2.1 The rings

We introduce the following objects.

• – Let A be a p-adically complete $\mathbf{Z}_{p}$ -algebra which is topologically of finite type.

• – Set $\widetilde{{\mathbf A}}^{+}=\mathrm{{\mathbf A}}_{\mathrm{inf}}:=W(\mathcal{O}_{{\mathbf{C}}}^{\flat})$ and $\widetilde{{\mathbf {{\mathbf A}}}}:=W({\mathbf{C}}^{\flat})$ . We endow $\widetilde{\mathrm{{\mathbf A}}}^{+}$ with its $(p,[\varpi])$ -topology. We endow $\widetilde{{\mathbf {{\mathbf A}}}}=\varprojlim_{i}\widetilde{{\mathbf {{\mathbf A}}}}/p^{i}$ with the inverse limit topology, where the topology on $\widetilde{{\mathbf {{\mathbf A}}}}/p^{i}=(\widetilde{{\mathbf {{\mathbf A}}}}^{+}/p^{i})[1/[\varpi]]$ is characterized by having $\widetilde{{\mathbf {A}}}^{+}/p^{i}$ as an open subring.

• – For $1/r\in\mathbf{Z}[1/p]_{>0}$ we write

  \[\widetilde{\mathrm{{\mathbf A}}}^{(0,r],\circ}:=\bigg\{\sum_{k\geq0}p^{k}[x_{k}]\in\widetilde{\mathrm{{\mathbf A}}}:0\leq\mathrm{val}_{{\mathbf{C}}^{\flat}}(x_{k})+\bigg(\frac{pr}{p-1}\bigg)k\rightarrow\infty\bigg\}.\]
  According to [Reference Cherbonnier and ColmezCC98, Remark II.1.3], this ring may be equivalently defined as $\widetilde{\mathrm{{\mathbf A}}}^{+}\langle p/[\varpi]^{1/r}\rangle$ , the p-adic completion of $\widetilde{\mathrm{{\mathbf A}}}^{+}[p/[\varpi]^{1/r}]$ . We endow the ring $\widetilde{\mathrm{{\mathbf A}}}^{(0,r],\circ}$ with its $[\varpi]$ -adic topology, or what is the same, the topology defined by the valuation
  \[\mathrm{val}^{(0,r]}(x):=\bigg(\frac{p}{p-1}\bigg)\mathrm{sup}\{t\in\mathbf{Z}[1/p]:x\in[\varpi]^{t}\widetilde{\mathrm{{\mathbf A}}}^{(0,r],\circ}\} .\]
  One checks that
  \[\mathrm{val}^{(0,r]}\bigg(\sum_{k\geq0}p^{k}[x_{k}]\bigg)=\mathrm{inf}_{k}\bigg[\mathrm{val}_{{\mathbf{C}}^{\flat}}(x_{k})+\bigg(\frac{pr}{p-1}\bigg)k\bigg].\]

• – For $1/r\in\mathbf{Z}[1/p]_{>0}$ we set

  \[\widetilde{\mathrm{{\mathbf A}}}^{(0,r]}:=\bigg\{x=\sum_{k\gg-\infty}p^{k}[x_{k}]\in\widetilde{\mathrm{{\mathbf A}}}:\mathrm{val}_{{\mathbf{C}}^{\flat}}(x_{k})+\bigg(\frac{pr}{p-1}\bigg)k\rightarrow\infty\bigg\}.\]
  Equivalently, $\widetilde{\mathrm{{\mathbf A}}}^{(0,r]}=\widetilde{\mathrm{{\mathbf A}}}^{(0,r],\circ}[1/[\varpi]]$ . We endow $\widetilde{\mathrm{{\mathbf A}}}^{(0,r]}$ with the topology which makes $\widetilde{\mathrm{{\mathbf A}}}^{(0,r],\circ}$ into an open subring. This topology is the same as that defined by the valuation
  \[\mathrm{val}^{(0,r]}(x):=\bigg(\frac{p}{p-1}\bigg)\mathrm{sup}\{t\in\mathbf{Z}[1/p]:x\in[\varpi]^{t}\widetilde{\mathrm{{\mathbf A}}}^{(0,r],\circ}\} .\]

• – For $r=\infty$ we set $\widetilde{\mathrm{{\mathbf A}}}^{(0,\infty),\circ}=\widetilde{\mathrm{{\mathbf A}}}^{+}$ with its $(p,[\varpi])$ -topology and $\widetilde{\mathrm{{\mathbf A}}}^{(0,\infty)}=\widetilde{\mathrm{{\mathbf A}}}^{+}[1/[\varpi]]$ with its p-adic topology.

For each of the rings introduced above, there are versions with coefficients in A, which we introduce next. We have the following.

• – For $1/r\in\mathbf{Z}[1/p]_{>0}$ ,

  \[\widetilde{\mathrm{{\mathbf A}}}_{A}^{(0,r],\circ}=\widetilde{\mathrm{{\mathbf A}}}^{(0,r],\circ}\widehat{\otimes}_{\mathbf{Z}_{p}}A:=\varprojlim_{i}(\widetilde{\mathrm{{\mathbf A}}}^{(0,r],\circ}\otimes_{\mathbf{Z}_{p}}A)/[\varpi^{i}]\]
  and $\widetilde{\mathrm{{\mathbf A}}}_{A}^{(0,r]}=\widetilde{\mathrm{{\mathbf A}}}_{A}^{(0,r],\circ}[1/[\varpi]]$ .

• – The image of $\widetilde{\mathrm{{\mathbf A}}}_{A}^{(0,r],\circ}$ in $\widetilde{\mathrm{{\mathbf A}}}_{A}^{(0,r]}$ , endowed with its subspace topology, is $\widetilde{\mathrm{{\mathbf A}}}_{A}^{(0,r],+}$ . (The map $\widetilde{\mathrm{{\mathbf A}}}_{A}^{(0,r],\circ}\rightarrow\widetilde{\mathrm{{\mathbf A}}}_{A}^{(0,r]}$ may not be injective if A has p-torsion.)

• – For $a\geq1$ the a-typical Witt vectors $W_{a}(\mathcal{O}_{{\mathbf{C}}}^{\flat})=W(\mathcal{O}_{{\mathbf{C}}}^{\flat})/p^{a}$ , with the $[\varpi]$ -adic topology.

• – For $a\geq1$ ,

  \[W_{a}(\mathcal{O}_{{\mathbf{C}}}^{\flat})_{A}=W_{a}(\mathcal{O}_{{\mathbf{C}}}^{\flat})\widehat{\otimes}_{\mathbf{Z}_{p}}A:=\varprojlim_{i}(W_{a}(\mathcal{O}_{{\mathbf{C}}}^{\flat})\otimes_{\mathbf{Z}_{p}}A)/[\varpi^{i}].\]

• – For $r=\infty$ , $\widetilde{\mathrm{{\mathbf A}}}_{A}^{+}=\widetilde{\mathrm{{\mathbf A}}}_{A}^{(0,\infty),+}=\widetilde{\mathrm{{\mathbf A}}}_{A}^{(0,\infty),\circ}:=W(\mathcal{O}_{{\mathbf{C}}}^{\flat})_{A}=\varprojlim_{a}W_{a}(\mathcal{O}_{{\mathbf{C}}}^{\flat})_{A}$ and $\widetilde{\mathrm{{\mathbf A}}}_{A}^{(0,\infty)}=W(\mathcal{O}_{{\mathbf{C}}}^{\flat})_{A}[1/[\varpi]]$ .

• – We have $\widetilde{\mathrm{{\mathbf A}}}_{A}=W({\mathbf{C}}^{\flat})_{A}:=\varprojlim_{a}W_{a}(\mathcal{O}_{{\mathbf{C}}}^{\flat})_{A}[1/[\varpi]]$ .

• – We have $\widetilde{\mathrm{{\mathbf A}}}_{A}^{\dagger}:=\varinjlim_{r}\widetilde{\mathrm{{\mathbf A}}}_{A}^{(0,r]}$ .

For each $R_{A}\in\{\widetilde{\mathrm{{\mathbf A}}}_{A}^{+},\widetilde{\mathrm{{\mathbf A}}}_{A},\widetilde{\mathrm{{\mathbf A}}}_{A}^{(0,r]},\widetilde{\mathrm{{\mathbf A}}}_{A}^{(0,\infty]},\widetilde{\mathrm{{\mathbf A}}}_{A}^{\dagger}\}$ we introduce versions relative to K, by setting $R_{K,A}:=(R_{A})^{H_{K}}$ . The following lemma shows that each of these can be defined in terms of $\widehat{K}_{\infty}$ . Define $W(\mathcal{O}_{\widehat{K}_{\infty}}^{\flat})_{A}$ and $W(\widehat{K}_{\infty}^{\flat})_{A}$ in a similar way to the definition of $W(\mathcal{O}_{{\mathbf{C}}}^{\flat})_{A}$ and $W({\mathbf{C}}^{\flat})_{A}$ , respectively.

Here, in parts (iii) and (v), the tensor products are completed with respect to the $[\varpi]$ -adic topology.

Finally, we have imperfect versions of the rings relative to K as above. They are defined as follows.

• – We have standard rings $\mathrm{{\mathbf A}}_{K}^{+}$ and $\mathrm{{\mathbf A}}_{K}$ , which are defined in § 2.1 of [Reference Emerton and GeeEG19] (where they are denoted by $(\mathrm{{\mathbf A}}_{K}^{\prime})^{+}$ and $\mathbf{\mathrm{{\mathbf A}}}_{K}^{\prime}$ ). We have a certain element $T\in\mathrm{{\mathbf A}}_{K}^{+}$ lifting $\varpi$ . Note that by the theory of the field of norms, we have canonical embeddings $\mathrm{{\mathbf A}}_{K}^{+}\hookrightarrow\widetilde{\mathrm{{\mathbf A}}}$ and $\mathrm{{\mathbf A}}_{K}\hookrightarrow\widetilde{\mathrm{{\mathbf A}}}$ which map T to $[\varepsilon]-1$ .

• – We setFootnote ²

  \[\mathrm{{\mathbf A}}_{K,A}^{+}=\mathrm{{\mathbf A}}_{K}^{+}\widehat{\otimes}_{\mathbf{Z}_{p}}A:=\varprojlim_{m}\bigg(\varprojlim_{n}\mathrm{{\mathbf A}}_{K}^{+}/(p^{m},T^{n})\otimes_{\mathbf{Z}_{p}}A\bigg),\]
  endowed with the inverse limit topology, and
  \[\mathrm{{\mathbf A}}_{K,A}=\mathrm{{\mathbf A}}_{K}\widehat{\otimes}_{\mathbf{Z}_{p}}A:=\varprojlim_{m}\bigg(\varprojlim_{n}(\mathrm{{\mathbf A}}_{K}^{+}/(p^{m},T^{n})\otimes_{\mathbf{Z}_{p}}A)[1/T]\bigg).\]

• – We let $\mathrm{{\mathbf A}}_{K}^{(0,r],\circ}=\mathrm{{\mathbf A}}_{K}\cap\widetilde{\mathrm{{\mathbf A}}}^{(0,r],\circ}$ and

  \[\mathrm{{\mathbf A}}_{K,A}^{(0,r],\circ}=\varprojlim_{i}\big(\mathrm{{\mathbf A}}_{K}^{(0,r],\circ}\otimes_{\mathbf{Z}_{p}}A\big)/T^{i}.\]

• – Let $\mathrm{{\mathbf A}}_{K,A}^{(0,r]}:=\mathrm{{\mathbf A}}_{K,A}^{(0,r],\circ}[1/T]$ and write $\mathrm{{\mathbf A}}_{K,A}^{(0,r],+}$ for the image of $\mathrm{{\mathbf A}}_{K,A}^{(0,r],\circ}$ in $\mathrm{{\mathbf A}}_{K,A}^{(0,r]}$ .

• – Finally, let $\mathrm{{\mathbf A}}_{K,A}^{\dagger}:=\varinjlim_{r}\mathrm{{\mathbf A}}_{K,A}^{(0,r]}$ .

We have natural maps $\widetilde{\mathrm{{\mathbf A}}}_{A}^{(0,r]}\rightarrow\widetilde{\mathrm{{\mathbf A}}}_{A}$ , $\mathrm{{\mathbf A}}_{K,A}\rightarrow\widetilde{\mathrm{{\mathbf A}}}_{A}$ , $\widetilde{\mathrm{{\mathbf A}}}_{A}^{(0,s]}\rightarrow\widetilde{\mathrm{{\mathbf A}}}_{A}^{(0,r]}$ and $\mathrm{{\mathbf A}}_{K,A}^{(0,s]}\rightarrow\mathrm{{\mathbf A}}_{K,A}^{(0,r]}$ for $s>r$ . One can show they are all injective (this claim is not trivial: it is the main content of Appendix A; cf. Theorem A.9 and what follows). This implies that in the definition of $\widetilde{\mathrm{{\mathbf A}}}_{A}^{\dagger}$ , the colimit is, in fact, a union. Similar comments apply to $\widetilde{\mathrm{{\mathbf A}}}_{K,A}^{\dagger}$ and $\mathrm{{\mathbf A}}_{K,A}^{\dagger}$ .

We shall need the following fact on compatibility with reduction.

Proposition 2.2 (Proposition 7.17). For $N\in\mathbf{Z}_{\geq1}$ and $1/r\in\mathbf{Z}[1/p]_{>0}$ we have natural isomorphisms

\[\widetilde{\mathrm{{\mathbf A}}}_{K,A}/p^{N}\widetilde{\mathrm{{\mathbf A}}}_{K,A}\cong\widetilde{\mathrm{{\mathbf A}}}_{K,A/p^{N}}\cong\widetilde{\mathrm{{\mathbf A}}}_{K,A/p^{N}}^{(0,r]}\cong\widetilde{\mathrm{{\mathbf A}}}_{K,A/p^{N}}^{(0,\infty)}.\]

The following result is used later in § 5.

Proposition 2.3 (Corollary 7.4). For $1/r\in\mathbf{Z}[1/p]_{>0}$ the topology on $\widetilde{\mathrm{{\mathbf A}}}_{A}^{(0,r]}$ is defined by the valuation given by

\[\mathrm{val}^{(0,r]}(x)=(p/p-1)\,\mathrm{sup}\{t\in\mathbf{Z}[1/p]:x\in[\varpi]^{t}\widetilde{\mathrm{{\mathbf A}}}_{A}^{(0,r],+}\}.\]

Finally, there are continuous $(\varphi,G_{K})$ -actions on $\widetilde{\mathrm{{\mathbf A}}}_{A}$ by [Reference Emerton and GeeEG19, Lemma 2.2.18]. This gives rise to continuous $(\varphi,\Gamma_{K})$ -actions on all of the rings introduced here.

Namely, given $r\in{\mathbf{R}}_{>0}\cup\{\infty\}$ we have a continuous map

\[\varphi:\widetilde{\mathrm{{\mathbf A}}}_{A}^{(0,r]}\rightarrow\widetilde{\mathrm{{\mathbf A}}}_{A}^{(0,r/p]}\]

induced by extending the action of $\varphi$ on $\mathrm{{\mathbf A}}_{\mathrm{inf}}$ . It is continuous since $\varphi([\varpi])=[\varpi]^{p}$ and the topology is $[\varpi]$ -adic on both the source and the target.

Similarly, we have a continuous inverse

\[\varphi^{-1}:\widetilde{\mathrm{{\mathbf A}}}_{A}^{(0,r/p]}\rightarrow\widetilde{\mathrm{{\mathbf A}}}_{A}^{(0,r]}.\]

This immediately extends to give continuous $\varphi$ and $\varphi^{-1}$ actions on $\widetilde{\mathrm{{\mathbf A}}}_{A}^{\dagger}$ . The $(\varphi^{\pm1},G_{K})$ -actions (respectively, $(\varphi^{\pm1},\Gamma_{K})$ -actions, respectively, $(\varphi,\Gamma_{K})$ -actions) on $\widetilde{\mathrm{{\mathbf A}}}_{A}^{\dagger}$ and $\widetilde{\mathrm{{\mathbf A}}}_{A}$ (respectively, $\widetilde{\mathrm{{\mathbf A}}}_{K,A}^{\dagger}$ and $\widetilde{\mathrm{{\mathbf A}}}_{K,A}$ , respectively, $\mathrm{{\mathbf A}}_{K,A}^{\dagger}$ and $\mathrm{{\mathbf A}}_{K,A}$ ) are continuous.

3.1 Descent of $\varphi$ -modules

The topology on $\widetilde{\mathrm{{\mathbf A}}}_{K,A}^{(0,r]}$ is defined by a valuation $\mathrm{val}^{(0,r]}$

\[\mathrm{val}_{A}^{(0,r]}(x):=\bigg(\frac{p}{p-1}\bigg)\mathrm{sup}\{t\in\mathbf{Z}[1/p]:x\in[\varpi]^{t}\widetilde{\mathrm{{\mathbf A}}}_{K,A}^{(0,r],+}\}.\]

We also define, for $x\in\widetilde{\mathrm{{\mathbf A}}}_{K,A}^{(0,\infty)}$ ,

\[\mathrm{val}_{A}^{(0,\infty)}(x):=\bigg(\frac{p}{p-1}\bigg)\mathrm{sup}\{t\in\mathbf{Z}[1/p]:x\in[\varpi]^{t}\widetilde{\mathrm{{\mathbf A}}}_{K,A}^{(0,\infty),+}\}.\]

We extend $\mathrm{val}_{A}^{(0,r]}$ and $\mathrm{val}_{A}^{(0,\infty)}$ to all of $\widetilde{\mathrm{{\mathbf A}}}_{K,A}$ , by allowing the value $-\infty$ , so that $\widetilde{\mathrm{{\mathbf A}}}_{K,A}^{(0,r]}$ (respectively, $\widetilde{\mathrm{{\mathbf A}}}_{K,A}^{(0,\infty)}$ ) is the subset of elements $x\in\widetilde{\mathrm{{\mathbf A}}}_{K,A}$ with $\mathrm{val}_{A}^{(0,r]}(x)>-\infty$ (respectively, $\mathrm{val}_{A}^{(0,\infty)}(x)>-\infty$ ). The following lemma serves as a generalization of the usual Teichmüller digits. Recall from Proposition 2.2 that

\[\widetilde{\mathrm{{\mathbf A}}}_{K,A}/p^{N}\cong\widetilde{\mathrm{{\mathbf A}}}_{K,A/p^{N}}\cong\widetilde{\mathrm{{\mathbf A}}}_{K,A/p^{N}}^{(0,\infty)}\cong\widetilde{\mathrm{{\mathbf A}}}_{K,A/p^{N}}^{(0,r]}.\]

Proof. Multiplication by p induces surjections

\[A/p\twoheadrightarrow pA/p^{2}\twoheadrightarrow p^{2}A/p^{3}\twoheadrightarrow\cdots\]

of ${\mathbf{F}}_{p}$ -vector spaces. Let $W_{n}$ be the kernel of $\pi_{n}:A/p\twoheadrightarrow p^{n}A/p^{n+1}$ , so that the $W_{n}$ are increasing with n. Choose a complement U to the union of the $W_{n}$ so that

\[A/p=U\oplus\bigcup_{n\geq0}W_{n}.\]

For U choose an ${\mathbf{F}}_{p}$ -basis $\{e_{i}\}_{i\in I_{U}}$ and choose, as we may, compatible bases $\{e_{i}\}_{i\in I_{n}}$ for $W_{n}$ so that $\{e_{i}\}_{i\in I_{n}}\subset\{e_{i}\}_{i\in I_{n+1}}$ , and set $I=\bigcup_{n\geq0}I_{n}$ . With this being given, we set $J_{n}=I_{U}\cup(I\backslash I_{n})$ for $n\geq0$ . We claim that $\{\pi_{n}(e_{i})\}_{i\in J_{n}}$ gives an ${\mathbf{F}}_{p}$ -basis of $p^{n}A/p^{n+1}$ . Indeed, by construction, the map $\pi_{n}:A/p\twoheadrightarrow p^{n}A/p^{n+1}$ gives a decomposition

\[A/p=\mathrm{span}\{e_{i}\}_{i\in I_{U}\cup I}=W_{n}\oplus\mathrm{span}\{e_{i}\}_{i\in J_{n}},\]

so that applying $\pi_{n}$ to the $e_{i}$ for $i\in J_{n}$ gives an ${\mathbf{F}}_{p}$ -basis of $p^{n}A/p^{n+1}$ . With this choice of $\{e_{i}\}_{i\in I\cup I_{U}}$ , choose an arbitrary lifting of them to A, which we denote by $\widetilde{e}_{i}$ .

Now each $x\in\mathcal{O}_{\widehat{K}_{\infty}}^{\flat}\otimes_{{\mathbf{F}}_{p}}A/p$ can be written uniquely as a finite sum of the form $x=\sum x_{i}\otimes e_{i}$ with $x_{i}\in\mathcal{O}_{\widehat{K}_{\infty}}^{\flat}$ . Let

\[[x]:=\sum[x_{i}]\otimes\widetilde{e_{i}}\in W(\mathcal{O}_{\widehat{K}_{\infty}}^{\flat})\otimes_{\mathbf{Z}_{p}}A,\]

where $[x_{i}]$ is the usual Teichmüller lift. This defines a map

\[[\bullet]:\mathcal{O}_{\widehat{K}_{\infty}}^{\flat}\otimes_{{\mathbf{F}}_{p}}A/p\rightarrow W(\mathcal{O}_{\widehat{K}_{\infty}}^{\flat})\otimes_{\mathbf{Z}_{p}}A.\]

It follows from the definition that

\[[\bullet](\varpi^{t}\mathcal{O}_{\widehat{K}_{\infty}}^{\flat}\otimes_{{\mathbf{F}}_{p}}A/p)\subset[\varpi]^{t}W(\mathcal{O}_{\widehat{K}_{\infty}}^{\flat})\otimes_{\mathbf{Z}_{p}}A;\]

hence, it is continuous for the $[\varpi]$ -topology on the source and the $(p,[\varpi])$ -topology on the target. It therefore extends to a map

\[[\bullet]:(\mathcal{O}_{\widehat{K}_{\infty}}^{\flat})_{A/p}\rightarrow W(\mathcal{O}_{\widehat{K}_{\infty}}^{\flat})_{A},\]

which we can extend further to a continuous map

\[[\bullet]:\widetilde{\mathrm{{\mathbf A}}}_{K,A/p}=(\mathcal{O}_{\widehat{K}_{\infty}}^{\flat})_{A/p}[1/\varpi]\rightarrow W(\widehat{K}_{\infty}^{\flat})_{A}=\widetilde{\mathrm{{\mathbf A}}}_{K,A}.\]

Clearly properties 1 and 2 are satisfied.

To show that property 3 holds, we need to check that $\varpi^{-t}x\in(\mathcal{O}_{\widehat{K}_{\infty}}^{\flat})_{A/p}$ implies $[\varpi]^{-t}[x]\in W(\mathcal{O}_{\widehat{K}_{\infty}}^{\flat})_{A}$ . Replacing x with $\varpi^{-t}x$ , we reduce to the case $t=0$ . Using the continuity of $[\bullet]$ , we may assume $x=\sum x_{i}\otimes e_{i}\in\mathcal{O}_{\widehat{K}_{\infty}}^{\flat}\otimes_{{\mathbf{F}}_{p}}A/p$ . But then it is clear that $[x]=\sum[x_{i}]\otimes e_{i}$ lies in $W(\mathcal{O}_{\widehat{K}_{\infty}}^{\flat})\otimes_{\mathbf{Z}_{p}}A\subset W(\mathcal{O}_{\widehat{K}_{\infty}}^{\flat})_{A}$ .

To show that property 4 holds, we may again twist, so it suffices to show that $p^{n}[x]\text{mod }p^{n+1}\in\widetilde{\mathrm{{\mathbf A}}}_{K,A/p^{n}}^{(0,r],+}$ implies $p^{n}[x]\in\widetilde{\mathrm{{\mathbf A}}}_{K,A}^{(0,r],+}$ . Again using the continuity of $[\bullet]$ , we may assume $x\in\widehat{K}_{\infty}^{\flat}\otimes_{\mathbf{Z}_{p}}A$ . Hence, [x], and consequently $p^{n}[x]$ , belongs to the image of $\widetilde{\mathrm{{\mathbf A}}}_{K}^{(0,r]}\otimes_{\mathbf{Z}_{p}}A$ , in $\widetilde{\mathrm{{\mathbf A}}}_{K,A}^{(0,r]}$ . Explicitly, writing $x=\sum x_{i}\otimes e_{i}$ with $x_{i}\in\widehat{K}_{\infty}^{\flat}$ , the sum being finite, we have that $p^{n}[x]$ is the image under the map $\widetilde{\mathrm{{\mathbf A}}}_{K}^{(0,r]}\otimes_{\mathbf{Z}_{p}}A\rightarrow\widetilde{\mathrm{{\mathbf A}}}_{K,A}^{(0,r]}$ of $\sum[x_{i}]\otimes p^{n}\widetilde{e}_{i}$ . Taking this modulo $p^{n+1}$ , we obtain that $p^{n}[x]$ mod $p^{n+1}$ is the image of $\sum[x_{i}]\otimes\pi_{n}(p^{n}\widetilde{e}_{i})$ under the map $\widetilde{\mathrm{{\mathbf A}}}_{K}^{(0,r]}\otimes_{\mathbf{Z}_{p}}p^{n}A/p^{n+1}\rightarrow\widetilde{\mathrm{{\mathbf A}}}_{K,A/p^{n+1}}^{(0,r]}$ . On the other hand, by assumption $p^{n}[x]$ mod $p^{n+1}\in\widetilde{\mathrm{{\mathbf A}}}_{K,A/p^{n}}^{(0,r],+}$ . Since the $\pi_{n}(p^{n}\widetilde{e}_{i})$ are linearly independent, the assumption $p^{n}[x]\text{ mod }p^{n+1}\in\widetilde{\mathrm{{\mathbf A}}}_{K,A/p^{n}}^{(0,r],+}$ implies that each $[x_{i}]$ lies in the image of $\widetilde{\mathrm{{\mathbf A}}}_{K}^{(0,r],\circ}$ . Hence, $p^{n}[x]$ , which is the image of $\sum[x_{i}]\otimes p^{n}\widetilde{e_{i}}$ under $\widetilde{\mathrm{{\mathbf A}}}_{K}^{(0,r]}\otimes_{\mathbf{Z}_{p}}A\rightarrow\widetilde{\mathrm{{\mathbf A}}}_{K,A}^{(0,r]}$ , has to lie in $\widetilde{\mathrm{{\mathbf A}}}_{K,A}^{(0,r],+}$ . This concludes the proof.

The next key lemma is essentially the same as [Reference de Shalit and PoratdSP19, Lemma 1.3], but we include the proof for completeness. Let R be a commutative ring endowed with a nonarchimedean valuation, where we allow $\mathrm{val}=-\infty$ . Suppose there exists $p\in\mathbf{Z}_{\geq2}$ and an invertible morphism $\varphi:R\rightarrow R$ such that $p\,\mathrm{val}(x)=\mathrm{val}(\varphi(x))$ . Recall our notation for valuation of matrices from § 1.4.

Lemma 3.3 Let $X\in\mathrm{GL}_{d}(R)$ . Then for any $c<(1/p-1)[\mathrm{val}(X)+\mathrm{val}(X^{-1})]$ and for any $Y\in\mathrm{M}_{d}(R)$ there exist $U,V\in\mathrm{M}_{d}(R)$ such that $\mathrm{val}(V)\geq c$ and

\[X^{-1}\varphi(U)X-U=Y-V.\]

Proof. Let

\[U=\sum_{i=1}^{N}\varphi^{-1}(X)\varphi^{-2}(X)\cdots\varphi^{-i}(X)\cdot\varphi^{-i}(Y)\cdot\varphi^{-i}(X)^{-1}\cdots\varphi^{-2}(X)^{-1}\varphi^{-1}(X)^{-1}\]

and

\[V=\varphi^{-1}(X)\varphi^{-2}(X)\cdots\varphi^{-N}(X)\cdot\varphi^{-N}(Y)\cdot\varphi^{-N}(X)^{-1}\cdots\varphi^{-2}(X)^{-1}\varphi^{-1}(X)^{-1}.\]

Then

\[X^{-1}\varphi(U)X-U=Y-V.\]

If $Y=0$ , then choose $V=0$ . Otherwise, by selecting N large enough, we can make $\mathrm{val}(\varphi^{-N}(Y))$ as close as we want to 0. In addition,

\[\mathrm{val}(\varphi^{-1}(X)\varphi^{-2}(X)\cdots\varphi^{-N}(X))\geq(p^{-1}+\cdots+p^{-N})\,\mathrm{val}(X)\]

and

\[\mathrm{val}(\varphi^{-N}(X)^{-1}\cdots\varphi^{-2}(X)^{-1}\varphi^{-1}(X)^{-1})\geq(p^{-1}+\cdots+p^{-N})\,\mathrm{val}(X^{-1}),\]

so by taking N sufficiently large we have $\mathrm{val}(V)\geq c$ .

Proposition 3.4 Let $\widetilde{M}$ be a free étale $\varphi$ -module over $\widetilde{\mathrm{{\mathbf A}}}_{K,A}$ . Then there exists a free étale $\varphi$ -module $\widetilde{M}^{\dagger}$ over $\widetilde{\mathrm{{\mathbf A}}}_{K,A}^{\dagger}$ , contained in $\widetilde{M}$ , such that the natural map

\[\widetilde{\mathrm{{\mathbf A}}}_{K,A}\otimes_{\widetilde{\mathrm{{\mathbf A}}}_{K,A}^{\dagger}}\widetilde{M}^{\dagger}\rightarrow\widetilde{M}\]

is an isomorphism.

To do this, write $X=\sum_{n\geq0}p^{n}[X_{n}]$ with

\[X_{n}\in\mathrm{M}_{d}(\widetilde{\mathrm{{\mathbf A}}}_{K,A}/p)=\mathrm{M}_{d}(\widetilde{\mathrm{{\mathbf A}}}_{K,A/p})=\mathrm{M}_{d}(\widetilde{\mathrm{{\mathbf A}}}_{K,A/p}^{(0,\infty)})\]

and

\[\mathrm{val}_{A}^{(0,\infty)}(p^{k}[X_{n}])=\mathrm{val}_{p^{k}A/p^{k+1}}^{(0,\infty)}(p^{k}[X_{n}]\text{mod }p^{k+1})\]

as we may according to Lemma 3.2. We shall construct $U=\sum_{n\geq0}p^{n}[U_{n}]$ with $U_{n}\in\mathrm{M}_{d}(\widetilde{\mathrm{{\mathbf A}}}_{K,A}/p)$ and $\mathrm{val}_{A}^{(0,\infty)}([U_{n}])=\mathrm{val}_{A/p}^{(0,\infty)}(U_{n})$ by constructing $U_{n}$ inductively.

or $n=0$ take $U_{0}=\mathrm{Id}$ . Now suppose $U_{0},\ldots,U_{n-1}$ have been defined. Let $U'=\sum_{i=0}^{n-1}p^{i}[U_{i}]$ . We shall also suppose we have chosen matrices $C_{0}=X_{0},C_{1},\ldots,C_{n-1}\in\mathrm{M}_{d}(\widetilde{\mathrm{{\mathbf A}}}_{K,A}/p)$ inductively such that

\[U^{\prime-1}X\varphi(U^{\prime})\equiv\sum_{i=0}^{n-1}p^{i}[C_{i}]\text{ mod}p^{n}\mathrm{M}_{d}(\widetilde{\mathrm{{\mathbf A}}}_{K,A}).\]

Write

\[U^{\prime-1}X\varphi(U^{\prime})-\sum_{i=0}^{n-1}p^{i}[C_{i}]=p^{n}Y\]

with $Y\in\mathrm{M}_{d}(\widetilde{\mathrm{{\mathbf A}}}_{K,A})$ . Now look for $U_{n}\in\mathrm{M}_{d}(\widetilde{\mathrm{{\mathbf A}}}_{K,A}/p)$ such that

\[(U^{\prime}+p^{n}[U_{n}])^{-1}X\varphi(U^{\prime}+p^{n}[U_{n}])\equiv\sum_{i=0}^{n}p^{i}[C_{i}]\text{mod }p^{n+1}\]

with $\mathrm{val}_{A/p}^{(0,\infty)}(C_{n})$ bounded below. Noting that $(U^{\prime}+p^{n}[U_{n}])^{-1}=U^{\prime-1}-p^{n}[U_{n}]$ mod $p^{n+1}$ , and letting $\overline{Y}$ denote the reduction of Y mod $p\mathrm{M}_{d}(\widetilde{\mathrm{{\mathbf A}}}_{K,A})$ , it suffices to solve the mod p equation

\[U_{n}-X_{0}\varphi(U_{n})X_{0}^{-1}=\overline{Y}X_{0}^{-1}-C_{n}X_{0}^{-1}\]

in $\widetilde{\mathrm{{\mathbf A}}}_{K,A/p}$ .

By Lemma 3.3, this equation can be solved for both $U_{n}$ and $C_{n}$ , with $C_{n}$ , hence also $p^{n}[C_{n}]$ , with $\mathrm{val}_{A}^{(0,\infty)}(p^{n}[C_{n}])$ bounded independently of n. More precisely, we can solve for $U_{n}$ and $C_{n}$ with

\[\mathrm{val}_{A/p}^{(0,\infty)}(C_{n}X_{0}^{-1})\geq c\]

for any $c<(\frac{1}{p-1})[\mathrm{val}(X_{0})+\mathrm{val}(X_{0}^{-1})]$ . Then we may and do choose the $C_{n}$ in a way such that for $n\geq1$ we have

\[\mathrm{val}_{A}^{(0,\infty)}(p^{n}[C_{n}])\geq\mathrm{val}_{A}^{(0,\infty)}([C_{n}])\geq\mathrm{val}_{A/p}^{(0,\infty)}(C_{n})\geq\mathrm{val}_{A/p}^{(0,\infty)}(X_{0})+c.\]

In particular, this bound is independent of n, the bound, in fact, depending only on $X_{0}$ . With these choices of the $C_{n}$ , letting $C=\sum_{n\geq0}[C_{n}]p^{n}$ and $U=\sum_{n\geq0}[U_{n}]p^{n}$ we therefore have $C=U^{-1}X\varphi(U),$ where $C\in\mathrm{GL}_{d}(\widetilde{\mathrm{{\mathbf A}}}_{K,A}^{(0,\infty)})$ because $\mathrm{val}_{A}^{(0,\infty)}(C)\geq c>-\infty$ .

3.2 Descending $\varphi$ -module morphisms

We need to regularize the action of $\varphi$ .

Proposition 3.5 Let $1/r\in\mathbf{Z}[1/p]_{>0}\cup\{\infty\}$ , and let $X\in\mathrm{M}_{d}(\widetilde{\mathrm{{\mathbf A}}}_{K,A}^{(0,r]}),Y\in\mathrm{M}_{e}(\widetilde{\mathrm{{\mathbf A}}}_{K,A}^{(0,r]})$ and $U\in\mathrm{M}_{d\times e}(\widetilde{\mathrm{{\mathbf A}}}_{K,A})$ satisfy

\[\varphi(U)=XUY.\]

Then $U\in\mathrm{M}_{d\times e}(\widetilde{\mathrm{{\mathbf A}}}_{K,A}^{(0,r]})$ .

In particular, if $X\in\mathrm{M}_{d}(\widetilde{\mathrm{{\mathbf A}}}_{K,A}^{\dagger}),Y\in\mathrm{M}_{e}(\widetilde{\mathrm{{\mathbf A}}}_{K,A}^{\dagger})$ and $U\in\mathrm{M}_{d\times e}(\widetilde{\mathrm{{\mathbf A}}}_{K,A})$ , then $U\in\mathrm{M}_{d\times e}(\widetilde{\mathrm{{\mathbf A}}}_{K,A}^{\dagger})$ .

Proof. Write $X=[\varpi]^{t}X'$ with $\mathrm{val}_{A}^{(0,r]}(X')\geq0$ , then we see

\[\mathrm{val}_{A}^{(0,r]}(\varphi^{-1}(X)\varphi^{-2}(X)\cdots\varphi^{-N}(X))\geq t(p^{-1}+\cdots+p^{-N})\]

is bounded independently of N. (This also happens with another bound if $r=\infty$ .) A similar analysis applies to $\varphi^{-N}(Y)\cdots\varphi^{-2}(Y)\varphi^{-1}(Y)$ .

From the equation $U=\varphi^{-1}(X)\varphi^{-1}(U)\varphi^{-1}(Y)$ we get by iteration

\[U=\varphi^{-1}(X)\varphi^{-2}(X)\cdots\varphi^{-N}(X)\cdot\varphi^{-N}(U)\cdot\varphi^{-N}(Y)\cdots\varphi^{-2}(Y)\varphi^{-1}(Y),\]

which we write as $U=X_{N}\varphi^{-N}(U)Y_{N}$ , with $\mathrm{val}_{A}^{(0,r]}(X_{N})$ and $\mathrm{val}_{A}^{(0,r]}(Y_{N})$ bounded independently of N.

Now write $U=\sum_{n\geq0}[U_{n}]p^{n}$ , where $[\bullet]$ denotes the generalized Teichmüller digit of Lemma 3.2, applied successively to the entries of the reduction of U mod $p^{n}$ . Let $U^{(k)}$ be the kth truncation of U, i.e. $U^{(k)}=\sum_{n=0}^{k-1}[U_{n}]p^{n}$ . We then have

\[U^{(k)}=X_{N}\varphi^{-N}(U^{(k)})Y_{N}\mod p^{k}.\]

Fixing k and choosing N large we can make $\mathrm{val}_{A}^{(0,r]}(X_{N}\varphi^{-N}(U^{(k)})Y_{N})\geq c$ where c is a constant depending only on X and Y, because $\mathrm{val}_{A}^{(0,r]}(\varphi^{-N}(U^{(k)}))=p^{-N}\,\mathrm{val}_{A}^{(0,r]}(U^{(k)})$ .

We then get that

\[\mathrm{val}_{A/p^{k}}^{(0,r]}(U^{(k)}\text{ mod }p^{k})\geq\mathrm{val}_{A}^{(0,r]}(U^{(k)})\]

is bounded below by c.

We shall now prove by induction that $\mathrm{val}_{A}^{(0,r]}([U_{k-1}]p^{k-1})\geq c$ . By possibly making c smaller, we may assume this is true for $k=1$ . Arguing by induction on k, using $U^{(k)}=[U_{k-1}]p^{k-1}+U^{(k-1)}$ , we deduce that

\[\mathrm{val}_{A/p^{k}}^{(0,r]}([U_{k-1}]p^{k-1}\text{ mod }p^{k})\geq c\]

is bounded below by c. But by Lemma 3.2, this implies that $\mathrm{val}_{A}^{(0,r]}([U_{k-1}]p^{k-1})\geq c$ .

Hence, for each k, we have $\mathrm{val}_{A}^{(0,r]}(U^{(k)})\geq c$ and, consequently, $\mathrm{val}_{A}^{(0,r]}(U)\geq c$ . This proves that $U\in\mathrm{M}_{d\times e}(\widetilde{\mathrm{{\mathbf A}}}_{A}^{(0,r]})$ , as required.

Corollary 3.6 Let $\widetilde{M}$ be a free étale $\varphi$ -module over $\widetilde{\mathrm{{\mathbf A}}}_{K,A}$ . Then there exists a unique free étale $\varphi$ -module $\widetilde{M}^{\dagger}$ over $\widetilde{\mathrm{{\mathbf A}}}_{K,A}^{\dagger}$ , contained in $\widetilde{M}$ , such that the natural map

\[\widetilde{M}^{\dagger}\otimes_{\widetilde{\mathrm{{\mathbf A}}}_{K,A}^{\dagger}}\widetilde{\mathrm{{\mathbf A}}}_{K,A}\rightarrow\widetilde{M}\]

is an isomorphism.

Corollary 3.7 Let $\widetilde{M}$ be a free étale $\varphi$ -module over $\widetilde{\mathrm{{\mathbf A}}}_{A}$ , and let $\widetilde{M}^{\dagger}$ be the unique free étale $\varphi$ -module over $\widetilde{\mathrm{{\mathbf A}}}_{A}^{\dagger}$ with $\widetilde{M}^{\dagger}\otimes_{\widetilde{\mathrm{{\mathbf A}}}_{A}^{\dagger}}\widetilde{\mathrm{{\mathbf A}}}_{A}\xrightarrow{\sim}\widetilde{M}$ as in Corollary 3.6. Then

\[(\widetilde{M}^{\dagger})^{\varphi=1}=\widetilde{M}^{\varphi=1}.\]

Proof. Let $e_{1},\ldots,e_{d}$ be a basis of $\widetilde{M}^{\dagger}$ and let $X=\mathrm{Mat}(\varphi)\in\mathrm{GL}_{d}(\widetilde{\mathrm{{\mathbf A}}}_{K,A}^{\dagger})$ be the matrix of $\varphi$ with respect to this basis. If $m\in\widetilde{M}^{\varphi=1}$ and $m=\sum a_{i}e_{i}$ with $a_{i}\in\widetilde{\mathrm{{\mathbf A}}}_{K,A}$ , the vector $a=(a_{i})$ satisfies the equation

\[\varphi(a)=X^{-1}a,\]

so by Proposition 3.5 we conclude that the $a_{i}$ belong to $\widetilde{\mathrm{{\mathbf A}}}_{K,A}^{\dagger}$ .

3.3 The equivalence of categories

The following lemma will allow us to reduce from the projective case to the free case.

Lemma 3.10 If M,N are projective étale $\varphi$ -modules over a ring R, then $M^{\vee}\otimes_{R}N$ is also a projective étale $\varphi$ -module, and we have a natural identification

\[\mathbb{\mathrm{Hom}}_{R,\varphi}(M,N)\xrightarrow{\sim}(M^{\vee}\otimes_{R}N)^{\varphi=1}.\]

Proposition 3.11 Let $\widetilde{M}^{\dagger},\widetilde{N}^{\dagger}$ be projective étale $\varphi$ -modules over $\widetilde{\mathrm{{\mathbf A}}}_{K,A}^{\dagger}$ , and let $\widetilde{M}=\widetilde{M}^{\dagger}\otimes_{\widetilde{\mathrm{{\mathbf A}}}_{K,A}^{\dagger}}\widetilde{\mathrm{{\mathbf A}}}_{K,A}$ , $\widetilde{N}=\widetilde{N}^{\dagger}\otimes_{\widetilde{\mathrm{{\mathbf A}}}_{K,A}^{\dagger}}\widetilde{\mathrm{{\mathbf A}}}_{K,A}$ . Then

\[\mathbb{\mathrm{Hom}}_{\widetilde{\mathrm{{\mathbf A}}}_{K,A}^{\dagger},\varphi}(\widetilde{M}^{\dagger},\widetilde{N}^{\dagger})\xrightarrow{\sim}\mathbb{\mathrm{Hom}}_{\widetilde{\mathrm{{\mathbf A}}}_{K,A},\varphi}(\widetilde{M},\widetilde{N}).\]

Combining Propositions 3.11 and 3.12 we get a full descent result for projective étale $\varphi$ -modules.

Finally, to prove the descent result for $(\varphi,\Gamma_{K})$ -modules, we only need to descend $\Gamma_{K}$ along the equivalence.

For essential surjectivity, suppose $\widetilde{M}$ is a projective étale $(\varphi,\Gamma_{K})$ -module over $\widetilde{\mathrm{{\mathbf A}}}_{K,A}$ , and let $\widetilde{M}^{\dagger}$ be the unique étale projective $\varphi$ -module over $\widetilde{\mathrm{{\mathbf A}}}_{K,A}^{\dagger}$ associated to it in Proposition 3.12. Then by uniqueness, $\widetilde{M}^{\dagger}$ is stable under the action of $\Gamma_{K}$ , hence is actually a $(\varphi,\Gamma_{K})$ -module over $\widetilde{\mathrm{{\mathbf A}}}_{K,A}^{\dagger}$ . This concludes the proof.

4.1 The Tate–Sen axioms

With the previous setting, we define them to be the following.Footnote ³

1. (TS1) There exists $c_{1}>0$ such that for each pair $H_{1}\subset H_{2}$ of open subgroups of $H_{0}$ there exists $\alpha\in\widetilde{\Lambda}^{H_{1}}$ such that $\mathrm{val}_{\Lambda}(\alpha)>-c_{1}$ and $\sum_{\tau\in H_{2}/H_{1}}\tau(\alpha)=1$ .

2. (TS2) There exists $c_{2}>0$ and for each open subgroup H of $H_{0}$ an integer n(H), as well as an increasing sequence $(\Lambda_{H,n})_{n\geq n(H)}$ of closed subalgebras of $\widetilde{\Lambda}^{H}$ , each containing $f^{\pm1}$ , and $\Lambda_{H,n}$ -linear maps

   \[\mathbb{\mathrm{R}}_{H,n}:\widetilde{\Lambda}^{H}\rightarrow\Lambda_{H,n}\]
   such that:

   1. (1) if $H_{1}\subset H_{2}$ , then $\Lambda_{H_{2},n}\subset\Lambda_{H_{1},n}$ and $\mathbb{\mathrm{R}}_{H_{1},n}|_{\widetilde{\Lambda}^{H_{2}}}=\mathbb{\mathrm{R}}_{H_{2},n}$ ;

   2. (2) $\mathbb{\mathrm{R}}_{H,n}(x)=x$ if $x\in\Lambda_{H,n}$ ;

   3. (3) $g(\Lambda_{H,n})=\Lambda_{gHg^{-1},n}$ and $g(\mathbb{\mathrm{R}}_{H,n}(x))=\mathbb{\mathrm{R}}_{gHg^{-1}}(gx)$ if $g\in G_{0}$ ;

   4. (4) if $n\geq n(H)$ and if $x\in\widetilde{\Lambda}^{H}$ , then $\mathrm{val}_{\Lambda}(\mathbb{\mathrm{R}}_{H,n}(x))\geq\mathrm{val}_{\Lambda}(x)-c_{2}$ ;

   5. (5) if $x\in\widetilde{\Lambda}^{H}$ , then $\lim_{n\rightarrow\infty}\mathbb{\mathrm{R}}_{H,n}(x)=x$ .

3. (TS3) There exists $c_{3}>0$ and, for each open subgroup G of $G_{0}$ an integer $n(G)\geq n_{1}(H)$ where $H=G\cap H_{0}$ , such that if $n(\gamma)\leq n\leq n(G)$ for $\gamma\in\widetilde{\Gamma}_{H}$ then $\gamma-1$ is invertible on $X_{H,n}=(1-\mathbb{\mathrm{R}}_{H,n})(\widetilde{\Lambda}^{H})$ and $\mathrm{val}_{\Lambda}((\gamma-1)^{-1}(x))\geq\mathrm{val}_{\Lambda}(x)-c_{3}$ .

For the rest of the section we shall assume (TS1), (TS2) and (TS3) are satisfied.

4.2 Descent to H-invariants

4.3 Decompletion

The following lemma needs to be modified slightly compared with the treatment of [Reference Berger and ColmezBC08].

Lemma 4.5 Let $\delta>0$ and let $a,b\in{\mathbf{R}}$ such that $a\geq c_{2}+c_{3}+\delta$ and $b\ge\mathrm{sup}(a+c_{2},2c_{2}+2c_{3}+\delta$ ). Let H be an open subgroup of $H_{0}$ , $n\geq n(H)$ and $\gamma\in\widetilde{\Gamma}_{H}$ such that $n(\gamma)\leq n$ . Finally, let

\[U=1+f^{k}U_{1}+f^{k}U_{2}\]

such that

\[\begin{array}{l}U_{1}\in\mathrm{M}_{d}(\Lambda_{H,n}),\mathrm{val}_{\Lambda}(U_{1})\geq a-\mathrm{val}_{\Lambda}(f^{k}),\\U_{2}\in\mathrm{M}_{d}(\widetilde{\Lambda}^{H}),\mathrm{val}_{\Lambda}(U_{2})\geq b-\mathrm{val}_{\Lambda}(f^{k}).\end{array}\]

Then there exists $M\in\mathrm{M}_{d}(\widetilde{\Lambda}^{H})$ with $\mathrm{val}(M-1)\geq b-c_{2}-c_{3}$ such that $M^{-1}U\gamma(M)=1+f^{k}V_{1}+f^{k}V_{2}$ such that

\[\begin{array}{l}V_{1}\in\mathrm{M}_{d}(\Lambda_{H,n}),\mathrm{val}_{\Lambda}(V_{1})\geq a-\mathrm{val}_{\Lambda}(f^{k}),\\V_{2}\in\mathrm{M}_{d}(\widetilde{\Lambda}^{H}),\mathrm{val}_{\Lambda}(V_{2})\geq b-\mathrm{val}_{\Lambda}(f^{k})+\delta.\end{array}\]

Proof. This again just follows the proof of [Reference Berger and ColmezBC08, Lemma 3.2.3], but there are a few small modifications required, so we give more details. One setsFootnote ⁴

\[\begin{array}{l}V=f^{-k}(1-\gamma)^{-1}(1-\mathbb{\mathrm{R}}_{H,n})(f^{k}U_{2}),\\V_{1}=U_{1}+\mathbb{\mathrm{R}}_{H,n}(U_{2}),\\V_{2}=f^{-k}[(1+f^{k}V)^{-1}U\gamma(1+f^{k}V)-(1+f^{k}U_{1}+f^{k}\mathbb{\mathrm{R}}_{H,n}(U_{2}))],\\M=1+f^{k}V.\end{array}\]

By explicit calculations, one checks using (TS2), (TS3), the $\Lambda_{H,n}$ -linearity of $\mathbb{\mathrm{R}}_{H,n}$ and the expansion

\[(1+f^{k}V)^{-1}=1-f^{k}V+f^{2k}V^{2}-\cdots\]

that all the terms are well defined and that the conditions are satisfied. (Here we have implicitly used the assumption $f\in\Lambda_{H,n}$ in use of the $\Lambda_{H,n}$ -linearity of $\mathbb{\mathrm{R}}_{H,n}$ .)

4.4 Descent

Let $M^{+}$ be a finite free $\widetilde{\Lambda}^{+}$ -semilinear representation of $G_{0}$ and for $H\subset H_{0}$ open let $\Lambda_{H,n}^{+}=\widetilde{\Lambda}^{+}\cap\Lambda_{H,n}$ .

Then for $n\geq n(G)$ there exists a unique free $\Lambda_{H,n}^{+}$ -submodule $\mathrm{\mathbf{D}}_{H,n}^{+}(M^{+})$ of $M^{+}$ such that:

1. (1) $\mathrm{\mathbf{D}}_{H,n}^{+}(M^{+})$ is fixed by H and stable by $G_{0}$ ;

2. (2) the natural map $\widetilde{\Lambda}^{+}\otimes_{\Lambda_{H,n}^{+}}\mathrm{\mathbf{D}}_{H,n}^{+}(M^{+})\rightarrow M^{+}$ is an isomorphism; in particular, $\mathrm{\mathbf{D}}_{H,n}^{+}(M^{+})$ is free of rank = $\mathrm{rank} M^{+}$ ;

3. (3) $\mathrm{\mathbf{D}}_{H,n}^{+}(M^{+})$ has a basis which is $c_{3}$ -fixed by $G/H$ , meaning that for $\gamma\in G/H$ we have $\mathrm{val}(\mathrm{Mat}(\gamma)-1)>c_{3}$ .

Let $v_{1},\ldots,v_{d}$ be a basis of $M^{+}$ over $\widetilde{\Lambda}^{+}$ . We get a cocycle U in $\mathrm{H}^{1}(G_{0},\mathrm{GL}_{d}(\widetilde{\Lambda}^{+}))$ . By assumption $\mathrm{val}(U_{\sigma}-1)>c_{1}+2c_{2}+2c_{3}$ if $\sigma\in G$ . By Proposition 4.8 there exists $M\in\mathrm{M}_{d}(\widetilde{\Lambda})$ such that $\mathrm{val}(M-1)>c_{2}+c_{3}$ and the cocycle $\sigma\mapsto V_{\sigma}=M^{-1}U_{\sigma}\sigma(M)$ is trivial on H and is valued in $\mathrm{GL}_{d}(\Lambda_{H,n(G)})$ . Now $\mathrm{val}(M-1)>c_{2}+c_{3}>0$ so $M\in\mathrm{GL}_{d}(\widetilde{\Lambda}^{+})$ . It follows that V is valued in $\mathrm{GL}_{d}(\Lambda_{H,n(G)})\cap\mathrm{GL}_{d}(\widetilde{\Lambda}^{+})=\mathrm{GL}_{d}(\Lambda_{H,n(G)}^{+})$ .

Now let $M=(m_{ij})$ and $U_{\sigma}=(u_{ij})$ . If we write $e_{i}=Mv_{i}$ for $i=1,\ldots,d$ , then

\[\sigma(e_{k})=\sum_{j}\sigma(m_{jk})\sigma(v_{j})=\sum_{i}\sum_{j}u_{ij}\sigma(m_{ij})v_{i}=e_{k}\]

for $\sigma\in H$ . Thus, $e_{1},\ldots,e_{d}$ is a basis of $M^{+}$ fixed by H, and if we write $\mathrm{\mathbf{D}}_{H,n}^{+}(M^{+})=\bigoplus\Lambda_{H,n}^{+}e_{i}$ , then the natural map $\widetilde{\Lambda}^{+}\otimes_{\Lambda_{H,n}^{+}}\mathrm{\mathbf{D}}_{H,n}^{+}(M^{+})\rightarrow M^{+}$ is an isomorphism.

Furthermore, if $\gamma\in G/H$ , then $W=\mathrm{Mat}(\gamma)$ in the basis $e_{1},\ldots,e_{d}$ is of the form $M^{-1}U_{\sigma}\sigma(M)$ where $\sigma\in G$ is a lift of $\gamma$ . Using the identity

\[W-1=M^{-1}U_{\sigma}\sigma(M)-1=(M^{-1}-1)U_{\sigma}\sigma(M)+(U_{\sigma}-1)\sigma(M)+\sigma(M)-1,\]

it follows that $\mathrm{val}(W-1)>c_{2}+c_{3}>c_{3}$ . This implies that the basis $\{e_{1},\ldots,e_{d}\}$ is $c_{3}$ -fixed.

Finally, we show that $\mathrm{\mathbf{D}}_{H,n}^{+}(M^{+})$ is unique. Choose $\gamma\in C_{H}$ with $n(\gamma)=n$ and let $e_{1}',\ldots,e_{d}'$ be another basis. Then $\mathrm{Mat}_{\{e_{i}\}}(\gamma)=W$ and $\mathrm{Mat}_{\{e_{i}'\}}(\gamma)=W'$ are both in $\mathrm{GL}_{d}(\Lambda_{H,n(G)}^{+})$ with $n\geq n(G)$ and $\mathrm{val}(W-1),\mathrm{val}(W'-1)>c_{3}$ . Let $B\in\mathrm{GL}_{d}(\widetilde{\Lambda}^{+})$ be the matrix expressing $e_{i}'$ in terms of $e_{i}$ . Then B is H-invariant and $W'=B^{-1}W\gamma(B)$ . According to Lemma 4.7 we have $B\in\mathrm{GL}_{d}(\Lambda_{H,n})$ , so $B\in\mathrm{GL}_{d}(\widetilde{\Lambda}^{+})\cap\mathrm{GL}_{d}(\Lambda_{H,n})=\mathrm{GL}_{d}(\Lambda_{H,n}^{+})$ . It follows that $e_{i}'$ and $e_{i}$ generate the same submodule.

Proof. Indeed, if we have a submodule generated by $c_{3}$ -fixed elements $f_{1},\ldots,f_{l}$ and if $e_{1},\ldots,e_{d}$ is a $c_{3}$ -fixed basis, let $B\in\mathrm{M}_{l\times d}(\widetilde{\Lambda}^{H,+})$ be a matrix expressing the $f_{i}$ in terms of the $e_{i}$ . We have

\[\mathrm{Mat}_{\{f_{i}\}}(\gamma)B=\gamma(B)\,\mathrm{Mat}_{\{e_{i}\}}(\gamma).\]

(Here $\mathrm{Mat}_{\{f_{i}\}}(\gamma)$ is not necessarily a uniquely determined matrix, since the submodule generated by the $f_{i}$ may not be free, but this does not matter.) We have $\mathrm{val}(\mathrm{Mat}_{\{f_{i}\}}(\gamma)-1)>c_{3}$ , and this implies that $\mathrm{Mat}_{\{f_{i}\}}(\gamma)$ is invertible. Thus, by Lemma 4.7,

\[B\in\mathrm{M}_{l\times d}(\Lambda_{H,n})\cap\mathrm{M}_{l\times d}(\widetilde{\Lambda}^{H,+})=\mathrm{M}_{l\times d}(\Lambda_{H,n}^{+}),\]

which means the submodule generated by $f_{i}$ is contained in $\mathrm{\mathbf{D}}_{H,n}^{+}(M^{+})$ .

We introduce two additive, $\otimes$ -categories:

• – $\mathrm{Mod}_{\widetilde{\Lambda}^{+}}^{G_{0}}(G)$ , the category of finite free $\widetilde{\Lambda}^{+}$ -semilinear representations of $G_{0}$ such that for some basis $\mathrm{val}(\mathrm{Mat}(g)-1)>c_{1}+2c_{2}+2c_{3}$ for $g\in G$ ;

• – $\mathrm{Mod}_{\Lambda_{H,n}^{+}}^{G_{0}}(G)$ , the category of finite free $\Lambda_{H,n}^{+}$ -semilinear representations of $G_{0}$ that are fixed by $H=G\cap H_{0}$ and which have $c_{3}$ -fixed basis.

For $n\geq n(G)$ , Propositions 4.9 and 4.10 give us a functor $M^{+}\mapsto\mathrm{\mathbf{D}}_{H,n}^{+}(M^{+})$ from $\mathrm{Mod}_{\widetilde{\Lambda}^{+}}(G)$ to $\mathrm{Mod}_{\Lambda_{H,n}^{+}}(G)$ . We also have a functor $N^{+}\mapsto\widetilde{\Lambda}^{+}\otimes_{\Lambda_{H,n}^{+}}N^{+}$ from $\mathrm{Mod}_{\Lambda_{H,n}^{+}}(G)$ to $\mathrm{Mod}_{\widetilde{\Lambda}^{+}}(G)$ . We can then express Proposition 4.9 in the following way.

5.1 Verifying the Tate–Sen axioms

We take $G_{0}=\Gamma_{K}$ , and $\chi:G_{0}\rightarrow\mathbf{Z}_{p}^{\times}$ is the cyclotomic character, so that $H_{0}=1$ .

Let $\widetilde{\Lambda}=\widetilde{\mathrm{{\mathbf A}}}_{K,A}^{(0,r]}$ for $1/r\in\mathbf{Z}[1/p]_{>0}$ with $r<1$ and $\widetilde{\Lambda}^{+}=\widetilde{\mathrm{{\mathbf A}}}_{K,A}^{(0,r],+}$ . Then:

1. (i) $\widetilde{\Lambda}$ is a Tate ring, with ring of definition $\widetilde{\Lambda}^{+}$ , with a pseudouniformizer f taken to be T, the element introduced in § 2.1; note that T was introduced as an element of $\mathrm{{\mathbf A}}_{K,A}^{+}$ , but it can be thought of as an element of $\mathrm{{\mathbf A}}_{K,A}^{(0,r]}$ , the latter being a subring of $\widetilde{\Lambda}$ ;

2. (ii) $\widetilde{\Lambda}^{+}$ is T-adically complete, by Proposition 2.3;

3. (iii) by § 6 of [Reference ColmezCo08], $T/[\varpi]$ is a unit in $\widetilde{\mathrm{{\mathbf A}}}^{(0,r],\circ}$ if $r<1$ , hence also in $\widetilde{\mathrm{{\mathbf A}}}_{K,A}^{(0,r],+}$ ; we endow $\widetilde{\mathrm{{\mathbf A}}}_{K,A}^{(0,r]}$ with the valuation $\mathrm{val}^{(0,r]}$ ,

   \[\mathrm{val}^{(0,r]}(a)=(p/p-1)\,\mathrm{sup}\{x\in\mathbf{Z}[1/p]:a\in[\varpi]^{x}\widetilde{\mathrm{{\mathbf A}}}_{K,A}^{(0,r],+}\};\]
   note that $[\varpi]^{x}\widetilde{\mathrm{{\mathbf A}}}_{A}^{(0,r],+}=T^{x}\widetilde{\mathrm{{\mathbf A}}}_{A}^{(0,r],+}$ whenever $T^{x}$ makes sense; it therefore induces the T-adic topology;

4. (iv) the group $G_{0}$ acts continuously on $\widetilde{\mathrm{{\mathbf A}}}_{K,A}^{(0,r]}$ , and is unitary for the valuation $\mathrm{val}^{(0,r]}$ .

As explained in Remark 4.2, the condition (TS1) is automatic in this setting since $H_{0}=1$ .

5.1.1 The axiom (TS2).

We shall check this axiom holds in the following setting. Recalling $H_{0}=1$ , we set

\[\Lambda_{n}=\Lambda_{H_{0},n}:=\varphi^{-n}\big(\mathrm{{\mathbf A}}_{K,A}^{(0,rp^{-n}]}\big),\]

which is a closed subalgebra of $\widetilde{\mathrm{{\mathbf A}}}_{K,A}^{(0,r]}$ , and

\[\mathbb{\mathrm{R}}_{n}:=\mathbb{\mathrm{R}}_{H_{0},n}:\widetilde{\mathrm{{\mathbf A}}}_{K,A}^{(0,r]}\rightarrow\varphi^{-n}\big(\mathrm{{\mathbf A}}_{K,A}^{(0,rp^{-n}]}\big)\]

is the continuous extension of the map $(\widetilde{\mathrm{{\mathbf A}}}_{K}^{(0,r]})\rightarrow\varphi^{-n}(\mathrm{{\mathbf A}}_{K}^{(0,rp^{-n}]})$ constructed in § 8 of [Reference ColmezCo08] for the case $A=\mathbf{Z}_{p}$ . This extension exists because this original map is T-adically continuous.

We shall now verify (TS2) holds. The only thing which is not immediate is condition (4), i.e. we need to check that there exists $c_{2}>0$ such that $\mathrm{val}_{\Lambda}(R_{n}(x))\geq\mathrm{val}_{\Lambda}(x)-c_{2}$ . To do this, choose $c_{2}>0$ which works in the case $A=\mathbf{Z}_{p}$ , which is known to exist by [Reference Berger and ColmezBC08, Prop. 4.2.1].

Suppose $x\in\widetilde{\mathrm{{\mathbf A}}}_{K,A}^{(0,r]}$ . If $\mathrm{val}_{\Lambda}(x)\geq(p/p-1)t$ for $t\in\mathbf{Z}[1/p]$ , then $x\in[\varpi]^{t}\widetilde{\mathrm{{\mathbf A}}}_{K,A}^{(0,r],+}$ . We may write x as the image of a (possibly infinite) sum

\[\sum x_{i}\otimes a_{i}\]

with

\[x_{i}\in[\varpi]^{t}\widetilde{\mathrm{{\mathbf A}}}_{K}^{(0,r],\circ},\quad a_{i}\in A,\]

so that $\mathrm{val}^{(0,r]}(x_{i})\geq(p/p-1)t$ .

Now $\mathbb{\mathrm{R}}_{n}(x)=\sum\mathbb{\mathrm{R}}_{n}(x_{i})\otimes a_{i},$ and we have from the case $A=\mathbf{Z}_{p}$ that

\[\mathrm{val}^{(0,r]}(\mathbb{\mathrm{R}}_{n}(x_{i}))\geq(p/p-1)t-c_{2}.\]

Hence,

\[\mathbb{\mathrm{R}}_{n}(x_{i})\in([\varpi]^{t-(\frac{p-1}{p})c_{2}}\widetilde{\mathrm{{\mathbf A}}}_{K}^{(0,r],\circ})\,\widehat{\otimes}\,A\]

for each i, which shows that

\[\mathbb{\mathrm{R}}_{n}(x)\in[\varpi]^{t-(\frac{p-1}{p})c_{2}}\widetilde{\mathrm{{\mathbf A}}}_{K,A}^{(0,r],+},\]

so that $\mathrm{val}_{\Lambda}(\mathbb{\mathrm{R}}_{n}(x))\geq(\frac{p}{p-1})t-c_{2}$ . Hence, (TS2) holds with the same $c_{2}$ .

5.1.2 The axiom (TS3).

We need to show that there exists $c_{3}>0$ and, for each open subgroup G of $G_{0}$ , an integer n(G) such that if $n\geq n(G)$ and if $n(\gamma)\leq n$ , then $\gamma-1$ is invertible on $X_{n}=(1-\mathbb{\mathrm{R}}_{n})(\widetilde{\Lambda})$ and $\mathrm{val}_{\Lambda}((\gamma-1)^{-1}(x))\geq\mathrm{val}_{\Lambda}(x)-c_{3}$ .

We shall show that if $c_{2}=c_{2}(\mathbf{Z}_{p})$ and $c_{3}=c_{3}(\mathbf{Z}_{p})$ work for the case $A=\mathbf{Z}_{p}$ as in [Reference Berger and ColmezBC08, Proposition 4.2.1], then any $c_{3}'>c_{2}(\mathbf{Z}_{p})+c_{3}(\mathbf{Z}_{p})$ works for general A.

Let $x\in X_{n}=(1-\mathbb{\mathrm{R}}_{n})(\widetilde{\Lambda})$ . Let $y\in\widetilde{\Lambda}\cong\widetilde{\mathrm{{\mathbf A}}}_{K,A}^{(0,r]}$ . We may write y as the image of

\[\sum y_{i}\otimes a_{i}\]

in $\widetilde{\mathrm{{\mathbf A}}}_{K,A}^{(0,r]}$ , with $y_{i}\in\widetilde{\mathrm{{\mathbf A}}}_{K}^{(0,r]}$ (the sum possibly infinite). If $\mathrm{val}^{(0,r]}(y)\geq(p/p-1)t$ for $t\in\mathbf{Z}[1/p]$ , then $y\in[\varpi]^{t}\widetilde{\mathrm{{\mathbf A}}}_{K,A}^{(0,r],+}$ , which is the image of $[\varpi]^{t}\widetilde{\mathrm{{\mathbf A}}}_{K}^{(0,r],\circ}\,\widehat{\otimes}\,A$ , so one can choose $y_{i}\in[\varpi]^{t}\widetilde{\mathrm{{\mathbf A}}}_{K}^{(0,r],\circ}$ . This shows that we may assume $\mathrm{val}^{(0,r]}(y_{i})\geq\mathrm{val}^{(0,r]}(y)$ for every i.

We let $x=(1-\mathbb{\mathrm{R}}_{n})(y)$ , so that x is the image of $\sum(1-\mathbb{\mathrm{R}}_{n})(y_{i})\otimes a_{i}$ . Writing $x_{i}=(1-\mathbb{\mathrm{R}}_{n})(y_{i})$ we have

\[\mathrm{val}_{\Lambda}((\gamma-1)^{-1}(x_{i}))\geq\mathrm{val}_{\Lambda}(x_{i})-c_{3}\geq\mathrm{val}^{(0,r]}(y_{i})-c_{3},\]

which approaches zero. The sum $\sum((\gamma-1)^{-1}(x_{i}))\otimes a_{i}$ therefore converges in $\widetilde{\mathrm{{\mathbf A}}}_{K,A}^{(0,r]}$ . This shows that $\gamma-1$ is invertible on $X_{n}$ .

Finally, suppose $x\in X_{n}$ with $\mathrm{val}^{(0,r]}(x)\geq(p/p-1)t$ , we shall show that

\[\mathrm{val}^{(0,r]}((\gamma-1)^{-1}(x))\geq\frac{p}{p-1}t-c_{2}-c_{3}.\]

By assumption, $x\in[\varpi]^{t}\widetilde{\mathrm{{\mathbf A}}}_{K,A}^{(0,r],+}$ , so we may write $x=\sum x_{i}\otimes a_{i}$ (the sum possibly infinite) with $x_{i}\in[\varpi]^{t}\widetilde{\mathrm{{\mathbf A}}}_{K}^{(0,r],\circ}$ . Then since $(1-\mathbb{\mathrm{R}}_{n})$ is idempotent we have $x=\sum(1-\mathbb{\mathrm{R}}_{n})(x_{i})\otimes a_{i}$ . Letting $y_{i}=(1-\mathbb{\mathrm{R}}_{n})(x_{i})$ we have $y_{i}\in[\varpi]^{t-(\frac{p-1}{p})c_{2}}\widetilde{\mathrm{{\mathbf A}}}_{K}^{(0,r],\circ}$ and so $(\gamma-1)^{-1}(x)=\sum(\gamma-1)^{-1}(y_{i})\otimes a_{i}$ . Then

\[(\gamma-1)^{-1}(y_{i})\in[\varpi]^{t-(\frac{p-1}{p})c_{2}-(\frac{p-1}{p})c_{3}}\widetilde{\mathrm{{\mathbf A}}}_{K}^{(0,r],\circ},\]

which shows

\[\mathrm{val}_{\Lambda}((\gamma-1)^{-1}(x))\geq\mathrm{val}_{\Lambda}(x)-c_{2}-c_{3},\]

which shows we can take $c_{3}'=c_{2}-c_{3}$ , as required.

Remark 5.1.

1. (i) With a little more work, once can prove that $\widetilde{\mathrm{{\mathbf A}}}_{A}^{(0,r]}$ also satisfies the Tate–Sen axioms. This recovers overconvergence results appearing in § 4.2 of [Reference Berger and ColmezBC08].

2. (ii) The following was pointed out to us by Rebecca Bellovin: if A is the ring of definition of a pseudoaffinoid algebra with pseudouniformizer u, the rings $\widetilde{\mathrm{{\mathbf A}}}_{A/u}^{(0,r]}$ are consistent with the reduction mod u of the rings $\widetilde{\Lambda}_{A,(0,r]}$ of [Reference BellovinBel20]. By taking u-adic limits, one should be able to recover the main result of [Reference BellovinBel20] from the results of § 4.

3. (iii) More generally, one could probably phrase the results of this section as some sort of stability of our version of the Tate–Sen axioms under base change, but we have not attempted to do so.

5.2 Descent

The following proposition is a variant of [Reference BergerBer08, Thm. I.3.3].

In particular:

1. (1) $\widetilde{\mathrm{{\mathbf A}}}_{K,A}^{(0,s]}\otimes_{\widetilde{\mathrm{{\mathbf A}}}_{K,A}^{(0,r]}}\widetilde{M}^{(0,r]}=\widetilde{M}^{(0,s]}$ for $s<r$ ;

2. (2) $\varphi$ induces an isomorphism $\varphi^{*}\widetilde{M}^{(0,r]}\xrightarrow{\sim}\widetilde{M}^{(0,r/p]}$ ;

3. (3) If $\widetilde{M}^{\dagger}$ is a $(\varphi,\Gamma_{K})$ -module, then each $\widetilde{M}^{(0,r]}$ is $\Gamma_{K}$ -stable.

Furthermore, if $\widetilde{M}^{\dagger}$ is free, then so is $\widetilde{M}^{(0,r]}$ .

To prove existence, choose any basis $e_{1},\ldots,e_{d}$ of $\widetilde{M}^{\dagger}$ over $\widetilde{\mathrm{{\mathbf A}}}_{K,A}^{\dagger}$ . Then $\mathrm{Mat}(\varphi)\in\mathrm{GL}_{d}(\widetilde{\mathrm{{\mathbf A}}}_{K,A}^{\dagger})$ , and so for some $r_{0}$ and all $r\leq r_{0}$ we have $\mathrm{Mat}(\varphi),\mathrm{Mat}(\varphi^{-1})\in\mathrm{GL}_{d}(\widetilde{\mathrm{{\mathbf A}}}_{K,A}^{(0,r_{0}]})$ . We take $\widetilde{M}^{(0,r]}=\bigoplus\widetilde{\mathrm{{\mathbf A}}}_{K,A}^{(0,r]}e_{i}$ , so that conditions (i) and (ii) are satisfied. For $r>r_{0}$ we may recursively define $\widetilde{M}^{(0,r]}:=(\varphi^{-1})^{*}(\widetilde{M}^{(0,r/p]})$ using the isomorphism $(\varphi^{-1})^{*}\widetilde{M}^{\dagger}\xrightarrow{\sim}\widetilde{M}^{\dagger}$ .

For uniqueness, suppose that $\widetilde{M}^{(0,r],(1)}$ and $\widetilde{M}^{(0,r],(2)}$ are two submodules satisfying conditions (i) and (ii). Let $X\in\mathrm{GL}_{d}(\widetilde{\mathrm{{\mathbf A}}}_{K,A}^{\dagger})$ be the transition matrix between the two bases of $\widetilde{M}^{(0,r],(1)}$ and $\widetilde{M}^{(0,r],(2)}$ and let $P_{1}$ and $P_{2}$ be the matrices in $\mathrm{GL}_{d}(\widetilde{\mathrm{{\mathbf A}}}_{K,A}^{(0,r/p]})$ of $\varphi$ in these bases. Then we have the equation

\[X=P_{1}^{-1}\varphi(X)P_{2},\]

which implies by Proposition 3.5 that $X\in\mathrm{GL}_{d}(\widetilde{\mathrm{{\mathbf A}}}_{K,A}^{(0,r/p]})$ . Hence,

\[\widetilde{\mathrm{{\mathbf A}}}_{K,A}^{(0,r/p]}\otimes_{\widetilde{\mathrm{{\mathbf A}}}_{K,A}^{(0,r]}}\widetilde{M}^{(0,r],(1)}=\widetilde{\mathrm{{\mathbf A}}}_{K,A}^{(0,r/p]}\otimes_{\widetilde{\mathrm{{\mathbf A}}}_{K,A}^{(0,r]}}\widetilde{M}^{(0,r],(2)},\]

and it follows from condition (ii) that $\varphi^{*}\widetilde{M}^{(0,r],(1)}=\varphi^{*}\widetilde{M}^{(0,r],(2)}$ . Since $\varphi:\widetilde{\mathrm{{\mathbf A}}}_{K,A}^{(0,r]}\rightarrow\widetilde{\mathrm{{\mathbf A}}}_{K,A}^{(0,r/p]}$ is an isomorphism, this gives $\widetilde{M}^{(0,r],(1)}=\widetilde{M}^{(0,r],(2)}$ .

We now prove existence and uniqueness when $\widetilde{M}^{\dagger}$ is only assumed projective. To show existence, given $\widetilde{M}^{\dagger}$ , embed it as a direct summand of a free étale $\varphi$ -module $\widetilde{F}^{\dagger}$ , as we may according to Lemma 3.9 and set $\widetilde{M}^{(0,r]}=\widetilde{M}^{\dagger}\cap\widetilde{F}^{(0,r]}$ . Let $\pi:\widetilde{F}^{\dagger}\rightarrow\widetilde{M}^{\dagger}$ be the projection.

To see this, choose a basis $e_{1},\ldots,e_{d}$ of $\widetilde{F}^{(0,r]}$ . Let $\mathrm{Mat}(\varphi)$ be the matrix of $\varphi$ with respect to this basis. As the proof of the existence in the free case has shown, if r is taken to be sufficiently small, we can actually arrange that $\mathrm{Mat}(\varphi)\in\mathrm{GL}_{d}(\widetilde{\mathrm{{\mathbf A}}}_{K,A}^{(0,r]})$ (and not just $\mathrm{Mat}(\varphi)\in\mathrm{GL}_{d}(\widetilde{\mathrm{{\mathbf A}}}_{K,A}^{(0,r/p]})$ ). The relation $\pi\circ\varphi=\varphi\circ\pi$ shows that if the claim is true for sufficiently small r it holds for any r, so we may restrict to the case where $\mathrm{Mat}(\varphi)\in\mathrm{GL}_{d}(\widetilde{\mathrm{{\mathbf A}}}_{K,A}^{(0,r]})$ . Now since $\widetilde{\mathrm{{\mathbf A}}}_{K,A}^{\dagger}\otimes_{\widetilde{\mathrm{{\mathbf A}}}_{K,A}^{(0,r]}}\widetilde{F}^{(0,r]}\xrightarrow{\sim}\widetilde{F}^{\dagger}$ , there is a matrix $\mathrm{Mat}(\pi)\in\mathrm{M}_{d}(\widetilde{\mathrm{{\mathbf A}}}_{K,A}^{\dagger})$ representing $\pi$ with respect to the basis $e_{i}$ . We need to show that $\mathrm{Mat}(\pi)\in\mathrm{M}_{d}(\widetilde{\mathrm{{\mathbf A}}}_{K,A}^{(0,r]})$ . But again using the relation $\pi\circ\varphi=\varphi\circ\pi$ we deduce

\[\mathrm{Mat}(\varphi)\varphi(\mathrm{Mat}(\pi))=\mathrm{Mat}(\pi)\,\mathrm{Mat}(\varphi),\]

which implies once more by Proposition 3.5 that $\mathrm{Mat}(\pi)\in\mathrm{M}_{d}(\widetilde{\mathrm{{\mathbf A}}}_{K,A}^{(0,r]})$ , as required.

Now write $\widetilde{P}^{\dagger}$ for the étale $\varphi$ -module which is the complement of $\widetilde{M}^{\dagger}$ in $\widetilde{F}^{\dagger}$ . Letting $\widetilde{P}^{(0,r]}=\widetilde{P}^{\dagger}\cap\widetilde{F}^{(0,r]},$ the claim implies there is a direct decomposition

\[\widetilde{F}^{(0,r]}=\widetilde{M}^{(0,r]}\oplus\widetilde{P}^{(0,r]},\]

which respects the $\varphi$ -action. With this given, the fact that conditions (i) and (ii) hold for $\widetilde{F}^{(0,r]}$ implies that they also hold for $\widetilde{M}^{(0,r]}$ . The existence of this decomposition also shows that $\widetilde{M}^{(0,r]}$ is projective. This finishes the proof of existence of $\widetilde{M}^{(0,r]}$ in general.

To prove uniqueness, it suffices to show that if $\widetilde{M}^{(0,r],\prime}$ satisfies conditions (i) and (ii), and if $\widetilde{F}^{(0,r]}$ is constructed as above, and if $\widetilde{M}^{(0,r]}=\widetilde{F}^{(0,r]}\cap\widetilde{M}^{\dagger}$ , then $\widetilde{M}^{(0,r],\prime}=\widetilde{M}^{(0,r]}$ .

But this follows from the uniqueness proved in the free case, because

\[\widetilde{M}^{(0,r],\prime}\oplus\widetilde{P}^{(0,r]}=\widetilde{F}^{(0,r]}=\widetilde{M}^{(0,r]}\oplus\widetilde{P}^{(0,r]}.\]

Applying the projection $\pi:\widetilde{F}^{\dagger}\rightarrow\widetilde{M}^{\dagger}$ we obtain $\widetilde{M}^{(0,r]}=\widetilde{M}^{(0,r],\prime}$ . This concludes the proof.

There exists an open normal subgroup $\Gamma_{L}=\mathrm{Gal}(K_{\infty}/L)$ of $\Gamma_{K}$ such that the following hold.

1. (1) There exists at most one free $\varphi^{-n}(\mathrm{{\mathbf A}}_{K,A}^{(0,r/p^{n}]})$ -submodule $M_{n}^{(0,r]}$ of $\widetilde{M}^{(0,r]}$ such that:

   1. (i) the natural map $\widetilde{\mathrm{{\mathbf A}}}_{K,A}^{(0,r]}\otimes_{\varphi^{-n}(\mathrm{{\mathbf A}}_{K,A}^{(0,r/p^{n}]})}M_{n}^{(0,r]}\rightarrow\widetilde{M}^{(0,r]}$ is an isomorphism;

   2. (ii) $M_{n}^{(0,r]}$ is $\Gamma_{K}$ -stable;

   3. (iii) $M_{n}^{(0,r]}$ has a $c_{3}$ -fixed basis for the $\Gamma_{L}$ -action.

2. (2) If $n\geq n(\Gamma_{L},\widetilde{M}^{(0,r]})$ , then $M_{n}^{(0,r]}$ exists.

Proof. We start by proving part (2). Choose a basis of $\widetilde{M}^{(0,r]}$ . Then since $\widetilde{\mathrm{{\mathbf A}}}_{K,A}^{(0,r],+}$ is open in $\widetilde{\mathrm{{\mathbf A}}}_{K,A}^{(0,r]}$ , there exists an open subgroup $\Gamma_{L}$ of $\Gamma_{K}$ such that $\mathrm{Mat}(g)\in\mathrm{GL}_{d}(\widetilde{\mathrm{{\mathbf A}}}_{K,A}^{(0,r],+})$ for $g\in\Gamma_{L}$ . By possibly shrinking $\Gamma_{L}$ , we may assume it to be normal, and we may further assume for $g\in\Gamma_{L}$ we have $\mathrm{val}(\mathrm{Mat}(g)-1)>c_{3}$ . Let $\widetilde{M}^{(0,r],+}$ be the $\widetilde{\mathrm{{\mathbf A}}}_{K,A}^{(0,r],+}$ -span of this basis. It is a free $\widetilde{\mathrm{{\mathbf A}}}_{K,A}^{(0,r],+}=\widetilde{\mathrm{{\mathbf A}}}_{L,A}^{(0,r],+}$ -semilinearFootnote ⁵ representation of $\Gamma_{L}$ , which satisfies the assumptions in Theorem 4.11. Hence, there exists a unique finite free $\varphi^{-n}(\mathrm{{\mathbf A}}_{K,A}^{(0,r/p^{n}],+})$ -submodule $M_{n}^{(0,r],+}:=\mathrm{\mathbf{D}}_{n}^{+}(\widetilde{M}^{(0,r],+})$ of $M^{(0,r],+}$ , satisfying that the natural map

\[\widetilde{\mathrm{{\mathbf A}}}_{K,A}^{(0,r],+}\otimes_{\varphi^{-n}(\mathrm{{\mathbf A}}_{K,A}^{(0,r/p^{n}],+})}M_{n}^{(0,r],+}\rightarrow\widetilde{M}^{(0,r],+}\]

is an isomorphism, that $M_{n}^{(0,r],+}$ is $\Gamma_{L}$ -stable, and which has a $\varphi^{-n}(\mathrm{{\mathbf A}}_{K,A}^{(0,r/p^{n}],+})$ -basis which is $c_{3}$ -fixed for the action of $\Gamma_{L}$ .

Set $M_{n}^{(0,r]}=M_{n}^{(0,r],+}[1/T]$ . In order to finish the proof of existence of $M_{n}^{(0,r]}$ , the only part which is not yet clear is the following.

Indeed, choose coset representatives $\{g_{i}\}_{i\in I}$ for $\Gamma_{K}/\Gamma_{L}$ . For each such $g=g_{i}$ , consider $g(M_{n}^{(0,r],+})$ . If $e_{1},\ldots,e_{d}$ is the $c_{3}$ -fixed basis of $M_{n}^{(0,r],+}$ , then $g(e_{1}),\ldots,g(e_{d})$ is a basis of $g(M_{n}^{(0,r],+})$ . It may not be $c_{3}$ -fixed, however. By continuity, we may find a nontrivial $\gamma\in\Gamma_{L}$ , with $\mathrm{val}(\mathrm{Mat}_{\{g(e_{i})\}}(\gamma)-1))>c_{3}$ , and by taking n larger we can arrange that $n\geq n(\gamma)$ . Lemma 4.7 then implies that $g(M_{n}^{(0,r],+})=\mathrm{\mathbf{D}}_{n}^{+}(g(\widetilde{M}^{(0,r],+}))$ . Choosing n large enough for all the $g_{i}$ simultaneously, we may arrange that $g(M_{n}^{(0,r],+})=\mathrm{\mathbf{D}}_{n}^{+}(g(\widetilde{M}^{(0,r],+}))$ for every $g\in\Gamma_{K}$ .

With this in mind, let $g\in\Gamma_{K}$ . Then for some $t\in\mathbf{Z}$ , we have by continuity

\[g(\widetilde{M}^{(0,r],+})\subset T^{-t}\widetilde{M}^{(0,r],+},\]

so that

\[g(M_{n}^{(0,r],+})=\mathrm{\mathbf{D}}_{n}^{+}(g(\widetilde{M}^{(0,r],+}))\subset\mathrm{\mathbf{D}}_{n}^{+}(T^{-t}\widetilde{M}^{(0,r],+})=T^{-t}M_{n}^{(0,r],+}.\]

Every element of $M_{n}^{(0,r]}$ can be written in the form $T^{t}m$ with $m\in M_{n}^{(0,r],+}$ , and since

\[g(T^{t}m)=[g(T^{t})/T^{t}]T^{t}g(m)\in M_{n}^{(0,r]},\]

we see that $M_{n}^{(0,r]}$ is $\Gamma_{K}$ -stable. This proves the claim.

Finally, we show uniqueness. Suppose $M_{n}^{(0,r],(1)}$ and $M_{n}^{(0,r],(2)}$ are two submodules satisfying these properties. Let $M_{n}^{(0,r],(1),+}$ be the $\varphi^{-n}(\mathrm{{\mathbf A}}_{K,A}^{(0,r/p^{n}],+})$ -span of a $c_{3}$ -fixed basis in $M_{n}^{(0,r],(1)}$ . Let $\widetilde{M}^{(0,r],(1),+}$ be the image of $\widetilde{\mathrm{{\mathbf A}}}_{K,A}^{(0,r],+}\otimes_{\varphi^{-n}(\mathrm{{\mathbf A}}_{K,A}^{(0,r/p^{n}],+})}M_{n}^{(0,r],(1)}$ in $\widetilde{M}^{(0,r]}$ . Define $M_{n}^{(0,r],(2),+}$ and $\widetilde{M}^{(0,r],(2),+}$ similarly. Then we have for some sufficiently large t the inclusions

\[T^{t}\widetilde{M}^{(0,r],(2),+}\subset\widetilde{M}^{(0,r],(1),+}\subset T^{-t}\widetilde{M}^{(0,r],(2),+},\]

which implies upon applying $\mathrm{\mathbf{D}}_{n}^{+}$ that

\[T^{t}M_{n}^{(0,r],(2),+}\subset M_{n}^{(0,r],(1),+}\subset T^{-t}M_{n}^{(0,r],(2),+}.\]

Hence, $M_{n}^{(0,r],(1)}=M_{n}^{(0,r],(2)}$ .

5.3 The equivalence of categories

The following is an analogue of Lemma 3.9, with the action of $\varphi$ replaced by the $\Gamma_{K}$ action.

Proof. Choose a topological generator $\gamma$ of $\Gamma_{K}$ . Then by the same argument proving Lemma 3.9, we may find a finite free R-module F endowed with an isomorphism $\gamma^{*}F\cong F$ and which contains M as a direct summand as a semilinear representation of $\gamma^{\mathbf{Z}}$ . Namely, to construct F, choose first a free R-module G and a projective R-module P together with an R-module isomorphism $M\oplus P\cong G$ . Choose a basis $e_{1},\ldots,e_{d}$ of G, and give G the structure of a semilinear representation of $\gamma^{\mathbf{Z}}$ by setting $\gamma(\sum r_{i}e_{i})=\sum\gamma(r_{i})e_{i}$ . Then $G\oplus P$ also admits such a structure, by taking the composite

\[\begin{aligned}\gamma^{*}(G\oplus P)&\cong\gamma^{*}G\oplus\gamma^{*}P\cong G\oplus\gamma^{*}P\\&=M\oplus P\oplus\gamma^{*}P\cong\gamma^{*}M\oplus P\oplus\gamma^{*}P\cong\gamma^{*}G\oplus P\cong G\oplus P.\end{aligned}\]

We then take $F:=(G\oplus P)\oplus M$ . This is a free R-semilinear representation of $\gamma^{\mathbf{Z}}$ , with M being a direct summand.

It remains to explain how to extend the action on $G\oplus P$ to all of $\Gamma_{K}$ . Let $\pi_{M}:G\rightarrow M,\pi_{P}:G\rightarrow P$ be the projections and given $\gamma^{k}\in\gamma^{\mathbf{Z}}$ let $\gamma_{\{e_{i}\}}^{k}:G\rightarrow G$ be the action obtained by fixing the basis $e_{1},\ldots,e_{d}$ . Unraveling the definitions, one checks that the action of $\gamma^{k}$ on elements $(\sum x_{i}e_{i},p)\in G\oplus P$ is given by

\[\gamma^{k}\Big(\Big(\sum x_{i}e_{i},p\Big)\Big)=\Big(\gamma_{\{e_{i}\}}^{k}\Big(\pi_{M}\Big(\sum\gamma^{k}(x_{i})e_{i}\Big)+p\Big),\pi_{P}\Big(\sum\gamma^{k}(x_{i})e_{i}\Big)\Big).\]

Clearly, this formula can be extended to all elements of $\Gamma_{K}$ , as required.

The injectivity of $(M^{\dagger})^{\varphi,\Gamma_{K}}\rightarrow(\widetilde{M}^{\dagger})^{\varphi,\Gamma_{K}}$ is easy, the reason being that we already know that $\mathrm{{\mathbf A}}_{K,A}^{\dagger}\rightarrow\widetilde{\mathrm{{\mathbf A}}}_{K,A}^{\dagger}$ is injective, so it remains injective after tensoring with the projective, hence flat, $\mathrm{{\mathbf A}}_{K,A}^{\dagger}$ -module $M^{\dagger}$ . It then still remains injective after taking fixed points.

For the surjectivity we argue as follows. Let $x\in(\widetilde{M}^{\dagger})^{\varphi,\Gamma_{K}}$ . Then $x\in(\widetilde{M}^{(0,r]})^{\varphi,\Gamma_{K}}$ for some $r>0$ . Take a finite free $\mathrm{{\mathbf A}}_{K,A}^{\dagger}$ -semilinear $\Gamma_{K}$ -representation $F^{\dagger}$ which contains $M^{\dagger}$ as a direct summand as we may according to Lemma 5.4. Choose a basis $e_{1},\ldots,e_{d}$ of $F^{\dagger}$ . We can write $\sum a_{i}e_{i}=x$ with $a_{i}\in\widetilde{\mathrm{{\mathbf A}}}_{K,A}^{(0,r]}$ . Choose a nontorsion $\gamma\in\Gamma_{K}$ . By possibly making r smaller, we can arrange that $\mathrm{Mat}_{\{e_{i}\}}(\gamma)$ also has coefficients in $\widetilde{\mathrm{{\mathbf A}}}_{K,A}^{(0,r]}$ . Since x is fixed by $\gamma$ , we obtain the equation of $\widetilde{\mathrm{{\mathbf A}}}_{K,A}^{(0,r]}$ -valued matrices

\[\mathrm{Mat}_{\{e_{i}\}}(\gamma)\gamma(a)=a,\]

where a is the vector of the $a_{i}$ . Replacing $\gamma$ by $\gamma^{p^{k}}$ for $k\gg0$ we may arrange, in addition, that $\mathrm{val}(\mathrm{Mat}_{\{e_{i}\}}(\gamma)-1)>c_{3}$ . Thus, by Lemma 4.7 we know that for $n\gg0$ , we have $a_{i}\in\varphi^{-n}(\mathrm{{\mathbf A}}_{K,A}^{(0,r/p^{-n}]})$ , which is contained in $\varphi^{-n}(\mathrm{{\mathbf A}}_{K,A}^{\dagger})$ . Hence, $x\in\varphi^{-n}(\mathrm{{\mathbf A}}_{K,A}^{\dagger})\otimes_{\mathrm{{\mathbf A}}_{K,A}^{\dagger}}M^{\dagger}$ , and since x is fixed by $\varphi$ , we see after n successive applications of $\varphi$ that $x\in M^{\dagger}$ . This shows that $x\in M^{\dagger}\cap(\widetilde{M}^{\dagger})^{\varphi,\Gamma_{K}}=(M^{\dagger})^{\varphi,\Gamma_{K}}$ , as required.

Next, we prove essential surjectivity. Let $\widetilde{M}^{\dagger}$ be a projective étale $\varphi$ -module over $\widetilde{\mathrm{{\mathbf A}}}_{K,A}^{\dagger}$ . Let $\widetilde{M}^{(0,r]}$ be as in Proposition 5.2. Then $\widetilde{M}^{(0,r]}$ is a projective $\widetilde{\mathrm{{\mathbf A}}}_{K,A}^{(0,r]}$ -semilinear representation of $\Gamma_{K}$ , so by Lemma 5.4, we may find a free $\widetilde{\mathrm{{\mathbf A}}}_{K,A}^{(0,r]}$ -semilinear $\Gamma_{K}$ -representation $\widetilde{F}^{(0,r]}$ and a projective $\widetilde{\mathrm{{\mathbf A}}}_{K,A}^{(0,r]}$ -semilinear $\Gamma_{K}$ -representation $\widetilde{P}^{(0,r]}$ such that $\widetilde{M}^{(0,r]}\oplus\widetilde{P}^{(0,r]}=\widetilde{F}^{(0,r]}$ . By Proposition 5.3, we can find for $n\gg0$ a free $\varphi^{-n}(\mathrm{{\mathbf A}}_{K,A}^{(0,r/p^{n}]})$ -submodule $F_{n}^{(0,r]}\subset\widetilde{F}^{(0,r]}$ which is $\Gamma_{K}$ -stable.

To see this, choose a basis $e_{1},\ldots,e_{d}$ of $F_{n}^{(0,r]}$ . Choose $\gamma\in\Gamma_{K}$ nontorsion and let $\mathrm{Mat}(\gamma)$ be the matrix of $\gamma$ with respect to this basis. Since $F_{n}^{(0,r]}$ spans $\widetilde{F}^{(0,r]}$ as an $\widetilde{\mathrm{{\mathbf A}}}_{K,A}^{(0,r]}$ -module, there is a matrix $\mathrm{Mat}(\pi)\in\mathrm{M}_{d}(\widetilde{\mathrm{{\mathbf A}}}_{K,A}^{(0,r]})$ representing $\pi$ with respect to the basis $e_{i}$ . The relation $\pi\circ\gamma=\gamma\circ\pi$ gives

\[\mathrm{Mat}(\gamma)\gamma(\mathrm{Mat}(\pi))=\mathrm{Mat}(\pi)\,\mathrm{Mat}(\gamma),\]

and after replacing $\gamma$ by $\gamma^{p^{k}}$ for $k\gg0$ we may assume $\mathrm{val}(\mathrm{Mat}(\gamma)-1)>c_{3}$ . This implies by Lemma 4.7 that $\mathrm{Mat}(\pi)\in\mathrm{M}_{d}(\varphi^{-n}(\mathrm{{\mathbf A}}_{K,A}^{(0,r/p^{n}]}))$ , as required.

Set $M_{n}^{(0,r]}=F_{n}^{(0,r]}\cap\widetilde{M}^{(0,r]}$ and $P_{n}^{(0,r]}=F_{n}^{(0,r]}\cap\widetilde{P}^{(0,r]}$ . Then the claim shows that $M_{n}^{(0,r]}\oplus P_{n}^{(0,r]}=F_{n}^{(0,r]}$ . The isomorphism $\widetilde{\mathrm{{\mathbf A}}}_{K,A}^{(0,r]}\otimes_{\varphi^{-n}(\mathrm{{\mathbf A}}_{K,A}^{(0,r/p^{n}]})}F_{n}^{(0,r]}\xrightarrow{\sim}\widetilde{F}^{(0,r]}$ implies that $\widetilde{\mathrm{{\mathbf A}}}_{K,A}^{(0,r]}\otimes_{\varphi^{-n}(\mathrm{{\mathbf A}}_{K,A}^{(0,r/p^{n}]})}M_{n}^{(0,r]}\xrightarrow{\sim}\widetilde{M}^{(0,r]}$ and, hence, also

\[\widetilde{\mathrm{{\mathbf A}}}_{K,A}^{\dagger}\otimes_{\varphi^{-n}(\mathrm{{\mathbf A}}_{K,A}^{(0,r/p^{n}]})}M_{n}^{(0,r]}\xrightarrow{\sim}\widetilde{M}^{\dagger}.\]

It is also clear that $M_{n}^{(0,r]}$ is $\Gamma_{K}$ -stable. We set

\[M^{\dagger}:=\mathrm{{\mathbf A}}_{K,A}^{\dagger}\otimes_{\mathrm{{\mathbf A}}_{K,A}^{(0,r/p^{n}]}}\varphi^{n}(M_{n}^{(0,r]}),\]

then $M^{\dagger}$ is a $\Gamma_{K}$ -stable, projective $\mathrm{{\mathbf A}}_{K,A}^{\dagger}$ -submodule of $\widetilde{M}^{\dagger}$ , and the natural map $\widetilde{\mathrm{{\mathbf A}}}_{K,A}^{\dagger}\otimes_{\mathrm{{\mathbf A}}_{K,A}^{\dagger}}M^{\dagger}\xrightarrow{\sim}\widetilde{M}^{\dagger}$ is an isomorphism. It remains to show that $M^{\dagger}$ is $\varphi$ -stable and an étale $\varphi$ -module. To do this, simply note that the uniqueness of $F_{n}^{(0,r]}$ implies the uniqueness of $M_{n}^{(0,r]}$ , and so if n is sufficiently large so that $M_{n-1}^{(0,r/p]}$ and $M_{n}^{(0,r]}$ are both defined, we get

\[\varphi(M_{n}^{(0,r]})=M_{n-1}^{(0,r/p]}=\varphi^{-(n-1)}\big(\mathrm{{\mathbf A}}_{K,A}^{(0,r/p^{n-1}]}\big)\otimes_{\varphi^{-n}(\mathrm{{\mathbf A}}_{K,A}^{(0,r/p^{n}]})}M_{n}^{(0,r]},\]

which implies both that $M^{\dagger}$ is $\varphi$ -stable and that the action of $\varphi$ is invertible. This finishes the proof.