3.1 Definition of the Robba ring $\mathcal {R}$

For any interval $I\subseteq (0,\infty )$ that is either compact or of the form $(0,r]$ , $r>0$ , we define

$$ \begin{align*} \mathcal{R}^{I}_F:=\left\{\sum_{k\in \mathbb{Z}}a_k\cdot Z^k\mid a_k\in F,\ \lim_{\lvert k \rvert \to\infty}v_{F}(a_k)+kt=\infty\text{ for all }t\in I \right\}. \end{align*} $$

We always assume that the boundary points of I are in the value group of $v_F$ , so that $\mathcal {R}_F^I$ is the ring of rigid analytic functions on the annulus

$$ \begin{align*} \{x\in F\mid v_F(x)\in I\}. \end{align*} $$

Furthermore, for $r>0$ , let

$$ \begin{align*} \mathcal{R}_F^r:=\mathcal{R}_F^{(0,r]}. \end{align*} $$

i.e. the ring of rigid analytic functions on the annulus with outer radius $1$ and inner radius depending on r. For any $s\in (0,r]$ , one has $\mathcal {R}_F^{[s,r]}\subseteq \mathcal {R}_F^r$ , and $\mathcal {R}_F^{[s,r]}$ is a Banach algebra over F with the norm

$$ \begin{align*} V_{[s,r]}\left(f \right)=\min_{t\in [s,r]}\left(\inf_{k\in\mathbb{Z}} (v_F(a_k)+kt)\right),\ \ \ \text{ where }f=\sum_{k\in\mathbb{Z}}a_k Z^k\in \mathcal{R}_F^{[s,r]}. \end{align*} $$

Thus $\mathcal {R}_F^r=\bigcap _{0<s\leq r}\mathcal {R}_F^{[s,r]}$ is a Fréchet space. There are natural inclusions $\mathcal {R}_F^r\subseteq \mathcal {R}_F^s$ for $s\leq r$ . Now the Robba ring over F in the variable Z is defined by

$$ \begin{align*} \mathcal{R}_F:=\bigcup_{r>0}\mathcal{R}_F^r. \end{align*} $$

We endow $\mathcal {R}_F$ with the locally convex direct limit topology of the $\mathcal {R}_F^r$ , making it an LF-space.

Moreover, let

$$ \begin{align*} \mathcal{R}_F^+:=\mathcal{R}_F\cap F[[ Z]]. \end{align*} $$

This is the ring of power series with coefficients in F that are convergent on the open unit disk. In particular, we have $\mathcal {R}_F^+\subseteq \mathcal {R}_F^r$ for all $r>0$ . For a complete field extension $F\subset F' \subset \mathbb {C}_p$ we have

$$ \begin{align*}F' \hat{\otimes}_{F,i}\mathcal{R}_F \cong \mathcal{R}_{F'}\end{align*} $$

(see. [Reference Berger, Schneider and Xie7, Corollary 2.1.8]). Their proof also shows $F' \hat {\otimes }_{F,\pi }\mathcal {R}_F^{r}\cong \mathcal {R}_{F'}^r.$

Inside $\mathcal {R}_F$ , we have the subring $\mathcal {R}_F^b$ of bounded elements, i.e., those Laurent series $f=\sum _{k\in \mathbb {Z}}a_k Z^k$ where the coefficients $a_k$ are bounded in F. It is well-known that $\mathcal {R}_F^\times =(\mathcal {R}_F^b)^\times $ . Furthermore, the map $f\mapsto \lVert f\rVert _1:=\sup _k\lvert a_k\rvert $ defines a multiplicative norm on $\mathcal {R}_F^b$ , see [Reference Berger, Schneider and Xie7, §1.3].

3.2 Frobenius and $\Gamma _L$ -action on $\mathcal {R}$

On $\mathcal {R}_F$ , we define a Frobenius $\varphi _L$ and a commuting $\Gamma _L$ -action by

$$ \begin{align*} \varphi_L(Z):=[\pi_L](Z)\ \ \ \text{ and }\ \ \ \gamma(Z):=[\chi_{LT}( \gamma)](Z)\ \text{for }\gamma\in \Gamma_L \end{align*} $$

on the variable and trivial actions on the coefficients. For $r>0$ , the Frobenius $\varphi _L$ and each $\gamma \in \Gamma _L$ restrict to maps

$$ \begin{align*} \varphi\colon\mathcal{R}_F^r\longrightarrow \mathcal{R}_F^{r/q}\ \ \ \text{ and }\ \ \ \gamma\colon\mathcal{R}_F^r \overset{\sim}{\longrightarrow}\mathcal{R}_F^r. \end{align*} $$

For r small enough, there is a left inverse

$$ \begin{align*}\psi_L\colon\mathcal{R}_F^{r/q}\longrightarrow\mathcal{R}_F^r\end{align*} $$

of $\varphi _L$ , given by $\Psi =\frac {\varphi _L^{-1}}{q}\circ \operatorname {\mathrm {Tr}}_{\mathcal {R}_F^{r/q}/\varphi _L(\mathcal {R}_F^r)}$ , see [Reference Fourquaux and Xie22, §2]. We have $\Psi = \frac {\pi _L}{q}\psi _L.$

3.3 $(\varphi _L,\Gamma _L)$ -Modules

Definition 3.1. A $\varphi _L$ -module over $\mathcal {R}_F$ is a finitely generated free $\mathcal {R}_F$ -module M, equipped with a continuousFootnote ³ , $\varphi _L$ -semilinear endomorphism $\varphi _M$ , such that the induced $\mathcal {R}_F$ -linear map

$$ \begin{align*} \mathcal{R}_F\otimes_{\mathcal{R}_F,\varphi_L}M\longrightarrow M,\ f\otimes x\longmapsto f\cdot \varphi_M(x) \end{align*} $$

is an isomorphism. Note that in the above tensor product, $\mathcal {R}_F$ is viewed as a left-module over itself in the usual way and as a right module via $ \varphi _L$ .

We will often simply write $\varphi $ instead of $\varphi _M$ .

Proposition 3.2. Let M be a $\varphi _L$ -module over $\mathcal {R}_F$ . Then there exists an $r(M)>0$ such that, for each $0<r\leq r(M)$ , there exists a unique finitely generated free $\mathcal {R}_F^r$ -submodule $M^r\subseteq M$ satisfying the following properties:

1. (i) $M=\mathcal {R}_F\otimes _{\mathcal {R}_F^r}M^r$ .

2. (ii) $\varphi _M$ induces an isomorphism $\mathcal {R}_F^{r/q}\otimes _{\mathcal {R}_F^r,\varphi _L}M^r\overset {\sim }{\longrightarrow } \mathcal {R}_F^{r/q}\otimes _{\mathcal {R}_F^r}M^r$ .

In particular, for $0<s\leq r\leq r(M)$ , one has

$$ \begin{align*} M^s=\mathcal{R}_F^s\otimes_{\mathcal{R}_F^r}M^r. \end{align*} $$

Remark 3.3. Let M be a $\varphi _L$ -module over $\mathcal {R}_F$ . Then for $0<s\leq r \leq r(M)$ and ${?\in \{s,[s,r],\varnothing \}}$ we write

$$ \begin{align*} M^?:=\mathcal{R}_F^?\otimes_{\mathcal{R}_F^{r(M)}}M^{r(M)}. \end{align*} $$

Composing the canonical map $M^r\longrightarrow \mathcal {R}_F^{r/q}\otimes _{\mathcal {R}_F^r,\varphi _L}M^r,\ m\longmapsto 1\otimes m$ with the isomorphism $\mathcal {R}_F^{r/q}\otimes _{\mathcal {R}_F^r,\varphi _L}M^r\cong M^{r/q}$ from Prop. 3.2(ii) above, we obtain $\varphi _L$ -semilinear maps

$$ \begin{align*}\varphi\colon M^r\longrightarrow M^{r/q}.\end{align*} $$

There is also an operator

$$ \begin{align*} \Psi_M\colon M^{r/q}\cong \mathcal{R}_F^{r/q}\otimes_{\mathcal{R}_F^r,\varphi_L}M^r \longrightarrow M^r \end{align*} $$

given by $f\otimes m\longmapsto \Psi (f)\cdot m$ .

Let M be a $(\varphi _L,\Gamma _L)$ -module over $\mathcal {R}_F$ . After fixing a basis of M, consider the matrix $P\in \operatorname {GL}_n(\mathcal {R}_F)$ representing $\varphi _M$ . Then we have $\det (P)\in \mathcal {R}_F^\times =(\mathcal {R}_F^b)^\times $ and may thus take the norm $\lVert \det (P) \rVert _1$ introduced at the end of Subsection 3.1. Define the degree $\deg (M)$ of M as the number satisfying $\lVert \det (P)\rVert _1=q^{-\deg (M)}$ ; one checks that this is independent of the initial choice of basis of M, see [Reference Berger, Schneider and Xie7, §3.3] for details. Furthermore, the slope of M is defined as $\mu (M):=\deg (M)/\operatorname {rk}(M)$ .

If F is not spherically complete, we do not know if there exist non-free, projective $(\varphi _L,\Gamma _L)$ -modules over $\mathcal {R}_F.$ In all cases considered by us, we will only need free modules. According to [Reference Berger, Schneider and Xie7] Prop. 2.25 the $\Gamma _L$ -action on a $(\varphi _L,\Gamma _L)$ -module M is differentiable so that the derived action of the Lie algebra $\operatorname {Lie}(o_L^\times )$ on M is available.

For the relation with L-analytic continuous Galois representations $Rep_L^{an}(G_L)$ of $G_L$ on finite dimensional vector spaces V, which are analytic, i.e., satisfying that, if $D_{dR}^{\mathbb {Q}_p}(V) := (V \otimes _{\mathbb {Q}_p} B_{dR})^{G_L}$ , the filtration on $D_{dR}^{\mathbb {Q}_p}(V)_{\mathfrak {m}}$ is trivial for each maximal ideal $\mathfrak {m}$ of $L \otimes _{\mathbb {Q}_p} L$ which does not correspond to the identity $\operatorname {id} : L \to L$ , Berger’s theorem is crucial.

Theorem 3.10. There is an equivalences of categories

$$ \begin{align*} Rep_L^{an}(G_L) & \longleftrightarrow \mathfrak{M}^{an,\acute{e}t}(\mathcal{R}_L) \\ V & \mapsto D_{\mathrm{rig}}^\dagger(V). \end{align*} $$

The embedding in (5) depends by construction on the choice of u. Any other choice does not change the image of the embedding because $Z_{au} = [a](Z_u)$ for $a \in o_L^\times $ by property (i) above (5). As explained in [Reference Schneider and Venjakob47, §8] the image $Z_u$ of the variable Z already lies in $W({\hat {L}_\infty }^\flat )_L$ , so that we actually have an embedding $o_L[[Z]]\to W({\hat {L}_\infty }^\flat )_L$ . Similarly, as in [Reference Kedlaya and Liu29, 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 $\mathcal {R}_L\to \tilde {\mathcal {R}}_L$ into the perfect Robba ring, see [Reference Schneider and Venjakob47, §5] for a definition and [Reference Witzelsperger57, Konstruktion 1.3.27] for a proof in the Lubin-Tate setting.

3.4 Rank one modules and characters

Let A be an affinoid algebra over $F.$ To each continuous character $\delta \colon L^\times \to A^\times $ we can attach a $(\varphi _L,\Gamma _L)$ -module of rank one $\mathcal {R}_A(\delta ) := \mathcal {R}_A\mathbf {e}_\delta $ by setting $\varphi _L(\mathbf {e}_\delta ) = \delta (\pi _L)\mathbf {e}_\delta $ and $\gamma (\mathbf {e}_\delta ) = \delta (\chi _{LT}(\gamma ))\mathbf {e}_\delta $ for $\gamma \in \Gamma _L.$ We say a module is of character type if it arises in this way. A $(\varphi _L,\Gamma _L)$ -module of character type is L-analytic (in the sense of Definition 3.9) if and only if $\delta $ is locally L-analytic (or equivalently $\delta _{|o_L^\times }$ is locally L-analytic). Over $\mathcal {R}_L$ any rank one module is of character type (cf. [Reference Fourquaux and Xie22, Proposition 1.9]). We write ${\Sigma =\Sigma (A)}$ for the set of continuous characters $\delta :L^\times \to A^\times $ . We denote by $\Sigma _{an}:=\Sigma _{an}(A)$ the set of locally L-analytic characters $\delta \colon L ^\times \to A^\times $ . Consider the following characters $\delta _{LT},\chi =x|x|, \delta ^{un}_c:L^\times \to L^\times $ for $c\in L^\times $ given by

$$ \begin{align*} \delta_{LT}(\pi_L)=1, &\; \delta_{LT|o_L^\times}=\operatorname{id} _{o_L^\times}, \\ \chi(\pi_L)=\frac{\pi_L}{q} ,&\; \chi_{|o_L^\times}=\operatorname{id} _{o_L^\times}, \\ \delta^{un}_c(\pi_L)=c, & \; (\delta^{un}_c)_{|o_L^\times}\equiv1. \end{align*} $$

In particular, $\chi =\delta ^{un}_{\frac {\pi _L}{q}}\delta _{LT}.$ Then $\delta _{LT}$ corresponds via class field theory to the character $\chi _{LT}\colon G_L\to o_L^\times .$ Let $\delta \colon L^\times \to L^\times $ be any continuous character; setting $\delta _0:=\delta ^{un}_{\delta ^{-1}(\pi _L)}\delta $ we may always decompose $\delta = \delta ^{un}_{\delta (\pi _L)}\delta _0$ satisfying $\delta _{|o_L^\times } = (\delta _0)_{|o_L^\times }$ and $\delta _0(\pi _L)=1.$ If $|\delta (\pi _L)|=1$ , the character $\delta $ corresponds to a Galois character $\chi _\delta $ via local class field theory. Then $D^\dagger _{rig}(L(\chi _\delta ))=\mathcal {R}_L(\delta )$ and we call $\delta $ étale.

Later, for descent calculations we will have to select out the sets of special characters $\Sigma _1:=\{x^{-i}| i\in \mathbb {N}\}$ and $\Sigma _2:=\{x^{i}\chi | i\in \mathbb {N}\}$ from the generic ones $\Sigma _{gen}:=\Sigma _{an}\setminus (\Sigma _1\cup \Sigma _2).$

Note that we have two $\psi $ -operators. While $\psi $ satisfies the identity $\psi \circ \varphi =\frac {q}{\pi _L}\operatorname {id}$ and makes sense even integrally, $\Psi $ denotes the left inverse of $\varphi ,$ i.e., satisfying $\Psi \circ \varphi = \operatorname {id}$ . In particular, we have $\psi =\frac {q}{\pi _L}\Psi .$ Note that $\psi (\mathbf {e}_\delta )=\frac {q}{\pi _L}\delta ^{-1}(\pi _L)\mathbf {e}_\delta .$

If $\delta \in \Sigma _{an}(K)$ and $a \in o_L^\times $ such that $\log (a) \neq 0,$ then one defines the weight of $\delta $ as $\omega _\delta := \log (\delta (a))/\log (a)$ (which is independent of a). We shall say that $\delta $ is de Rham, if the attached $(\varphi _L,\Gamma _L)$ -module $R_K(\delta )$ is de Rham in the sense that will be introduced in subsection 5.4 below. As shown in the Appendix A, Remark A.8, $\delta $ is de Rham if and only if there exist some locally constant character $\delta _{\mathrm {lc}}$ and $k(=\omega _\delta )\in \mathbb {Z,}$ such that

$$\begin{align*}\delta=\delta_{\mathrm{lc}}x^k \text{ (or equivalently }\delta=\delta_{\mathrm{lc}}\delta_{LT}^k \text{ for some other } \delta_{\mathrm{lc}}), \end{align*}$$

see also [Reference Schneider and Venjakob46, Rem. 3.2.3/4] for the étale case.

We fix some notation for the remainder of the article. Consider the differential operator $\partial := \partial _{\mathrm {inv}}= \frac {1}{\log _{LT}'(Z)}\frac {d}{dZ}$ acting on $\mathcal {R}_K.$ (This differs from [Reference Colmez15] by a constant.) Let ${\nabla \in Lie(\Gamma _L)\cong L}$ be the element corresponding to $1 \in L.$

3.5 The modules $\mathbf {D}_{\mathrm {dif}}^{(+)}(M)$

We set $t_{LT}=\log _{LT}(Z) \in L[[Z]]$ , so that

$$ \begin{align*} \varphi(t_{LT})=\pi_L\cdot t_{LT}\ \ \ \text{ and }\ \ \ \gamma(t_{LT})=\chi_{LT}(\gamma)\cdot t_{LT} \text{ for all }\gamma\in\Gamma_L \end{align*} $$

by (3). For $n\geq 1$ , we set

$$\begin{align*}[\pi_L^{-n}](Z):=u_n+_{LT}\exp_{LT}(\frac{t_{LT}}{\pi_L^n}) \in L_n[[Z]]. \end{align*}$$

Then $[\pi _L^n](u_n+_{LT}\exp _{LT}(\frac {t_{LT}}{\pi _L^n}))=Z,$ which is how Colmez justifies this notation in [Reference Colmez15, 1.4.2]. Note that the constant term of $[\pi _L^{-n}](Z)$ is equal to $u_n$ and hence is non-zero, so $[\pi _L^{-n}](Z)$ is a unit in $L_n[[Z]]$ .

Furthermore, let $\theta :K_n[[t_{LT}]]\to K_n$ denote the $K_n$ -linear map sending $t_{LT}$ to $0,$ i.e., the reduction modulo $t_{LT}.$ This is the completed base change to K of the restriction of $\theta :\mathbf {B}_{dR}^+\to \mathbb {C}_p$ to $L_n[[t_{LT}]]\to L_n.$

In the following, let F be a complete non-Archimedean field containing L.

Definition 3.13. The group $\Gamma _L$ acts diagonally on

$$ \begin{align*}F_n:= L_n\otimes_LF\end{align*} $$

(trivially on the right factor and naturally on the left), and we extend this to an action on $F_n[[Z]]$ via its usual action on Z.Footnote ⁴ Now define

$$ \begin{align*} \iota_n=\iota_n^{(F)}:\mathcal{R}_F^{r_n} & \to F_n[[t_{LT}]]=F_n[[Z]], \\ \sum_{k\in\mathbb{Z}} a_kZ^k & \mapsto \sum_{k\in\mathbb{Z}} a_k([\pi_L^{-n}](Z))^k, \end{align*} $$

where on the right-hand side $a_k$ denotes (by abuse of notation) the image under the canonical embedding and $[\pi _L^{-n}](Z)$ is viewed as a power series over $L_n\otimes _LF$ via the embedding .

Proof. To see that $\iota _n$ is well-defined for $F=L$ first recall that by [Reference Colmez14, Prop. 8.10], the ring $\mathbf {B}_{\mathrm {dR}}^+$ contains a period $t_L$ for the Lubin-Tate character, i.e. we have $g(t_L)= \chi _{LT}(g)t_L$ for all $g \in G_L$ and $t_L$ differs from the usual t by a unit. Thus $L_n[[t_{LT}]]$ embeds into $\mathbf {B}_{\mathrm {dR}}^+$ via $t_{LT} \longmapsto t_L$ and we endow it with the subspace topology, making it a closed subspace of $\mathbf {B}_{\mathrm {dR}}^+$ . It hence suffices to show that $\iota _n(f)$ converges in $\mathbf {B}_{\mathrm {dR}}^+.$ A series of the form $x=\sum _{k \gg -\infty } p^k [x_k] \in W(o_{\mathbb {C}_p^\flat })[1/p]$ converges in $\mathbf {B}_{\mathrm {dR}}^+$ if and only if $\theta (x)$ converges in $\mathbb {C}_p$ (which is the case precisely when $k+v(x_k) \to \infty $ for $k \to \infty $ ). As in [Reference Cherbonnier and Colmez12, Prop. III.2.1 (i)], the condition $x=\sum _{k \in \mathbb {Z}}a_k Z_u^k$ with $a_k \in o_L$ and $v_p(a_k) + k \cdot r_n \to \infty $ for $k \to -\infty $ implies that $\iota _n(x)$ converges in $\mathbf {B}_{\mathrm {dR}}^+$ . Even though the coefficients of an element $x \in \mathcal {R}_L^{r_n} $ are not bounded, they do satisfy the same growth condition, which ensures the convergence of $\iota _n(x).$ The case of general F is obtained via completed base change $\mathcal {R}_F^{r_n} = F \hat {\otimes }_{L,\pi }\mathcal {R}_L^{r_n} \to F_n[[t_{LT}]] = F \hat {\otimes }_{L,\pi }L_n[[t_{LT}]].$

For the second point we compute

$$ \begin{align*} \iota_n(t_{LT})=\log_{LT}([\pi_L^{-n}](Z)) =\underbrace{\log_{LT}(u_n)}_{=0}+\log_{LT}\exp_{LT}\left( \frac{t_{LT}}{\pi_L^n}\right)=\frac{t_{LT}}{ \pi_L^n}. \end{align*} $$

For the injectivity of $\iota _n$ we can assume $F=L$ because completed base change $F \hat {\otimes }_{L,\pi }-$ preserves injectivity by [Reference Emerton19, 1.1.26]. Consider the reduction modulo $t_{LT}$ such that $\theta \circ \iota _n(f)=f(u_n).$ If $\iota _n(f)=0$ then $f(u_n)=0$ and hence f is divisible by $Q_n(Z):=\frac {[\pi _L^n](Z)}{[\pi _L^{n-1}](Z)}.$ Because $\iota _n(t_{LT})\neq 0$ and $Q_n \mid t_{LT} = Z\prod _{\mu \geq 1}\frac {Q_\mu } {\pi _L}$ we conclude $\iota _n(Q_n)\neq 0$ and hence $\iota _n(f/Q_n)=0.$ Inductively $Q_n^k \mid f$ for every $k\geq 0.$ The choice of n ensures that $Q_n \in \mathcal {R}_L^{r_n}$ is a non-unit. By considering the image of f in the noetherian domain $\mathcal {R}_L^{[r_n,r_n]}$ under the inclusion $\mathcal {R}_L^{r_n}\subset \mathcal {R}_L^{[r_n,r_n]}$ we conclude $f=0$ by Krull’s Intersection Theorem.

The map $\iota _n$ commutes with the action of $\Gamma _L$ . Writing $\mathrm {Tr}=\operatorname {id}_F\otimes \frac {1}{q}\mathrm {Tr}_{L_{n+1}/L_n}$ we obtain the commutative diagrams

and

Definition 3.15. Let M be a $(\varphi _L,\Gamma _L)$ -module over $\mathcal {R}_F$ . Viewing $F_{n}[[t_{LT}]]$ as an $\mathcal {R}_F^{r_n}$ -module via $\iota _n$ , we define the $F_{n}[[t_{LT}]]$ - and $F_{n}((t_{LT}))$ -modules

$$ \begin{align*} \mathbf{D}_{\text{dif},n}^+(M):= F_{n}[[t_{LT}]] \otimes_{\mathcal{R}_F^{r_n}} M^{(n)} \quad \text{ and } \quad \mathbf{D}_{\text{dif},n}(M):=\mathbf{D}_{\text{dif},n}^+(M)_{t_{LT}}, \end{align*} $$

respectively, where $(-)_{t_{LT}}$ means localising an $F_n[[t_{LT}]]$ -module at the multiplicative subset generated by $t_{LT}$ . Furthermore, $\mathbf {D}_{\text {dif},n}^+(M)$ carries the diagonal action of $\Gamma _L$ , which also extends to $\mathbf {D}_{\text {dif},n}(M)$ . Under the isomorphism $\varphi ^{*}(M^{(n)}) = \mathcal {R}_F^{r_{n+1}} \otimes _{\varphi ,\mathcal {R}^{r_n}_F} M^{(n)} \cong M^{(n+1)}$ , the map $\varphi \colon M^{(n)} \to M^{(n+1)}$ corresponds to the canonical map $\operatorname {can_{n,n+1}}\colon M^{(n)}\to \varphi ^{*}(M^{(n)}); x\mapsto 1 \otimes x.$ The above diagrams then induce the diagrams (see [Reference Nakamura36, §2.B] for details)

where the map $\mathrm {can}_{n,n+1}$ is given by , and $\iota _n$ by $m\mapsto 1\otimes m$ , as well as, for $n\geq 1,$

with transitions maps $f(t)\otimes x\mapsto \mathrm {Tr}(f(t))\otimes \psi (x)$ on the right hand side. Finally, we define

$$ \begin{align*} \mathbf{D}_{\text{dif}}^{(+)}(M):= \varinjlim_{n \gg 0}\mathbf{D}_{\text{dif},n}^{(+)}(M) \end{align*} $$

with $\mathrm {can}_{n,n+1}$ as transition maps.

As in [Reference Nakamura36], we have $\mathbf {D}_{\text {dif},n}^{+}(M) \otimes _{F_n[[t_{LT}]]} F_{n+1}[[t_{LT}]] \overset {\sim }{\longrightarrow } \mathbf {D}_{\text {dif},n+1}^{+}(M)$ and hence

$$ \begin{align*} \mathbf{D}_{\text{dif}}^{(+)}(M) = \mathbf{D}_{\text{dif},n}^{(+)}(M) \otimes_{F_n[[t_{LT}]]} (\bigcup_{m \geq n} F_m[[t_{LT}]]) \end{align*} $$

for $n \gg 0$ .

Later on it will be crucial to form the cohomology groups $H^i_{\varphi , \mathfrak {Z}}(\mathbf {D}_{\mathrm {dif}}^+(M))$ from Section 4. For this we need a $D(\Gamma _L,F)$ -module-structure on $\mathbf {D}_{\mathrm {dif}}^{(+)}(M)$ , which we get from Proposition 3.18 below after showing that the action is pro-L-analytic. Let us first recall this notion.

Lemma 3.19. Let B be a Fréchet $\Gamma _L$ -ring over F and W a finitely generated free B-module with a compatible $\Gamma _L$ -action. Assume there is a basis ${\mathfrak {A}}:=(e_1,\ldots ,e_d)$ for W such that the map

$$ \begin{align*} \Gamma_L\longrightarrow \mathrm{GL}_d(B),\ \gamma\longmapsto \mathrm{Mat}_{{\mathfrak{A}}}(\gamma) \end{align*} $$

is pro-L-analytic. $W^{L-\mathrm {pa}}=\bigoplus _{j=1}^dB^{L-\mathrm {pa}}\cdot e_j$ .

Proof. We start with $\operatorname {{{\mathbf {D}}_{\mathrm {dif}}^+}}(M)=\varinjlim _{n\gg 0}\mathbf {D}_{\mathrm {dif},n}^+(M)$ . By definition, it suffices to check that the $\Gamma _L$ -action on the Fréchet space $\mathbf {D}_{\mathrm {dif},n}^+(M)$ is pro-L-analytic for $n\gg 0$ .

We wish to apply Lemma 3.19 with $B:=F_n[[t_{LT}]]$ and $W:=\mathbf {D}_{\mathrm {dif},n}^+(M)=B\otimes _{\mathcal {R}_F^{(n)}}M^{(n)}$ : Choose any $\mathcal {R}_F^{(n)}$ -module basis $x_1,\dots ,x_d$ of $M^{(n)}$ . Then ${\mathfrak {A}}:=(1\otimes x_1,\ldots ,1\otimes x_d)$ is a basis of the free B-module W, and the map $\gamma \longmapsto \mathrm {Mat}_{{\mathfrak {A}}}(\gamma )$ is given by the composite

$$ \begin{align*} \Gamma_L\longrightarrow \operatorname{GL}_d(\mathcal{R}_F^{(n)})\overset{\iota_n}{\longrightarrow} \operatorname{GL}_d(B) \end{align*} $$

where the first map is pro-L-analytic because $M^{(n)}$ is pro-L-analytic by assumption. Moreover, since $\iota _n$ is a continuous homomorphism of F-algebras, we conclude that 3.19 is applicable. Thus we obtain

$$ \begin{align*} W^{L-\mathrm{pa}}=\bigoplus_{j=1}^d B^{L-\mathrm{pa}}\cdot (1\otimes x_j)= F_n[[t_{LT}]]^{L-\mathrm{pa}}\otimes_{\mathcal{R}_F^{(n)}}M^{(n)}. \end{align*} $$

Finally, from [Reference Porat38, Prop. 2.6 2.] it follows that $F_n[[t_{LT}]]^{L-\mathrm {pa}}=F_n[[t_{LT}]]$ , which completes the proof for $\operatorname {{{\mathbf {D}}_{\mathrm {dif}}^+}}(M)$ .

Moving on to $\operatorname {{{\mathbf {D}}_{\mathrm {dif}}}}(M)$ , we write $\operatorname {{{\mathbf {D}}_{\mathrm {dif}}}}(M)=\varinjlim _{n,k}\mathbf {D}_{\mathrm {dif},n}^+(M)\cdot t_{LT}^{-k}$ as a direct limit of Fréchet spaces. By what we have just shown, one can express $\mathbf {D}_{\mathrm {dif},n}^+(M)$ for $n\gg 0$ as inverse limit $\mathbf {D}_{\mathrm {dif},n}^+(M)=\varprojlim _r B_{n,r}$ for certain F-Banach spaces $B_{n,r}$ on which $\Gamma _L$ acts L-analytically. So for any k one has

$$ \begin{align*}\mathbf{D}_{\mathrm{dif},n}^+(M)\cdot t_{LT}^{-k} =\varprojlim_r B_{n,r}\cdot t_{LT}^{-k},\end{align*} $$

where by $B_{n,r}\cdot t_{LT}^{-k}$ we denote the $\Gamma _L $ -module $B_{n,r}$ whose $\Gamma _L$ -action is twisted by $\chi _{LT}^{-k}$ . Since the inversion in $\Gamma _L$ is an L-analytic map, we see that the twisted action $b\longmapsto \chi _{LT}(\gamma ^{-k})\cdot \gamma (b)$ on $B_{n,r}$ is again L-analytic. Thus $\Gamma _L $ acts pro-L-analytically on $\mathbf {D}_{\mathrm {dif},n}^+(M)\cdot t_{LT}^{-k}$ for $n\gg 0$ and $k\geq 1$ , so the claim follows.

Note that $\mathbf {D}_{\mathrm {dif}}(M)$ depends on the coefficient field of $\mathcal {R}_F.$ For a complete field extension $F'/F$ and an L-analytic $(\varphi _L,\Gamma _L)$ -module M over $\mathcal {R}_F$ one checks that $F' \hat {\otimes }_{F,i}M$ is an L-analytic $(\varphi _L,\Gamma _L)$ -module over $\mathcal {R}_{F'}.$ Here $\hat {\otimes }_{F,i}$ denotes the inductive tensor product topology.

Remark 3.21. Let $F'/F$ be a complete field extension and let M be an L-analytic $(\varphi _L,\Gamma _L)$ -module over $\mathcal {R}_F.$ The natural maps

$$ \begin{align*}F'\hat{\otimes}_{F,i}\mathbf{D}_{\mathrm{dif}}(M) \to \mathbf{D}_{\mathrm{dif}}(M\hat{\otimes}_{F,i}F')\end{align*} $$

and

$$ \begin{align*}F' \hat{\otimes}_{F,i}\mathbf{D}_{\mathrm{dif},n}(M) \to \mathbf{D}_{\mathrm{dif},n}(M\hat{\otimes}_{F,i}F')\end{align*} $$

are $\Gamma _L$ -equivariant isomorphisms.

Proof. The completed inductive tensor product commutes with strict locally convex inductive limits by [Reference Emerton19, Theorem 1.1.30] together with the argument in the proof of [Reference Berger, Schneider and Xie7, 2.1.7(i)]. Hence the first statement follows from the second. For Fréchet spaces inductive and projective tensor products agree and commute with projective limits (of Hausdorff spaces) with dense transition maps (cf. [Reference Berger, Schneider and Xie7, 2.1.4] and [Reference Schneider40, 17.6]). This allows us to first reduce to the corresponding statement for $\mathbf {D}_{\operatorname {\mathrm {dif}},n}^+(M)t_{LT}^{-k_0}$ since $\mathbf {D}_{\operatorname {\mathrm {dif}},n}(M) = \varinjlim _k \mathbf {D}_{\operatorname {\mathrm {dif}},n}^+(M)t_{LT}^{-k}$ and by 3.16 we have $\mathbf {D}^+_{\mathrm {dif},n}(M) \cong \varprojlim _k \mathbf {D}^+_{\mathrm {dif},n}(M)/(t_{LT}^k)$ , hence we even have surjective transition maps which allow us to reduce to the corresponding statement for $\mathbf {D}^+_{\mathrm {dif},n}(M)/(t_{LT}^k)$ (assuming for simplicity $k_0=0$ , the general case being treated analogously). Since each $\mathbf {D}^+_{\mathrm {dif},n}(M)/(t_{LT}^k)$ is finite dimensional over F, we may omit the completion and see that

$$ \begin{align*}F' \otimes_{F}\mathbf{D}^+_{\mathrm{dif},n}(M)/(t_{LT}^k) \to \mathbf{D}^+_{\mathrm{dif},n}(M\hat{\otimes}_{F,i}F')/(t_{LT}^k)\end{align*} $$

is an isomorphism of finite dimensional $F'$ -vector spaces, which follows from the fact that any basis of $M^{(n)}$ gives rise on the one hand to a $F_n[[t_{LT}]]$ basis of $\mathbf {D}^+_{\mathrm {dif},n}(M)$ and on the other hand to a basis of $F'\hat {\otimes }M^{(n)}$ and thus to a $F^{\prime }_n[[t_{LT}]]$ basis of $\mathbf {D}^+_{\mathrm {dif},n}(F'\hat {\otimes }M).$ Note that $F_n[[t_{LT}]]/(t_{LT})^k \otimes _F F' \cong F^{\prime }_n[[t_{LT}]]/(t_{LT})^k$ by a dimension argument.

Lemma 3.22. Let V be a F-Banach space and let G be a group acting on V via continuous F-linear maps. Let W be an F-Banach space of countable type endowed with the trivial G-action. Then

$$ \begin{align*}(V \hat{\otimes}W)^G = V^G \hat{\otimes}W\end{align*} $$

Corollary 3.23. Let $F'/F$ be a complete field extension contained in $\mathbb {C}_p$ and let M be an L-analytic $(\varphi _L,\Gamma _L)$ -module over $\mathcal {R}_F.$ We have

$$ \begin{align*}\mathbf{D}^{(+)}_{\mathrm{dif}}(M)^{\Gamma_L} \hat{\otimes}_{F,i}F'= \mathbf{D}^{(+)}_{\mathrm{dif}}(M\hat{\otimes}_{F,i}F')^{\Gamma_L}.\end{align*} $$

4.1 Finiteness of analytic Cohomology

For a commutative ring R and an object $C \in \mathbf {D}^b(R)$ whose cohomology groups are of finite rank over $R,$ we denote by $\chi _R(C) = \sum _i (-1)^i \operatorname {rank}H^i(C)$ the Euler-Poincaré-characteristic of $C.$ Recall that a $(\varphi _L,\Gamma _L)$ -module is called trianguline if it can be written as a successive extension of rank one modules of character type in the sense of section 3.4.

Remark 4.7. Let $A/K$ be affinoid and let $M $ be a trianguline L-analytic $(\varphi _L,\Gamma _L)$ -module over $\mathcal {R}_A.$ Then the Euler-Poincaré Formula holds, i.e.,

$$ \begin{align*}\chi(M):= \chi(K_{\varphi_L,\mathfrak{Z_n}}(M))=\sum (-1)^i\mathrm{rk}_{\mathcal{R}_A}(H^i_{\varphi_L,{\mathfrak{Z}}_n}(M)) = [\Gamma_L:\Gamma_n]\mathrm{rk}_{\mathcal{R}_A}(M).\end{align*} $$

We will require a slight generalization of 4.6. Recall that $K_{\varphi _L,\mathfrak {Z}}(M)$ is (up to shift) quasi isomorphic to the cone of $1-\varphi $ on $\operatorname {RHom}_{D(U,K)}(K,M).$ As a consequence of [Reference Steingart51, Lemma 2.5] K admits a finite projective resolution consisting of finitely generated projective $D(\Gamma _L,K)$ -modules. In particular the complex computing $\operatorname {RHom}_{D(U,K)}(K,M)$ (and hence also $K_{\varphi _L,\mathfrak {Z}}(M)$ ) can be represented by a complex of $A[\Gamma _L/U]$ -modules the terms of which are all of the form $\operatorname {Hom}_{D(U,K)}(P,M),$ where P is the restriction of scalars of a projective $D(\Gamma _L,K)$ -module with $\Gamma _L$ acting via $(\gamma f)(x) = \gamma (f(\gamma ^{-1}x))$ and A acting by multiplication on $M.$

Remark 4.8. In the situation of 4.6, if we view $K_{\varphi _L,{\mathfrak {Z}}}(M)$ as an object in $\mathbf {D}(A[\Gamma _L/U])$ we have

$$ \begin{align*}K_{\varphi_L,\mathfrak{Z}}(M) \in \mathbf{D}^{[0,2]}_{\mathrm{perf}}(A[\Gamma_L/U]).\end{align*} $$

4.2 Perfectness of analytic Iwasawa cohomology and the Lubin-Tate deformation

For M any $(\varphi _L,\Gamma _L)$ -module over any basis consider the complex

$$\begin{align*}\mathcal{T}_{\Psi}(M):=[M\xrightarrow{\Psi-1}M]\end{align*}$$

concentrated in degrees $1$ and $2$ , whose cohomology we call (analytic) Iwasawa cohomology due to Fontaine’s classical result, which relates these groups in the étale case to usual Iwasawa cohomology defined in terms of Galois cohomology. We set $D:=D(\Gamma _L,K).$ The following result [Reference Steingart51, Thm. 4.8] will be central for the whole article:

For the rest of this subsection we assume that $M\in \mathfrak {M}^{an}(K)$ is trianguline.

Later for our approach it will be important to interpret Iwasawa cohomology as analytic cohomology of a deformation $\mathbf {Dfm}(M)$ of M via generalized Herr complexes. This deformation lives over the character variety $\mathfrak {X}_{\Gamma _L}$ (base changed to K) of the locally L-analytic group $\Gamma _L$ ([Reference Schneider and Teitelbaum42]) and will allow to use density arguments to deduce many properties of the Epsilon-isomorphism for rank one modules just from properties over its de Rham points.

We pick an affinoid cover $X_n\cong \operatorname {Sp}(D_n)$ of $\mathfrak {X}_{\Gamma _L}$ with $D_n:=K[\Gamma _L]\otimes _{K[U]}D_{r_n}(U,K)$ for a decreasing sequence $r_n$ such that each $D_{r_n}(U,K) $ corresponds to the ring of rigid analytic functions on the annulus $[r_n,\infty ]$ via the Fourier isomorphism for $D(U,K)$ . Over the space $\mathfrak {X}_{\Gamma _L}$ we have the sheaf of Robba rings $\mathcal {R}_{\mathfrak {X}_{\Gamma _L}}$ given by mapping $X_n$ to $\mathcal {R}_{\mathcal {O}_{\mathfrak {X}_{\Gamma _L}}(X_n)}$ and $\mathbf {Dfm}(M)$ should be thought of as a $(\varphi _L, \Gamma _L)$ -module (sheaf) over $\mathcal {R}_{\mathfrak {X}_{\Gamma _L}}$ (but unfortunately, Schneider’s and Teitelbaum’s formalism of coadmissible modules does not apply here as $\mathcal {R}_{\mathfrak {X}_{\Gamma _L}}(\mathfrak {X}_{\Gamma _L})$ does not form a Fréchet-Stein algebra in any obvious sense):

For an L-analytic $(\varphi ,\Gamma _L)$ -module M over $\mathcal {R}_L$ we define

$$ \begin{align*}\mathbf{Dfm}(M)(X_n):=\mathbf{Dfm}_n(M):=\mathcal{O}_{\mathfrak{X}_{\Gamma_L}}(X_n) \hat{\otimes}_L M,\end{align*} $$

where $\Gamma _L$ acts diagonally, on the left factor via the inversion and on M via its given action. For each n this is a $(\varphi ,\Gamma _L)$ -module M over $\mathcal {R}_{\mathcal {O}_{\mathfrak {X}_{\Gamma _L}}(X_n)}$ by [Reference Steingart51, Prop. 3.2].

As definition for the generalized Herr complex for the sheaf $\mathbf {Dfm}(M)$ , philosophically, we would like to take the complex in $\mathbf {D}(D(\Gamma _L,K))$ Footnote ⁹ defined as total derived sheaf cohomology of the complex of sheaves

$$\begin{align*}K_{\Psi,D(\Gamma_L,K)}(\mathbf{Dfm}(M)) =\mathcal{T}_{\Psi}(\mathbf{Dfm}(M))\otimes^{\mathbb{L}}_{D(\Gamma_L,K),\mathrm{diag}}K \ ("\cong K_{\Psi,{\mathfrak{Z}}}(\mathbf{Dfm}(M))\otimes_{K[\Gamma_L/U],\mathrm{diag}}^{\mathbb{L}}K"),\end{align*}$$

where for the last (quasi-)isomorphism in quotation marks we used implicitly the free resolution

(17)

which induces an isomorphism $\mathcal {T}_{\Psi }(\mathbf {Dfm}(M))\otimes ^{\mathbb {L}}_{D(\Gamma _L,K),\mathrm {diag}}K[\Gamma _L/U]\cong K_{\Psi ,{\mathfrak {Z}}}(\mathbf {Dfm}(M)).$

But strictly speaking one needs a resolution of $D(\Gamma _L,K)\otimes _K K[\Gamma _L/U]$ -modules in order to define the $(D_n,K[\Gamma _L/U])$ -bimodule structure on $K_{\Psi ,{\mathfrak {Z}}}(\mathbf {Dfm}_n(M))\cong \mathcal {T}_{\Psi }(\mathbf {Dfm}_n(M))\otimes ^{\mathbb {L}}_{D(\Gamma _L,K)}K[\Gamma _L/U]$ . To this end we can formally work with the resolution

(18)

for an explicit construction of $K_{\Psi ,{\mathfrak {Z}}}(\mathbf {Dfm}(M))$ in $\mathbf {D}((D_n,K[\Gamma _L/U])\mathrm {-bimod}),$ with the last non-trivial map given by $a\otimes b\mapsto H(a)\cdot b,$ where $H\colon D(\Gamma _L,K)\to K[\Gamma _L/U]$ denotes the augmentation map sending the Dirac distributions of $u\in U$ to $1.$ Indeed, the kernel C is projective (hence flat) as a D-module by the same reasoning as for [Reference Steingart51, Lem. 2.5]. This sequence is related to (the direct sum of) the sequences

for the characters $\chi _i$ of $\Gamma _L$ which factor through $\Gamma _L/U.$

Instead of verifying that we really have a complex of (coherent) sheaves we just use the facts as a motivation that on a Stein space $\Gamma (\mathfrak {X}_{\Gamma _L},-)=\varprojlim _n \Gamma (X_n,-)$ and that higher sheaf cohomology of coherent sheaves vanishes on affinoids. Thus we rather take the total derived inverse limit as a formal definition, i.e.,

$$\begin{align*}C^\bullet:=R\Gamma_{\Psi,D(\Gamma_L,K)}(\mathfrak{X}_{\Gamma_L}, \mathbf{Dfm}(M)):=\mathbf{Rlim}\big( K_{\Psi,D(\Gamma_L,K)}(\mathbf{Dfm}_n(M)) \big).\end{align*}$$

The following results are variants of those in [Reference Steingart51, §3.3]; among others they are based on the observation that for the sheaf of cohomology groups sending $X_n$ to $H^i_{\Psi , D(\Gamma _L,K)}(\mathbf {Dfm}_n(M))$ the formalism of coadmissible modules over $D(\Gamma _L,K)$ does apply.

For the proof of Theorem 4.10 we need the following lemma for which we recall some notation from [Reference Steingart51, Def. 3.20]: We define $D_n\hat {\otimes }_D M^r$ as the completion of $D_n\otimes _D M^r$ with respect to the quotient topology of the projective tensor product $D_n\otimes _{K,\pi }M^r$ . Then we set $D_n\tilde {\otimes }_D M:=\varinjlim _r D_n\hat {\otimes }_D M^r.$

Proof. For (i) the same proof as for [Reference Steingart51, Lem. 3.23] works and the assumptions are satisfied by Theorem 4.9, but note that there $D_n$ , D have a slightly different meaning. The augmentation ideal $I_{\Gamma _L}$ is a finitely generated submodule of $D(\Gamma _L,K)$ and thus projective as a $D(U,K)$ -module since the latter is a Prüfer Domain. Using [Reference Steingart51, Lemma 2.5] one can conclude projectivity as a $D(\Gamma _L,K)$ -module. Using that $D{\mathfrak {Z}}$ is contained in $I_{\Gamma _L}$ gives rise to the projective resolution of D-modules

We can represent the complex in question in (ii) by the complex

$$\begin{align*}\mathbf{Dfm}_n(M)\otimes_{D,\mathrm{diag}} I_{\Gamma_L}\to \mathbf{Dfm}_n(M)\otimes_{D,\mathrm{diag}} D\end{align*}$$

with cokernel

$$ \begin{align*} \mathbf{Dfm}_n(M)/I_{\Gamma_L}\mathbf{Dfm}_n(M)&\cong (\mathbf{Dfm}_n(M)/{\mathfrak{Z}}\mathbf{Dfm}_n(M))/(I_{\Gamma_L}/D{\mathfrak{Z}})\\ &=(D_n\tilde{\otimes}_{D(U,K)}M)_{\Gamma_L/U}\\ &=D_n \tilde{\otimes}_DM, \end{align*} $$

where $ \gamma \in \Gamma _L\subseteq D(\Gamma _L,K)$ acts diagonally (via $\gamma (a \otimes b) = \delta _{\gamma ^{-1}} a \otimes \gamma b)$ ) on $D_n\tilde {\otimes }_{D(U,K)}M$ and this action factors over $\Gamma _L/U$ . For the second equality we use an obvious variant of [Reference Steingart51, (31)], while by the exactness of colimits the last one is easily reduced to the claim that on the level of models $M^r$ we have

$$\begin{align*}(D_m\hat{\otimes}_{D(U,K)} M^r)_{\Gamma/U}\cong D_m\hat{\otimes}_{D(\Gamma_L,K)} M^r.\end{align*}$$

Since $\Gamma /U$ is finite and taking $\Gamma _L/U$ -invariants in this situation is exact by Maschke’s theorem, this follows in the context of Fréchet spaces by completion from the well-known fact that

$$\begin{align*}(D_m{\otimes}_{D(U,K)} M^r)_{\Gamma/U}\cong D_m{\otimes}_{D(\Gamma_L,K)} M^r.\end{align*}$$

The injectivity of the non-trivial differential in the above complex can be checked by calculating instead the cohomology in degree $-1$ of $\mathbf {Dfm}_n(M) \otimes ^{\mathbb {L}}_{D(\Gamma _L,K),\mathrm {diag}}K[\Gamma _L/U]$ , because taking $\Gamma _L/U$ -(co)invariants is exact and leads to the original complex

$$\begin{align*}\big(\mathbf{Dfm}_n(M) \otimes^{\mathbb{L}}_{D(\Gamma_L,K),\mathrm{diag}}K[\Gamma_L/U]\big)\otimes_{K[\Gamma_L/U]}K\cong\mathbf{Dfm}_n(M) \otimes^{\mathbb{L}}_{D(\Gamma_L,K),\mathrm{diag}}K.\end{align*}$$

For this composition of functors it is crucial that $\mathbf {Dfm}_n(M) \otimes ^{\mathbb {L}}_{D(\Gamma _L,K),\mathrm {diag}}K[\Gamma _L/U]$ belongs to $\mathbf {D}((D_n,K[\Gamma _L/U])\mathrm {-bimod})$ as in (18) in order to allow an action by $\Gamma _L/U$ . But then the vanishing in degree $-1$ can be checked just as complex of K-vector spaces and therefore it suffices to calculate the derived functor by a projective resolution of D-modules (instead of bi-modules). To this end we use the resolution (17), which leads to the complex

$$\begin{align*}\mathbf{Dfm}_n(M)\xrightarrow{{\mathfrak{Z}}} \mathbf{Dfm}_n(M),\end{align*}$$

which is left exact by an obvious variant of [Reference Steingart51, (31)], again.

Proof of Theorem 4.10

Using Lemma 4.12 we obtain isomorphisms in $\mathbf {D}(D_n)$

$$ \begin{align*} K_{\Psi,D(\Gamma_L,K)}&(\mathbf{Dfm}_n(M))\\ & \cong \mathrm{cone} \big( \mathbf{Dfm}_n(M)\otimes^{\mathbb{L}}_{D(\Gamma_L,K),\mathrm{diag}}K\xrightarrow{\Psi-1} \mathbf{Dfm}_n(M)\otimes^{\mathbb{L}}_{D(\Gamma_L,K),\mathrm{diag}}K \big)[-2] \\ & \cong \mathcal{T}_\Psi(D_n\tilde{\otimes}_{D}M ) \\ & \cong D_n{\otimes}_{D}\mathcal{T}_\Psi(M ) \end{align*} $$

compatible for the variation in n by an obvious variant of Theorem 4.6 (2). Thus, combining [Reference Steingart51, Prop. 3.15] with Theorem 4.9 we obtain in $\mathbf {D}^b_{\mathrm {perf}}(D)$ an isomorphism

$$\begin{align*}\mathcal{T}_{\Psi}(M)\cong\mathbf{Rlim} \big(D_n\otimes_D\mathcal{T}_{\Psi}(M)\big)\cong \mathbf{Rlim} \big(K_{\Psi,D(\Gamma_L,K)}(\mathbf{Dfm}_n(M))\big).\end{align*}$$

This proves (ii) while (i) follows by the same arguments as in [Reference Steingart51, Rem. 3.16] using that the projective system $( K_{\Psi ,D(\Gamma _L,K)}(\mathbf {Dfm}_n(M)) )_m $ defines a consistent object in $\mathbf {D}(\mathrm {mod}(\mathbb {N},D))$ (using the notation of (loc. cit.)) together with the fact that D is a Fréchet-Stein algebra.

4.3 Replacing Local Tate duality

In this subsection we develop local duality analogous to local Tate duality for Galois cohomology, see [Reference Herr25, Reference Liu33] for an approach purely in terms of $(\varphi ,\Gamma )$ -modules. We focus technically on the complexes $K_{f,{\mathfrak {Z}}}$ and shall then apply (12) to deal with $K_{f,U}$ . Assume henceforth that M is an analytic $(\varphi _L,\Gamma _L)$ -module over $\mathcal {R}=\mathcal {R}_K$ . For an analytic character $\delta : L^\times \to K^\times $ we define the twisted module $M(\delta ) \in \mathfrak {M}^{an}(\mathcal {R}),$ where $M(\delta ) := M\otimes _{\mathcal {R}}\mathcal {R}(\delta )$ endowed with the diagonal $\varphi _L$ - and $\Gamma _L$ -action. Recall the residue map (at Z)

$$\begin{align*}Res: \Omega^1_{\mathcal{R}} := \mathcal{R} dZ\otimes_{\mathcal{R}}\mathcal{R}(\delta^{un}_{\chi(\pi)}) \to K,\;\;\; \sum_i a_iZ^idZ\otimes \mathbf{e}_{\delta^{un}_{\chi(\pi)}} \mapsto a_{-1},\end{align*}$$

and that the $(\varphi _L,\Gamma _L)$ -action on $ \mathcal {R} dZ $ with respect to the basis $d\log _{LT} = g_{LT}dZ$ is given by the character $\chi _{LT}.$ Footnote ¹⁰ As a formal consequence, we have the following:

Lemma 4.13. The map

$$ \begin{align*} \mathcal{R}(\chi) & \xrightarrow{\;\cong\;} \Omega^1_{\mathcal{R}}, \\ f {\mathbf{e}_{\chi}} & \longmapsto fd\log_{LT} \otimes \mathbf{e}_{\delta^{un}_{\chi(\pi)}} , \end{align*} $$

is an isomorphism of $(\varphi _L,\Gamma _L)$ -modules.

Setting $\tilde {M}:= \operatorname {Hom}_{\mathcal {R}}(M,{\mathcal {R}})(\chi ) \cong \operatorname {Hom}_{\mathcal {R}}(M,\Omega ^1_{\mathcal {R}}) $ , for any finitely generated projective $\mathcal {R}$ -module M, we obtain more generally the pairing

(19) $$ \begin{align} \langle\;,\;\rangle:=\langle\;,\;\rangle_M: \tilde{M} \times M \to K,\;\;\; (g,f)\mapsto Res(g(f)), \end{align} $$

where by abuseFootnote ¹¹ of notation we also write $Res:\mathcal {R}(\chi )\to K$ for the map sending $\sum _i a_iZ^i \otimes \mathbf {e}_{\chi }$ to $a_{-1}.$ This map identifies M and $\tilde {M}$ with the (strong) topological duals of $\tilde {M}$ and M, respectively. Moreover, the isomorphism $\tilde {M} \cong \operatorname {Hom}_{K,cts}(M,K)$ (induced by $ \langle \;,\;\rangle $ ) is $D(\Gamma _L,K)$ -linear by [Reference Schneider and Venjakob46, Corollary 4.5.4].

For compatibility questions we renormalise the residuum map to obtain the trace map $Tr=C_{Tr}({{\mathfrak {Z}}_n})Res:H_{\varphi _L,D(\Gamma _L)}^2(\Omega ^1_{\mathcal {R}} )=H_{\varphi _L,{\mathfrak {Z}}_n}^2(\Omega ^1_{\mathcal {R}} )^{\Gamma _L}\cong K$ by setting

(20) $$ \begin{align} C_{Tr}({{\mathfrak{Z}}_n}):=\frac{q}{q-1}\frac{\Omega}{\pi_L^n}. \end{align} $$

Note that for $L={\Bbb Q}_p$ and $\pi _L=p$ this trace map is compatible with Tate’s trace map in Galois cohomology by [Reference Nakamura36, Prop. 5.2]. Independence of n follows by the same argument as for Definition 5.11 below. The principle is explained as follows:

The map of complexes, for $m\geq n$ ,

induces the restriction maps $res^n_m:H^i_{{{\mathfrak {Z}}_n}}(M)\to H^i_{{{\mathfrak {Z}}_m}}(M),$ where $\mathfrak {Q}_{m-n}({{\mathfrak {Z}}_n}):=\frac {{{\mathfrak {Z}}_m}}{{{\mathfrak {Z}}_n}}=\frac {\varphi _L^{m-n}({{\mathfrak {Z}}_n})}{{{\mathfrak {Z}}_n}}$ with $\mathfrak {Q}_{m-n}(0)=\pi _L^{m-n}.$ Since $C_{Tr}({{\mathfrak {Z}}_n})= \pi _L^{m-n}C_{Tr}({{\mathfrak {Z}}_m})$ by (48) the isomorphism

(21) $$ \begin{align} \vartheta_n: H^1_{{{\mathfrak{Z}}_n}}(M)^{\Gamma_L}& \xrightarrow{\cong}M_{\Gamma_L}, [x]\mapsto [C_{Tr}({{\mathfrak{Z}}_n})x] \end{align} $$

into the $\Gamma _L$ -coinvariants is compatible with $res^n_m,$ i.e., the diagram

commutes.

For a complex $(X^\bullet , d_X )$ of topological K-vector spaces we define its K-dual $((X^*)^\bullet , d_{X^*})$ to be the complex with

$$\begin{align*}(X^*)^i:=\mathrm{Hom}_{K,cts}(X^{-i},K)\end{align*}$$

and

$$\begin{align*}d_{X^*}(f):=(-1)^{\mathrm{deg}(f)-1} f\circ d_X.\end{align*}$$

The following lemma is taken from [Reference Schneider and Venjakob46, Lemma 5.2.4 and Remark 5.2.6].

The translation $X[n]$ of a complex X is given by $X[n]^i:=X^{i+n}$ and $d^i_{X[n]}:=(-1)^nd_X^{i+n}.$

Let $\iota $ denote the involution on $D(o_L,K)$ induced by the inversion on the group $o_L$ . We observe that

(22) $$ \begin{align} {\mathfrak{Z}}^\iota=\lambda {\mathfrak{Z}} \end{align} $$

for a unit $\lambda \in D(o_L,K)$ as they both generate the augmentation ideal: more explicitly, ${\mathfrak {Z}}^\iota =[-1]({{\mathfrak {Z}}}), \lambda ^{-1}=\lambda ^\iota $ .

Remark 4.17. Identify M with $\tilde {\tilde {M}}$ via $m \mapsto m^{**}$ and consider the pairing in degree $(1,1)$ from Theorem 4.16 obtained by exchanging the roles of M and $\tilde {M},$ i.e.,

$$ \begin{align*} \langle -,- \rangle_{\tilde{M}} \colon H^1_{\varphi,\mathfrak{Z}}(\tilde{M}) \times H^1_{\varphi,\mathfrak{Z}}(\tilde{\tilde{M}})&\to K,\\ ((f,g),(m^{**},n^{**})) &\mapsto -\operatorname{Res}(\varphi(n^{**})(f)+(\lambda^{\iota}m^{**})(g)). \end{align*} $$

We have

(26) $$ \begin{align} \langle \overline{(m,n)},\overline{(f,g)}\rangle_{M} = -\langle \overline{(f,g)},\overline{(m^{**},n^{**})} \rangle_{\tilde{M}}. \end{align} $$

In the other degrees consider

$$ \begin{align*} \langle -,- \rangle_{\tilde{M}}\colon H^2_{\varphi,\mathfrak{Z}}(\tilde{M}) \times H^0_{\varphi,\mathfrak{Z}}(\tilde{\tilde{M}})&\to K,\\ { (\overline{f}, m^{**})} &\mapsto \operatorname{Res}(m^{**}(-\lambda^\iota(\varphi(f)))), \end{align*} $$

satisfying $\langle \overline {f}, {m^{**}}\rangle _{\tilde {M}} = \langle {m},\overline {f} \rangle _{M}$ and

$$ \begin{align*} H^0_{\varphi,\mathfrak{Z}}(\tilde{M}) \times H^2_{\varphi,\mathfrak{Z}}(\tilde{\tilde{M}})&\to K,\\ (g,\overline{n^{**}}) &\mapsto \operatorname{Res}({ n^{**}(g)}), \end{align*} $$

satisfying $\langle g,\overline {n^{**}}\rangle _{\tilde {M}} = \langle \overline {n},g \rangle _{M}.$ Footnote ¹³

Proof. By viewing $K_{\varphi ,\mathfrak {Z}}(\tilde {M})$ as a Koszul complex attached to the automorphisms $\varphi -1,\mathfrak {Z}$ of M one can see that $\mathfrak {Z}$ and $\varphi _L-1$ act as $0$ on the cohomology groups. Since

$$ \begin{align*}\lambda = -1 + \text{ terms divisible by } \mathfrak{Z}\end{align*} $$

we see that the class $\overline {(f,g)} \in H^1_{\varphi ,\mathfrak {Z}}(\tilde {M})$ is equal to the class of $\overline {(-\lambda f,- \lambda g)}.$ Now let $ \overline {(m,n)} \in H^1_{\varphi ,\mathfrak {Z}}(M).$ Using $\mathfrak {Z}f = (\varphi -1)g$ and $\mathfrak {Z}m = (\varphi -1)n$ we compute

$$ \begin{align*} \langle \overline{(m,n)},\overline{(f,g)}\rangle_{M} &= Res\Big(-\varphi(g)(m)-(\lambda^\iota f)(n)\Big)\\ &= Res\Big(-[g+\mathfrak{Z}f](m)-(\lambda^\iota f)(\varphi(n)-\mathfrak{Z}m)\Big)\\ &= Res\Big(-g(m)-(\lambda^\iota f)\varphi(n))- \underbrace{(\mathfrak{Z}f)(m) + (\lambda^\iota \mathfrak{Z}^\iota f)(m)}_{=0}\Big)\\ & = Res\Big((\lambda g)(m)+(\lambda \lambda^\iota f)(\varphi(n))\Big)\\ & = Res\Big(m^{**}(\lambda g)+\varphi(n^{**})( f)\Big)\\ & = - \langle \overline{(f,g)},\overline{(m^{**},n^{**})} \rangle_{\tilde{M}}, \end{align*} $$

where in the fifth equation we replace $\overline {(f,g)}$ with $(-\lambda f,-\lambda g).$ Now consider the degree $(0,2)$ case with regard to $\langle -,-\rangle _M$ . Since $\overline {\varphi (f)} = \overline {f}$ we get $\operatorname {Res}(f(m)) = \operatorname {Res}(\varphi (f)(m)) = - \operatorname {Res}(\varphi (f)(\lambda m))$ using that $\mathfrak {Z}{m} = 0$ in $H^0$ and hence $\lambda {m} =-{m}.$ The computation in degree $(2,0)$ is similar.

Later in explicit calculations we will need to work partly with $\Psi $ -versions, which we therefore establish in the next remark.

Remark 4.18. As a obvious variant of (i) in Theorem 4.16 there is also a canonical quasi-isomorphism

(27)

In particular, we obtain an isomorphism $\Upsilon ^{\prime }_M: H^1_{\varphi _L,{\mathfrak {Z}}}(M)\cong H^1_{\Psi _L,{\mathfrak {Z}}}(M)$ sending a class $[(x,y)]$ to the class $[(-\Psi (x),y)].$

Using this one derives from $\langle -, - \rangle _M$ in Theorem 4.16 the (asymmetric) perfect pairings, denoted by $\{\!\!\{ -, - \}\!\!\}_M,$

$$\begin{align*}H^1_{\Psi, {\mathfrak{Z}}}(M) \times H^1_{\varphi, {{\mathfrak{Z}}} }(\tilde{M}) \to K, (\overline{(m,n)},\overline{(f,g)})\mapsto Res\Big( g( m)- (\lambda^\iota f)( n)\Big),\end{align*}$$

$$\begin{align*}H^2_{\Psi, {\mathfrak{Z}}}(M) \times H^0_{\varphi, {{\mathfrak{Z}}} }(\tilde{M}) \to K, (\overline{m},\tilde{n})\mapsto Res \Big( \tilde{n}(\lambda m)\Big) ,\end{align*}$$

$$\begin{align*}H^0_{\Psi, {\mathfrak{Z}}}(M) \times H^2_{\varphi, {{\mathfrak{Z}}} }(\tilde{M}) \to K, ({m},\overline{\tilde{n}})\mapsto Res \Big( \tilde{n}(m)\Big),\end{align*}$$

for which by construction we have

$$\begin{align*}\langle x,y \rangle_M=\{\!\!\{ \Upsilon^{\prime}_M(x), y \}\!\!\}_{M}.\end{align*}$$

Moreover, we obtain, for $x\in H^i_{\varphi , {{\mathfrak {Z}}} }({M}),\ y\in H^{2-i}_{\varphi , {{\mathfrak {Z}}} }(\tilde {M}),$

(28) $$ \begin{align} (-1)^i\langle x,y \rangle_M= \langle y,x^{**} \rangle_{\tilde{M}} =\{\!\!\{ \Upsilon^{\prime}_{\tilde{M}}(y), x^{**} \}\!\!\}_{\tilde{M}}, \end{align} $$

by Remark 4.17.