Compairing Categories of Lubin–Tate $(\varphi _L,\Gamma _L)$-Modules

Abstract

In the Lubin–Tate setting we compare different categories of $(\varphi _L,\Gamma )$-modules over various perfect or imperfect coefficient rings. Moreover, we study their associated Herr-complexes. Finally, we show that a Lubin Tate extension gives rise to a weakly decompleting, but not decompleting tower in the sense of Kedlaya and Liu.

1 Introduction

Since its invention by Fontaine in [17] the concept of $(\varphi ,\Gamma )$-modules (for the p-cyclotomic extension) has become a powerful tool in the study of p-adic Galois representations of local fields. In particular, it could be fruitfully applied in Iwasawa theory [2,3,4, 33, 34, 36,37,38, 48] and in the p-adic local Langlands programme [12]. A good introduction to the subject regarding the state of the art around 2010 can be found in [9, 18].

Afterwards a couple of generalisations have been developed. Firstly, Berger and Colmez [6] as well as Kedlaya, Pottharst and Xiao [25] extended the theory to (arithmetic) families of $(\varphi ,\Gamma )$-modules, in which representations of the absolute Galois group of a local field on modules over affinoid algebras over $\mathbb {Q}_p$ instead of finite dimensional vector spaces are studied. Secondly, parallel to and influenced by Scholze’s point of view of perfectoid spaces as well as the upcoming of the Fargues-Fontaine curve [16] Kedlaya and Liu developed a (geometric) relative p-adic Hodge theory [26, 27], in which the Galois group of a local field is replaced by the étale fundamental group of affinoid spaces over $\mathbb {Q}_p$ thereby extending an earlier approach by Andreatta and Brinon. In particular, Kedlaya and Liu have introduced systematically $(\varphi ,\Gamma )$-moduels over perfect coefficient rings, i.e., for which the Frobenius endomorphism is surjective, and they have studied their decent to imperfect coefficient rings, which is needed for Iwasawa theoretic applications and which generalized the work of Cherbonnier and Colmez [11].

Recently there has been a growing interest and activity in introducing and studying $(\varphi _L,\Gamma _L)$-modules for Lubin–Tate extensions of a finite extension L of $\mathbb {Q}_p$, motivated again by requirements from or potential applications to the p-adic local Langlands programme [8, 13, 19] or Iwasawa theory [7, 35, 40, 43, 44]. The textbook [42] contains a very detailed and thorough approach to the analogue of Fontaine’s original equivalence of categories between Galois representations and étale $(\varphi ,\Gamma )$-modules to the case of Lubin–Tate extensions as had been proposed, but only sketched in [28], see Theorem 4.1. In this setting it has been shown in [30, 31] that - as in the cyclotomic case due to Herr [20] - the Galois cohomology of a L-representation V of the absolute Galois group $G_L$ of L can again be obtained as cohomology of a generalized Herr complex for the $(\varphi _L,\Gamma _L)$-module attached to V, see Theorem 7.1.

The purpose of this article is to spell out in the Lubin–Tate case concretely the various categories of (classical) $(\varphi _L,\Gamma _L)$-modules over perfect and imperfect coefficient rings (analogously to those considered in [26, 27] who do not cover the Lubin–Tate situation) such as ${\textbf{A}}_L, {\textbf{A}}_L^\dagger , \tilde{\textbf{A}}_L, \tilde{\textbf{A}}_L^\dagger , \textbf{B}_L, {\textbf{B}}_L^\dagger , {\textbf{B}}_L, \tilde{\textbf{B}}_L^\dagger , {\mathcal {R}}_L, \tilde{\mathcal {R}}_L$ to be defined in the course of the main text and to compare them among each other. Moreover, we investigate for which versions the generalized Herr complex calculates again the Galois cohomology of a given representation. The results are summarized in diagrams (6) and (7). Finally, we study in the last section how Lubin–Tate extensions fit into Kedlaya’s and Liu’s concept of (weakly) decompleting towers. We show that for $L\ne \mathbb {Q}_p$ they are weakly decompleting, but not decompleting.

See [46] for some results regarding arithmetic families of $(\varphi _L,\Gamma _L)$-modules in the Lubin–Tate setting.

2 Notation

Let $\mathbb {Q}_p \subseteq L \subset \mathbb {C}_p$ be a field of finite degree d over $\mathbb {Q}_p$, $o_L$ the ring of integers of L, $\pi _L \in o_L$ a fixed prime element, $k_L = o_L/\pi _L o_L$ the residue field, $q := |k_L|$ and e the absolute ramification index of L. We always use the absolute value $|\ |$ on $\mathbb {C}_p$ which is normalized by $|\pi _L| = q^{-1}$. We warn the reader, though, that we will use the references [19] and [32] in which the absolute value is normalized differently from this paper by $|p| = p^{-1}$. Our absolute value is the dth power of the one in these references. The transcription of certain formulas to our convention will usually be done silently.

We fix a Lubin–Tate formal $o_L$-module $LT = LT_{\pi _L}$ over $o_L$ corresponding to the prime element $\pi _L$. We always identify LT with the open unit disk around zero, which gives us a global coordinate Z on LT. The $o_L$-action then is given by formal power series $[a](Z) \in o_L[[Z]]$. For simplicity the formal group law will be denoted by $+_{LT}$.

Let $T_{\pi }$ be the Tate module of LT. Then $T_{\pi }$ is a free $o_L$-module of rank one, say with generator $\eta$, and the action of $G_L := {\text {Gal}}(\overline{L}/L)$ on $T_{\pi }$ is given by a continuous character $\chi _{LT} : G_L \longrightarrow o_L^\times$.

For $n \ge 0$ we let $L_n/L$ denote the extension (in $\mathbb {C}_p$) generated by the $\pi _L^n$-torsion points of LT, and we put $L_\infty := \bigcup _n L_n$. The extension $L_\infty /L$ is Galois. We let $\Gamma _L := {\text {Gal}}(L_\infty /L)$ and $H_L := {\text {Gal}}(\overline{L}/L_\infty )$. The Lubin–Tate character $\chi _{LT}$ induces an isomorphism $\Gamma _L \xrightarrow {\cong } o_L^\times$.

Henceforth we use the same notation as in [43]. In particular, the ring endomorphisms induced by sending Z to $[\pi _L](Z)$ are called $\varphi _L$ where applicable; e.g. for the ring $\mathscr {A}_L$ defined to be the $\pi _L$-adic completion of $o_L[[Z]][Z^{-1}]$ or $\mathscr {B}_L := \mathscr {A}_L[\pi _L^{-1}]$ which denotes the field of fractions of $\mathscr {A}_L$. Recall that we also have introduced the unique additive endomorphism $\psi _L$ of $\mathscr {B}_L$ (and then $\mathscr {A}_L$) which satisfies

$$\begin{aligned} \varphi _L \circ \psi _L = \pi _L^{-1} \cdot trace_{\mathscr {B}_L/\varphi _L(\mathscr {B}_L)} \ . \end{aligned}$$

Moreover, projection formula

$$\begin{aligned} \psi _L(\varphi _L(f_1)f_2) = f_1 \psi _L(f_2) \qquad \text {for any }f_i \in \mathscr {B}_L \end{aligned}$$

as well as the formula

$$\begin{aligned} \psi _L \circ \varphi _L = \frac{q}{\pi _L} \cdot {\text {id}}\ \end{aligned}$$

hold. An étale $(\varphi _L,\Gamma _L)$-module M comes with a Frobenius operator $\varphi _M$ and an induced operator denoted by $\psi _M$.

Let $\widetilde{\textbf{E}}^+ := \varprojlim o_{\mathbb {C}_p}/p o_{\mathbb {C}_p}$ with the transition maps being given by the Frobenius $\varphi (a) = a^p$. We may also identify $\widetilde{\textbf{E}}^+$ with $\varprojlim o_{\mathbb {C}_p}/\pi _L o_{\mathbb {C}_p}$ with the transition maps being given by the q-Frobenius $\varphi _q (a) = a^q$. Recall that $\widetilde{\textbf{E}}^+$ is a complete valuation ring with residue field $\overline{\mathbb {F}_p}$ and its field of fractions $\widetilde{\textbf{E}} = \varprojlim \mathbb {C}_p$ being algebraically closed of characteristic p. Let ${\mathfrak {m}}_{\widetilde{\textbf{E}}}$ denote the maximal ideal in $\widetilde{\textbf{E}}^+$.

The q-Frobenius $\varphi _q$ first extends by functoriality to the rings of the Witt vectors $W(\widetilde{\textbf{E}})$ and then $o_L$-linearly to $W(\widetilde{\textbf{E}})_L := W(\widetilde{\textbf{E}}) \otimes _{o_{L_0}} o_L$, where $L_0$ is the maximal unramified subextension of L. The Galois group $G_L$ obviously acts on $\widetilde{\textbf{E}}$ and $W(\widetilde{\textbf{E}})_L$ by automorphisms commuting with $\varphi _q$. This $G_L$-action is continuous for the weak topology on $W(\widetilde{\textbf{E}})_L$ (cf. [42, Lem. 1.5.3]).

By sending the variable Z to $\omega _{LT}\in W(\widetilde{\textbf{E}})_L$ (see directly after [43, Lem. 4.1]) we obtain an $G_L$-equivariant, Frobenius compatible embedding of rings

$$\begin{aligned} \mathscr {A}_L \longrightarrow W(\widetilde{\textbf{E}})_L \ \end{aligned}$$

the image of which we call $\textbf{A}_L$. The latter ring is a complete discrete valuation ring with prime element $\pi _L$ and residue field the image $\textbf{E}_L$ of $k_L ((Z)) \hookrightarrow \widetilde{\textbf{E}}$ sending Z to $\omega :=\omega _{LT} \mod \pi _L.$ We form the maximal integral unramified extension ($=$ strict Henselization) $\textbf{A}_L^{nr}$ of $\textbf{A}_L$ inside $W(\widetilde{\textbf{E}})_L$. Its p-adic completion $\textbf{A}$ still is contained in $W(\widetilde{\textbf{E}})_L$. Note that $\textbf{A}$ is a complete discrete valuation ring with prime element $\pi _L$ and residue field the separable algebraic closure $\textbf{E}_L^{sep}$ of $\textbf{E}_L$ in $\widetilde{\textbf{E}}$. By the functoriality properties of strict Henselizations the q-Frobenius $\varphi _q$ preserves $\textbf{A}$. According to [28, Lemma 1.4] the $G_L$-action on $W(\widetilde{\textbf{E}})_L$ respects $\textbf{A}$ and induces an isomorphism $H_L = \ker (\chi _{LT}) \xrightarrow {\cong } {\text {Aut}}^{cont}(\textbf{A}/\textbf{A}_L)$.

Sometimes we omit the index q, L,  or M from the Frobenius operator.

Finally, for a valued field K we denote as usual by $\hat{K}$ its completion.

3 An Analogue of Tate’s Result

Let $\mathbb {C}_p^\flat$ together with its absolute value $|\cdot |_\flat$ be the tilt of $\mathbb {C}_p$. The aim of this section is to prove an analogue of Tate’s classical result [47, Prop. 10] for $\mathbb {C}_p^\flat$ instead of $\mathbb {C}_p$ itself and in the Lubin Tate situation instead of the cyclotomic one. In the following we always consider continuous group cohomology.

Proposition 1.1

$H^n(H,\mathbb {C}_p^\flat )=0$ for all $n\ge 1$ and $H\subseteq H_L$ any closed subgroup.

Since the proof is formally very similar to that of loc. cit. or [9, Prop. 14.3.2.] we only sketch the main ingredients. To this aim we fix H and write sometimes W for $\mathbb {C}_p^\flat$ as well as $W_{\ge m}:=\{x\in W| |x|_\flat \le \frac{1}{p^m}\}.$

Lemma 1.2

The Tate-Sen axiom (TS1) is satisfied for $\mathbb {C}_p^\flat$ with regard to H, i.e., there exists a real constant $c>1$ such that for all open subgroups $H_1\subseteq H_2$ in H there exists $\alpha \in (\mathbb {C}_p^\flat )^{H_1}$ with $|\alpha |_\flat < c$ and $Tr_{H_2|H_1}(\alpha ):=\sum _{\tau \in H_2|H_1 }\tau (\alpha )=1.$ Moreover, for any sequence $(H_m)_m$ of open subgroups $H_{m+1}\subseteq H_m$ of H there exists a trace compatible system $(y_{H_m})_m$ of elements $y_{H_m}\in (\mathbb {C}_p^\flat )^{H_m}$ with $|y_{H_m}|_\flat < c$ and $Tr_{H|H_m}(y_{H_m} )=1$.

Proof

Note that for a perfect field K (like $(\mathbb {C}_p^\flat )^{H}$) of characteristic p complete for a multiplicative norm with maximal ideal ${\mathfrak {m}}_K$ and a finite extension F one has $\textrm{Tr}_{F/K}({\mathfrak {m}}_F)={\mathfrak {m}}_K$ by [23, Thm. 1.6.4]. Fix some $x\in (\mathbb {C}_p^\flat )^{H}$ with $0<|x|_\flat <1$ and set $c:=|x|_\flat ^{-1}>1.$ Then we find $\tilde{\alpha }$ in the maximal ideal of $(\mathbb {C}_p^\flat )^{H_1}$ with $Tr_{H|H_1}(\tilde{\alpha })=x$ and $\alpha :=(Tr_{H_2|H_1}(\tilde{\alpha }))^{-1}\tilde{\alpha }$ satisfies the requirement as $|Tr_{H_2|H_1}(\tilde{\alpha })|_\flat ^{-1}\le |x|_\flat ^{-1}=c.$

For the second claim we successively choose elements $\tilde{\alpha }_m$ in the maximal ideal of $(\mathbb {C}_p^\flat )^{H_m}$ such that $Tr_{H|H_1}(\tilde{\alpha }_1)=x$ and $Tr_{H_{m+1}|H_m}(\tilde{\alpha }_{m+1})=\tilde{\alpha }_{m}$ for all $m\ge 1.$ Renormalization $\alpha _m:=x^{-1}\tilde{\alpha }_m$ gives the desired system. $\square$

Remark 3.3

Since H is also a closed subgroup of the absolute Galois group $G_L$ of L it possesses a countable fundamental system $(H_m)_m$ of open neighbourhoods of the identity, as for any $n>0$ the local field L of characteristic 0 has only finitely many extensions of degree smaller than n.

Proof

The latter statement reduces easily to finite Galois extensions $L'$ of L, which are known to be solvable, i.e. $L'$ has a series of at most n intermediate fields $L\subseteq L_1\subseteq \cdots \subseteq L_n=L'$ such that each subextension is abelian. Now its known by class field theory that each local field in characteristic 0 only has finitely many abelian extensions of a given degree. $\square$

We write $\mathcal {C}^n(G,V)$ for the abelian group of continuous n-cochains of a profinite group G with values in a topological abelian group V carrying a continuous G-action and $\partial$ for the usual differentials. In particular, we endow $\mathcal {C}^n(H,W)$ with the maximum norm $\Vert -\Vert$ and consider the subspace $\mathcal {C}^n(H,W)^\delta :=\bigcup _{H'\unlhd H \text{ open } }\mathcal {C}^n(H/H',W)\subseteq \mathcal {C}^n(H,W)$ of those cochains with are even continuous with respect to the discrete topology of W.

Lemma 1.4

• (i) The completion of ${\mathcal {C}^n(H,W)^\delta }$ with respect to the maximum norm equals $\mathcal {C}^n(H,W).$

• (ii) There exist $(\mathbb {C}_p^\flat )^H$-linear continuous maps

  $$\begin{aligned} \sigma ^n:\mathcal {C}^n(H,W)\rightarrow \mathcal {C}^{n-1}(H,W) \end{aligned}$$

  satisfying $\Vert f-\partial \sigma ^n f\Vert \le c\Vert \partial f\Vert .$

Proof

Since the space $\mathcal {C}^n(H,W)$ is already complete we only have to show that an arbitrary cochain f in it can be approximated by a Cauchy sequence $f_m$ in $\mathcal {C}^n(H,W)^\delta$. To this end we observe that, given any m, the induced cochain $H^n\xrightarrow {f}W\xrightarrow {pr_m} W/W_{\ge m}$ comes, for some open normal subgroup $H_m$, from a cochain in $\mathcal {C}^n(H/H_m,W/W_{\ge m}),$ which in turn gives rise to $f_m \in \mathcal {C}^n(H,W)^\delta$ when composing with any set theoretical section $W/W_{\ge m}\xrightarrow {s_m}W$ of the canonical projection $W\xrightarrow {pr_m} W/W_{\ge m}.$ Note that $s_m$ is automatically continuous, since $W/W_{\ge m}$ is discrete. By construction we have $\Vert f-f_m\Vert \le \frac{1}{p^m}$ and $(f_m)_m$ obviously is a Cauchy sequence. This shows (i).

For (ii) recall from Lemma 3.2 together with Remark 3.3 the existence of a trace compatible system $(y_{H'})_{H'}$ of elements $y_{H'}\in (\mathbb {C}_p^\flat )^{H'}$ with $|y_{H'}|_\flat < c$ and $Tr_{H|H'}(y_{H'} )=1$, where $H'$ runs over the open normal subgroups of H. Now we first define $(\mathbb {C}_p^\flat )^H$-linear maps

$$\begin{aligned} \sigma ^n: \mathcal {C}^n(H,W)^\delta \rightarrow \mathcal {C}^{n-1}(H,W) \end{aligned}$$

satisfying $\Vert f-\partial \sigma ^n f\Vert \le c\Vert \partial f\Vert$ and $\Vert \sigma ^n f\Vert \le c \Vert f\Vert$ by setting for $f\in \mathcal {C}^n(H/H',W)$

$$\begin{aligned} \sigma ^n(f):=y_{H'}\cup f \end{aligned}$$

(by considering $y_{H'}$ as a $-1$-cochain), i.e.,

$$\begin{aligned} \sigma ^n(f)(h_1,\ldots , h_{n-1})=(-1)^n\sum _{\tau \in H/H'} (h_1 \ldots h_{n-1}\tau )(y_{H'}) f(h_1,\ldots , h_{n-1},\tau ). \end{aligned}$$

The inequality $\Vert y_{H'}\cup f\Vert \le c\Vert f\Vert$ follows immediately from this description, see the proof of [9, Lem. 14.3.1.]. Upon noting that $\partial y_{H'}=Tr_{H|H'}(y_{H'})=1$, the Leibniz rule for the differential $\partial$ with respect to the cup-product then implies that

$$\begin{aligned} f-\partial (y_{H'}\cup f)=y_{H'}\cup \partial f, \end{aligned}$$

hence

$$\begin{aligned} \Vert f-\partial (y_{H'}\cup f)\Vert \le c \Vert \partial f\Vert \end{aligned}$$

by the previous inequality, see again (loc. cit.). In order to check that this map $\sigma ^n$ is well defined we assume that f arises also from a cochain in $\mathcal {C}^n(H/H'',W)$. Since we may make the comparison within $\mathcal {C}^n(H/(H'\cap H''),W)$ we can assume without loss of generality that $H''\subseteq H'$. Then

$$\begin{aligned}&(y_{H''}\cup f)( h_1,\ldots , h_{n-1})\\&\quad =(-1)^n\sum _{\tau \in H/H''} (h_1 \ldots h_{n-1}\tau )(y_{H''}) f(h_1,\ldots , h_{n-1},\tau ) \\&\quad =(-1)^n\sum _{\tau \in H/H'} \left( h_1 \ldots h_{n-1}\sum _{\tau '\in H'/H''}\tau '\right) (y_{H''}) f(h_1,\ldots , h_{n-1},\tau )\\&\quad =(-1)^n\sum _{\tau \in H/H'} \left( h_1 \ldots h_{n-1}\right) (\sum _{\tau '\in H'/H''}\tau '(y_{H''})) f(h_1,\ldots , h_{n-1},\tau )\\&\quad =(-1)^n\sum _{\tau \in H/H'} \left( h_1 \ldots h_{n-1}\right) ( y_{H'}) f(h_1,\ldots , h_{n-1},\tau )\\&\quad = (y_{H'}\cup f)( h_1,\ldots , h_{n-1}) \end{aligned}$$

using the trace compatibility in the fourth equality. Finally the inequality $\Vert \sigma ^n f\Vert \le c \Vert f\Vert$ implies that $\sigma ^n$ is continuous on $\mathcal {C}^n(H,W)^\delta$ and therefore extends continuously to its completion $\mathcal {C}^n(H,W)$. $\square$

The proof of Prop.  3.1 is now an immediate consequence of Lemma  3.4(ii).

4 The Functors D, $\tilde{D}$ and $\tilde{D}^\dagger$

Let ${\text {Rep}}_{o_L}(G_L),$ ${\text {Rep}}_{o_L,f}(G_L)$ and ${\text {Rep}}_{L}(G_L)$ denote the category of finitely generated $o_L$-modules, finitely generated free $o_L$-modules and finite dimensional L-vector spaces, respectively, equipped with a continuous linear $G_L$-action. The following result is established in [28, Thm. 1.6] (see also [42, Thm. 3.3.10]) and [43, Prop. 4.4 (ii)].

Theorem 1.5

The functors

$$\begin{aligned} T \longmapsto D(T) := (\textbf{A} \otimes _{o_L} T)^{H_{L}} \qquad \text {and}\qquad M \longmapsto (\textbf{A} \otimes _{\textbf{A}_L} M)^{\varphi _q \otimes \varphi _M =1} \end{aligned}$$

are exact quasi-inverse equivalences of categories between ${\text {Rep}}_{o_L}(G_L)$ and the category ${\mathfrak {M}}^{et}(\textbf{A}_L)$ of finitely generated étale $(\varphi _L,\Gamma _L)$-modules over $\textbf{A}_L.$ Moreover, for any T in ${\text {Rep}}_{o_L}(G_L)$ the natural map

$$\begin{aligned} \textbf{A} \otimes _{\textbf{A}_L} D(T) \xrightarrow {\; \cong \;} \textbf{A} \otimes _{o_L} T \end{aligned}$$

(1)

is an isomorphism (compatible with the $G_L$-action and the Frobenius on both sides).

In the following we would like to establish a version of the above for $\tilde{\textbf{A}}$ and prove similar properties for it. In the classical situation such versions have been studied by Kedlaya et al. using the unramified rings of Witt vectors W(R). In our Lubin–Tate situation we have to work with ramified Witt vectors $W(R)_L$. Many results and their proofs transfer almost literally from the classical setting. Often we will try to at least sketch the proofs for the convenience of the reader, but when we just quote results from the classical situation, e.g. from [26], this usually means that the transfer is purely formal.

We start defining $\tilde{\textbf{A}}:=W(\mathbb {C}_p^\flat )_L$ and

$$\begin{aligned} \tilde{\textbf{A}}^\dagger :=\left\{ x=\sum _{n\ge 0}\pi _L^n[x_n]\in \tilde{\textbf{A}}: |\pi _L^{n}\Vert x_n|_\flat ^r \xrightarrow {n \rightarrow \infty } 0 \text{ for } \text{ some } r>0\right\} \end{aligned}$$

as well as $\tilde{D}(T):=(\tilde{\textbf{A}}\otimes _{o_L} T)^{H_L}$ and $\tilde{D}^\dagger (T):=(\tilde{\textbf{A}}^\dagger \otimes _{o_L} T)^{H_L}.$

More generally, let K be any perfectoid field containing L and let $K^\flat$ denote its tilt. For $r>0$ let $W^r(K^\flat )_L$ be the set of $x=\sum _{n=0}^{\infty } \pi _L^n[x_n]\in W(K^\flat )_L$ such that $|\pi _L|^n|x_n|^r_\flat$ tends to zero as n goes to $\infty .$ This is a subring by [26, Prop. 5.1.2] on which the function

$$\begin{aligned} |x|_{r}:=\sup _n \{|\pi _L^{n}\Vert x_n|^r_\flat \}=\sup _n \{q^{-n}| x_n|^r_\flat \} \end{aligned}$$

is a complete multiplicative norm; it extends multiplicatively to $W^r(K^\flat )_L[\frac{1}{\pi _L}].$ Furthermore, $W^\dagger (K^\flat )_L:=\bigcup _{r>0}W^r(K^\flat )_L$ is a henselian discrete valuation ring by [21, Lem. 2.1.12], whose $\pi _L$-adic completion equals $W(K^\flat )_L$ since they coincide modulo $\pi _L^n$. Then $\tilde{\textbf{A}}^{\dagger }=W^\dagger (\mathbb {C}_p^\flat )_L$, and we write $\tilde{\textbf{A}}_L$ and $\tilde{\textbf{A}}_L^{\dagger }$ for $W(\hat{L}_\infty ^\flat )_L$ and $W^\dagger (\hat{L}_\infty ^\flat )_L$, respectively. We set $\tilde{\textbf{B}}_L=\tilde{\textbf{A}}_L[\frac{1}{\pi _L}]$, $\tilde{\textbf{B}}=\tilde{\textbf{A}}[\frac{1}{\pi _L}]$, $\tilde{\textbf{B}}_L^{\dagger }=\tilde{\textbf{A}}_L^{\dagger }[\frac{1}{\pi _L}]$ and $\tilde{\textbf{B}}^{\dagger }=\tilde{\textbf{A}}^{\dagger }[\frac{1}{\pi _L}]$ for the corresponding fields of fractions.

Remark 4.2

By the Ax-Tate-Sen theorem [1] and since $\mathbb {C}_p^\flat$ is the completion of an algebraic closure $\overline{\hat{L}_\infty ^\flat }$ he have that $(\mathbb {C}_p^\flat )^H=((\overline{\hat{L}_\infty ^\flat })^H)^\wedge$ for any closed subgroup $H\subseteq H_L$, in particular $(\mathbb {C}_p^\flat )^{H_L} = \hat{L}_\infty ^\flat$. As completion of an algebraic extension of the perfect field $\hat{L}_\infty ^\flat$ the field $(\mathbb {C}_p^\flat )^H$ is perfect, too. Moreover, we have $\tilde{\textbf{A}}^{H_L}=\tilde{\textbf{A}}_L$, $(\tilde{\textbf{A}}^\dagger )^{H_L}=\tilde{\textbf{A}}_L^\dagger$ and analogously for the rings $\tilde{\textbf{B}}$ and $\tilde{\textbf{B}}^{\dagger }$. It also follows that $\tilde{\textbf{A}}$ is the $\pi _L$-adic completion of a maximal unramified extension of $\tilde{\textbf{A}}_L$.

Lemma 1.7

The rings $\textbf{A}_L$ and $\textbf{A}$ embed into $\tilde{\textbf{A}}_L$ and $\tilde{\textbf{A}}$, respectively.

Proof

The embedding $\textbf{A}_L \hookrightarrow \tilde{\textbf{A}}_L$ is explained in [42, p. 94]. Moreover, $\textbf{A}$ is the $\pi _L$-adic completion of the maximal unramified extension of $\textbf{A}_L$ inside $\tilde{\textbf{A}} = W(\mathbb {C}_p^\flat )_L$ (cf. [42, §3.1]). $\square$

On $\tilde{\textbf{A}}=W(\mathbb {C}_p^\flat )_L$ the weak topology is defined to be the product topology of the valuation topologies on the components $\mathbb {C}_p^\flat .$ The induced topology on any subring R of it is also called weak topology of R. If M is a finitely generated R-module, then we call the canonical topology of M (with respect to the weak topology of R) the quotient topology with respect to any surjection $R^n\twoheadrightarrow M$ where the free module carries the product topology; this is independent of any choices. We recall that a $(\varphi _L,\Gamma _L)$-module M over $R\in \{{\textbf{A}}_L, \tilde{\textbf{A}}_L, \tilde{\textbf{A}}_L^\dagger \}$ is a finitely generated R-module M together with

–:

a $\Gamma _L$-action on M by semilinear automorphisms which is continuous for the weak topology and

–:

a $\varphi _L$-linear endomorphism $\varphi _M$ of M which commutes with the $\Gamma _L$-action.

We let ${\mathfrak {M}}(R)$ denote the category of $(\varphi _L,\Gamma _L)$-modules M over R. Such a module M is called étale if the linearized map

$$\begin{aligned} \varphi _M^{lin} : R \otimes _{R,\varphi _L} M&\xrightarrow {\; \cong \;} M \\ f \otimes m&\longmapsto f \varphi _M (m) \end{aligned}$$

is bijective. We let ${\mathfrak {M}}^{\acute{e}t}(R)$ denote the full subcategory of étale $(\varphi _L,\Gamma _L)$-modules over R.

Definition 1.8

For $*={\textbf{B}}_L, \tilde{\textbf{B}}_L, \tilde{\textbf{B}}_L^\dagger$ we write ${\mathfrak {M}}^{\acute{e}t}(*):= {\mathfrak {M}}^{\acute{e}t}(*')\otimes _{o_L}L$ with $*'={\textbf{A}}_L, \tilde{\textbf{A}}_L, \tilde{\textbf{A}}_L^\dagger$, respectively, and call the objects étale $(\varphi _L,\Gamma _L)$-modules over $*.$

Lemma 1.9

Let G be a profinite group and $R\rightarrow S$ be a topological monomorphism of topological $o_L$-algebras, for which there exists a system of open neighbourhoods of 0 consisting of $o_L$-submodules. Consider a finitely generated R-module M, for which the canonical map $M\rightarrow S\otimes _R M$ is injective (e.g. if S is faithfully flat over R or M is free), and endow it with the canonical topology with respect to R. Assume that G acts continuously, $o_L$-linearly and compatible on R and S as well as continuously and R-semilinearly on M. Then the diagonal G-action on $S\otimes _R M$ is continuous with regard to the canonical topology with respect to S.

Proof

Imitate the proof of [42, Lem. 3.1.11]. $\square$

Proposition 1.10

The canonical map

$$\begin{aligned} \tilde{\textbf{A}}_L\otimes _{ {\textbf{A}}_L} {D}(T)\xrightarrow {\cong } \tilde{D}(T) \end{aligned}$$

(2)

is an isomorphism and the functor $\tilde{D}(-):{\text {Rep}}_{o_L}(G_L) \rightarrow {\mathfrak {M}}^{\acute{e}t}(\tilde{\textbf{A}}_L)$ is exact. Moreover, we have a comparison isomorphism

$$\begin{aligned} \tilde{\textbf{A}}\otimes _{ \tilde{\textbf{A}}_L} \tilde{D}(T) \xrightarrow {\cong } \tilde{\textbf{A}}\otimes _{o_L} T. \end{aligned}$$

(3)

Proof

The isomorphism (2) implies formally the isomorphism (3) after base change of the comparison isomorphism (1). Secondly, the isomorphism (2), resp. (3), implies easily that $\tilde{D}(T)$ is finitely generated, resp. étale. Thirdly, since the ring extension $\tilde{\textbf{A}}_L/ { {\textbf{A}}_L}$ is faithfully flat as local extension of (discrete) valuation rings, the exactness of $\tilde{D}$ follows from that of D. Moreover, the isomorphism (2) implies by Lemma 4.5 that $\Gamma _L$ acts continuously on $\tilde{D}(T)$, i.e., the functor $\tilde{D}$ is well-defined. Thus we only have to prove that

$$\begin{aligned} \tilde{\textbf{A}}_L\otimes _{ {\textbf{A}}_L} (\textbf{A}\otimes _{ o_L}T)^{H_L}\xrightarrow {\cong } (\tilde{\textbf{A}}\otimes _{ o_L}T)^{H_L} \end{aligned}$$

is an isomorphism. To this aim let us assume first that T is finite. Then we find an open normal subgroup $H\unlhd H_L$ which acts trivially on T. Application of the subsequent Lemma 4.7 to $M=(\textbf{A}\otimes _{ o_L}T)^{H}$ and $G=H_L/H$ interprets the left hand side as $\left( \tilde{\textbf{A}}_L\otimes _{ {\textbf{A}}_L}(\textbf{A}\otimes _{ o_L}T)^{H} \right) ^{H_L/H}$ while the right hand side equals $\left( (\tilde{\textbf{A}}\otimes _{ o_L}T)^{H} \right) ^{H_L/H}$. Hence it suffices to establish the isomorphism

$$\begin{aligned} \tilde{\textbf{A}}_L\otimes _{ {\textbf{A}}_L} (\textbf{A}\otimes _{ o_L}T)^{H}\xrightarrow {\cong } (\tilde{\textbf{A}}\otimes _{ o_L}T)^H. \end{aligned}$$

By Lemma 4.8 below this is reduced to showing that the canonical map

$$\begin{aligned} \tilde{\textbf{A}}_L\otimes _{ {\textbf{A}}_L} \textbf{A}^{H}\otimes _{ o_L}T\xrightarrow {\cong } \tilde{\textbf{A}}^H\otimes _{ o_L}T \end{aligned}$$

is an isomorphism, which follows from Lemma 4.9 below. Finally let T be arbitrary. Then we have isomorphisms

$$\begin{aligned} \tilde{\textbf{A}}_L\otimes _{ {\textbf{A}}_L} {D}(T)&\cong \tilde{\textbf{A}}_L\otimes _{ {\textbf{A}}_L} \varprojlim _{n}{D}(T/\pi _L^nT) \\&\cong \tilde{\textbf{A}}_L\otimes _{ {\textbf{A}}_L} \varprojlim _{n}{D}(T)/\pi _L^n{D}(T) \\&\cong \varprojlim _{n}\tilde{\textbf{A}}_L\otimes _{ {\textbf{A}}_L}{D}(T)/\pi _L^n{D}(T) \\&\cong \varprojlim _{n}\tilde{\textbf{A}}_L\otimes _{ {\textbf{A}}_L}{D}(T/\pi _L^n T) \\&\cong \varprojlim _{n} \tilde{D}(T/\pi _L^n T) \\&\cong \tilde{D}(T), \end{aligned}$$

where we use for the second and fourth equation exactness of D, for the second last one the case of finite T and for the first, third and last equation the elementary divisor theory for the discrete valuation rings $o_L$, $\textbf{A}_L$ and $\tilde{\textbf{A}}_L,$ respectively. $\square$

Lemma 1.11

Let $A\rightarrow B$ be a flat extension of rings and M an A-module with an A-linear action by a finite group G. Then $B\otimes _A M$ carries a B-linear G-action and we have

$$\begin{aligned} \left( B\otimes _A M\right) ^G=B\otimes _A M^G. \end{aligned}$$

Proof

Apply the exact functor $B\otimes _A -$ to the exact sequence

which gives the desired description of $\left( B\otimes _A M\right) ^G.$ $\square$

Lemma 1.12

Let A be $\textbf{A}$, $\textbf{A}_L^{nr}$, $\tilde{\textbf{A}}^\dagger$ or $\tilde{\textbf{A}}$ and T be a finitely generated $o_L$-module with trivial action by an open subgroup $H\subseteq H_L$. Then $(A\otimes _{o_L}T)^H=A^H\otimes _{o_L} T.$ Moreover, $\textbf{A}^{H}$ and $\tilde{\textbf{A}}^{H}$ are free $\textbf{A}_{L}$- and $\tilde{\textbf{A}}_{L}$-modules of finite rank, respectively.

Proof

Since $T\cong \bigoplus _{i=1}^r o_L/\pi _L^{n_i}o_L$ with $n_i \in \mathbb {N} \cup \{\infty \}$ we may assume that $T=o_L/\pi _L^no_L$ for some $n \in \mathbb {N} \cup \{\infty \}$. We then we have to show that

$$\begin{aligned} (A/\pi _L^n A)^H=&A^H/\pi _L^nA^H \end{aligned}$$

(4)

For $n = \infty$ there is nothing to prove.

The case $n = 1$: First of all we have $\textbf{A}/\pi _L\textbf{A} =\textbf{A}_L^{nr}/\pi _L\textbf{A}_L^{nr}=\textbf{E}_L^{sep}$. On the other hand, by the Galois correspondence between unramified extensions and their residue extensions, we have that $(\textbf{E}_L^{sep})^H$ is the residue field of $(\textbf{A}_L^{nr})^{H}$. Hence the case $n=1$ holds true for $A = \textbf{A}_L^{nr}$. After having finished all cases for $A = \textbf{A}_L^{nr}$ we will see at the end of the proof that $(\textbf{A}_L^{nr})^{H} =\textbf{A}^{H}$. Therefore the case $n=1$ for $A = \textbf{A}$ will be settled, too.

For $A = \tilde{\textbf{A}}$ we only need to observe that $\tilde{\textbf{A}}/\pi _L\tilde{\textbf{A} } =W(\mathbb {C}_p^\flat )_L/\pi _L W(\mathbb {C}_p^\flat )_L=\mathbb {C}_p^\flat$ and that $(\mathbb {C}_p^\flat )^H$ is the residue field of $(W(\mathbb {C}_p^\flat )_L)^{H}=W((\mathbb {C}_p^\flat )^{H})_L.$

For $A = \tilde{\textbf{A}}^\dagger$ we argue by the following commutative diagram

The case $1< n < \infty$: This follows by induction using the commutative diagram with exact lines

in which the outer vertical arrows are isomorphism by the case $n=1$ and the induction hypothesis.

Finally we can check, using the above equality (4) for $A = \textbf{A}_L^{nr}$ in the third equation:

$$\begin{aligned} \textbf{A}^{H}&= \left( \varprojlim _{n} \textbf{A}_L^{nr}/\pi _L^n\textbf{A}_L^{nr}\right) ^H\\&= \varprojlim _{n} \left( \textbf{A}_L^{nr}/\pi _L^n\textbf{A}_L^{nr}\right) ^H \\&= \varprojlim _{n} \left( \textbf{A}_L^{nr})^H/\pi _L^n(\textbf{A}_L^{nr}\right) ^H \\&= (\textbf{A}_L^{nr})^H. \end{aligned}$$

Note that $(\textbf{A}_L^{nr})^H$ is a finite unramified extension of $\textbf{A}_L$ and therefore is $\pi _L$-adically complete. We also see that $\textbf{A}^{H}$ is a free $\textbf{A}_{L}$-module of finite rank. Similarly, $W(\mathbb {C}_p^\flat )_L^{H} \cong (W(\hat{L}_\infty ^\flat )_L^{nr})^H$ is a free $W(\hat{L}_\infty ^\flat )_L$-module of finite rank. $\square$

Lemma 1.13

For any open subgroup H of $H_L$ the canonical maps

$$\begin{aligned} W(\hat{L}_\infty ^\flat )_L\otimes _{ {\textbf{A}}_L} \textbf{A}^{H}&\xrightarrow {\cong } W((\mathbb {C}_p^\flat )^H)_L, \\ W(\hat{L}_\infty ^\flat )_L\otimes _{ \tilde{\textbf{A}}_L^\dagger } (\tilde{\textbf{A}}^\dagger )^{H}&\xrightarrow {\cong } W((\mathbb {C}_p^\flat )^H)_L \end{aligned}$$

are isomorphisms.

Proof

We begin with the first isomorphism. Since $\textbf{A}^{H}$ is finitely generated free over $\textbf{A}_{L}$ by Lemma 4.8, we have

$$\begin{aligned} W(\hat{L}_\infty ^\flat )_L\otimes _{ {\textbf{A}}_L} \textbf{A}^{H} \cong \left( \varprojlim _{n} W_n(\hat{L}_\infty ^\flat )_L\right) \otimes _{ {\textbf{A}}_L} \textbf{A}^{H} \cong \varprojlim _{n} \left( W_n(\hat{L}_\infty ^\flat )_L\otimes _{ {\textbf{A}}_L} \textbf{A}^{H}\right) . \end{aligned}$$

It therefore suffices to show the corresponding assertion for Witt vectors of finite length:

$$\begin{aligned} W_n(\hat{L}_\infty ^\flat )_L\otimes _{ {\textbf{A}}_L} \textbf{A}^{H}/ \pi _L^n \textbf{A}^H = W_n(\hat{L}_\infty ^\flat )_L\otimes _{ {\textbf{A}}_L} \textbf{A}^{H} \xrightarrow {\cong } W_n((\mathbb {C}_p^\flat )^H)_L . \end{aligned}$$

To this aim we first consider the case $n=1$. From (4) we know that $\textbf{A}^{H}/ \pi _L^n \textbf{A}^H = (\textbf{E}_L^{sep})^H$. Hence we need to check that

$$\begin{aligned} \hat{L}_\infty ^\flat \otimes _{\textbf{E}_L} (\textbf{E}_L^{sep})^H \xrightarrow {\cong } (\mathbb {C}_p^\flat )^H \end{aligned}$$

is an isomorphism. Since the perfect hull $\textbf{E}_L^{perf}$ of $\textbf{E}_L$ (being purely inseparable and normal) and $(\textbf{E}_L^{sep})^H$ (being separable) are linear disjoint extensions of $\textbf{E}_L$ their tensor product is equal to the composite of fields $\textbf{E}_L^{perf} (\textbf{E}_L^{sep})^H$ (cf. [10, Thm. 5.5, p. 188]), which moreover has to have degree $[H_L : H]$ over $\textbf{E}_L^{perf}$. Since the completion of the tensor product is $\hat{L}_\infty ^\flat \otimes _{\textbf{E}_L} (\textbf{E}_L^{sep})^H$, we see that the completion of the field $\textbf{E}_L^{perf} (\textbf{E}_L^{sep})^H$ is the composite of fields $\hat{L}_\infty ^\flat (\textbf{E}_L^{sep})^H$, which has degree $[H_L : H]$ over $\hat{L}_\infty ^\flat$. But $\hat{L}_\infty ^\flat (\textbf{E}_L^{sep})^H \subseteq (\mathbb {C}_p^\flat )^H$. By the Ax-Tate-Sen theorem $(\mathbb {C}_p^\flat )^H$ has also degree $[H_L : H]$ over $\hat{L}_\infty ^\flat$. Hence the two fields coincide, which establishes the case $n=1$.

The commutative diagram

shows that also the lower map is an isomorphism. Using that Verschiebung V on $W_n((\mathbb {C}_p^\flat )^H)_L$ and $W_n(\hat{L}_\infty ^\flat )_L$ is additive and satisfies the projection formula $V^m(x)\cdot y=V^m(x\cdot \varphi _q^m(y))$ we see that we obtain a commutative exact diagram

from which the claim follows by induction because the outer vertical maps are isomorphisms by the above and the induction hypothesis. Here the first non-trivial horizontal morphisms map onto the highest Witt vector component.

The second isomorphism is established as follows: We choose a subgroup $N \subseteq H \subseteq H_L$ which is open normal in $H_L$ and obtain the extensions of henselian discrete valuation rings

$$\begin{aligned} \tilde{\textbf{A}}^\dagger _L \subseteq (\tilde{\textbf{A}}^\dagger )^H = W^\dagger ((\mathbb {C}_p^\flat )^H)_L \subseteq (\tilde{\textbf{A}}^\dagger )^N = W^\dagger ((\mathbb {C}_p^\flat )^N)_L . \end{aligned}$$

The corresponding extensions of their field of fractions

$$\begin{aligned} \tilde{\textbf{B}}^\dagger _L \subseteq E := (\tilde{\textbf{A}}^\dagger )^H [\tfrac{1}{\pi _L}] \subseteq F := (\tilde{\textbf{A}}^\dagger )^N [\tfrac{1}{\pi _L}] \end{aligned}$$

satisfy $F^{H/N} = E$ and $F^{H_L/N} = \tilde{\textbf{B}}^\dagger _L$. Hence F/E and $F/\tilde{\textbf{B}}^\dagger _L$ are Galois extensions of degree [H : N] and $[H_L:N]$, respectively. It follows that $E/\tilde{\textbf{B}}^\dagger _L$ is a finite extension of degree $[H_L:H]$. The henselian condition then implies¹ that $(\tilde{\textbf{A}}^\dagger )^H = W^\dagger ((\mathbb {C}_p^\flat )^H)_L$ is free of rank $[H_L:H]$ over $\tilde{\textbf{A}}^\dagger _L = W^\dagger (\hat{L}_\infty ^\flat )_L$. The $\pi _L$-adic completion $(-)^{\widehat{}}$ of the two rings therefore can be obtained by the tensor product with $\tilde{\textbf{A}}_L = W(\hat{L}_\infty ^\flat )_L$. This gives the wanted

$$\begin{aligned} W(\hat{L}_\infty ^\flat )_L \otimes _{\tilde{\textbf{A}}^\dagger _L} (\tilde{\textbf{A}}^\dagger ) ^H = W^\dagger (\hat{L}_\infty ^\flat )_L^{\widehat{}} \otimes _{\tilde{\textbf{A}}^\dagger _L} (\tilde{\textbf{A}}^\dagger )^H = W^\dagger ((\mathbb {C}_p^\flat )^H)_L^{\widehat{}} = W((\mathbb {C}_p^\flat )^H)_L . \end{aligned}$$

$\square$

Proposition 1.14

The sequences

$$\begin{aligned}&0 \rightarrow o_L \rightarrow {\textbf{A}} \xrightarrow {\varphi _q - 1} {\textbf{A}} \rightarrow 0, \end{aligned}$$

(5)

$$\begin{aligned}&0 \rightarrow o_L \rightarrow \tilde{\textbf{A}} \xrightarrow {\varphi _q - 1} \tilde{\textbf{A}} \rightarrow 0, \end{aligned}$$

(6)

$$\begin{aligned}&0 \rightarrow o_L \rightarrow \tilde{\textbf{A}}^\dagger \xrightarrow {\varphi _q - 1} \tilde{\textbf{A}}^\dagger \rightarrow 0. \end{aligned}$$

(7)

are exact.

Proof

The first sequence is [43, (26), Rem. 5.1]. For the second sequence one proves by induction the statement for finite length Witt vectors using that the Artin-Schreier equation has a solution in $\mathbb {C}_p^\flat$. Taking projective limits then gives the claim. For the third sequence only the surjectivity has to be shown. This can be achieved by the same calculation as in the proof of [27, Lem. 4.5.3] with $R=\mathbb {C}_p^\flat$.²$\square$

Lemma 1.15

For any finite T in $Rep_{o_L}(G_L)$ the map $\tilde{\textbf{A}}\otimes _{o_L} T \xrightarrow {\varphi _q\otimes {\text {id}}- 1} \tilde{\textbf{A}}\otimes _{o_L} T$ has a continuous set theoretical section.

Proof

Since $T\cong \bigoplus _{i=1}^r o_L/\pi _L^{n_i}o_L$ for some natural numbers $r, n_i$ we may assume that $T=o_L/\pi _L^no_L$ for some n and then we have to show that the surjective map $W_n(\mathbb {C}_p^\flat )_L \xrightarrow {\varphi _q-{\text {id}}} W_n(\mathbb {C}_p^\flat )_L$ has a continuous set theoretical section. Thus me may neglect the additive structure and identify source and target with $X=(\mathbb {C}_p^\flat )^n$. In order to determine the components of the map $\varphi _q-{\text {id}}=: f = (f_0,\ldots , f_{n-1}):X\rightarrow X$ with respect to these coordinates we recall that the addition in Witt rings is given by polynomials

$$\begin{aligned} S_j(X_0,\ldots X_{j},Y_0,\ldots ,Y_j)=X_j+Y_j + \text{ terms } \text{ in } X_0,\ldots ,X_{j-1},Y_0,\ldots ,Y_{j-1} \end{aligned}$$

while the additive inverse is given by

$$\begin{aligned} I_j(X_0,\ldots , X_{j} )= -X_j+ \text{ terms } \text{ in } X_0,\ldots ,X_{j-1}. \end{aligned}$$

Indeed, the polynomials $I_j$ are defined by the property that $\Phi _j(I_0,\ldots , I_j)=-\Phi _j(X_0,\ldots , X_j)$ where the Witt polynomials have the form $\Phi _j(X_0,\ldots ,X_j)=X_0^{q^j}+\pi _LX_1^{q^{j-1}}+\cdots +\pi _L^jX_j.$ Modulo $(X_0,\ldots , X_{j-1})$ we derive that $\pi _L^j I_j(X_0,\ldots , X_j) \equiv -\pi _L^j X_j$ and the claim follows. Since $\varphi _q$ acts componentwise rising the entries to their qth power, we conclude that

$$\begin{aligned} f_j=S_j(X_0^q,\ldots , X_{j}^q,I_0(X_0),\ldots ,I_j(X_0,\ldots , X_{j} )). \end{aligned}$$

Hence the Jacobi matrix of f at a point $x\in X$ looks like

$$\begin{aligned} D_x(f)=\begin{pmatrix} -1 &{} &{} 0 \\ &{} \ddots &{} \\ *&{} &{} -1 \\ \end{pmatrix} ,\end{aligned}$$

i.e., is invertible in every point. As a polynomial map f is locally analytic. It therefore follows from the inverse function theorem [41, Prop. 6.4] that f restricts to a homeomorphism $f|U_0:U_0\xrightarrow {\cong }U_1$ of open neighbourhoods of x and f(x),  respectively. By the surjectivity of f every $x\in X$ has an open neighbourhood $U_x$ and a continuous map $s_x:U_x\rightarrow X$ with $f\circ s_x={\text {id}}_{|U_x}$. But X is strictly paracompact by Remark 8.6 (i) in (loc. cit.), i.e., the covering $(U_x)_x$ has a disjoint refinement. There the restrictions of the $s_x$ glue to a continuous section of f. $\square$

Corollary 1.16

For T in $Rep_{o_L}(G_L)$, the nth cohomology groups of the complexes concentrated in degrees 0 and 1

(8)

(9)

are isomorphic to $H^n(H_L,T)$ for any $n\ge 0$.

Proof

Assume first that T is finite. For (9) see [43, Lem. 5.2]. For (8) we use Lemma 4.11, which says that the right hand map in the exact sequence

has a continuous set theoretical section and thus gives rise to the long exact sequence of continuous cohomology groups

$$\begin{aligned} 0 \rightarrow H^0(H_L,T) \rightarrow \tilde{D}(T) \xrightarrow {\varphi -1} \tilde{D}(T) \rightarrow H^1(H_L,T) \rightarrow H^1(H_L,\tilde{\textbf{A}} \otimes _{o_L} T) \rightarrow \ldots \nonumber \\ \end{aligned}$$

(10)

Using the comparison isomorphism (3) and the subsequent Prop.  4.13 we see that all terms from the fifth on vanish.

For the general case (for $\tilde{D}(T)$ as well as D(T)) we take inverse limits in the exact sequences for the $(T/\pi _L^mT)$ and observe that $H^n(H_L,T) \cong \varprojlim _{m} H^n(H_L,T/\pi _L^m T)$. This follows for $n \ne 2$ from [39, Cor. 2.7.6]. For $n=2$ we use [39, Thm. 2.7.5] and have to show that the projective system $(H^1(H_L,T/\pi ^m_L T))_m$ is Mittag-Leffler. Since it is a quotient of the projective system $(D(T/\pi _L^m T))_m$, it suffices for this to check that the latter system is Mittag-Leffler. But due to the exactness of the functor D this latter system is equal to the projective system of artinian $\textbf{A}_L$-modules $(D(T)/\pi _L^m D(T))_m$ and hence is Mittag-Leffler. We conclude by observing that taking inverse limits of the system of sequences (10) remains exact. The reasoning being the same for $\tilde{D}(T)$ and D(T) we consider only the former. Indeed, we split the 4-term exact sequences into two short exact sequences of projective systems

$$\begin{aligned} 0 \rightarrow H^0(H_L,V/\pi _L^m T) \rightarrow \tilde{D}(T/\pi _L^m T) \rightarrow (\varphi -1)\tilde{D}(T/\pi _L^m T) \rightarrow 0 \end{aligned}$$

and

$$\begin{aligned} 0 \rightarrow (\varphi -1)\tilde{D}(T/\pi _L^m T) \rightarrow \tilde{D}(T/\pi _L^m T) \rightarrow H^1(H_L,T/\pi _L^m T) \rightarrow 0 . \end{aligned}$$

Passing to the projective limits remains exact provided the left most projective systems have vanishing $\varprojlim ^1$. For the system $H^0(H_L,T/\pi _L^m T)$ this is the case since it is Mittag-Leffler. The system $(\varphi -1)\tilde{D}(T/\pi _L^m T)$ even has surjective transition maps since the system $\tilde{D}(T/\pi _L^m T)$ has this property by the exactness of the functor $\tilde{D}$ (cf. Prop. 4.6). $\square$

Proposition 1.17

$H^n(H,\tilde{\textbf{A}}/\pi _L^m \tilde{\textbf{A}})=0$ for all $n,m\ge 1$ and $H\subseteq H_L$ any closed subgroup.

Proof

For $j<i$ the canonical projection $W_i(\mathbb {C}_p^\flat )\cong \tilde{\textbf{A}}/\pi _L^i \tilde{\textbf{A}}\twoheadrightarrow \tilde{\textbf{A}}/\pi _L^j \tilde{\textbf{A}}\cong W_j(\mathbb {C}_p^\flat )$ corresponds to the projection $(\mathbb {C}_p^\flat )^i\twoheadrightarrow (\mathbb {C}_p^\flat )^j$ and hence have set theoretical continuous sections. Using the associated long exact cohomology sequence (after adding the kernel) allows to reduce the statement to Prop.  3.1. $\square$

For any commutative ring R with endomorphism $\varphi$ we write $\Phi (R)$ for the category of $\varphi$-modules consisting of R-modules equipped with a semi-linear $\varphi$-action. We write $\Phi ^{\acute{e}t}(R)$ for the subcategory of étale $\varphi$-modules, i.e., such that M is finitely generated over R and $\varphi$ induces an R-linear isomorphism $\varphi ^*M \xrightarrow {\cong } M$. Finally, we denote by $\Phi ^{\acute{e}t}_{f}(R)$ the subcategory consisting of finitely generated free R-modules.

For $M_1,M_2\in \Phi (R)$ with $M_1$ being étale the R-module ${\text {Hom}}_R(M_1,M_2)$ has a natural structure as a $\varphi$-module satisfying

$$\begin{aligned} \varphi _{{\text {Hom}}_{R}(M_1,M_2)}(\alpha )(\varphi _{M_1}(m))=\varphi _{M_2}( \alpha (m)) \ , \end{aligned}$$

(11)

hence in particular

$$\begin{aligned} {\text {Hom}}_{R}(M_1,M_2)^{\varphi ={\text {id}}}={\text {Hom}}_{\Phi (R)}(M_1,M_2). \end{aligned}$$

(12)

Note that with $M_1,M_2$ also ${\text {Hom}}_{R}(M_1,M_2)$ is étale.

Remark 4.14

We recall from [26, §1.5] that the cohomology groups $H^i_\varphi (M)$ of the complex $M\xrightarrow {\varphi -1}M$ can be identified with the Yoneda extension groups $\textrm{Ext}^i_{\Phi (R)}(R,M)$. Indeed, if $S:=R[X;\varphi ]$ denotes the twisted polynomial ring satisfying $Xr=\varphi (r)X$ for all $r\in R,$ then we can identify $\Phi (R)$ with the category S-Mod of (left) S-modules by letting X act via $\varphi _M$ on X. Using the free resolution

the result follows.

Remark 4.15

Note that $\tilde{\textbf{A}}_L^{\dagger }\subseteq \tilde{\textbf{A}}_L$ is a faithfully flat ring extension as both rings are discrete valuation rings and the bigger one is the completion of the previous one.

Proposition 1.20

Base extension induces

• (i) an equivalence of categories

  $$\begin{aligned} \Phi ^{\acute{e}t}_{f}(\tilde{\textbf{A}}_L^{\dagger } ) \leftrightarrow \Phi ^{\acute{e}t}_{f}(\tilde{\textbf{A}}_L^{}) \end{aligned}$$

• (ii) and an isomorphism of Yoneda extension groups

  $$\begin{aligned} \textrm{Ext}^1_{\Phi (\tilde{\textbf{A}}_L^{\dagger } )}(\tilde{\textbf{A}}_L^{\dagger },M) \cong \textrm{Ext}^1_{\Phi (\tilde{\textbf{A}}_L )}(\tilde{\textbf{A}}_L, \tilde{\textbf{A}}_L \otimes _{\tilde{\textbf{A}}_L^{\dagger }} M) \end{aligned}$$

  for all $M\in \Phi ^{\acute{e}t}_{f}(\tilde{\textbf{A}}_L^{\dagger } ).$

Proof

For the first item we imitate the proof of [26, Thm. 8.5.3], see also [23, Lem. 2.4.2,Thm. 2.4.5]: First we will show that for every $M\in \Phi ^{\acute{e}t}_{f}(\tilde{\textbf{A}}_L^\dagger )$ it holds that $(\tilde{\textbf{A}}_L\otimes _{}M)^{\varphi ={\text {id}}}\subseteq M^{\varphi ={\text {id}}}$ and hence equality. Applied to $M:={\text {Hom}}_{\tilde{\textbf{A}}_L^\dagger }(M_1,M_2)$ this implies that the base change is fully faithful by the equation (12). We observe that the analogue of [26, Lem. 3.2.6] holds in our setting and that S in loc. cit. can be chosen to be a finite separable field extension of the perfect field $R=\hat{L}_\infty ^\flat$. Thus we may choose S in the analogue of [26, Prop. 7.3.6] (with $a=1$, $c=0$ and $M_0$ being our M) as completion of a (possibly infinite) separable field extension of R. This means in our situation that there exists a closed subgroup $H\subseteq H_L$ such that $(\tilde{\textbf{A}}^\dagger )^H \otimes _{\tilde{\textbf{A}}_L^\dagger } M = \bigoplus (\tilde{\textbf{A}}^\dagger )^H e_i$ for a basis $e_i$ invariant under $\varphi .$ Now let $v=\sum x_i e_i$ be an arbitrary element in

$$\begin{aligned} \tilde{\textbf{A}}_L \otimes _{\tilde{\textbf{A}}_L^\dagger } M \subseteq \tilde{\textbf{A}}^H \otimes _{\tilde{\textbf{A}}_L^\dagger } M = \tilde{\textbf{A}}^H \otimes _{(\tilde{\textbf{A}}^\dagger )^H} (\tilde{\textbf{A}}^\dagger )^H \otimes _{\tilde{\textbf{A}}_L^\dagger } M =\bigoplus \tilde{\textbf{A}}^H e_i \end{aligned}$$

with $x_i \in \tilde{\textbf{A}}^H$ and such that $\varphi (v)=v.$ The latter condition implies that $x_i\in \tilde{\textbf{A}}^{H,\varphi _q={\text {id}}}=o_L,$ i.e., v belongs to $(M\otimes _{\tilde{\textbf{A}}_L^\dagger }(\tilde{\textbf{A}}^\dagger )^H)\cap ( M\otimes _{\tilde{\textbf{A}}_L^\dagger }\tilde{\textbf{A}}_L)=M$, because M is free and one has $\tilde{\textbf{A}}_L\cap (\tilde{\textbf{A}}^\dagger )^H=(\tilde{\textbf{A}}^\dagger )^{H_L}= \tilde{\textbf{A}}^\dagger _L.$ To show essential surjectivity one proceeds literally as in the proof of [26, Thm. 8.5.3] adapted to ramified Witt vectors.

For the second statement choose a quasi-inverse functor $F: \Phi ^{\acute{e}t}_{f}(\tilde{\textbf{A}}_L^{})\rightarrow \Phi ^{\acute{e}t}_{f}(\tilde{\textbf{A}}_L^{\dagger } )$ with $F(\tilde{\textbf{A}}_L)=\tilde{\textbf{A}}_L^{\dagger }.$ Given an extension over $\Phi (\tilde{\textbf{A}}_L )$ with $M\in \Phi ^{\acute{e}t}_{f}(\tilde{\textbf{A}}_L )$ first observe that $E\in \Phi ^{\acute{e}t}_{f}(\tilde{\textbf{A}}_L )$, too. Indeed, $\tilde{\textbf{A}}_L\xrightarrow {\varphi _q}\tilde{\textbf{A}}_L$ is a flat ring extension, whence $\varphi ^*E\rightarrow E$ is an isomorphism, if the corresponding outer maps are. The analogous statement holds over $\tilde{\textbf{A}}_L^{\dagger }$. Therefore the sequence is exact by Remark 4.15, because its base extension - being isomorphic to the original extension - is, by assumption. $\square$

We denote by ${\mathfrak {M}}^{\acute{e}t}_{f}(\tilde{\textbf{A}}_L^{\dagger } )$ and ${\mathfrak {M}}^{\acute{e}t}_{f}(\tilde{\textbf{A}}_L^{})$ the full subcategories of ${\mathfrak {M}}^{\acute{e}t}(\tilde{\textbf{A}}_L^{\dagger } )$ and ${\mathfrak {M}}^{\acute{e}t}(\tilde{\textbf{A}}_L^{})$, respectively, consisting of finitely generated free modules over the base ring.

Remark 4.17

Let M be in ${\mathfrak {M}}^{\acute{e}t}_{f}(\tilde{\textbf{A}}_L^{})$ and endow $N:=\tilde{\textbf{A}}_L\otimes _{\tilde{\textbf{A}}_L^{\dagger }} M$ with the canonical topology with respect to the weak topology of $\tilde{\textbf{A}}_L$. Then the induced subspace topology of $M\subseteq N$ coincides with the canonical topology with respect to the weak topology of $\tilde{\textbf{A}}_L^{\dagger }$. Indeed for free modules this is obvious while for torsion modules this can be reduced by the elementary divisor theory to the case $M=\tilde{\textbf{A}}_L^{\dagger }/\pi _L^n \tilde{\textbf{A}}_L^{\dagger }\cong \tilde{\textbf{A}}_L/\pi _L^n \tilde{\textbf{A}}_L$. But the latter spaces are direct product factors of $\tilde{\textbf{A}}_L^{\dagger }$ and $\tilde{\textbf{A}}_L,$ respectively, as topological spaces, from wich the claim easily follows.

Proposition 1.22

For $T\in {\text {Rep}}_{o_L}(G_L)$ and $V\in {\text {Rep}}_L(G_L)$ we have natural isomorphisms

$$\begin{aligned} \tilde{\textbf{A}}_L^{}\otimes _{\tilde{\textbf{A}}_L^{\dagger }}\tilde{D}^\dagger (T)&\cong \tilde{D}(T) \text{ and } \end{aligned}$$

(13)

$$\begin{aligned} \tilde{\textbf{B}}_L^{}\otimes _{\tilde{\textbf{B}}_L^{\dagger }}\tilde{D}^\dagger (V)&\cong \tilde{D}(V), \end{aligned}$$

(14)

as well as

$$\begin{aligned} \tilde{\textbf{A}}^{\dagger }\otimes _{\tilde{\textbf{A}}_L^{\dagger }}\tilde{D}^\dagger (T)&\cong \tilde{\textbf{A}}^{\dagger }\otimes _{o_L} T \text{ and } \end{aligned}$$

(15)

$$\begin{aligned} \tilde{\textbf{B}}^{\dagger }\otimes _{\tilde{\textbf{B}}_L^{\dagger }}\tilde{D}^\dagger (V)&\cong \tilde{\textbf{B}}^{\dagger }\otimes _L V, \end{aligned}$$

(16)

respectively. In particular, the functor $\tilde{D}^\dagger (-):{\text {Rep}}_{o_L}(G_L) \rightarrow {\mathfrak {M}}^{\acute{e}t}(\tilde{\textbf{A}}_L^\dagger )$ is exact.

Moreover, base extension induces equivalences of categories

$$\begin{aligned} {\mathfrak {M}}^{\acute{e}t}_f(\tilde{\textbf{A}}_L^{\dagger } ) \leftrightarrow {\mathfrak {M}}^{\acute{e}t}_f(\tilde{\textbf{A}}_L^{}), \end{aligned}$$

and hence also an equivalence of categories

$$\begin{aligned} {\mathfrak {M}}^{\acute{e}t}(\tilde{\textbf{B}}_L^{\dagger } ) \leftrightarrow {\mathfrak {M}}^{\acute{e}t}(\tilde{\textbf{B}}_L^{}). \end{aligned}$$

Proof

Note that the base change functor is well-defined - regarding the continuity of the $\Gamma _L$-action - by Lemma 4.5 and Remark 4.15 while $\tilde{D}^\dagger$ is well-defined by Remark 4.17, once (13) will have been shown. We first show the equivalence of categories for free modules: By Prop.  4.16 we already have, for $M_1,M_2\in {\mathfrak {M}}^{\acute{e}t}_{f}(\tilde{\textbf{A}}_L^{\dagger } )$, an isomorphism

$$\begin{aligned} {\text {Hom}}_{\Phi (\tilde{\textbf{A}}_L^{\dagger } )} (M_1,M_2) \cong {\text {Hom}}_{\Phi (\tilde{\textbf{A}}_L )} (\tilde{\textbf{A}}_L \otimes _{\tilde{\textbf{A}}_L^{\dagger }} M_1,\tilde{\textbf{A}}_L \otimes _{\tilde{\textbf{A}}_L^{\dagger }} M_2). \end{aligned}$$

Taking $\Gamma _L$-invariants gives that the base change functor in question is fully faithful.

In order to show that this base change functor is also essentially surjective, consider an arbitrary $N \in {\mathfrak {M}}^{\acute{e}t}_{f}(\tilde{\textbf{A}}_L^{}).$ Again by  4.16 we know that there is a free étale $\varphi$-module M over $\tilde{\textbf{A}}_L^{\dagger }$ whose base change is isomorphic to N. By the fully faithfulness the $\Gamma _L$-action descends to M³. Since the weak topology of M is compatible with that of N by Remark 4.17, this action is again continuous.

To prepare for the proof of the isomorphism (13) we first observe the following fact. The isomorphism (3) implies that T and $\tilde{D}(T)$ have the same elementary divisors, i.e.: If $T \cong \oplus _{i=1}^r o_L/\pi _L^{n_i} o_L$ as $o_L$-module (with $n_i \in \mathbb {N} \cup \{\infty \}$) then $\tilde{D}(T) \cong \oplus _{i=1}^r \tilde{\textbf{A}}_L / \pi _L^{n_i} \tilde{\textbf{A}}_L$ as $\tilde{\textbf{A}}_L$-module.

We shall prove (13) in several steps: First assume that T is finite. Then T is annihilated by some $\pi _L^n$. We have $\tilde{D}^\dagger (T)= \tilde{D}(T)$ and $\tilde{\textbf{A}}_L^\dagger / \pi _L^n \tilde{\textbf{A}}_L^\dagger = \tilde{\textbf{A}}_L / \pi _L^n \tilde{\textbf{A}}_L$ so that there is nothing to prove. Secondly we suppose that T is free and that $\tilde{D}^\dagger (T)$ is free over $\tilde{\textbf{A}}_L^\dagger$ of the same rank $r := {\text {rk}}_{o_L} T$. On the other hand, as the functor $\tilde{D}^\dagger$ is always left exact, we obtain the injective maps

$$\begin{aligned} \tilde{D}^\dagger (T) / \pi _L^n \tilde{D}^\dagger (T) \rightarrow \tilde{D}^\dagger (T/\pi _L^n T) = \tilde{D}(T/ \pi _L^n T) . \end{aligned}$$

for any $n \ge 1$. We observe that both sides are isomorphic to $(\tilde{\textbf{A}}_L^\dagger / \pi _L^n \tilde{\textbf{A}}_L^\dagger )^r = (\tilde{\textbf{A}}_L / \pi _L^n \tilde{\textbf{A}}_L)^r$. Hence the above injective maps are bijections. We deduce that

$$\begin{aligned} \tilde{\textbf{A}}_L\otimes _{ {\textbf{A}}_L^\dagger } \tilde{D}^\dagger (T)&\cong \varprojlim _{n} \tilde{D}^\dagger (T)/\pi _L^n\tilde{D}^\dagger (T) \\&\cong \varprojlim _{n} \tilde{D}(T /\pi _L^n T) \\&\cong \varprojlim _{n} \tilde{D}(T)/\pi _L^n\tilde{D}(T) \\&\cong \tilde{D}(T) \end{aligned}$$

using that the above tensor product means $\pi _L$-adic completion for finitely generated $\tilde{\textbf{A}}^\dagger _L$-modules.

Thirdly let $T \in {\text {Rep}}_{o_L,f}(G_L)$ be arbitrary and $M \in {\mathfrak {M}}^{\acute{e}t}_{f}(\tilde{\textbf{A}}_L^{\dagger } )$ such that $\tilde{\textbf{A}}_L \otimes _{\tilde{\textbf{A}}_L^\dagger } M \cong \tilde{D}(T)$ according the equivalence of categories. Without loss of generality we may treat this isomorphism as an equality. Similarly as in the proof of Prop.  4.16 and with the same notation one shows that $(\tilde{\textbf{A}}^\dagger \otimes _{\tilde{\textbf{A}}^\dagger _L } M)^{\varphi =1} = \bigoplus _{i=1}^ro_Le_i$ for some appropriate $\varphi$-invariant basis $e_1,\ldots , e_r$ of $\tilde{\textbf{A}}^\dagger \otimes _{\tilde{\textbf{A}}^\dagger _L } M$. Note that $r = {\text {rk}}_{o_L} T$. Using (3), it follows that

$$\begin{aligned} T&= (\tilde{\textbf{A}} \otimes _{o_L} T)^{\varphi =1} \cong ( \tilde{\textbf{A}} \otimes _{\tilde{\textbf{A}}_L} \tilde{D}(T))^{\varphi =1} = (\tilde{\textbf{A}} \otimes _{\tilde{\textbf{A}}_L^\dagger } M)^{\varphi =1} \\&= \bigoplus _{i=1}^r\tilde{\textbf{A}}^{\varphi _q=1} e_i = \bigoplus _{i=1}^r o_Le_i = (\tilde{\textbf{A}}^\dagger \otimes _{\tilde{\textbf{A}}^\dagger _L } M)^{\varphi =1} . \end{aligned}$$

It shows that the comparison isomorphism (3) restricts to an injective map $T \hookrightarrow \tilde{\textbf{A}}^\dagger \otimes _{\tilde{\textbf{A}}^\dagger _L } M$, which extends to a homomorphism $\tilde{\textbf{A}}^\dagger \otimes _{o_L} T \xrightarrow {\alpha } \tilde{\textbf{A}}^\dagger \otimes _{\tilde{\textbf{A}}^\dagger _L } M$ of free $\tilde{\textbf{A}}^\dagger$-modules of the same rank r. Further base extension by $\tilde{\textbf{A}}$ gives back the isomorphism (3). Since $\tilde{\textbf{A}}$ is faithfully flat over $\tilde{\textbf{A}}^\dagger$ the map $\alpha$ was an isomorphism already. By passing to $H_L$-invariants we obtain an isomorphism $\tilde{D}^\dagger (T) \cong M$ and see that $\tilde{D}^\dagger (T)$ is free of the same rank as T. Hence the second case applies and gives (13) for free T and (14). Finally, let T be just finitely generated over $o_L.$ Write $0\rightarrow T_{\textrm{fin}}\rightarrow T\rightarrow T_{\textrm{free}}\rightarrow 0$ with finite $T_{\textrm{fin}}$ and free $T_{\textrm{free}}.$ We then have the commutative exact diagram

in which we use the first and third step for the vertical isomorphisms. In order to show that the middle perpendicular arrow is an isomorphism it suffices to prove that $H^1(H_L,\tilde{\textbf{A}}^\dagger \otimes _{o_L} T_{\textrm{fin}}) =0$. But since $T_{\textrm{fin}}$ is annihilated by some $\pi _L^n$ we have

$$\begin{aligned} \tilde{\textbf{A}}^\dagger \otimes _{o_L} T_{\textrm{fin}}\cong \tilde{\textbf{A}}/\pi _L^n\tilde{\textbf{A}}\otimes _{o_L} T_{\textrm{fin}}\cong \tilde{\textbf{A}}/\pi _L^n\tilde{\textbf{A}}\otimes _{\tilde{\textbf{A}}_L}\tilde{D}( T_{\textrm{fin}}), \end{aligned}$$

the last isomorphism by (3). Thus it suffices to prove the vanishing of $H^1(H_L,\tilde{\textbf{A}}/\pi _L^n\tilde{\textbf{A}}),$ which is established in Prop.  4.13 and finishes the proof of the isomorphism (13).

Note that this base change isomorphism implies the exactness of $\tilde{D}^\dagger$ as $\tilde{D}$ is exact by Prop.  4.6 and using that the base extension is faithfully flat by Remark 4.15.

For free T the statement (15) (and hence (16)) is already implicit in the above arguments while for finite T the statement coincides with (3). The general case follows from the previous ones by exactness of $\tilde{D}^\dagger$ and the five lemma as above.

$\square$

Corollary 1.23

For a T in ${\text {Rep}}_{o_L,f}(G_L)$ and V in ${\text {Rep}}_{L}(G_L)$, the nth cohomology group, for any $n \ge 0$, of the complexes concentrated in degrees 0 and 1

(17)

(18)

is isomorphic to $H^n(H_L,T)$ and $H^n(H_L,V),$ respectively.

Proof

The integral result reduces, by (13), Remark 4.14, and Prop.  4.16, to Corollary 4.12. Since inverting $\pi _L$ is exact and commutes with taking cohomology [39, Prop. 2.7.11], the second statement follows. $\square$

Set $\textbf{A}^\dagger :=\tilde{\textbf{A}}^{\dagger }\cap \textbf{A}$ and $\textbf{B}^\dagger :=\textbf{A}^\dagger [\frac{1}{\pi _L}]$ as well as $\textbf{A}^\dagger _L:=(\textbf{A}^\dagger )^{H_L}.$ Note that $\textbf{B}^\dagger _L:=(\textbf{B}^\dagger )^{H_L}\subseteq \textbf{B}^\dagger \subseteq {\tilde{\textbf{B}}}^\dagger .$ For $V\in {\text {Rep}}_L(G_L)$ we define $D^\dagger (V):=(\textbf{B}^\dagger \otimes _L V)^{H_L}.$ The categories ${\mathfrak {M}}^{\acute{e}t}(\textbf{A}^\dagger _L)$ and ${\mathfrak {M}}^{\acute{e}t}(\textbf{B}^\dagger _L)$ are defined analogously as in Definition 4.4.

We now introduce the Robba ring $\mathcal {R} = \mathcal {R}_K = \mathcal {R}_K(\textbf{B})$ of the open unit disk $\textbf{B}_{/K}$, where $L \subseteq K \subseteq \mathbb {C}_p$ denotes a complete intermediate field. The ring of K-valued global holomorphic functions $\mathcal {O}_K(\textbf{B})$⁴ on $\textbf{B}$ is the Fréchet algebra of all power series in the variable Z with coefficients in K which converge on the open unit disk $\textbf{B}(\mathbb {C}_p)$. The Fréchet topology on ${\mathcal {O}_K(\textbf{B})}$ is given by the family of norms

$$\begin{aligned} \Big |\sum _{i \ge 0} c_i Z^i\Big |_r := \max _i |c_i| r^i \qquad \text {for}\ 0< r < 1 \ . \end{aligned}$$

In the commutative integral domain ${\mathcal {O}_K(\textbf{B})}$ we have the multiplicative subset $Z^{\mathbb {N}} = \{Z^j : j \in \mathbb {N}\}$, so that we may form the corresponding localization ${\mathcal {O}_K(\textbf{B})}_{Z^{\mathbb {N}}}$. Each norm $|\ |_r$ extends to this localization ${\mathcal {O}_K(\textbf{B})}_{Z^{\mathbb {N}}}$ by setting $|\sum _{i \gg -\infty } c_iZ^i|_r := \max _i |c_i| r^i$.

The Robba ring $\mathcal {R} \supseteq {\mathcal {O}_K(\textbf{B})}$ is constructed as follows. For any $s > 0$, resp. any $0 < r \le s$, in $p^{\mathbb {Q}}$ let $\textbf{B}_{[0,s]}$, resp. $\textbf{B}_{[r,s]}$, denote the affinoid disk around 0 of radius s, resp. the affinoid annulus of inner radius r and outer radius s, over K. For $I=[0,s]$ or [r, s] we denote by

$$\begin{aligned} \mathcal {R}^I := \mathcal {R}_K^I(\textbf{B}) := \mathcal {O}_K(\textbf{B}_{I}) \end{aligned}$$

the affinoid K-algebra of $\textbf{B}_{I}$. The Fréchet algebra $\mathcal {R}^{[r,1)} := \varprojlim _{r< s < 1} \mathcal {R}^{[r,s]}$ is the algebra of (infinite) Laurent series in the variable Z with coefficients in K which converge on the half-open annulus $\textbf{B}_{[r,1)} := \bigcup _{r< s < 1} \textbf{B}_{[r,s]}$. The Banach algebra $\mathcal {R}^{[0,s]}$ is the completion of ${\mathcal {O}_K(\textbf{B})}$ with respect to the norm $|\ |_s$. The Banach algebra $\mathcal {R}^{[r,s]}$ is the completion of ${\mathcal {O}_K(\textbf{B})}_{Z^{\mathbb {N}}}$ with respect to the norm $|\ |_{r,s} := \max (|\ |_r,|\ |_s)$. It follows that the Fréchet algebra $\mathcal {R}^{[r,1)}$ is the completion of ${\mathcal {O}_K(\textbf{B})}_{Z^{\mathbb {N}}}$ in the locally convex topology defined by the family of norms $(|\ |_{r,s})_{r< s < 1}$. Finally, the Robba ring is $\mathcal {R} = \bigcup _{0< r < 1} \mathcal {R}^{[r,1)}$.

Remark 4.20

There is also the following more concrete description for $\textbf{A}^\dagger _L$ in terms of Laurent series in $\omega _{LT}:$

$$\begin{aligned} \textbf{A}^\dagger _L=\{F(\omega _{LT})\in \textbf{A}_L|&F(Z) \text{ converges } \text{ on } \rho \le |Z|<1 \text{ for } \text{ some } \rho \in (0,1)\}\subseteq \textbf{A}_L. \end{aligned}$$

Indeed this follows from the analogue of [11, Lem. II.2.2] upon noting that the latter holds with and without the integrality condition: ”$rv_p(a_n)+n\ge 0$ for all $n\in \mathbb {Z}$” (for $r\in \overline{\textbf{R}}\setminus \textbf{R}$) in the notation of that article.

In particular we obtain canonical embeddings $\textbf{A}^\dagger _L\subseteq \textbf{B}^\dagger _L \hookrightarrow \mathcal {R}_L$ of rings.

Definition 1.25

V in ${\text {Rep}}_L(G_L)$ is called overconvergent, if $\dim _{\textbf{B}^\dagger _L} D^\dagger (V) = \dim _LV.$ We denote by $\textrm{Rep}_L^\dagger (G_{L})\subseteq {\text {Rep}}_L(G_L)$ the full subcategory of overconvergent representations.

Remark 4.22

We always have $\dim _{\textbf{B}^\dagger _L} D^\dagger (V) \le \dim _LV$. If $V \in {\text {Rep}}_L(G_L)$ is overconvergent then we have the natural isomorphism

$$\begin{aligned} {\textbf{B}}_L \otimes _{{\textbf{B}}^\dagger _L}D^\dagger (V) \xrightarrow {\cong } {D}(V). \end{aligned}$$

(19)

Proof

Since $\textbf{B}_L$ and $\textbf{B}^\dagger _L$ are fields this is immediate from [18, Thm. 2.13]. $\square$

Remark 4.23

In [5, §10] Berger uses the following condition to define overconvergence of V: There exists a $\textbf{B}_L$-basis $x_1,\ldots ,x_n$ of D(V) such that $M := \bigoplus _{i=1}^n \textbf{B}^\dagger _L x_i$ is a $(\varphi _L,\Gamma _L)$-module over $\textbf{B}_L^\dagger .$ This then implies a natural isomorphism

$$\begin{aligned} \textbf{B}_L \otimes _{\textbf{B}^\dagger _L } M \cong D(V). \end{aligned}$$

(20)

Lemma 1.28

V in ${\text {Rep}}_L(G_L)$ is overconvergent if and only if V satisfies the above condition of Berger. In this case $M=D^\dagger (V).$

Proof

If V is overconvergent, we can take a basis within $M:=D^\dagger (V).$ Conversely let V satisfy Berger’s condition, i.e. we have the isomorphism (20). One easily checks by faithfully flat descent that with D(V) also M is étale. By [19, Prop. 1.5 (a)]⁵ we obtain the identity $V=\left( \textbf{B}^\dagger \otimes _{\textbf{B}^\dagger _L} M\right) ^{\varphi =1}$ induced from the comparison isomorphism

$$\begin{aligned} \textbf{B}\otimes _L V\cong \textbf{B}\otimes _{\textbf{B}_L}D(V)\cong \textbf{B}\otimes _{\textbf{B}_L^\dagger }M. \end{aligned}$$

(21)

We shall prove that $M\subseteq D^\dagger (V)=(\textbf{B} ^\dagger \otimes _LV)^{H_L}$ as then $M=D^\dagger (V)$ by dimension reasons. To this aim we may write a basis $v_1,\ldots , v_n$ of V over L as $v_i=\sum c_{ij}x_j$ with $c_{ij}\in \textbf{B}^\dagger$. Then (21) implies that the matrix $C=(c_{ij})$ belongs to $M_n(\textbf{B}^\dagger )\cap GL_n( \textbf{B} )=GL_n(\textbf{B}^\dagger ).$ Thus M is contained in $\textbf{B}^\dagger \otimes _LV$ and - as subspace of D(V) - also $H_L$-invariant, whence the claim. $\square$

Remark 4.25

Note that the imperfect version of Prop.  4.18 is not true: the base change ${\mathfrak {M}}^{\acute{e}t}({\textbf{B}}_L^\dagger ) \rightarrow {\mathfrak {M}}^{\acute{e}t}({\textbf{B}}_L)$ is not essentially surjective in general, whence not an equivalence of categories, by [19]. By definition, its essential image consists of overconvergent $(\varphi _L,\Gamma _L)$-modules, i.e., whose corresponding Galois representations are overconvergent.

Lemma 1.30

Assume that $V\in {\text {Rep}}_L(G_L)$ is overconvergent. Then there is natural isomorphism

$$\begin{aligned} \tilde{\textbf{B}}^\dagger _L\otimes _{\tilde{\textbf{B}}^\dagger _L}D^\dagger (V)\cong \tilde{D}^\dagger (V). \end{aligned}$$

Proof

By construction we have a natural map $\tilde{\textbf{B}}^\dagger _L\otimes _{\tilde{\textbf{B}}^\dagger _L}D^\dagger (V)\rightarrow \tilde{D}^\dagger (V),$ whose base change to $\tilde{\textbf{B}}_L$

$$\begin{aligned} \tilde{\textbf{B}}_L\otimes _{\tilde{\textbf{B}}^\dagger _L}D^\dagger (V)\rightarrow \tilde{\textbf{B}}_L\otimes _{\tilde{\textbf{B}}^\dagger _L}\tilde{D}^\dagger (V)\cong \tilde{D}(V) \end{aligned}$$

arises also as the base change of the isomorphism (19), whence is an isomorphism itself. Here we have used the (base change of the) isomorphisms (14), (2). By faithfully flatness the original map is an isomorphism, too. $\square$

5 The Perfect Robba Ring

Again let K be any perfectoid field containing L and $r>0.$ For $0<s\le r,$ let $\tilde{\mathcal {R}}^{[s,r]}(K)$ be the completion of $W^r(K^\flat )_L[\frac{1}{\pi _L}]$ with respect to the norm $\max \{|\ |_{s} , |\ |_{r}\},$ and put

$$\begin{aligned} \tilde{\mathcal {R}}^r(K) = \varprojlim _{s\in (0,r]} \tilde{\mathcal {R}}^{[s,r]}(K) \end{aligned}$$

equipped with the Fréchet topology. Let $\tilde{\mathcal {R}}(K) = \varinjlim _{r>0} \tilde{\mathcal {R}}^r(K)$, equipped with the locally convex direct limit topology (LF topology). We set $\tilde{\mathcal {R}}=\tilde{\mathcal {R}}({\mathbb {C}_p})$ and $\tilde{\mathcal {R}}_{L}:=\tilde{\mathcal {R}}({\hat{L}_\infty })$. For geometric interpretation of these definitions, see [15]. As in [26, Thm. 9.2.15] we have

$$\begin{aligned} \tilde{\mathcal {R}}^{H_L}=\tilde{\mathcal {R}}_{L}. \end{aligned}$$

Recall from Sect. 2 the embedding $o_L[[Z]]\rightarrow W(\tilde{\textbf{E}})_L$. As we will explain in Sect. 8 the image $\omega _{LT}$ of the variable Z already lies in $W({\hat{L}_\infty }^\flat )_L$, so that we actually have an embedding $o_L[[Z]]\rightarrow W({\hat{L}_\infty }^\flat )_L$. Similarly as in [26, Def. 4.3.1] for the cyclotomic situation one shows that the latter embedding extends to a $\Gamma _L$- and $\varphi _L$-equivariant topological monomorphism $\mathcal {R}_L\rightarrow \tilde{\mathcal {R}}_L,$ see also [49, Konstruktion 1.3.27] in the Lubin–Tate setting.

Let R be either $\mathcal {R}_L$ or $\tilde{\mathcal {R}}_L$. A $(\varphi _L,\Gamma _L)$-module over R is a finitely generated free R-module M equipped with commuting semilinear actions of $\varphi _M$ and $\Gamma _L$, such that the action is continuous for the LF topology and such that the semi-linear map $\varphi _M:M\rightarrow M$ induces an isomorphism $\varphi _M^{lin}:R\otimes _{R,\varphi _R}M\xrightarrow {\cong } M.$ Such M is called étale, if there exists an étale $(\varphi _L,\Gamma _L)$-module N over $\textbf{A}_L^\dagger$ and $\tilde{\textbf{A}}_L^\dagger$ (see before Definition 4.4), such that $\mathcal {R}_L\otimes _{\textbf{A}_L^\dagger } N\cong M$ and $\tilde{\mathcal {R}}_L\otimes _{\tilde{\textbf{A}}_L^\dagger } N\cong M,$ respectively.

By ${\mathfrak {M}}(R)$ and ${\mathfrak {M}}^{\acute{e}t}(R)$ we denote the category of $(\varphi _L,\Gamma _L)$-modules and étale $(\varphi _L,\Gamma _L)$-modules over R,  respectively.

We call the topologies on $\tilde{\textbf{A}}_L^\dagger$ and $\tilde{\textbf{A}}^\dagger$, which make the inclusions $\tilde{\textbf{A}}_L^\dagger \subseteq \tilde{\textbf{A}}^\dagger \subseteq \tilde{\mathcal {R}}$ topological embeddings, the LF-topologies.

Lemma 1.31

For $M\in {\mathfrak {M}}^{\acute{e}t}_f(\tilde{\textbf{A}}_L^\dagger )$ the $\Gamma _L$-action is also continuous with respect to the canonical topology with respect to the LF-topology of $\tilde{\textbf{A}}_L^\dagger$.

Proof

The proof in fact works in the following generality: Suppose that $\tilde{\textbf{A}}^\dagger$ is equipped with an $o_L$-linear ring topology which induces the $\pi _L$-adic topology on $o_L$. Consider on $\tilde{\textbf{A}}^\dagger _L$ the corresponding induced topology. We claim that then the $\Gamma _L$-action on M is continuous with respect to the corresponding canonical topology. By Prop.  6.1 we may choose $T\in {\text {Rep}}_{o_L,f}(G_L)$ such that $M\cong \tilde{D}^\dagger (T).$ Then we have a homeomorphism $\tilde{\textbf{A}}^\dagger \otimes _{o_L}T\cong \tilde{\textbf{A}}^\dagger \otimes _{\tilde{\textbf{A}}^\dagger _L}M$ with respect to the canonical topology by (15) (as any R-module homomorphism of finitely generated modules is continuous with respect to the canonical topology with regard to any topological ring R). Since $o_L\subseteq \tilde{\textbf{A}}^\dagger$ is a topological embedding with respect to the $\pi _L$-adic and the given topology, respectively, Lemma 4.5 implies that $G_L$ is acting continuously on $\tilde{\textbf{A}}^\dagger \otimes _{\tilde{\textbf{A}}^\dagger _L}M$, whence $\Gamma _L$ acts continuously on $M=\left( \tilde{\textbf{A}}^\dagger \otimes _{\tilde{\textbf{A}}^\dagger _L}M\right) ^{H_L}$ with respect to the induced topology as subspace of the previous module. Since all involved modules are free and hence carry the product topologies and since $\tilde{\textbf{A}}^\dagger _L\subseteq \tilde{\textbf{A}}^\dagger$ is a topological embedding, it is clear that the latter topology of M coincides with its canonical topology. $\square$

We define the functor

$$\begin{aligned} \tilde{D}^\dagger _{rig}(-) : {\text {Rep}}_{L}(G_L)&\longrightarrow {\mathfrak {M}}(\tilde{\mathcal {R}}_L) \\ V&\longmapsto (\tilde{\mathcal {R}} \otimes _{L} V)^{H_L} , \end{aligned}$$

where the fact, that $\Gamma _L$ acts continuously on the image with respect to the LF-topology can be seen as follows, once we have shown the next lemma. Indeed, (22) implies that for any $G_L$-stable $o_L$-lattice T of V we also have an isomorphism $\tilde{\mathcal {R}}_L \otimes _{\tilde{\textbf{A}}^\dagger _L} \tilde{D}^\dagger (T)\xrightarrow {\cong } \tilde{D}^\dagger _{rig}$. Now again Lemma 4.5 applies to conclude the claim.

Lemma 1.32

The canonical map

$$\begin{aligned} \tilde{\mathcal {R}}_L \otimes _{\tilde{\textbf{B}}^\dagger _L} \tilde{D}^\dagger (V)\xrightarrow {\cong } \tilde{D}^\dagger _{rig}(V) \end{aligned}$$

(22)

is an isomorphism and the functor $\tilde{D}^\dagger _{rig}(-) :{\text {Rep}}_{L}(G_L) \rightarrow {\mathfrak {M}}(\tilde{\mathcal {R}}_L)$ is exact. Moreover, we have a comparison isomorphism

$$\begin{aligned} \tilde{\mathcal {R}}\otimes _{ \tilde{\mathcal {R}}_L} \tilde{D}^\dagger _{rig}(V) \xrightarrow {\cong } \tilde{\mathcal {R}}\otimes _{o_L} V. \end{aligned}$$

(23)

Proof

The comparison isomorphism in the proof of (an analogue of) [24, Thm. 2.13] implies the comparison isomorphism

$$\begin{aligned} \tilde{\mathcal {R}}\otimes _{ \tilde{\mathcal {R}}_L } \tilde{D}^\dagger _{rig}(V)&\cong \tilde{\mathcal {R}}\otimes _{o_L} V \end{aligned}$$

together with the identity $V=(\tilde{\mathcal {R}}\otimes _{ \tilde{\mathcal {R}}_L } \tilde{D}^\dagger _{rig}(V))^{\varphi _L=1}.$ On the other hand the comparison isomorphism (16) induces by base change an isomorphism

$$\begin{aligned} \tilde{\mathcal {R}}\otimes _{ \tilde{\textbf{B}}^\dagger _L} \tilde{D}^\dagger (V) \xrightarrow {\cong } \tilde{\mathcal {R}}\otimes _{o_L} V. \end{aligned}$$

Taking $H_L$-invariants gives the first claim. The exactness of the functor $\tilde{D}^\dagger _{rig}(-)$ follows from the exactness of the functor $\tilde{D}^\dagger (-)$ by Prop.  4.6. $\square$

Let R be ${\textbf{B}}_L,$ ${\textbf{B}}_L^{\dagger }$, ${\mathcal {R}}_L$, $\tilde{\textbf{B}}_L,$ $\tilde{\textbf{B}}_L^{\dagger }$, $\tilde{\mathcal {R}}_L$ and let correspondingly $R^{int}$ be ${\textbf{A}}_L,$ ${\textbf{A}}_L^{\dagger }$, ${\textbf{A}}_L^{\dagger }$, $\tilde{\textbf{A}}_L,$ $\tilde{\textbf{A}}_L^{\dagger }$, $\tilde{\textbf{A}}_L^{\dagger }$. We denote by $\Phi (R)^{\acute{e}t}$ the essential image of the base change functor $R\otimes _{R^{int}} -:\Phi ^{\acute{e}t,f}(R^{int} )\rightarrow \Phi ^{\acute{e}t,f}(R)$ (sic!).

Proposition 1.33

Base change induces an equivalence of categories

$$\begin{aligned} \Phi (\tilde{\textbf{B}}_L^{\dagger } )^{\acute{e}t} \leftrightarrow \Phi (\tilde{\mathcal {R}}_L^{})^{\acute{e}t} \end{aligned}$$

and an isomorphism of Yoneda extension groups

$$\begin{aligned} \textrm{Ext}^1_{\Phi (\tilde{\textbf{B}}_L^{\dagger } )}(\tilde{\textbf{B}}_L^{\dagger },M) \cong \textrm{Ext}^1_{\Phi (\tilde{\mathcal {R}}_L )}(\tilde{\mathcal {R}}_L, \tilde{\mathcal {R}}_L \otimes _{\tilde{\textbf{B}}_L^{\dagger }} M ) \end{aligned}$$

for all $M \in \Phi (\tilde{\textbf{B}}_L^{\dagger } )^{\acute{e}t}.$

Proof

The first claim is an analogue of [26, Thm. 8.5.6]. The second claim follows as in the proof of Prop. (4.16) using the fact that by Lemma 8.6.3 in loc. cit. any extension of étale $\varphi$-modules over $\tilde{R}_L$ is again étale. Note that $\tilde{\mathcal {R}}_L/\tilde{\textbf{B}}^\dagger _L$ is a faithfully flat ring extension, $\tilde{\textbf{B}}^\dagger _L$ being a field. $\square$

Corollary 1.34

If V belongs to $Rep_{L}(G_L)$, the following complex concentrated in degrees 0 and 1 is acyclic

(24)

In particular, we have that the nth cohomology groups of the complex concentrated in degrees 0 and 1

are isomorphic to $H^n(H_L,V)$ for $n\ge 0.$

Proof

Compare with [26, Thm. 8.6.4] and its proof (Note that the authors meant to cite Thm. 8.5.12 (taking c=0, d=1) instead of Thm. 6.2.9 - a reference which just does not exist within that book). Using the interpretation of the $H^i_{\varphi }$ as ${\text {Hom}}$- and ${\text {Ext}}^1$-groups, respectively, the assertion is immediate from Prop.  5.3. The last statement now follows from Corollary 4.19. $\square$

Proposition 1.35

Base extension gives rise to an equivalence of categories

$$\begin{aligned} {\mathfrak {M}}^{\acute{e}t}({\textbf{B}}_L^\dagger ) \leftrightarrow {\mathfrak {M}}^{\acute{e}t}({\mathcal {R}}_L) . \end{aligned}$$

Proof

[19, Prop. 1.6]. $\square$

Lemma 1.36

• (i) $\textbf{B}_L^{\dagger }\subseteq {\mathcal {R}}_{L}$ are Bézout domains and the strong hypothesis in the sense of [22, Hypothesis 1.4.1] holds, i.e., for any $n\times n$ matrix A over $\textbf{A}_L^{\dagger }$ the map $({\mathcal {R}}_{L}/\textbf{B}_L^{\dagger })^n \xrightarrow {1-A\varphi _L} ({\mathcal {R}}_{L}/\textbf{B}_L^{\dagger })^n$ is bijective.

Proof

[22, Prop. 1.2.6]. $\square$

Proposition 1.37

If V belongs to $Rep_{L}^\dagger (G_L)$, the following complex concentrated in degrees 0 and 1 is acyclic

(25)

where ${D}^\dagger _{rig}(V):={\mathcal {R}}_L\otimes _{{\textbf{B}}_L^\dagger }{D}^\dagger (V).$ In particular, the complexes

(concentrated in degrees 0 and 1) have the same cohomology groups of for $n\ge 0$.

Proof

This follows from the strong hypothesis in Lemma 5.6 as the Frobenius endomorphism on $M\in {\mathfrak {M}}^{\acute{e}t}({\textbf{B}}_L^\dagger )$ is of the form $A\varphi _L$ by definition. $\square$

Lemma 1.38

Base change induces fully faithful embeddings $\Phi (\textbf{A}_L^\dagger )^{\acute{e}t}\subseteq \Phi (\textbf{A}_L)^{\acute{e}t}$ and $\Phi (\textbf{B}_L^\dagger )^{\acute{e}t}\subseteq \Phi (\textbf{B}_L)^{\acute{e}t}$.

Proof

As in the proof of Prop.  4.16 this reduces to checking that $\left( \textbf{A}_L\otimes _{\textbf{A}_L^\dagger }M\right) ^{\varphi ={\text {id}}}\subseteq M.$ By that proposition we know that

$$\begin{aligned} \left( \textbf{A}_L\otimes _{\textbf{A}_L^\dagger }M\right) ^{\varphi ={\text {id}}}\subseteq \left( \tilde{\textbf{A}}_L\otimes _{\textbf{A}_L^\dagger }M\right) ^{\varphi ={\text {id}}} \subseteq \tilde{\textbf{A}}_L^\dagger \otimes _{\textbf{A}_L^\dagger } M. \end{aligned}$$

Since $\textbf{A}_L\cap \tilde{\textbf{A}}_L^\dagger =\textbf{A}_L^\dagger$ within $\tilde{\textbf{A}}_L$ by definition, the claim follows for the integral version, whence also for the other one by tensoring the integral embedding with L over $o_L.$ $\square$

Remark 5.9

Note that $H^0_\dagger (H_L,V)\!=\!H^0(H_L,V)$ and $H^1_\dagger (H_L,V)\!\subseteq \! H^1(H_L,V)$. For the latter relation use the previous lemma, which implies that an extension which splits after base change already splits itself, together with Corollary 4.12 and Remark 4.14. In general the inclusion for $H^1$ is strict as follows indirectly from [19]. Indeed, otherwise the complex

(26)

would be always acyclic, which would imply by the same observation as in Prop.  7.2 below together with [44, Thm. 5.2.10(ii)] that $H^1_\dagger (G_L,V)\!=\!\! H^1(G_L,V)$ in contrast to Remark 5.2.13 in (loc. cit.).

6 The Web of Equivalences

We summarize the various equivalences of categories, for which we only sketch proofs or indicate analogue results whose proofs can be transferred to our setting.

Proposition 1.40

The following categories are equivalent:

• (i) $Rep_{o_L}(G_L)$,

• (ii) ${\mathfrak {M}}^{\acute{e}t}({\textbf{A}}_L) ,$

• (iii) ${\mathfrak {M}}^{\acute{e}t}(\tilde{\textbf{A}}_L)$ and

• (iv) ${\mathfrak {M}}^{\acute{e}t}(\tilde{\textbf{A}}_L^{\dagger } ).$

The equivalences from (ii) and (iv) to (iii) are induced by base change.

Proof

This can be proved in the same way as in [23, Thm. 2.3.5], although it seems to be only a sketch. Another way is to check that the very detailed proof for the equivalence between (i) and (ii) in [42] almost literally carries over to a proof for the equivalence between (i) and (iii). Alternatively, this is a consequence of Prop.  8.2 by [27, Thm. 5.4.6]. See also [29]. For the equivalence between (iii) and (iv) consider the 2-commutative diagram

which is induced by the isomorphism (13) and immediately implies (essential) surjectivity on objects and morphisms while the faithfulness follows from faithfully flat base change. $\square$

Corollary 1.41

The following categories are equivalent:

• (i) $Rep_{L}(G_L)$,

• (ii) ${\mathfrak {M}}^{\acute{e}t}({\textbf{B}}_L)$,

• (iii) ${\mathfrak {M}}^{\acute{e}t}(\tilde{\textbf{B}}_L)$ and

• (iv) ${\mathfrak {M}}^{\acute{e}t}(\tilde{\textbf{B}}_L^\dagger )$.

The equivalences from (ii) and (iv) to (iii) are induced by base change.

Proof

This follows from Propositions 4.18 and 6.1 by inverting $\pi _L.$ $\square$

Proposition 1.42

The categories in Corollary 6.2 are - via base change from (iv) - also equivalent to

(v):

${\mathfrak {M}}^{\acute{e}t}(\tilde{\mathcal {R}}_L) .$

Proof

By definition base change is essentially surjective and it is well-defined - regarding the continuity of the $\Gamma _L$-action - by Lemma 5.1 and Lemma 4.5. Since for étale $\varphi _L$-modules we know fully faithfulness already, taking $\Gamma _L$-invariants gives fully faithfulness for $(\varphi _L,\Gamma _L)$-modules, too.⁶$\square$

Altogether we may visualize the relations between the various categories by the following diagram:

Here all arrows represent functors which are fully faithful, i.e., embeddings of categories. Arrows without label denote base change functors. Under them the functors $D, \tilde{D}, {D}^\dagger , \tilde{D}^\dagger , {D}^\dagger _{rig}$, and $\tilde{D}^\dagger _{rig}$ are compatible. The arrows ${=>}$ represent equivalences of categories, while the arrows ${-\hspace{-0.15cm}>}$ represent embeddings which are not essentially surjective in general. We recall that the quasi-inverse functors are given as follows⁷,⁸,⁹

$$\begin{aligned} V(M)=&(\textbf{B}\otimes _{\textbf{B}_L}M)^{\varphi =1},\;\; \tilde{V}(M)=(\tilde{\textbf{B}}\otimes _{\tilde{\textbf{B}}_L}M)^{\varphi =1} ,\;\; V^\dagger (M)=(\textbf{B}^\dagger \otimes _{\textbf{B}_L^\dagger }M)^{\varphi =1},\\ \tilde{V}^\dagger (M)=&(\tilde{\textbf{B}}^\dagger \otimes _{\tilde{\textbf{B}}_L^\dagger }M)^{\varphi =1},\;\; \tilde{V}^\dagger _{rig}(M)= (\tilde{\mathcal {R}}\otimes _{ \tilde{\mathcal {R}}_L } M)^{\varphi =1} \text{ and } {V}^\dagger _{rig}(M)= (\tilde{\mathcal {R}}\otimes _{ {\mathcal {R}}_L } M)^{\varphi =1}. \end{aligned}$$

7 Cohomology: Herr Complexes

The aim of this section is to compare the Herr complexes of the various $(\varphi _L,\Gamma _L)$-modules attached to a given Galois representation.

We fix some open subgroup $U\subseteq \Gamma _L$ and let $L'=L_\infty ^U.$

Let $M_0$ be a complete linearly topologised $o_L$-module with continuous U-action and with continuous U-equivariant endomorphism f. We define

$$\begin{aligned} \mathcal {T} := \mathcal {T}_{f,U}(M_0):=\textrm{cone}\left( \mathcal {C}^\bullet (U,M_0)\xrightarrow {(f)_*-1} \mathcal {C}^\bullet (U,M_0)\right) [-1] \end{aligned}$$

the mapping fibre of $\mathcal {C}^\bullet (U,f-1)$. The importance of this generalized Herr-complex is given by the fact that it computes Galois cohomology when applied to $M_0=D(V)$ and $f=\varphi _{D(V)}:$

Theorem 1.43

Let V be in ${\text {Rep}}_{L}(G_L)$ For D(V) the corresponding $(\varphi _L,\Gamma _L)$-module over $\textbf{B}_L$ we have canonical isomorphisms

$$\begin{aligned} h^* = h^*_{U,V}: H^*(L',V) \xrightarrow {\ \cong \ } h^*(\mathcal {T}_{\varphi ,U}(D(V)) ) \end{aligned}$$

(27)

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

Proof

To this aim let T be a $G_L$-stable lattice of V. In [31, Thm. 5.1.11.], [30, Thm. 5.1.11.] it is shown that the cohomology groups of $\mathcal {T}_{\varphi ,U}({D}(T))$ are canonically isomorphic to $H^i(L',T)$ for all $i\ge 0,$ whence the cohomology groups of $\mathcal {T}_{\varphi ,U}({D}(T))[\frac{1}{\pi _L}]$ are canonically isomorphic to $H^i(L',V)$ for all $i\ge 0$.

$\square$

Note that we obtain a decomposition $U\cong \Delta \times U'$ with a subgroup $U'\cong \mathbb {Z}_p^d$ of U and $\Delta$ the torsion subgroup of U. We now fix topological generators $\gamma _1,\ldots \gamma _d$ of $U'$ and we set $\Lambda := \Lambda (U').$ By [32, Thm. II.2.2.6] the $U'$-actions extends to continuous $\Lambda$-action and one has $\textrm{Hom}_{\Lambda , cts}(\Lambda ,M_0)=\textrm{Hom}_{\Lambda }(\Lambda ,M_0).$ Consider the (homological) complexes $K_\bullet (\gamma _i):=[\Lambda \xrightarrow {\gamma _i-1}\Lambda ]$ concentrated in degrees 1 and 0 and define the Koszul complexes

Following [14, §4.2] and [44, (169)] we obtain a quasi-isomorphism

$$\begin{aligned} K^\bullet _{U'}(M_0)\xrightarrow {\simeq } \mathcal {C}^\bullet (U',M_0) \end{aligned}$$

(28)

inducing the quasi-isomorphism

$$\begin{aligned} K_{f,U'}(M_0)\xrightarrow {\simeq } \mathcal {T}_{f,U'}(M_0), \end{aligned}$$

(29)

where we denote by $K_{f,U'}(M_0):=\textrm{cone}\left( K^\bullet (M_0)\xrightarrow {f-{\text {id}}}K^\bullet (M_0) \right) [-1]$ the mapping fibre of $K^\bullet (f)$. More generally, by [44, Lem. A.0.1] we obtain a canonical quasi-isomorphism

$$\begin{aligned} K_{f,U'}(M^\Delta )\xrightarrow {\simeq } \mathcal {T}_{f,U}(M), \end{aligned}$$

(30)

i.e., by Theorem 7.1 we also have canonical isomorphisms

$$\begin{aligned} h^* = h^*_{U,V}: H^*(L',V) \xrightarrow {\ \cong \ } h^*(K_{f,U'}(D(V)^\Delta ) ). \end{aligned}$$

(31)

The next proposition extends this result to $\tilde{D}(V), \tilde{D}^\dagger (V)$ and $\tilde{D}^\dagger _{rig}(V)$ instead of D(V).

Proposition 1.44

If V belongs to $Rep_{L}(G_L)$, the canonical inclusions of Herr complexes

$$\begin{aligned} K^\bullet _{\varphi ,U'}(\tilde{D}^\dagger (V)^\Delta )&\subseteq K^\bullet _{\varphi ,U'}(\tilde{D}^\dagger _{rig}(V)^\Delta ) ,\\ K^\bullet _{\varphi ,U'}(\tilde{D}^\dagger (V)^\Delta )&\subseteq K^\bullet _{\varphi ,U'}(\tilde{D}(V)^\Delta ) \text{ and } \\ K^\bullet _{\varphi ,U'}({D}(V)^\Delta )&\subseteq K^\bullet _{\varphi ,U'}(\tilde{D}(V)^\Delta ) \end{aligned}$$

are quasi-isomorphisms and their cohomology groups are canonically isomorphic to $H^i(L',V)$ for all $i\ge 0.$

Proof

Forming Koszul complexes with regard to $U'$ we obtain the following diagram of (double) complexes with exact columns

in which the bottom line is an isomorphism of complexes by 4.12, as the action of $\Delta$ commutes with $\varphi$. Hence, going over to total complexes gives an exact sequence

$$\begin{aligned} 0\rightarrow K_{\varphi ,U}^\bullet ({D}(V)^\Delta ) \rightarrow K_{\varphi ,U}^\bullet (\tilde{D}(V)^\Delta )\rightarrow K_{\varphi ,U}^\bullet ((\tilde{D}(V)/{D}(V))^\Delta )\rightarrow 0, \end{aligned}$$

in which $K_{\varphi ,U}^\bullet ((\tilde{D}(V)/{D}(V))^\Delta )$ is acyclic. Thus we have shown the statement regarding the last inclusion. The other two cases are dealt with similarly, now using (24) and 4.19 combined with (8). It follows in particular that all six Koszul complexes in the statement are quasi-isomorphic. Therefore the second part of the assertion follows from (31). $\square$

In accordance with diagram at the end of Sect. 6 we may visualize the relations between the various Herr complexes by the following diagram:

Here all arrows represent injective maps of complexes, among which the arrows ${=>}$ represent quasi-isomorphisms, while the arrows ${-\hspace{-0.15cm}>}$ need not induce isomorphisms on cohomology, in general. The interrupted arrow ${-\;-\hspace{-0.15cm}>}$ means a map in the derived category while ${<-\;-\hspace{-0.15cm}>}$ means a quasi-isomorphism in the derived category. By [44, Lem. A.0.1] we have a analogous diagram for $\mathcal {T}_{\varphi ,U}(?(V))$ with $?\in \{D, \tilde{D}, {D}^\dagger , \tilde{D}^\dagger , {D}^\dagger _{rig}, \tilde{D}^\dagger _{rig}\}.$

Remark 7.3

The image of

$$\begin{aligned} h^i(\mathcal {T}_{\varphi ,U}(D^\dagger _{rig}(V)))\cong h^i(K^\bullet _{\varphi ,U'}({D}^\dagger _{rig}(V)^\Delta ))\cong h^i(K^\bullet _{\varphi ,U'}({D}^\dagger (V)^\Delta ))\cong h^i(\mathcal {T}_{\varphi ,U}(D^\dagger (V))) \end{aligned}$$

in $H^i(L',V)$ is independent of the composite ($=$ path) in above diagram.

8 Weakly Decompleting Towers

Kedlaya and Liu’s developed in [27, §5] the concept of perfectoid towers and studied their properties in an axiomatic way. The aim of this section is to show that the Lubin–Tate extensions considered in this article form a weakly decompleting, but not a decompleting tower, properties which we will recall or refer to in the course of this section. Moreover, we have to show that the axiomatic period rings coincide with those introduced earlier.

In the sense of Def. 5.1.1 in (loc. cit.) the sequence $\Psi =(\Psi _n: (L_n, o_{L_n})\rightarrow (L_{n+1},o_{L_{n+1}}))_{n=0}^\infty$ forms a finite étale tower over $(L,o_L)$ or $X:=\textrm{Spa}(L,o_L)$, which is perfectoid as $\hat{L}_\infty$ is by [42, Prop. 1.4.12].¹⁰

Therefore we can use the perfectoid correspondence [27, Thm. 3.3.8] to associate with $(\hat{L}_\infty , o_{\hat{L}_\infty })$ the pair

$$\begin{aligned} (\tilde{R}_\Psi ,\tilde{R}_\Psi ^+):=(\hat{L}_\infty ^\flat , o_{\hat{L}_\infty }^\flat ). \end{aligned}$$

Now we recall the variety of period rings, which Kedlaya and Liu attach to the tower, in our notation, starting with

Perfect period rings:

$$\begin{aligned} \tilde{\textbf{A}}_\Psi&:=\tilde{\textbf{A}}_L=W(\hat{L}_\infty ^\flat )_L, \\ \tilde{\textbf{A}}_\Psi ^+&:=W(o_{\hat{L}_\infty }^\flat )_L\subseteq \tilde{\textbf{A}}_\Psi ^{\dagger ,r}:=\tilde{\textbf{A}}_L^{\dagger ,r}=\left\{ x=\sum _{i\ge 0} \pi _L^i[x_i]\in W({\hat{L}_\infty }^\flat )_L| |\pi _L^{i}\Vert x_i|_\flat ^r\xrightarrow {i\rightarrow \infty } 0\right\} , \\ \tilde{\textbf{A}}_\Psi ^{\dagger }&:=\bigcup _{r>0} \tilde{\textbf{A}}_\Psi ^{\dagger ,r}=\tilde{\textbf{A}}_L^{\dagger } \end{aligned}$$

Imperfect period rings:

To introduce these we first recall the map $\Theta : W(o_{\mathbb {C}_p}^\flat )_L\rightarrow o_{\mathbb {C}_p},\; \sum _{i\ge 0} \pi _L^i [x_i]\mapsto \sum \pi _L^ix_i^\sharp ,$ which extends to a map $\Theta : \tilde{\textbf{A}}_\Psi ^{\dagger ,s}\rightarrow {\mathbb {C}_p}$ for all $s\ge 1;$ for arbitrary $r>0$ and $n\ge -\log _q r$ the composite $\tilde{\textbf{A}}_\Psi ^{\dagger ,r}\xrightarrow {\varphi _L^{-n}} \tilde{\textbf{A}}_\Psi ^{\dagger ,1}\xrightarrow {\Theta } \mathbb {C}_p$ is well defined and continuous as it is easy to check. It is a homomorphism of $o_L$-algebras by [42, Lem. 1.4.18].

Following [27, §5] we set $\textbf{A}_\Psi ^{\dagger ,r}:=\{x \in \tilde{\textbf{A}}_\Psi ^{\dagger ,r}| \Theta (\varphi _q^{-n}(x))\in L_n \text{ for } \text{ all } n\ge -\log _q r\},$ $\textbf{A}_\Psi ^{\dagger }:=\bigcup _{r>0} \textbf{A}_\Psi ^{\dagger ,r},$ its completion $\textbf{A}_\Psi :=(\textbf{A}_\Psi ^{\dagger })^{\wedge \pi _L-adic}$, and residue field $R_\Psi := \textbf{A}_\Psi /(\pi _L)=(\textbf{A}_\Psi ^{\dagger })/(\pi _L)\subseteq \tilde{R}_\Psi$, $R_\Psi ^+:=R_\Psi \cap \tilde{R}_\Psi ^+.$

Note that $\omega _{LT}=\{[\iota (t)]\}\in \tilde{\textbf{A}}_\Psi ^+:=W(o_{\hat{L}_\infty }^\flat )_L\subseteq \tilde{\textbf{A}}_\Psi ^{\dagger ,r}$ for all $r>0$ (in the notation of [42]). [42, Lem. 2.1.12] shows

$$\begin{aligned} \Theta (\varphi _q^{-n}(\omega _{LT}))=\Theta (\{[\varphi _q^{-n}(\omega )]\})=\lim _{i\rightarrow \infty } [\pi _L^i]_\varphi (z_{i+n})=z_n\in L_n, \end{aligned}$$

where $t=(z_n)_{n\ge 1}$ is a fixed generator of the Tate module $T_\pi$ of the formal Lubin–Tate group and $\omega =\iota (t)\in W(o_{\mathbb {C}_p}^\flat )_L$ is the reduction of $\omega _{LT}$ modulo $\pi _L$ satisfying with $\textbf{E}_L=k((\omega ))$. Therefore $\omega _{LT}$ belongs to $\textbf{A}_\Psi ^+:=\textbf{A}_\Psi \cap \tilde{\textbf{A}}_\Psi ^+$. Then it is clear that first $\textbf{A}_L^+:=o_L[[\omega _{LT}]] \subseteq \tilde{\textbf{A}}_\Psi ^{\dagger }$ and by the continuity of $\Theta \circ \varphi _L^{-n}$ even $\textbf{A}_L^+ \subseteq \textbf{A}_\Psi ^{\dagger }$ holds. Since $\omega _{LT}^{-1}\in \tilde{\textbf{A}}_\Psi ^{\dagger ,\frac{q-1}{q}}$ by [45, Lem. 3.10] (in analogy with [11, Cor. II.1.5]) and $\Theta \circ \varphi _L^{-n}$ is a ring homomorphism, it follows that $\omega _{LT}^{-1}\in \textbf{A}_\Psi ^{\dagger ,\frac{q-1}{q}}$ and $o_L[[\omega _{LT}]][\frac{1}{\omega _{LT}}]\subseteq \textbf{A}_\Psi ^{\dagger }.$

Lemma 1.46

We have $R_\Psi ^+=\textbf{E}_L^+$ and $R_\Psi =\textbf{E}_L.$

Proof

From the above it follows that $\textbf{E}_L\subseteq R_\Psi$, whence $\textbf{E}_L^{perf}\subseteq R_\Psi ^{perf} \subseteq \tilde{R}_\Psi =\hat{L}_\infty ^\flat$ the latter being perfect. Since $\widehat{\textbf{E}_L^{perf}}=\hat{L}_\infty ^\flat$ by [42, Prop. 1.4.17] we conclude that

$$\begin{aligned} \text {R}^\text {perf}_\Psi \hbox { is dense in }\tilde{R}_\Psi . \end{aligned}$$

(32)

By [27, Lem. 5.2.2] have the inclusion

$$\begin{aligned} R_\Psi ^+\subseteq \{x\in \tilde{R}_\Psi | x=(\bar{x}_n)\hbox { with }\bar{x}_n\in o_{L_n}/(z_1)\hbox { for }n>>1\}{\mathop {=}\limits ^{\text {(*)}}}{} \textbf{E}_L^+=k[[\omega ]] \end{aligned}$$

where the equality (*) follows from work of Wintenberger as recalled in [42, Prop. 1.4.29]. Since $\textbf{E}_L^+\subseteq \tilde{R}^+_\Psi$ by its construction in (loc. cit.), we conclude that $R_\Psi ^+=\textbf{E}_L^+$.

Since each element of $R_\Psi$ is of the form $\frac{a}{\omega ^m}$ with $a\in R_\Psi ^+$ and $m\ge 0$ by [42, Lem. 1.4.6]¹¹, we conclude that $R_\Psi =\textbf{E}_L.$ $\square$

Thus for each $r>0$ such that $\omega _{LT}^{-1}\in \textbf{A}_\Psi ^{\dagger ,r},$ reduction modulo $\pi _L$ induces a surjection $\textbf{A}_\Psi ^{\dagger ,r}\twoheadrightarrow R_\Psi$. Recall that $\Psi$ is called weakly decompleting, if

• (i) $R^{perf}_\Psi$ is dense in $\tilde{R}_\Psi .$

• (ii) for some $r>0$ we have a strict surjection $\textbf{A}_\Psi ^{\dagger ,r}\twoheadrightarrow R_\Psi$ induced by the reduction modulo $\pi _L$ for the norms $|-|_{r}$ defined by $|x|_{r}:=\sup _i \{|\pi _L^{i}\Vert x_i|^r_\flat \}$ for $x=\sum _{i\ge 0} \pi _L^i[x_i]$, and $|-|^r_\flat .$

We recall from [16, Prop. 1.4.3.] or [26, Prop. 5.1.2 (a)] that $|-|_{r}$ is multiplicative.

Proposition 1.47

The above tower $\Psi$ is weakly decompleting.

Proof

Since (32) gives (i), only (ii) is missing: Since $\omega _{LT}$ has $[\omega ]$ in degree zero of its Teichmüller series, we may and do choose $r>0$ such that $|\omega _{LT}-[\omega ]|_{r}<|\omega |_\flat ^r.$ Then $|\omega _{LT}|_{r}=\max \{|\omega _{LT}-[\omega ]|_{r}, |\omega |_\flat ^{r}\}=|\omega |_\flat ^{r}.$ Consider the quotient norm $\Vert b\Vert ^{(r)}=\inf _{a\in \textbf{A}_\Psi ^{\dagger ,r},a\equiv b \mod \pi _L}|a|_{r}.$

Now let $b=\sum _{n\ge n_0} a_n\omega ^n\in R_\Psi =k((\omega ))$ with $a_{n_0}\ne 0$. Lift each $a_n\ne 0$ to $\breve{a}_n\in o_L^\times$ and set $\breve{a}_n=0$ otherwise. Then, for the lift $x:=\sum _{n\ge n_0} \breve{a}_n\omega _{LT}^n$ of b we have by the multiplicativity of $|-|_{r}$ that

$$\begin{aligned} \Vert b\Vert ^{(r)}\le |x|_{r}=(|\omega _{LT}|_{r})^{n_0}=(|\omega |_\flat ^{r})^{n_0}=|b|_\flat ^r. \end{aligned}$$

Since, the other inequality $|b|_\flat ^r\le \Vert b\Vert ^{(r)}$ giving by continuity is clear, the claim follows. $\square$

Proposition 1.48

$\textbf{A}_L=\textbf{A}_\Psi .$

Proof

Both rings have the same reduction modulo $\pi _L.$ And using that the latter element is not a zero-divisor in any of these rings we conclude inductively, that $\textbf{A}_L/\pi _L^n\textbf{A}_L =\textbf{A}_\Psi /\pi _L^n\textbf{A}_\Psi$ for all n. Taking projective limits gives the result. $\square$

Proposition 1.49

$\textbf{A}_L^\dagger =\textbf{A}_\Psi ^\dagger .$

Proof

By [27, Lem. 5.2.10] we have that $\textbf{A}_\Psi ^\dagger =\tilde{\textbf{A}}_L^\dagger \cap \mathcal {R}_L.$ On the other hand $\textbf{A}_L^\dagger = (\tilde{\textbf{A}}^\dagger \cap \textbf{A})^{H_L}=\tilde{\textbf{A}}^\dagger _L \cap \textbf{A}$ is contained in $\mathcal {R}_L$ by Remark 4.20, whence $\textbf{A}_L^\dagger \subseteq \textbf{A}_\Psi ^\dagger$ while the inclusion $\textbf{A}_\Psi ^\dagger \subseteq \tilde{\textbf{A}}^\dagger \cap \textbf{A}_L=\textbf{A}_L^\dagger$ follows from Prop. 8.3. $\square$

In Definition 5.6.1 in (loc. cit.) they define the property decompleting for a tower $\Psi ,$ which we are not going to recall here as it is rather technical. The cyclotomic tower over $\mathbb {Q}_p$ is of this kind for instance. If our $\Psi$ would be decompleting, the machinery of (loc. cit.), in particular Theorems 5.7.3/4, adapted to the Lubin–Tate setting would imply that all the categories at the end of Sect. 6 are equivalent, which contradicts Remark 4.25.