ_{n} satisfies an Eisenstein polynomial over K′_{n} as well as over F. Hence 𝔬_{K_{n}} = 𝔬_{K′_{n}}[ϖ_{n}] = 𝔬_{n} ⊗_{𝔬} 𝔬_{K}[{t_{α,n}}], and so

𝔬_{K_{n}}/ξ = 𝔬_{n}/ξ ⊗_{k} k_{K}[t̄_{1,n}, …, t̄_{d,n}] = 𝔬_{n}/ξ ⊗_{k} k_{K}^{1/p^{n}}

and we have a commutative diagram

[figure: 𝔬_{n+1}/ξ ⊗_{k} k_{K}^{1/p^{n+1}}; 1⊗f^{n+1}; ∼; f; 𝔬_{n+1}/ξ ⊗_{f^{−n−1}, k} k_{K}; f⊗1; 𝔬_{n}/ξ ⊗_{k} k_{K}^{1/p^{n}}; 1⊗f^{n}; ∼; 𝔬_{n}/ξ ⊗_{f^{−n}, k} k_{K}; ≀; k_{n}[ϖ_{n}]/(ϖ_{n}^{rp^{n}}) ⊗_{f^{−n}, k} k_{K}]

Therefore

E_{K}^{+} = X_{K}^{+} = \underleftarrow{lim}_{n≥n_{0}} k_{n}[ϖ_{n}]/(ϖ_{n}^{rp^{n}}) ⊗_{f^{−n}, k} k_{K} = k_{F}[[π̄]] ˆ⊗_{f^{−∞}, k} k_{K}

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where f^{−∞}: k ⟶ k_{F} is the homomorphism making the diagram

[figure: k_{F}; ∼; …; ∼; f; k_{n+1}; ∼; f; k_{n}; k; ∼; f; k; ∼; f^{n}; k; f^{−∞}]

commute. In other words, if we view k_{F} as an extension of k via the map f^{−∞} just defined, we have E_{K}^{+} = k_{F}[[π̄]]ˆ⊗_{k}k_{K}. Set K′_{∞} = ⋃ K′_{n} ⊂ K_{∞}. Define the various Galois groups

Γ_{K} = Gal(K_{∞}/K) = Γ_{F} × Δ_{K/F}
Γ_{F} = Gal(K_{∞}/K′_{∞}) = Gal(F_{∞}/F) ↪ ℤ_{p}^{∗}
Δ_{K/F} = Gal(K_{∞}/F_{∞}) ≃ ℤ_{p}^{d}

acting on K_{∞} as follows: if a ∈ ℤ_{p}^{∗} is the image of γ_{a} ∈ Γ_{F} and \underline{b} ∈ ℤ_{p}^{d} the image of δ\underline{b} ∈ Δ_{K/F} then

γ_{a}: ε_{n} ↦ ε_{n}^{a}  δ\underline{b}: ε_{n} ↦ ε_{n}
t_{α,n} ↦ t_{α,n}  t_{α,n} ↦ ε_{n}^{b_{α}}t_{α,n}.

To be more precise we suppose from now on that F/ℚ_{p} is unramified, so that 𝔬_{n} = 𝔬[ε_{n}], and we may choose ϖ_{n} = ε_{n} − 1. Then the projections k_{F} → k, k_{K} → k_{K} are isomorphisms, and Γ_{K} acts on E_{K}^{+} = k_{K}[[π̄]] as follows: for a ∈ ℤ_{p}^{∗},

γ_{a}: Π ↦ (1 + Π)^{a} − 1, γ_{a} = identity on k_{c}K

and for \underline{b} ∈ ℤ_{p}^{d}, δ\underline{b} is the unique automorphism of E_{K}^{+} whose reduction mod (π̄) is the identity, and which satisfies

δ\underline{b}: π̄ ↦ π̄, t̄_{α} ↦ (1 + π̄)^{b_{α}} t̄_{α}.

Such a unique automorphism exists since k_{K} is formally étale over 𝔽_{p}(t̄_{1}, …, t̄_{d}). To lift to characteristic 0, set A_{K}^{+} = 𝔬_{K}[[π]], with the obvious surjection to E_{K}^{+} = k_{K}[[π̄]]. The lifting ϕ of Frobenius is given as follows: on 𝔬_{K} it is the unique lifting of Frobenius for which ϕ(t_{i}) = t_{i}^{p}; and ϕ(π) = (1 + π)^{p} − 1. It is then immediate that the conditions (2.1.1), (2.1.2) hold, and the action of Γ_{K} on A_{K}^{+} satisfies

γ_{a}: π ↦ (1 + π)^{a} − 1  δ\underline{b}: π ↦ π
γ_{a} = identity on 𝔬_{K}  t_{α} ↦ (1 + π)^{b_{α}}t_{α}.

Remark 2.3.1. There is a natural generalisation of this construction for a Lubin-Tate formal group G over 𝔬_{F} associated to a distinguished polynomial g ∈ 𝔬_{F} [X]. One takes F_{∞}/F to be the Lubin-Tate extension generated by the division points of G, and K′_{n} = K({t_{α,n}}) where g(t_{α,n+1}) = t_{α,n}. Then A_{K}^{+} is the affine algebra of G over 𝔬_{K}; the lifting of Frobenius is given by g. For some details when d = 0, and indications of what does and what does not extend, see Lionel Fourquaux’s Ph.D. thesis [14, §1.4.1].

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[14] L. Fourquaux: Ph.D. thesis, Université Paris 6, December 2005.

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Anthony J. Scholl

DPMMS, University of Cambridge Centre for Mathematical Sciences Wilberforce Road Cambridge CB3 0WB, England a.j.scholl@dpmms.cam.ac.uk 710

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