2.1 The ring $A_{\text{inf}}$

We denote the tilt of ${\mathcal{O}}_{C}$ by

as multiplicative monoids (see [Reference ScholzeSch12, 3.4(i)]). We regard $p^{1/p^{\infty }}$ fixed in §1.5 as an element of ${\mathcal{O}}_{C}^{\flat }$ . Due to the fixed embedding $p^{\mathbb{Q}_{{\geqslant}0}}\subset {\mathcal{O}}_{C}$ , this element comes equipped with well-defined powers $(p^{1/p^{\infty }})^{q}\in {\mathcal{O}}_{C}^{\flat }$ for $q\in \mathbb{Q}_{{\geqslant}0}$ . For each $x\in {\mathcal{O}}_{C}^{\flat }$ , we let $(\ldots ,x^{(1)},x^{(0)})$ denote its preimage in $\mathop{\varprojlim }\nolimits_{y\mapsto y^{p}}{\mathcal{O}}_{C}$ . The map $x\mapsto \text{val}_{{\mathcal{O}}_{C}}(x^{(0)})$ makes ${\mathcal{O}}_{C}^{\flat }$ a complete valuation ring of height $1$ whose fraction field $C^{\flat }:=\operatorname{Frac}({\mathcal{O}}_{C}^{\flat })$ is algebraically closed (see [Reference ScholzeSch12, 3.4(iii), 3.7(ii)]). We let $\mathfrak{m}^{\flat }$ denote the maximal ideal of ${\mathcal{O}}_{C}^{\flat }$ .

The basic period ring $A_{\text{inf}}$ of Fontaine is defined by

We equip the local domain $A_{\text{inf}}$ with the product of the valuation topologies via the Witt coordinate bijection $W({\mathcal{O}}_{C}^{\flat })\cong \prod _{n=1}^{\infty }{\mathcal{O}}_{C}^{\flat }$ . Then $A_{\text{inf}}$ is complete and its topology agrees with the $(p,[x])$ -adic topology for any nonzero nonunit $x\in {\mathcal{O}}_{C}^{\flat }$ . We fix (once and for all) a compatible system $\unicode[STIX]{x1D716}=(\ldots ,\unicode[STIX]{x1D701}_{p^{2}},\unicode[STIX]{x1D701}_{p},1)$ of $p$ -power roots of unity in ${\mathcal{O}}_{C}$ , so that $\unicode[STIX]{x1D716}\in {\mathcal{O}}_{C}^{\flat }$ , and set

(2.1.1) $$\begin{eqnarray}\unicode[STIX]{x1D707}:=[\unicode[STIX]{x1D716}]-1\in A_{\text{inf}}.\end{eqnarray}$$

Since $(p,\unicode[STIX]{x1D707})=(p,[\unicode[STIX]{x1D716}-1])$ , the topology of $A_{\text{inf}}$ is $(p,\unicode[STIX]{x1D707})$ -adic. By forming the limit of the sequences

(2.1.2) $$\begin{eqnarray}0\rightarrow W_{n}({\mathcal{O}}_{C}^{\flat })\xrightarrow[{}]{\unicode[STIX]{x1D707}}W_{n}({\mathcal{O}}_{C}^{\flat })\rightarrow W_{n}({\mathcal{O}}_{C}^{\flat })/\unicode[STIX]{x1D707}\rightarrow 0,\end{eqnarray}$$

we see that $A_{\text{inf}}/\unicode[STIX]{x1D707}$ is $p$ -adically complete and that the ideal $(\unicode[STIX]{x1D707})\subset A_{\text{inf}}$ does not depend on the choice of $\unicode[STIX]{x1D716}$ (use the fact that the valuation of $\unicode[STIX]{x1D701}_{p}-1$ does not depend on $\unicode[STIX]{x1D701}_{p}$ ).

The assignment $[x]\mapsto x^{(0)}$ extends uniquely to a ring homomorphism

(2.1.3)

which is surjective, as indicated, and intertwines the Frobenius $\unicode[STIX]{x1D711}$ of $A_{\text{inf}}$ with the absolute Frobenius of ${\mathcal{O}}_{C}/p$ . Its kernel $\operatorname{Ker}(\unicode[STIX]{x1D703})\subset A_{\text{inf}}$ is principal and generated by the element

(2.1.4) $$\begin{eqnarray}\textstyle \unicode[STIX]{x1D709}:=\mathop{\sum }_{i=0}^{p-1}[\unicode[STIX]{x1D716}^{i/p}]\end{eqnarray}$$

(see [Reference Bhatt, Morrow and ScholzeBMS18, 3.16]). Analogues of the sequences (2.1.2) show that each $A_{\text{inf}}/\unicode[STIX]{x1D709}^{n}$ is $p$ -adically complete. In fact, the map $\unicode[STIX]{x1D703}$ identifies $A_{\text{inf}}/\unicode[STIX]{x1D709}^{n}$ with the initial $p$ -adically complete infinitesimal thickening of ${\mathcal{O}}_{C}$ of order $n-1$ , see [Reference Szamuely and ZábrádiSZ18, 3.13]. The composition

and its kernel is generated by the element $\unicode[STIX]{x1D711}(\unicode[STIX]{x1D709})=\sum _{i=0}^{p-1}[\unicode[STIX]{x1D716}^{i}]$ .

Due to the nature of our $C$ (see §1.5), the ring ${\mathcal{O}}_{C}/p$ is a $k$ -algebra, so $A_{\text{inf}}$ is a $W(k)$ -algebra.

2.2 The object $A\unicode[STIX]{x1D6FA}_{\mathfrak{X}}$

The operations that define ${\mathcal{O}}_{C}^{\flat }$ and $A_{\text{inf}}$ make sense on the proétale site $(\mathfrak{X}_{C}^{\text{ad}})_{\text{pro}\acute{\text{e}}\text{t}}$ : namely, as in [Reference ScholzeSch13a, 4.1, 5.10, and 6.1], we have the integral completed structure sheaf

(2.2.1) $$\begin{eqnarray}\textstyle \widehat{{\mathcal{O}}}_{\mathfrak{ X}_{C}^{\text{ad}}}^{+}:=\mathop{\varprojlim }\nolimits_{n}({\mathcal{O}}_{\mathfrak{X}_{C}^{\text{ad}},\,\text{pro}\acute{\text{e}}\text{t}}^{+}/p^{n}),\quad \text{its tilt}\quad \widehat{{\mathcal{O}}}_{\mathfrak{X}_{C}^{\text{ad}}}^{+,\,\flat }:=\mathop{\varprojlim }\nolimits_{y\mapsto y^{p}}({\mathcal{O}}_{\mathfrak{X}_{C}^{\text{ad}},\,\text{pro}\acute{\text{e}}\text{t}}^{+}/p),\end{eqnarray}$$

and the basic period sheaf

$$\begin{eqnarray}\mathbb{A}_{ inf,\,\mathfrak{X}_{C}^{\text{ad}}}:=W(\widehat{{\mathcal{O}}}_{\mathfrak{ X}_{C}^{\text{ad}}}^{+,\,\flat }).\end{eqnarray}$$

For brevity, we often denote these sheaves simply by $\widehat{{\mathcal{O}}}^{+}$ , $\widehat{{\mathcal{O}}}^{+,\,\flat }$ , and $\mathbb{A}_{\text{inf}}$ . Affinoid perfectoids form a basis for $(\mathfrak{X}_{C}^{\text{ad}})_{\text{pro}\acute{\text{e}}\text{t}}$ (see [Reference ScholzeSch13a, 4.7]) and the construction of the map $\unicode[STIX]{x1D703}$ of (2.1.3) makes sense for any perfectoid ${\mathcal{O}}_{C}$ -algebra (see [Reference Bhatt, Morrow and ScholzeBMS18, §3]). In particular, $\mathbb{A}_{\text{inf},\,\mathfrak{X}_{C}^{\text{ad}}}$ comes equipped with the map

(2.2.2) $$\begin{eqnarray}\textstyle \unicode[STIX]{x1D703}_{\mathfrak{X}_{C}^{\text{ad}}}:\mathbb{A}_{\text{inf},\,\mathfrak{X}_{C}^{\text{ad}}}\rightarrow \widehat{{\mathcal{O}}}_{\mathfrak{ X}_{C}^{\text{ad}}}^{+},\end{eqnarray}$$

which, by construction, is compatible with the map $\unicode[STIX]{x1D703}:A_{\text{inf}}\rightarrow {\mathcal{O}}_{C}$ , intertwines the Witt vector Frobenius $\unicode[STIX]{x1D711}$ of $\mathbb{A}_{ inf,\,\mathfrak{X}_{C}^{\text{ad}}}$ with the absolute Frobenius of $\widehat{{\mathcal{O}}}_{\mathfrak{ X}_{C}^{\text{ad}}}^{+}/p$ , and, by [Reference ScholzeSch13a, 6.3 and 6.5], is surjective with $\operatorname{Ker}(\unicode[STIX]{x1D703}_{\mathfrak{ X}_{C}^{\text{ad}}})=\unicode[STIX]{x1D709}\,\cdot \,\mathbb{A}_{\text{inf},\,\mathfrak{X}_{C}^{\text{ad}}}$ (in addition, $\unicode[STIX]{x1D709}$ is not a zero divisor in $\mathbb{A}_{\text{inf},\,\mathfrak{X}_{C}^{\text{ad}}}$ ).

The key object that we are going to study in this paper is

(2.2.3) $$\begin{eqnarray}A\unicode[STIX]{x1D6FA}_{\mathfrak{X}}:=L\unicode[STIX]{x1D702}_{(\unicode[STIX]{x1D707})}(R\unicode[STIX]{x1D708}_{\ast }(\mathbb{A}_{\text{inf},\,\mathfrak{X}_{C}^{\text{ad}}}))\in D^{{\geqslant}0}(\mathfrak{X}_{\acute{\text{e}}\text{t}},A_{\text{inf}}),\end{eqnarray}$$

where the décalage functor $L\unicode[STIX]{x1D702}$ of [Reference Bhatt, Morrow and ScholzeBMS18, §6] is formed with respect to the ideal $(\unicode[STIX]{x1D707})$ of the constant sheaf $A_{\text{inf}}$ of $\mathfrak{X}_{\acute{\text{e}}\text{t}}$ (the definition of $L\unicode[STIX]{x1D702}_{(\unicode[STIX]{x1D707})}$ builds on the formula (1.7.3) for $\unicode[STIX]{x1D702}_{(\unicode[STIX]{x1D707})}$ ). The formula (2.2.3) may also be executed with the Zariski site $\mathfrak{X}_{\text{Zar}}$ as the target of $\unicode[STIX]{x1D708}$ , and it then defines the object

(2.2.4) $$\begin{eqnarray}A\unicode[STIX]{x1D6FA}_{\mathfrak{X}_{\text{Zar}}}\in D^{{\geqslant}0}(\mathfrak{X}_{\text{Zar}},A_{\text{inf}}),\end{eqnarray}$$

which is the $A\unicode[STIX]{x1D6FA}_{\mathfrak{X}}$ that was used in [Reference Bhatt, Morrow and ScholzeBMS18]. We will only use $A\unicode[STIX]{x1D6FA}_{\mathfrak{X}_{\text{Zar}}}$ in Corollary 4.21 (and in some results that lead to it) for comparison with $A\unicode[STIX]{x1D6FA}_{\mathfrak{X}}$ .

Since $\unicode[STIX]{x1D711}(\unicode[STIX]{x1D707})=\unicode[STIX]{x1D711}(\unicode[STIX]{x1D709})\unicode[STIX]{x1D707}$ , by [Reference Bhatt, Morrow and ScholzeBMS18, 6.11], we have $L\unicode[STIX]{x1D702}_{(\unicode[STIX]{x1D711}(\unicode[STIX]{x1D707}))}\cong L\unicode[STIX]{x1D702}_{(\unicode[STIX]{x1D711}(\unicode[STIX]{x1D709}))}\circ L\unicode[STIX]{x1D702}_{(\unicode[STIX]{x1D707})}$ , so the Frobenius automorphism of $\mathbb{A}_{\text{inf},\,\mathfrak{X}_{C}^{\text{ad}}}$ gives the Frobenius morphism

(2.2.5)

which, by [Reference Bhatt, Morrow and ScholzeBMS18, 6.14], induces an isomorphism

(2.2.6) $$\begin{eqnarray}\textstyle (A\unicode[STIX]{x1D6FA}_{\mathfrak{X}}\otimes _{A_{\text{inf}},\,\unicode[STIX]{x1D711}}^{\mathbb{L}}A_{\text{inf}})\textstyle [\frac{1}{\unicode[STIX]{x1D711}(\unicode[STIX]{x1D709})}]\overset{{\sim}}{\longrightarrow }(A\unicode[STIX]{x1D6FA}_{\mathfrak{X}})[\frac{1}{\unicode[STIX]{x1D711}(\unicode[STIX]{x1D709})}].\end{eqnarray}$$

In addition, by loc. cit., we also have

(2.2.7) $$\begin{eqnarray}\textstyle \textstyle A\unicode[STIX]{x1D6FA}_{\mathfrak{X}}\otimes _{A_{\text{inf}}}^{\mathbb{L}}A_{\text{inf}}[\frac{1}{\unicode[STIX]{x1D707}}]\cong (R\unicode[STIX]{x1D708}_{\ast }(\mathbb{A}_{\text{inf},\,\mathfrak{X}_{C}^{\text{ad}}}))\otimes _{A_{\text{inf}}}^{\mathbb{L}}A_{\text{inf}}[\frac{1}{\unicode[STIX]{x1D707}}],\end{eqnarray}$$

so a result of Scholze [Reference Bhatt, Morrow and ScholzeBMS18, 5.6] supplies the following relation to integral $p$ -adic étale cohomology.

Theorem 2.3. If $\mathfrak{X}$ is proper over ${\mathcal{O}}_{C}$ , then there is an identification

(2.3.1) $$\begin{eqnarray}\textstyle \textstyle R\unicode[STIX]{x1D6E4}(\mathfrak{X}_{\acute{\text{e}}\text{t}},A\unicode[STIX]{x1D6FA}_{\mathfrak{X}})\otimes _{A_{\text{inf}}}^{\mathbb{L}}A_{\text{inf}}[\frac{1}{\unicode[STIX]{x1D707}}]\cong R\unicode[STIX]{x1D6E4}_{\acute{\text{e}}\text{t}}(\mathfrak{X}_{C}^{\text{ad}},\mathbb{Z}_{p})\otimes _{\mathbb{Z}_{p}}^{\mathbb{ L}}A_{\text{inf}}[\frac{1}{\unicode[STIX]{x1D707}}].\end{eqnarray}$$

In broad strokes, the proof of Theorem 2.3 given in [Reference Bhatt, Morrow and ScholzeBMS18, 5.6] goes as follows: one considers the map

(2.3.2) $$\begin{eqnarray}R\unicode[STIX]{x1D6E4}_{\acute{\text{e}}\text{t}}(\mathfrak{X}_{C}^{\text{ad}},\mathbb{Z}_{p})\otimes _{\mathbb{Z}_{p}}^{\mathbb{ L}}A_{\text{inf}}\cong R\unicode[STIX]{x1D6E4}_{\text{pro}\acute{\text{e}}\text{t}}(\mathfrak{X}_{C}^{\text{ad}},\mathbb{Z}_{p})\otimes _{\mathbb{Z}_{p}}^{\mathbb{ L}}A_{\text{inf}}\rightarrow R\unicode[STIX]{x1D6E4}_{\text{pro}\acute{\text{e}}\text{t}}(\mathfrak{X}_{C}^{\text{ad}},\mathbb{A}_{\text{inf},\,\mathfrak{X}_{C}^{\text{ad}}})\end{eqnarray}$$

induced by the inclusion $A_{\text{inf}}{\hookrightarrow}\mathbb{A}_{\text{inf},\,\mathfrak{X}_{C}^{\text{ad}}}$ and deduces from the almost purity theorem with, for instance, Lemma 3.17 below that the ideal

(2.3.3) $$\begin{eqnarray}W(\mathfrak{m}^{\flat }):=\operatorname{Ker}(W({\mathcal{O}}_{C}^{\flat }){\twoheadrightarrow}W(k))\quad \text{of}\quad A_{\text{inf}}\end{eqnarray}$$

kills the cohomology of its cone. Since $\unicode[STIX]{x1D707}$ lies in $W(\mathfrak{m}^{\flat })$ and we have the identification (2.2.7), it follows that the map (2.3.2) induces the identification (2.3.1).

Remark 2.4. In practice, $\mathfrak{X}$ often arises as the formal $p$ -adic completion of a proper, finitely presented ${\mathcal{O}}_{C}$ -scheme ${\mathcal{X}}$ . In this situation, $\mathfrak{X}_{C}^{\text{ad}}$ agrees with the adic space associated to ${\mathcal{X}}_{C}$ (see [Reference ConradCon99, 5.3.1 4.], [Reference HuberHub94, 4.6(i)], and [Reference HuberHub96, 1.9.2 ii)]) and, by [Reference HuberHub96, 3.7.2], we have

$$\begin{eqnarray}R\unicode[STIX]{x1D6E4}_{\acute{\text{e}}\text{t}}(\mathfrak{X}_{C}^{\text{ad}},\mathbb{Z}_{p})\cong R\unicode[STIX]{x1D6E4}_{\acute{\text{e}}\text{t}}({\mathcal{X}}_{C},\mathbb{Z}_{p}).\end{eqnarray}$$

3.1 The local setup

We assume throughout §3 that $\mathfrak{X}=\operatorname{Spf}(R)$ and for some $d\geqslant 0$ , some $0\leqslant r\leqslant d$ , and some $q\in \mathbb{Q}_{{>}0}$ , there is an étale $\operatorname{Spf}({\mathcal{O}}_{C})$ -morphism as in (1.5.1):

(3.1.1) $$\begin{eqnarray}\hspace{-3.00003pt}\mathfrak{X}=\operatorname{Spf}(R)\rightarrow \operatorname{Spf}(R^{\Box })=:\mathfrak{X}^{\Box }\quad \!\!\text{with}\!\!\quad R^{\Box }:={\mathcal{O}}_{C}\{t_{0},\ldots ,t_{r},t_{r+1}^{\pm 1},\ldots ,t_{d}^{\pm 1}\}/(t_{0}\cdots t_{r}-p^{q}).\end{eqnarray}$$

Due to our assumptions from §1.5, a general $\mathfrak{X}$ is of this form on a basis for its étale topology.

3.2 The perfectoid cover $\mathfrak{X}_{C,\,\infty }^{\text{ad}}$

For each $m\geqslant 0$ , we consider the $R^{\Box }$ -algebra

$$\begin{eqnarray}R_{m}^{\Box }:={\mathcal{O}}_{C}\{t_{0}^{1/p^{m}},\ldots ,t_{r}^{1/p^{m}},t_{r+1}^{\pm 1/p^{m}},\ldots ,t_{d}^{\pm 1/p^{m}}\}/(t_{0}^{1/p^{m}}\cdots t_{r}^{1/p^{m}}-p^{q/p^{m}}),\quad \text{and}\quad R_{\infty }^{\Box }:=(\varinjlim R_{m}^{\Box })\widehat{~},\end{eqnarray}$$

where, as always unless mentioned otherwise (see §1.7), the completion is $p$ -adic. Explicitly, we have the $p$ -adically completed direct sum decomposition

(3.2.1)

which shows that $R_{\infty }^{\Box }$ is perfectoid (see §1.7) and that, for each $m\geqslant 0$ , the map $R_{m}^{\Box }\rightarrow R_{\infty }^{\Box }$ is an inclusion of an $R_{m}^{\Box }$ -module direct summand composed of those summands ${\mathcal{O}}_{C}\,\cdot \,t_{0}^{a_{0}}\cdots t_{d}^{a_{d}}$ of (3.2.1) for which $p^{m}a_{j}\in \mathbb{Z}$ for all $j$ .

The corresponding $R$ -algebras are

$$\begin{eqnarray}R_{m}:=R\otimes _{R^{\Box }}R_{m}^{\Box }\quad \text{and}\quad R_{\infty }:=(\varinjlim R_{m})\widehat{~}\,\cong (R\otimes _{R^{\Box }}R_{\infty }^{\Box })\widehat{\,~}.\end{eqnarray}$$

Each $R_{m}$ (respectively,  $R_{\infty }$ ) is a $p$ -torsion-free $p$ -adically formally étale $R_{m}^{\Box }$ -algebra (respectively, $R_{\infty }^{\Box }$ -algebra). In particular, $R_{\infty }$ is perfectoid (see §1.7). By [Reference Gabber and RameroGR03, 7.1.6(ii)], each $R_{m}$ is $p$ -adically complete.

The summands in (3.2.1) with $a_{j}\not \in \mathbb{Z}$ for some $0\leqslant j\leqslant d$ comprise an $R^{\Box }$ -submodule $M_{\infty }^{\Box }$ of $R_{\infty }^{\Box }$ , and we set $M_{\infty }:=R\,\widehat{\otimes }_{R^{\Box }}\,M_{\infty }^{\Box }$ . Thus, we have the $R^{\Box }$ -module (respectively,  $R$ -module) decomposition

(3.2.2) $$\begin{eqnarray}R_{\infty }^{\Box }\cong R^{\Box }\oplus M_{\infty }^{\Box }\quad (\text{respectively,}\quad R_{\infty }\cong R\oplus M_{\infty }).\end{eqnarray}$$

The profinite group

acts $R^{\Box }$ -linearly on $R_{m}^{\Box }$ by scaling each $t_{j}^{1/p^{m}}$ by the $\unicode[STIX]{x1D707}_{p^{m}}$ -component of $\unicode[STIX]{x1D716}_{j}$ . The induced actions of $\unicode[STIX]{x1D6E5}$ on $R_{\infty }^{\Box }$ and $R_{\infty }$ are continuous, compatible, and preserve the decompositions (3.2.1) and (3.2.2). In terms of the element $\unicode[STIX]{x1D716}$ fixed in §2.1, $\unicode[STIX]{x1D6E5}$ is topologically freely generated by the following $d$ elements:

After inverting $p$ , for each $m\geqslant 0$ , we have

$$\begin{eqnarray}\textstyle \textstyle R_{m}^{\Box }[\frac{1}{p}]\cong \bigoplus _{a_{1},\ldots ,a_{d}\in \{0,\frac{1}{p^{m}},\ldots ,\frac{p^{m}-1}{p^{m}}\}}R^{\Box }[\frac{1}{p}]\,\cdot \,t_{1}^{a_{1}}\cdots t_{d}^{a_{d}},\end{eqnarray}$$

so $R_{m}^{\Box }[\frac{1}{p}]$ is the $R^{\Box }[\frac{1}{p}]$ -algebra obtained by adjoining the $(p^{m})^{\text{th}}$ roots of $t_{1},\ldots ,t_{d}\in (R^{\Box }[\frac{1}{p}])^{\times }$ , and hence is finite étale over $R^{\Box }[\frac{1}{p}]$ . Therefore, $\mathop{\varinjlim }\nolimits_{m}(R_{m}^{\Box }[\frac{1}{p}])$ is a pro-(finite étale) $\unicode[STIX]{x1D6E5}$ -cover of $R^{\Box }[\frac{1}{p}]$ . The explicit description (3.2.1) implies that $R_{m}^{\Box }=(R_{m}^{\Box }[\frac{1}{p}])^{\circ }$ , so the pro-object

$$\begin{eqnarray}\textstyle \textstyle (\mathfrak{X}^{\Box })_{C,\,\infty }^{\text{ad}}:=\varprojlim \operatorname{Spa}(R_{m}^{\Box }[\frac{1}{p}],R_{m}^{\Box })\quad \text{which determines the perfectoid space}\quad \operatorname{Spa}(R_{\infty }^{\Box }[\frac{1}{p}],R_{\infty }^{\Box }),\end{eqnarray}$$

is an affinoid perfectoid pro-(finite étale) $\unicode[STIX]{x1D6E5}$ -cover of the adic generic fiber $(\mathfrak{X}^{\Box })_{C}^{\text{ad}}$ of $\operatorname{Spf}(R^{\Box })$ ; in particular, $(\mathfrak{X}^{\Box })_{C,\,\infty }^{\text{ad}}$ is an affinoid perfectoid object of the proétale site $((\mathfrak{X}^{\Box })_{C}^{\text{ad}})_{\text{pro}\acute{\text{e}}\text{t}}$ . Consequently, the $\mathfrak{X}_{C}^{\text{ad}}$ -base change of $(\mathfrak{X}^{\Box })_{C,\,\infty }^{\text{ad}}$ , namely, the tower

$$\begin{eqnarray}\textstyle \textstyle \mathfrak{X}_{C,\,\infty }^{\text{ad}}:=\varprojlim \operatorname{Spa}(R_{m}[\frac{1}{p}],R_{m})\quad \text{which determines the perfectoid space}\quad \operatorname{Spa}(R_{\infty }[\frac{1}{p}],R_{\infty }),\end{eqnarray}$$

is an affinoid perfectoid pro-(finite étale) $\unicode[STIX]{x1D6E5}$ -cover of $\mathfrak{X}_{C}^{\text{ad}}$ , so, in particular, is an affinoid perfectoid object of $(\mathfrak{X}_{C}^{\text{ad}})_{\text{pro}\acute{\text{e}}\text{t}}$ .

By [Reference ScholzeSch13a, 4.10(iii)], the value on $\mathfrak{X}_{C,\,\infty }^{\text{ad}}$ of the sheaf $\widehat{{\mathcal{O}}}_{\mathfrak{ X}_{C}^{\text{ad}}}^{+}$ reviewed in (2.2.1) is $R_{\infty }$ .

3.3 The cohomology of $\widehat{{\mathcal{O}}}^{+}$ and continuous group cohomology

By [Reference ScholzeSch13a, 3.5, 3.7(iii) and its proof, 6.6] (see also [Reference ScholzeSch16]), the Čech complex of the sheaf $\widehat{{\mathcal{O}}}_{\mathfrak{ X}_{C}^{\text{ad}}}^{+}$ with respect to the pro-(finite étale) affinoid perfectoid cover

$$\begin{eqnarray}\mathfrak{X}_{C,\,\infty }^{\text{ad}}{\twoheadrightarrow}\mathfrak{X}_{C}^{\text{ad}}\end{eqnarray}$$

is identified with the continuous cochain complex $R\unicode[STIX]{x1D6E4}_{\text{cont}}(\unicode[STIX]{x1D6E5},R_{\infty })$ . In particular, by using [SP, 01GY], we obtain the edge map to the proétale cohomology of $\widehat{{\mathcal{O}}}_{\mathfrak{ X}_{C}^{\text{ad}}}^{+}$ :

(3.3.1) $$\begin{eqnarray}e:R\unicode[STIX]{x1D6E4}_{\text{cont}}(\unicode[STIX]{x1D6E5},R_{\infty })\rightarrow R\unicode[STIX]{x1D6E4}_{\text{pro}\acute{\text{e}}\text{t}}(\mathfrak{X}_{C}^{\text{ad}},\widehat{{\mathcal{O}}}^{+}),\end{eqnarray}$$

which on the level of cohomology is described by the Cartan–Leray spectral sequence (see [SP, 01GY] or [Reference Artin, Grothendieck and VerdierSGA 4_II , V.3.3]). By the almost purity theorem [Reference ScholzeSch13a, 4.10 (v)], the maximal ideal $\mathfrak{m}\subset {\mathcal{O}}_{C}$ kills the cohomology groups of $\operatorname{Cone}(e)$ .

We will show in Theorem 3.9 that $L\unicode[STIX]{x1D702}_{(\unicode[STIX]{x1D701}_{p}-1)}(e)$ is an isomorphism, so that

$$\begin{eqnarray}L\unicode[STIX]{x1D702}_{(\unicode[STIX]{x1D701}_{p}-1)}(R\unicode[STIX]{x1D6E4}_{\text{pro}\acute{\text{e}}\text{t}}(\mathfrak{X}_{C}^{\text{ad}},\widehat{{\mathcal{O}}}^{+}))\end{eqnarray}$$

is computed in terms of continuous group cohomology. For this, we will use the following lemma.

Lemma 3.4 [Reference Bhatt, Morrow and ScholzeBMS18, 8.11(i)].

An ${\mathcal{O}}_{C}$ -module map $f:M\rightarrow M^{\prime }$ with

$$\begin{eqnarray}\textstyle M[\mathfrak{m}]=(\frac{M}{(\unicode[STIX]{x1D701}_{p}-1)M})[\mathfrak{m}]=0\end{eqnarray}$$

and both $\operatorname{Ker}f$ and $\operatorname{Coker}f$ killed by $\mathfrak{m}$ induces an isomorphism

$$\begin{eqnarray}\textstyle \frac{M}{M[\unicode[STIX]{x1D701}_{p}-1]}\overset{{\sim}}{\longrightarrow }\frac{M^{\prime }}{M^{\prime }[\unicode[STIX]{x1D701}_{p}-1]}.\end{eqnarray}$$

In order to apply Lemma 3.4, we will check in Proposition 3.8 that the cohomology modules $H_{\text{cont}}^{i}(\unicode[STIX]{x1D6E5},R_{\infty })$ have no nonzero $\mathfrak{m}$ -torsion. This will use the following general lemmas.

Lemma 3.6. Fix an $i\in \mathbb{Z}_{{\geqslant}0}$ , let $H$ be a profinite group, let $\{M_{j}\}\text{}_{j\in J}$ be $p$ -adically complete, $p$ -torsion-free, continuous $H$ -modules, and suppose that either:

1. (i) the group $H_{\text{cont}}^{i}(H,M_{j})$ is $p$ -torsion free for every $j$ ; or

2. (ii) some $p^{n}$ kills $H_{\text{cont}}^{i}(H,M_{j})$ for every $j$ .

Then the following map is injective:

In particular, in the case (i) (respectively, (ii)), $H_{\text{cont}}^{i}(H,\widehat{\bigoplus }_{j\in J}M_{j})$ is $p$ -torsion free (respectively, killed by $p^{n}$ ).

Lemma 3.7 [Reference Bhatt, Morrow and ScholzeBMS18, 7.3(ii)].

Let $H$ be a profinite group isomorphic to $\mathbb{Z}_{p}^{\oplus d}$ for some $d>0$ , and let $M\cong \mathop{\varprojlim }\nolimits_{n\geqslant 1}M_{n}$ be a continuous $H$ -module with each $M_{n}$ a discrete, $p^{n}$ -torsion, continuous $H$ -module. For any $\unicode[STIX]{x1D6FE}_{1},\ldots ,\unicode[STIX]{x1D6FE}_{d}\in H$ that topologically freely generate $H$ , there is a natural identification

$$\begin{eqnarray}R\unicode[STIX]{x1D6E4}_{\text{cont}}(H,M)\cong K_{M}(\unicode[STIX]{x1D6FE}_{1}-1,\ldots ,\unicode[STIX]{x1D6FE}_{d}-1),\quad \text{so also}\quad H_{\text{cont}}^{j}(H,M)\cong H^{j}(K_{M}(\unicode[STIX]{x1D6FE}_{1}-1,\ldots ,\unicode[STIX]{x1D6FE}_{d}-1)),\end{eqnarray}$$

in the derived category (see §1.7 for the notation).◻

Proof. Let $S:={\mathcal{O}}_{C}\,\cdot \,t_{0}^{a_{0}}\cdots t_{d}^{a_{d}}$ be a summand of (3.2.1). By Lemma 3.7, the ${\mathcal{O}}_{C}$ -module $H_{\text{cont}}^{i}(\unicode[STIX]{x1D6E5},S)$ is the $i$ th cohomology of the ${\mathcal{O}}_{C}$ -tensor product of $d$ complexes of the form ${\mathcal{O}}_{C}\xrightarrow[{}]{\unicode[STIX]{x1D701}-1}{\mathcal{O}}_{C}$ for suitable $p$ -power roots of unity $\unicode[STIX]{x1D701}$ . Moreover, since the $d$ complexes may be defined over some discrete valuation subring of ${\mathcal{O}}_{C}$ , Lemma 3.5 ensures that

(3.8.1)

If $S$ contributes to $M_{\infty }$ , that is, if $a_{j}\not \in \mathbb{Z}$ for some $j$ , then some $\unicode[STIX]{x1D701}$ is not $1$ , and the corresponding factor complex is quasi-isomorphic to ${\mathcal{O}}_{C}/(\unicode[STIX]{x1D701}-1)$ . Thus, in this case,

(3.8.2) $$\begin{eqnarray}\unicode[STIX]{x1D701}-1\quad \text{and hence also}\quad \unicode[STIX]{x1D701}_{p}-1,\quad \text{kills}\quad H_{\text{cont}}^{i}(\unicode[STIX]{x1D6E5},S).\end{eqnarray}$$

For $m>0$ , let $M_{m}^{\Box }$ denote the $p$ -adically completed direct sum of those summands ${\mathcal{O}}_{C}\,\cdot \,t_{0}^{a_{0}}\cdots t_{d}^{a_{d}}$ of (3.2.1) for which $m$ is the smallest nonnegative integer with $p^{m}\,\cdot \,(a_{0},\ldots ,a_{d})\in \mathbb{Z}^{\oplus (d+1)}$ . Lemma 3.6 and (3.8.1)–(3.8.2) imply that the ${\mathcal{O}}_{C}$ -module

(3.8.3)

Since $R$ is $R^{\Box }$ -flat and $R\otimes _{R^{\Box }}M_{m}^{\Box }$ is $p$ -adically complete (see §§1.5 and 3.2), Lemma 3.7 gives

(3.8.4) $$\begin{eqnarray}H_{\text{cont}}^{i}(\unicode[STIX]{x1D6E5},R\otimes _{R^{\Box }}M_{m}^{\Box })\cong R\otimes _{R^{\Box }}H_{\text{cont}}^{i}(\unicode[STIX]{x1D6E5},M_{m}^{\Box }).\end{eqnarray}$$

Since $M_{\infty }\cong \widehat{\bigoplus }_{m}(R\otimes _{R^{\Box }}M_{m}^{\Box })$ , equations (3.8.3)–(3.8.4) and Lemma 3.6 imply that $\unicode[STIX]{x1D701}_{p}-1$ kills $H_{\text{cont}}^{i}(\unicode[STIX]{x1D6E5},M_{\infty })$ .

Since $R_{\infty }/b$ is $p$ -adically complete and each of the summands of the decomposition

may be defined over a suitably large discrete valuation subring of ${\mathcal{O}}_{C}$ , Lemma 3.5 ensures that $R_{\infty }/b$ has no nonzero $\mathfrak{m}$ -torsion. In addition, the $\unicode[STIX]{x1D6E5}$ -action on each summand may be defined over a possibly larger such subring, so, by Lemmas 3.5 and 3.7, in the case $b\neq 0$ each

This conclusion extends to the case $b=0$ because the $(\unicode[STIX]{x1D701}_{p}-1)$ -annihilation of $H_{\text{cont}}^{i}(\unicode[STIX]{x1D6E5},M_{\infty })$ supplies the injection $H_{\text{cont}}^{i}(\unicode[STIX]{x1D6E5},M_{\infty }){\hookrightarrow}H_{\text{cont}}^{i}(\unicode[STIX]{x1D6E5},M_{\infty }/(\unicode[STIX]{x1D701}_{p}-1))$ . It remains to observe that the ${\mathcal{O}}_{C}$ -module $H_{\text{cont}}^{i}(\unicode[STIX]{x1D6E5},R/b)$ also has no nonzero $\mathfrak{m}$ -torsion: $\unicode[STIX]{x1D6E5}$ acts trivially on $R/b$ , so Lemma 3.7 ensures that $H_{\text{cont}}^{i}(\unicode[STIX]{x1D6E5},R/b)$ is a direct sum of copies of $R/b$ .◻

Theorem 3.9. The edge map $e$ defined in (3.3.1) induces the isomorphism

$$\begin{eqnarray}L\unicode[STIX]{x1D702}_{(\unicode[STIX]{x1D701}_{p}-1)}(e):L\unicode[STIX]{x1D702}_{(\unicode[STIX]{x1D701}_{p}-1)}(R\unicode[STIX]{x1D6E4}_{\text{cont}}(\unicode[STIX]{x1D6E5},R_{\infty }))\overset{{\sim}}{\longrightarrow }L\unicode[STIX]{x1D702}_{(\unicode[STIX]{x1D701}_{p}-1)}(R\unicode[STIX]{x1D6E4}_{\text{pro}\acute{\text{e}}\text{t}}(\mathfrak{X}_{C}^{\text{ad}},\widehat{{\mathcal{O}}}^{+})).\end{eqnarray}$$

Proof. Proposition 3.8 ensures that the ${\mathcal{O}}_{C}$ -modules $H_{\text{cont}}^{i}(\unicode[STIX]{x1D6E5},R_{\infty })$ have no nonzero $\mathfrak{m}$ -torsion and that

$$\begin{eqnarray}\textstyle \frac{H_{\text{cont}}^{i}(\unicode[STIX]{x1D6E5},\,R_{\infty })}{H_{\text{cont}}^{i}(\unicode[STIX]{x1D6E5},\,R_{\infty })[\unicode[STIX]{x1D701}_{p}-1]}\cong \frac{H_{\text{cont}}^{i}(\unicode[STIX]{x1D6E5},\,R)}{H_{\text{cont}}^{i}(\unicode[STIX]{x1D6E5},\,R)[\unicode[STIX]{x1D701}_{p}-1]}.\end{eqnarray}$$

Since $\unicode[STIX]{x1D6E5}$ acts trivially on $R$ , this last quotient is a finite direct sum of copies of $R$ (see Lemma 3.7), so, by Proposition 3.8, it has no nonzero $\mathfrak{m}$ -torsion. Consequently, since $\mathfrak{m}$ kills the kernel and the cokernel of each map

$$\begin{eqnarray}H^{i}(e):H_{\text{cont}}^{i}(\unicode[STIX]{x1D6E5},R_{\infty })\rightarrow H^{i}(\mathfrak{X}_{C}^{\text{ad}},\widehat{{\mathcal{O}}}^{+})\end{eqnarray}$$

(see §3.3), Lemma 3.4 applies to these maps and gives the desired conclusion. ◻

Remark 3.10. Theorem 3.9 extends as follows: for any profinite group $\unicode[STIX]{x1D6E5}^{\prime }$ equipped with a continuous surjection $\unicode[STIX]{x1D6E5}^{\prime }{\twoheadrightarrow}\unicode[STIX]{x1D6E5}$ and any pro-(finite étale) affinoid perfectoid $\unicode[STIX]{x1D6E5}^{\prime }$ -cover

compatibly with the surjection $\unicode[STIX]{x1D6E5}^{\prime }{\twoheadrightarrow}\unicode[STIX]{x1D6E5}$ , the edge map $e^{\prime }$ defined analogously to (3.3.1) induces the isomorphism

$$\begin{eqnarray}L\unicode[STIX]{x1D702}_{(\unicode[STIX]{x1D701}_{p}-1)}(e^{\prime }):L\unicode[STIX]{x1D702}_{(\unicode[STIX]{x1D701}_{p}-1)}(R\unicode[STIX]{x1D6E4}_{\text{cont}}(\unicode[STIX]{x1D6E5}^{\prime },R_{\infty }^{\prime }))\overset{{\sim}}{\longrightarrow }L\unicode[STIX]{x1D702}_{(\unicode[STIX]{x1D701}_{p}-1)}(R\unicode[STIX]{x1D6E4}_{\text{pro}\acute{\text{e}}\text{t}}(\mathfrak{X}_{C}^{\text{ad}},\widehat{{\mathcal{O}}}^{+})).\end{eqnarray}$$

Indeed, by the almost purity theorem [Reference ScholzeSch13a, 4.10(v)], the ideal $\mathfrak{m}$ kills the cohomology of $\operatorname{Cone}(e^{\prime })$ (in addition to that of $\operatorname{Cone}(e)$ ), so the octahedral axiom (see [Reference Beilinson, Bernstein and DeligneBBD82, 1.1.7.1]) ensures that it also kills the cohomology of the cone of the map $R\unicode[STIX]{x1D6E4}_{\text{cont}}(\unicode[STIX]{x1D6E5},R_{\infty })\rightarrow R\unicode[STIX]{x1D6E4}_{\text{cont}}(\unicode[STIX]{x1D6E5}^{\prime },R_{\infty }^{\prime })$ ; Lemma 3.4 then applies to this map and combines with Theorem 3.9 to give the claim.

The main goal of this section is an analogue of Theorem 3.9 for the sheaf $\mathbb{A}_{\text{inf},\,\mathfrak{X}_{C}^{\text{ad}}}$ (see Theorem 3.20). To prepare for it, in §§3.11 and 3.14 we describe the values of the sheaves $\widehat{{\mathcal{O}}}_{\mathfrak{ X}_{C}^{\text{ad}}}^{+,\,\flat }$ and $\mathbb{A}_{\text{inf},\,\mathfrak{X}_{C}^{\text{ad}}}$ on  $\mathfrak{X}_{C,\,\infty }^{\text{ad}}$ .

3.11 The tilt $R_{\infty }^{\flat }$

Thanks to the explicit description (3.2.1) of the perfectoid ring $R_{\infty }^{\Box }$ , its tilt

$$\begin{eqnarray}(R_{\infty }^{\Box })^{\flat }:=\mathop{\varprojlim }\nolimits_{y\mapsto y^{p}}(R_{\infty }^{\Box }/p)\end{eqnarray}$$

is described explicitly by the identification

where $x_{i}^{1/p^{m}}$ corresponds to the $p$ -power compatible sequence $(\ldots ,t_{i}^{1/p^{m+1}},t_{i}^{1/p^{m}})$ of elements of $R_{\infty }^{\Box }$ , the completions are $p^{1/p^{\infty }}$ -adic, and the decomposition is as ${\mathcal{O}}_{C}^{\flat }$ -modules. Thus,

$$\begin{eqnarray}\textstyle \text{the tilt}\quad R_{\infty }^{\flat }:=\mathop{\varprojlim }\nolimits_{y\mapsto y^{p}}(R_{\infty }/p)\quad \text{of the perfectoid ring}\quad R_{\infty }\end{eqnarray}$$

is identified with the $p^{1/p^{\infty }}$ -adic completion of any lift of the étale $R_{\infty }^{\Box }/p$ -algebra $R_{\infty }/p$ to an étale $(R_{\infty }^{\Box })^{\flat }$ -algebra (such a lift exists, see [SP, 04D1]). By [Reference ScholzeSch13a, 5.11(i)], the value on $\mathfrak{X}_{C,\,\infty }^{\text{ad}}$ of the sheaf $\widehat{{\mathcal{O}}}_{\mathfrak{ X}_{C}^{\text{ad}}}^{+,\,\flat }$ reviewed in (2.2.1) is the ring $R_{\infty }^{\flat }$ .

By functoriality, the group $\unicode[STIX]{x1D6E5}$ acts continuously and ${\mathcal{O}}_{C}^{\flat }$ -linearly on $(R_{\infty }^{\Box })^{\flat }$ and $R_{\infty }^{\flat }$ . Explicitly, $\unicode[STIX]{x1D6E5}$ respects the completed direct sum decomposition and an $(\unicode[STIX]{x1D716}_{0},\ldots ,\unicode[STIX]{x1D716}_{d})\in \unicode[STIX]{x1D6E5}$ scales $x_{j}^{a_{j}}$ by $\unicode[STIX]{x1D716}_{j}^{a_{j}}\in {\mathcal{O}}_{C}^{\flat }$ .

Our analysis in §3.14 of the value on  $\mathfrak{X}_{C,\,\infty }^{\text{ad}}$ of the sheaf $\mathbb{A}_{\text{inf},\,\mathfrak{X}_{C}^{\text{ad}}}$ will hinge on the following lemmas.

Lemma 3.13. For any affinoid perfectoid $\operatorname{Spa}(R_{\infty }^{\prime }[\frac{1}{p}],R_{\infty }^{\prime })$ over $\operatorname{Spa}(C,{\mathcal{O}}_{C})$ , the ring

$$\begin{eqnarray}\mathbb{A}_{\text{inf}}(R_{\infty }^{\prime }):=W((R_{\infty }^{\prime })^{\flat })\quad (\text{respectively,}\quad \mathbb{A}_{\text{inf}}(R_{\infty }^{\prime })/\unicode[STIX]{x1D707})\end{eqnarray}$$

is $(p,\unicode[STIX]{x1D707})$ -adically complete (respectively,  $p$ -adically complete). Moreover, for any $n,n^{\prime }>0$ , the sequence $(p^{n},\unicode[STIX]{x1D707}^{n^{\prime }})$ is $\mathbb{A}_{\text{inf}}(R_{\infty }^{\prime })$ -regular and the $A_{\text{inf}}/(p^{n},\unicode[STIX]{x1D707}^{n^{\prime }})$ -algebra $\mathbb{A}_{\text{inf}}(R_{\infty }^{\prime })/(p^{n},\unicode[STIX]{x1D707}^{n^{\prime }})$ is flat.

3.14 The ring $\mathbb{A}_{\text{inf}}(R_{\infty })$

By [Reference ScholzeSch13a, 6.5(i)], the value on  $\mathfrak{X}_{C,\,\infty }^{\text{ad}}$ of the sheaf $\mathbb{A}_{\text{inf},\,\mathfrak{X}_{C}^{\text{ad}}}$ is the ring

$$\begin{eqnarray}\mathbb{A}_{\text{inf}}(R_{\infty }):=W(R_{\infty }^{\flat }).\end{eqnarray}$$

By Lemma 3.13 and the formal criterion of flatness, $\mathbb{A}_{\text{inf}}(R_{\infty })$ is $(p,\unicode[STIX]{x1D707})$ -adically formally flat as an $A_{\text{inf}}$ -algebra and $(p,\unicode[STIX]{x1D707})$ -adically formally étale as an $\mathbb{A}_{\text{inf}}(R_{\infty }^{\Box })$ -algebra. By using, in addition, Lemma 3.12, we see that each quotient

(3.14.1)

In general, for a perfect $\mathbb{F}_{p}$ -algebra $A$ , the Witt ring $W(A)$ is the unique $p$ -adically complete $p$ -torsion-free $\mathbb{Z}_{p}$ -algebra $\widetilde{A}$ equipped with an isomorphism $\widetilde{A}/p\simeq A$ (see [Reference BhattBha18, 2.5]). For an $a\in A$ , the Teichmüller $[a]\in \widetilde{A}$ is $\displaystyle \lim _{n\rightarrow \infty }(\widetilde{a}_{n}^{p^{n}})$ where $\widetilde{a}_{n}\in \widetilde{A}$ is any lift of $a^{1/p^{n}}$ (see [Reference BhattBha18, 2.4]). Therefore,

where the completions are $(p,\unicode[STIX]{x1D707})$ -adic, the decomposition is as $A_{\text{inf}}$ -modules, and, in terms of §3.11, we have $X_{i}^{1/p^{m}}=[x_{i}^{1/p^{m}}]$ . The summands for which $a_{i}\in \mathbb{Z}$ for all $i$ comprise a subring

(3.14.2) $$\begin{eqnarray}\hspace{-3.00003pt}A(R^{\Box })\cong A_{\text{inf}}\{X_{0},\ldots ,X_{r},X_{r+1}^{\pm 1},\ldots ,X_{d}^{\pm 1}\}/(X_{0}\cdots X_{r}-[(p^{1/p^{\infty }})^{q}])\quad \text{inside}\quad \mathbb{A}_{\text{inf}}(R_{\infty }^{\Box }),\end{eqnarray}$$

where the convergence is $(p,\unicode[STIX]{x1D707})$ -adic. The remaining summands, that is, those for which $a_{i}\not \in \mathbb{Z}$ for some $i$ , comprise an $A(R^{\Box })$ -submodule $N_{\infty }^{\Box }\subset \mathbb{A}_{\text{inf}}(R_{\infty }^{\Box })$ .

On sections over $\mathfrak{X}_{C,\,\infty }^{\text{ad}}$ , the map $\unicode[STIX]{x1D703}$ from (2.2.2) is identified with the unique ring homomorphism

$$\begin{eqnarray}\unicode[STIX]{x1D703}:\mathbb{A}_{\text{inf}}(R_{\infty }){\twoheadrightarrow}R_{\infty }\quad \text{such that}\quad [x]\mapsto x^{(0)},\end{eqnarray}$$

is surjective with the kernel generated by the regular element $\unicode[STIX]{x1D709}$ (see [Reference Bhatt, Morrow and ScholzeBMS18, 3.10, 3.11]), and intertwines the Witt vector Frobenius of $\mathbb{A}_{\text{inf}}(R_{\infty })$ with the absolute Frobenius of $R_{\infty }/p$ . Thus,

(3.14.3) $$\begin{eqnarray}\unicode[STIX]{x1D703}:A(R^{\Box }){\twoheadrightarrow}R^{\Box }\quad \text{is described by}\quad X_{i}\mapsto t_{i}.\end{eqnarray}$$

We use the surjection (3.14.3) to uniquely lift the étale $R^{\Box }/p$ -algebra $R/p$ to a $(p,\unicode[STIX]{x1D707})$ -adically complete, formally étale $A(R^{\Box })$ -algebra $A(R)$ . By construction, we have the identification

(3.14.4) $$\begin{eqnarray}\mathbb{A}_{\text{inf}}(R_{\infty })\cong \mathbb{A}_{\text{inf}}(R_{\infty }^{\Box })\,\widehat{\otimes }_{A(R^{\Box })}A(R),\end{eqnarray}$$

where the completion is $(p,\unicode[STIX]{x1D707})$ -adic. Therefore, by setting $N_{\infty }:=N_{\infty }^{\Box }\,\widehat{\otimes }_{A(R^{\Box })}\,A(R)$ , we arrive at the decompositions of $\mathbb{A}_{\text{inf}}(R_{\infty }^{\Box })$ and $\mathbb{A}_{\text{inf}}(R_{\infty })$ into ‘integral’ and ‘nonintegral’ parts:

(3.14.5) $$\begin{eqnarray}\mathbb{A}_{\text{inf}}(R_{\infty }^{\Box })\cong A(R^{\Box })\oplus N_{\infty }^{\Box }\quad \text{and}\quad \mathbb{A}_{\text{inf}}(R_{\infty })\cong A(R)\oplus N_{\infty }.\end{eqnarray}$$

Modulo $\operatorname{Ker}\unicode[STIX]{x1D703}$ (that is, modulo $\unicode[STIX]{x1D709}$ ), these decompositions reduce to the decompositions (3.2.2).

The Witt vector Frobenius of $\mathbb{A}_{\text{inf}}(R_{\infty }^{\Box })$ preserves $A(R^{\Box })$ ; explicitly: it is semilinear with respect to the Frobenius of $A_{\text{inf}}$ and raises each $X_{i}^{1/p^{m}}$ to the $p$ th power. By construction, $A(R)$ inherits a Frobenius ring endomorphism from $A(R^{\Box })$ , and the identification (3.14.4) is Frobenius-equivariant.

The natural $\unicode[STIX]{x1D6E5}$ -action on $\mathbb{A}_{\text{inf}}(R_{\infty })$ is continuous and commutes with the Frobenius. Explicitly, $\unicode[STIX]{x1D6E5}$ respects the completed direct sum decomposition and an $(\unicode[STIX]{x1D716}_{0},\ldots ,\unicode[STIX]{x1D716}_{d})\in \unicode[STIX]{x1D6E5}$ scales $X_{j}^{a_{j}}$ by $[\unicode[STIX]{x1D716}_{j}^{a_{j}}]\in A_{\text{inf}}$ . The $\unicode[STIX]{x1D6E5}$ -action on $A(R^{\Box })$ lifts uniquely to a necessarily Frobenius-equivariant $\unicode[STIX]{x1D6E5}$ -action on $A(R)$ . In particular, $\unicode[STIX]{x1D6E5}$ acts trivially on $A(R)/\unicode[STIX]{x1D707}$ . The identifications (3.14.4) and (3.14.5) are $\unicode[STIX]{x1D6E5}$ -equivariant.

3.15 The cohomology of $\mathbb{A}_{\text{inf}}$ and continuous group cohomology

Similarly to §3.3, the Čech complex of the sheaf $\mathbb{A}_{\text{inf},\,\mathfrak{X}_{C}^{\text{ad}}}$ with respect to the pro-(finite étale) affinoid perfectoid cover $\mathfrak{X}_{C,\,\infty }^{\text{ad}}\rightarrow \mathfrak{X}_{C}^{\text{ad}}$ is identified with the continuous cochain complex $R\unicode[STIX]{x1D6E4}_{\text{cont}}(\unicode[STIX]{x1D6E5},\mathbb{A}_{\text{inf}}(R_{\infty }))$ . Thus, by using [SP, 01GY], we obtain the edge map to the proétale cohomology of $\mathbb{A}_{\text{inf},\,\mathfrak{X}_{C}^{\text{ad}}}$ :

(3.15.1) $$\begin{eqnarray}e:R\unicode[STIX]{x1D6E4}_{\text{cont}}(\unicode[STIX]{x1D6E5},\mathbb{A}_{\text{inf}}(R_{\infty }))\rightarrow R\unicode[STIX]{x1D6E4}_{\text{pro}\acute{\text{e}}\text{t}}(\mathfrak{X}_{C}^{\text{ad}},\mathbb{A}_{\text{inf}}).\end{eqnarray}$$

By the almost purity theorem, more precisely, by [Reference ScholzeSch13a, 6.5(ii)], the subset $[\mathfrak{m}^{\flat }]\subset A_{\text{inf}}$ that consists of the Teichmüller lifts of the elements in the maximal ideal $\mathfrak{m}^{\flat }\subset {\mathcal{O}}_{C}^{\flat }$ kills all the cohomology groups of $\operatorname{Cone}(e)$ . Since $\unicode[STIX]{x1D707}\in W(\mathfrak{m}^{\flat })$ (see (2.3.3)), it will be useful to strengthen this annihilation as follows.

Proof. By the derived $p$ -adic completeness, any free $A_{\text{inf}}$ -module resolution $F^{\bullet }$ of $H$ satisfies

$$\begin{eqnarray}\textstyle H\cong \operatorname{Coker}\Big(\mathop{\varprojlim }\nolimits_{n}(F^{-1}/p^{n})\rightarrow \mathop{\varprojlim }\nolimits_{n}(F^{0}/p^{n})\Big).\end{eqnarray}$$

Moreover, for every $n\geqslant 1$ the ideals $[\mathfrak{m}^{\flat }]\cdot W_{n}({\mathcal{O}}_{C}^{\flat })$ and $W_{n}(\mathfrak{m}^{\flat }):=\operatorname{Ker}(W_{n}({\mathcal{O}}_{C}^{\flat })\rightarrow W_{n}(k))$ of $W_{n}({\mathcal{O}}_{C}^{\flat })$ agree. Thus, the $([\mathfrak{m}^{\flat }]A_{\text{inf}})$ -annihilation of $H$ implies that $W_{n}(\mathfrak{m}^{\flat })$ kills both

$$\begin{eqnarray}H/p^{n}\cong H^{0}(F^{\bullet }\otimes _{A_{\text{inf}}}A_{\text{inf}}/p^{n})\quad \text{and}\quad \operatorname{Tor}_{1}^{A_{\text{inf}}}(H,A_{\text{inf}}/p^{n})\cong H^{-1}(F^{\bullet }\otimes _{A_{\text{inf}}}A_{\text{inf}}/p^{n}).\end{eqnarray}$$

Thus, since $[\mathfrak{m}^{\flat }]^{2}=[\mathfrak{m}^{\flat }]$ and $F_{0}/p^{n}$ has no nonzero $m$ -torsion for every nonzero $m\in [\mathfrak{m}^{\flat }]$ , any element $x\in W_{n+1}(\mathfrak{m}^{\flat })\,\cdot \,(F_{0}/p^{n+1})$ may be lifted to $W_{n+1}(\mathfrak{m}^{\flat })\,\cdot \,(F_{-1}/p^{n+1})$ , compatibly with a specified lift of its image $\overline{x}\in W_{n}(\mathfrak{m}^{\flat })\,\cdot \,(F_{0}/p^{n})$ to $W_{n}(\mathfrak{m}^{\flat })\,\cdot \,(F_{-1}/p^{n})$ . In particular, $W(\mathfrak{m}^{\flat })\,\cdot \,(\mathop{\varprojlim }\nolimits_{n}(F^{0}/p^{n}))$ lies in the image of $\mathop{\varprojlim }\nolimits_{n}(F^{-1}/p^{n})$ , that is, $W(\mathfrak{m}^{\flat })$ kills $H$ , as desired.◻

We will show in Theorem 3.20 that $L\unicode[STIX]{x1D702}_{(\unicode[STIX]{x1D707})}(e)$ is an isomorphism, so that continuous group cohomology computes $L\unicode[STIX]{x1D702}_{(\unicode[STIX]{x1D707})}(R\unicode[STIX]{x1D6E4}_{\text{pro}\acute{\text{e}}\text{t}}(\mathfrak{X}_{C}^{\text{ad}},\mathbb{A}_{\text{inf}}))$ . For this, we will use the following lemma.

Proof. Since $L\unicode[STIX]{x1D702}$ is not a triangulated functor, the fact that $L\unicode[STIX]{x1D702}_{(\unicode[STIX]{x1D707})}(\operatorname{Cone}(b))\cong 0$ does not a priori suffice. Instead, the ideal $(W(\mathfrak{m}^{\flat }))^{2}$ kills the cohomology of $\operatorname{Cone}(b)\otimes _{A_{\text{inf}}}^{\mathbb{L}}A_{\text{inf}}/\unicode[STIX]{x1D707}$ , so the sequences

$$\begin{eqnarray}0\rightarrow H^{i}(B\otimes _{A_{\text{inf}}}^{\mathbb{L}}A_{\text{inf}}/\unicode[STIX]{x1D707})\rightarrow H^{i}(B^{\prime }\otimes _{A_{\text{inf}}}^{\mathbb{L}}A_{\text{inf}}/\unicode[STIX]{x1D707})\rightarrow H^{i}(\operatorname{Cone}(b)\otimes _{A_{\text{inf}}}^{\mathbb{L}}A_{\text{inf}}/\unicode[STIX]{x1D707})\rightarrow 0\end{eqnarray}$$

are short exact. By the Bockstein construction (see [Reference Bhatt, Morrow and ScholzeBMS18, 6.12]), as $i$ varies, they comprise a short exact sequence whose terms are complexes that compute $L\unicode[STIX]{x1D702}_{(\unicode[STIX]{x1D707})}(B)\otimes _{A_{\text{inf}}}^{\mathbb{L}}A_{\text{inf}}/\unicode[STIX]{x1D707}$ , etc. Thus, the vanishing of $L\unicode[STIX]{x1D702}_{(\unicode[STIX]{x1D707})}(\operatorname{Cone}(b))$ implies that $(L\unicode[STIX]{x1D702}_{(\unicode[STIX]{x1D707})}(b))\otimes _{A_{\text{inf}}}^{\mathbb{L}}A_{\text{inf}}/\unicode[STIX]{x1D707}$ is an isomorphism. It follows that $\operatorname{Cone}(L\unicode[STIX]{x1D702}_{(\unicode[STIX]{x1D707})}(b))\otimes _{A_{\text{inf}}}^{\mathbb{L}}A_{\text{inf}}/\unicode[STIX]{x1D707}\cong 0$ , so $\unicode[STIX]{x1D707}$ acts invertibly on the cohomology of $\operatorname{Cone}(L\unicode[STIX]{x1D702}_{(\unicode[STIX]{x1D707})}(b))$ . But then, as we see after applying $-\otimes _{A_{\text{inf}}}^{\mathbb{L}}A_{\text{inf}}[\frac{1}{\unicode[STIX]{x1D707}}]$ , this cohomology vanishes.◻

We now verify that the edge map $e$ defined in (3.15.1) also meets the first assumption of Lemma 3.18.

Proposition 3.19. For each $i\in \mathbb{Z}$ , the $A_{\text{inf}}$ -module $H_{\text{cont}}^{i}(\unicode[STIX]{x1D6E5},\mathbb{A}_{\text{inf}}(R_{\infty })/\unicode[STIX]{x1D707})$ is $p$ -torsion free and $p$ -adically complete; moreover, the following natural maps are isomorphisms:

(3.19.1) $$\begin{eqnarray}H_{\text{cont}}^{i}(\unicode[STIX]{x1D6E5},\mathbb{A}_{\text{inf}}(R_{\infty })/\unicode[STIX]{x1D707})\otimes _{A_{\text{inf}}}A_{\text{inf}}/p^{n}\overset{{\sim}}{\longrightarrow }H_{\text{cont}}^{i}(\unicode[STIX]{x1D6E5},\mathbb{A}_{\text{inf}}(R_{\infty })/(\unicode[STIX]{x1D707},p^{n}))\quad \text{for}\quad n>0\end{eqnarray}$$

and

(3.19.2) $$\begin{eqnarray}\textstyle H_{\text{cont}}^{i}(\unicode[STIX]{x1D6E5},\mathbb{A}_{\text{inf}}(R_{\infty })/\unicode[STIX]{x1D707})\overset{{\sim}}{\longrightarrow }\mathop{\varprojlim }\nolimits_{n}(H_{\text{cont}}^{i}(\unicode[STIX]{x1D6E5},\mathbb{A}_{\text{inf}}(R_{\infty })/(\unicode[STIX]{x1D707},p^{n}))).\end{eqnarray}$$

In addition, $H_{\text{cont}}^{i}(\unicode[STIX]{x1D6E5},\mathbb{A}_{\text{inf}}(R_{\infty })/(\unicode[STIX]{x1D707},p^{n}))$ and $H_{\text{cont}}^{i}(\unicode[STIX]{x1D6E5},\mathbb{A}_{\text{inf}}(R_{\infty })/\unicode[STIX]{x1D707})$ have no nonzero $W(\mathfrak{m}^{\flat })$ -torsion.

Proof. Since $A(R)/\unicode[STIX]{x1D707}$ is $p$ -adically complete and has a trivial $\unicode[STIX]{x1D6E5}$ -action (see Lemma 3.13 and §3.14), Lemma 3.7 implies that $H_{\text{cont}}^{i}(\unicode[STIX]{x1D6E5},A(R)/\unicode[STIX]{x1D707})$ is a direct sum of copies of $A(R)/\unicode[STIX]{x1D707}$ , and likewise for $H_{\text{cont}}^{i}(\unicode[STIX]{x1D6E5},A(R)/(\unicode[STIX]{x1D707},p^{n}))$ . Consequently, since, by (3.14.1), the rings $A(R)/(\unicode[STIX]{x1D707},p^{n})$ and $A(R)/\unicode[STIX]{x1D707}$ have no nonzero $W(\mathfrak{m}^{\flat })$ -torsion, the analogues of all the claims with $A(R)$ in place of $\mathbb{A}_{\text{inf}}(R_{\infty })$ follow. Thus, due to (3.14.5), we only need to establish these analogues with $N_{\infty }$ in place of $\mathbb{A}_{\text{inf}}(R_{\infty })$ .

To prepare for treating $N_{\infty }$ , we start by building on the ideas of [Reference BhattBha18, proof of Lemma 4.6] to analyze a single summand $S:=A_{\text{inf}}\,\cdot \,X_{0}^{a_{0}}\cdots X_{d}^{a_{d}}$ that, as in §3.14, contributes to $N_{\infty }^{\Box }$ . We set

(3.19.3) $$\begin{eqnarray}\quad b_{j}:=a_{j}-a_{0}\quad \text{for}\quad 1\leqslant j\leqslant r\quad \text{and}\quad b_{j}:=a_{j}\quad \text{for}\quad r+1\leqslant j\leqslant d,\end{eqnarray}$$

and let $m\in \mathbb{Z}_{{>}0}$ be the minimal such that $p^{m}b_{j}\in \mathbb{Z}$ for all $j$ . Lemma 3.7 applied with the topological generators $\unicode[STIX]{x1D6FF}_{1},\ldots ,\unicode[STIX]{x1D6FF}_{d}$ of $\unicode[STIX]{x1D6E5}$ defined in §3.2 gives an $A_{\text{inf}}$ -isomorphism

$$\begin{eqnarray}H_{\text{cont}}^{i}(\unicode[STIX]{x1D6E5},S/\unicode[STIX]{x1D707})\simeq H^{i}(C^{\bullet }),\end{eqnarray}$$

where $C^{\bullet }$ is the $(A_{\text{inf}}/\unicode[STIX]{x1D707})$ -tensor product of the $d$ complexes

(3.19.4) $$\begin{eqnarray}[A_{\text{inf}}/\unicode[STIX]{x1D707}\xrightarrow[{}]{[\unicode[STIX]{x1D716}^{b_{j}}]-1}A_{\text{inf}}/\unicode[STIX]{x1D707}]\cong A_{\text{inf}}/([\unicode[STIX]{x1D716}^{b_{j}}]-1)\otimes _{A_{\text{inf}}}^{\mathbb{L}}A_{\text{inf}}/\unicode[STIX]{x1D707}.\end{eqnarray}$$

By reordering the $b_{j}$ , we may assume that for all $j$ we have $b_{j}/b_{1}\in \mathbb{Z}_{(p)}$ , so that $b_{1}\not \in \mathbb{Z}$ and both $[\unicode[STIX]{x1D716}^{b_{1}}]-1\,\mid \,[\unicode[STIX]{x1D716}^{b_{j}}]-1$ and $[\unicode[STIX]{x1D716}^{b_{1}}]-1\,\mid \,\unicode[STIX]{x1D707}$ . Then the object (3.19.4) with $j=1$ is given by the complex $[A_{\text{inf}}/([\unicode[STIX]{x1D716}^{b_{1}}]-1)\xrightarrow[{}]{0}A_{\text{inf}}/([\unicode[STIX]{x1D716}^{b_{1}}]-1)]$ and, by using the left-hand sides of (3.19.4) for the factors with $j\neq 1$ , we see that $C^{\bullet }$ is quasi-isomorphic to a direct sum of shifts of

$$\begin{eqnarray}A_{\text{inf}}/([\unicode[STIX]{x1D716}^{b_{1}}]-1)\cong A_{\text{inf}}/\unicode[STIX]{x1D711}^{-m}(\unicode[STIX]{x1D707}).\end{eqnarray}$$

Thus, for $i\in \mathbb{Z}$ ,

(3.19.5)

By Lemma 3.7 and [SP, 061Z, 0662], this implies that

(3.19.6) $$\begin{eqnarray}H_{\text{cont}}^{i}(\unicode[STIX]{x1D6E5},S/\unicode[STIX]{x1D707})\otimes _{A_{\text{inf}}}A_{\text{inf}}/p^{n}\overset{{\sim}}{\longrightarrow }H_{\text{cont}}^{i}(\unicode[STIX]{x1D6E5},S/(\unicode[STIX]{x1D707},p^{n})).\end{eqnarray}$$

We now analyze $N_{\infty }^{\Box }$ . Since $\mathbb{A}_{\text{inf}}(R_{\infty }^{\Box })/\unicode[STIX]{x1D707}$ is $p$ -adically complete, §3.14 gives the $\unicode[STIX]{x1D6E5}$ -decomposition

in which the completion is $p$ -adic. Lemma 3.6 (i) then combines with (3.19.5) to prove that

$$\begin{eqnarray}H_{\text{cont}}^{i}(\unicode[STIX]{x1D6E5},N_{\infty }^{\Box }/\unicode[STIX]{x1D707})[p]=0\quad \text{for each}\quad i\in \mathbb{Z}.\end{eqnarray}$$

Analogously to (3.19.6), this, in turn, implies that

(3.19.7) $$\begin{eqnarray}H_{\text{cont}}^{i}(\unicode[STIX]{x1D6E5},N_{\infty }^{\Box }/\unicode[STIX]{x1D707})\otimes _{A_{\text{inf}}}A_{\text{inf}}/p^{n}\overset{{\sim}}{\longrightarrow }H_{\text{cont}}^{i}(\unicode[STIX]{x1D6E5},N_{\infty }^{\Box }/(\unicode[STIX]{x1D707},p^{n})).\end{eqnarray}$$

Finally, we analyze $N_{\infty }$ . The identification

$$\begin{eqnarray}\textstyle N_{\infty }/(\unicode[STIX]{x1D707},p^{n})\cong N_{\infty }^{\Box }/(\unicode[STIX]{x1D707},p^{n})\otimes _{A(R^{\Box })}A(R)\end{eqnarray}$$

is $\unicode[STIX]{x1D6E5}$ -equivariant and $A(R)/(\unicode[STIX]{x1D707},p^{n})$ is $(A(R^{\Box })/(\unicode[STIX]{x1D707},p^{n}))$ -flat, so Lemma 3.7 gives the identifications

(3.19.8) $$\begin{eqnarray}H_{\text{cont}}^{i}(\unicode[STIX]{x1D6E5},N_{\infty }/(\unicode[STIX]{x1D707},p^{n}))\cong H_{\text{cont}}^{i}(\unicode[STIX]{x1D6E5},N_{\infty }^{\Box }/(\unicode[STIX]{x1D707},p^{n}))\otimes _{A(R^{\Box })}A(R)\quad \text{for}\quad n\geqslant 1,\end{eqnarray}$$

which are compatible as $n$ varies. Consequently, for $n>1$ , the sequences

(3.19.9) $$\begin{eqnarray}0\rightarrow H_{\text{cont}}^{i}(\unicode[STIX]{x1D6E5},N_{\infty }/(\unicode[STIX]{x1D707},p^{n}))[p]\rightarrow H_{\text{cont}}^{i}(\unicode[STIX]{x1D6E5},N_{\infty }/(\unicode[STIX]{x1D707},p^{n}))\rightarrow H_{\text{cont}}^{i}(\unicode[STIX]{x1D6E5},N_{\infty }/(\unicode[STIX]{x1D707},p^{n-1}))\rightarrow 0\end{eqnarray}$$

are short exact because, by (3.19.5) and (3.19.7), so are their analogues with $N_{\infty }^{\Box }$ in place of $N_{\infty }$ . By taking the inverse limit of these sequences for varying $n$ and using [SP, 0D6K], we obtain

(3.19.10) $$\begin{eqnarray}\textstyle H_{\text{cont}}^{i}(\unicode[STIX]{x1D6E5},N_{\infty }/\unicode[STIX]{x1D707})\overset{{\sim}}{\longrightarrow }\mathop{\varprojlim }\nolimits_{n}(H_{\text{cont}}^{i}(\unicode[STIX]{x1D6E5},N_{\infty }/(\unicode[STIX]{x1D707},p^{n}))),\end{eqnarray}$$

which is the sought analogue of (3.19.2). The $p$ -torsion-freeness of $H_{\text{cont}}^{i}(\unicode[STIX]{x1D6E5},N_{\infty }/\unicode[STIX]{x1D707})$ follows from (3.19.9)–(3.19.10) and, as in (3.19.6), it implies that

$$\begin{eqnarray}H_{\text{cont}}^{i}(\unicode[STIX]{x1D6E5},N_{\infty }/\unicode[STIX]{x1D707})\otimes _{A_{\text{inf}}}A_{\text{inf}}/p^{n}\overset{{\sim}}{\longrightarrow }H_{\text{cont}}^{i}(\unicode[STIX]{x1D6E5},N_{\infty }/(\unicode[STIX]{x1D707},p^{n})).\end{eqnarray}$$

It remains to show that each $H_{\text{cont}}^{i}(\unicode[STIX]{x1D6E5},N_{\infty }/(\unicode[STIX]{x1D707},p^{n}))$ has no nonzero $W(\mathfrak{m}^{\flat })$ -torsion.

The surjectivity aspect of the short exact sequences (3.19.9) implies that the sequences

$$\begin{eqnarray}0\rightarrow N_{\infty }/(\unicode[STIX]{x1D707},p)\xrightarrow[{}]{p^{n-1}}N_{\infty }/(\unicode[STIX]{x1D707},p^{n})\rightarrow N_{\infty }/(\unicode[STIX]{x1D707},p^{n-1})\rightarrow 0\end{eqnarray}$$

remain short exact after applying $H_{\text{cont}}^{i}(\unicode[STIX]{x1D6E5},-)$ . Thus, $H_{\text{cont}}^{i}(\unicode[STIX]{x1D6E5},N_{\infty }/(\unicode[STIX]{x1D707},p^{n}))$ is a successive extension of copies of $H_{\text{cont}}^{i}(\unicode[STIX]{x1D6E5},N_{\infty }/(\unicode[STIX]{x1D707},p))$ . Consequently, it has no nonzero $W(\mathfrak{m}^{\flat })$ -torsion because, by Lemma 3.12, neither does $H_{\text{cont}}^{i}(\unicode[STIX]{x1D6E5},N_{\infty }/(\unicode[STIX]{x1D707},p))$ (note that $N_{\infty }/(\unicode[STIX]{x1D707},p)$ is a direct summand of $\mathbb{A}_{\text{inf}}(R_{\infty })/(\unicode[STIX]{x1D707},p)\cong R_{\infty }^{\flat }/\unicode[STIX]{x1D707}$ ).◻

Theorem 3.20. The edge map $e$ defined in (3.15.1) induces the isomorphism

$$\begin{eqnarray}L\unicode[STIX]{x1D702}_{(\unicode[STIX]{x1D707})}(e):L\unicode[STIX]{x1D702}_{(\unicode[STIX]{x1D707})}(R\unicode[STIX]{x1D6E4}_{\text{cont}}(\unicode[STIX]{x1D6E5},\mathbb{A}_{\text{inf}}(R_{\infty })))\overset{{\sim}}{\longrightarrow }L\unicode[STIX]{x1D702}_{(\unicode[STIX]{x1D707})}(R\unicode[STIX]{x1D6E4}_{\text{pro}\acute{\text{e}}\text{t}}(\mathfrak{X}_{C}^{\text{ad}},\mathbb{A}_{\text{inf},\,\mathfrak{X}_{C}^{\text{ad}}})).\end{eqnarray}$$

Proof. By the projection formula [SP, 0944],

(3.20.1) $$\begin{eqnarray}R\unicode[STIX]{x1D6E4}_{\text{cont}}(\unicode[STIX]{x1D6E5},\mathbb{A}_{\text{inf}}(R_{\infty }))\otimes _{A_{\text{inf}}}^{\mathbb{L}}A_{\text{inf}}/\unicode[STIX]{x1D707}\cong R\unicode[STIX]{x1D6E4}_{\text{cont}}(\unicode[STIX]{x1D6E5},\mathbb{A}_{\text{inf}}(R_{\infty })/\unicode[STIX]{x1D707}),\end{eqnarray}$$

so Proposition 3.19 implies that the cohomology modules of $R\unicode[STIX]{x1D6E4}_{\text{cont}}(\unicode[STIX]{x1D6E5},\mathbb{A}_{\text{inf}}(R_{\infty }))\otimes _{A_{\text{inf}}}^{\mathbb{L}}A_{\text{inf}}/\unicode[STIX]{x1D707}$ have no nonzero $W(\mathfrak{m}^{\flat })$ -torsion. Thus, the claim follows from Lemmas 3.16 and 3.18.◻

Remark 3.21. Analogously to Remark 3.10, Theorem 3.20 extends as follows: for any pro-(finite étale) affinoid perfectoid $\unicode[STIX]{x1D6E5}^{\prime }$ -cover

$$\begin{eqnarray}\textstyle \textstyle \operatorname{Spa}(R_{\infty }^{\prime }[\frac{1}{p}],R_{\infty }^{\prime })\rightarrow \operatorname{Spa}(R[\frac{1}{p}],R)\cong \mathfrak{X}_{C}^{\text{ad}}\quad \text{that refines}\quad \mathfrak{X}_{C,\,\infty }^{\text{ad}}\rightarrow \mathfrak{X}_{C}^{\text{ad}}\end{eqnarray}$$

subject to the same conditions as in Remark 3.10, the edge map $e^{\prime }$ defined analogously to (3.15.1) induces the isomorphism

$$\begin{eqnarray}L\unicode[STIX]{x1D702}_{(\unicode[STIX]{x1D707})}(e^{\prime }):L\unicode[STIX]{x1D702}_{(\unicode[STIX]{x1D707})}(R\unicode[STIX]{x1D6E4}_{\text{cont}}(\unicode[STIX]{x1D6E5}^{\prime },\mathbb{A}_{\text{inf}}(R_{\infty }^{\prime })))\overset{{\sim}}{\longrightarrow }L\unicode[STIX]{x1D702}_{(\unicode[STIX]{x1D707})}(R\unicode[STIX]{x1D6E4}_{\text{pro}\acute{\text{e}}\text{t}}(\mathfrak{X}_{C}^{\text{ad}},\mathbb{A}_{\text{inf},\,\mathfrak{X}_{C}^{\text{ad}}})).\end{eqnarray}$$

Indeed, as in Remark 3.10, by the almost purity theorem and the octahedral axiom, $[\mathfrak{m}^{\flat }]A_{\text{inf}}$ kills the cohomology modules of the cone of the map

$$\begin{eqnarray}e_{0}:R\unicode[STIX]{x1D6E4}_{\text{cont}}(\unicode[STIX]{x1D6E5},\mathbb{A}_{\text{inf}}(R_{\infty }))\rightarrow R\unicode[STIX]{x1D6E4}_{\text{cont}}(\unicode[STIX]{x1D6E5}^{\prime },\mathbb{A}_{\text{inf}}(R_{\infty }^{\prime }))\end{eqnarray}$$

and, by [Reference Bhatt and ScholzeBS15, 3.4.4 and 3.4.14], these modules are derived $p$ -adically complete; thus, by Lemma 3.17, even $W(\mathfrak{m}^{\flat })$ kills them, to the effect that Lemma 3.18 applies to the map $e_{0}$ and proves the claim.

As a final goal of §3, we wish to show in Theorem 3.34 that even the maps $L\unicode[STIX]{x1D702}_{(\unicode[STIX]{x1D707})}(e\,\widehat{\otimes }_{A_{\text{inf}}}^{\mathbb{L}}\,A_{\text{cris}}^{(m)})$ are isomorphisms for $A_{\text{inf}}$ -algebras $A_{\text{cris}}^{(m)}$ reviewed in §3.26 below. This extension of Theorem 3.20 will be important for relating $A\unicode[STIX]{x1D6FA}_{\mathfrak{X}}$ to logarithmic crystalline cohomology in §5. Our analysis of $L\unicode[STIX]{x1D702}_{(\unicode[STIX]{x1D707})}(e\,\widehat{\otimes }_{A_{\text{inf}}}^{\mathbb{L}}\,A_{\text{cris}}^{(m)})$ will use the following further consequences of the proof of Proposition 3.19.

3.22 The decomposition of $N_{\infty }$

For $m\geqslant 0$ , let $N_{m}^{\Box }$ be the $(p,\unicode[STIX]{x1D707})$ -adically completed direct sum of those summands $A_{\text{inf}}\,\cdot \,X_{0}^{a_{0}}\cdots X_{d}^{a_{d}}$ that contribute to $\mathbb{A}_{\text{inf}}(R_{\infty }^{\Box })$ in §3.14 for which $m$ is the smallest nonnegative integer such that $p^{m}a_{j}\in \mathbb{Z}$ for all $j$ (equivalently, in the notation of (3.19.3), such that $p^{m}b_{j}\in \mathbb{Z}$ for all $j$ ). For varying $m>0$ , the $A(R^{\Box })$ -modules $N_{m}^{\Box }$ and the $A(R)$ -modules $N_{m}:=N_{m}^{\Box }\,\widehat{\otimes }_{A(R^{\Box })}\,A(R)$ comprise the $(p,\unicode[STIX]{x1D707})$ -adically completed direct sum decompositions

(3.22.1) $$\begin{eqnarray}\textstyle \textstyle N_{\infty }^{\Box }\cong \widehat{\bigoplus }_{m>0}\,N_{m}^{\Box }\quad \text{and}\quad N_{\infty }\cong \widehat{\bigoplus }_{m>0}\,N_{m}.\end{eqnarray}$$

For a fixed $i$ , Lemma 3.7 and (3.19.5)–(3.19.6) imply that

(3.22.2)

Corollary 3.23. For all $i$ and $n,m\geqslant 0$ ,

We wish to supplement Proposition 3.19 with Proposition 3.25 that analyzes the cohomology of $N_{\infty }$ without reducing modulo $\unicode[STIX]{x1D707}$ . Its proof will use the following base change result for $L\unicode[STIX]{x1D702}$ .

Lemma 3.24 [Reference BhattBha18, 5.16].

For a ring $A$ , elements $f,g\in A$ with $g$ a nonzero-divisor, and a $K\in D(A)$ , if the modules $H^{i}(K\otimes _{A}^{\mathbb{L}}A/f)$ have no nonzero $g$ -torsion, then the natural map

is an isomorphism.◻

Proof. Let $\unicode[STIX]{x1D6FF}_{1},\ldots ,\unicode[STIX]{x1D6FF}_{d}$ be the free generators of $\unicode[STIX]{x1D6E5}$ fixed in §3.2. By Lemma 3.7, we need to prove that

(3.25.1) $$\begin{eqnarray}L\unicode[STIX]{x1D702}_{(\unicode[STIX]{x1D707})}(K_{N_{\infty }}(\unicode[STIX]{x1D6FF}_{1}-1,\ldots ,\unicode[STIX]{x1D6FF}_{d}-1))\cong 0.\end{eqnarray}$$

The key point, with which we start, is to prove the vanishing (3.25.1) modulo $\unicode[STIX]{x1D711}(\unicode[STIX]{x1D709})$ . The isomorphism

$$\begin{eqnarray}K_{N_{\infty }}(\unicode[STIX]{x1D6FF}_{1}-1,\ldots ,\unicode[STIX]{x1D6FF}_{d}-1)\otimes _{A_{\text{inf}}}^{\mathbb{L}}A_{\text{inf}}/\unicode[STIX]{x1D707}\cong K_{N_{\infty }/\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D6FF}_{1}-1,\ldots ,\unicode[STIX]{x1D6FF}_{d}-1),\end{eqnarray}$$

Lemma 3.7, and Proposition 3.19 show that the cohomology of $K_{N_{\infty }}(\unicode[STIX]{x1D6FF}_{1}-1,\ldots ,\unicode[STIX]{x1D6FF}_{d}-1)\otimes _{A_{\text{inf}}}^{\mathbb{L}}A_{\text{inf}}/\unicode[STIX]{x1D707}$ is $p$ -torsion free. Therefore, Lemma 3.24 supplies the identification

(3.25.2) $$\begin{eqnarray}L\unicode[STIX]{x1D702}_{(\unicode[STIX]{x1D707})}(K_{N_{\infty }}(\unicode[STIX]{x1D6FF}_{1}-1,\ldots ,\unicode[STIX]{x1D6FF}_{d}-1))\otimes _{A_{\text{inf}}}^{\mathbb{L}}A_{\text{inf}}/\unicode[STIX]{x1D711}(\unicode[STIX]{x1D709})\cong L\unicode[STIX]{x1D702}_{(\unicode[STIX]{x1D701}_{p}-1)}(K_{N_{\infty }/\unicode[STIX]{x1D711}(\unicode[STIX]{x1D709})}(\unicode[STIX]{x1D6FF}_{1}-1,\ldots ,\unicode[STIX]{x1D6FF}_{d}-1)).\end{eqnarray}$$

The inverse Frobenius $\unicode[STIX]{x1D711}^{-1}$ maps $N_{\infty }^{\Box }$ isomorphically onto a direct summand of $N_{\infty }^{\Box }$ , so it maps $N_{\infty }$ isomorphically onto a direct summand of $N_{\infty }$ . Thus, $\unicode[STIX]{x1D711}^{-1}$ maps $N_{\infty }/\unicode[STIX]{x1D711}(\unicode[STIX]{x1D709})$ isomorphically onto a direct summand of $N_{\infty }/\unicode[STIX]{x1D709}\cong M_{\infty }$ (see (3.14.5)). In particular, by Lemma 3.7 and Proposition 3.8, $\unicode[STIX]{x1D701}_{p}-1$ kills the cohomology of $K_{N_{\infty }/\unicode[STIX]{x1D711}(\unicode[STIX]{x1D709})}(\unicode[STIX]{x1D6FF}_{1}-1,\ldots ,\unicode[STIX]{x1D6FF}_{d}-1)$ , so both sides of (3.25.2) are acyclic.

Since $N_{\infty }$ is $(p,\unicode[STIX]{x1D707})$ -adically complete, it is also $\unicode[STIX]{x1D711}(\unicode[STIX]{x1D709})$ -adically complete (see [SP, 090T]). Thus, $K_{N_{\infty }}(\unicode[STIX]{x1D6FF}_{1}-1,\ldots ,\unicode[STIX]{x1D6FF}_{d}-1)$ is derived $\unicode[STIX]{x1D711}(\unicode[STIX]{x1D709})$ -adically complete, and [Reference Bhatt, Morrow and ScholzeBMS18, 6.19] implies the same for $L\unicode[STIX]{x1D702}_{(\unicode[STIX]{x1D707})}(K_{N_{\infty }}(\unicode[STIX]{x1D6FF}_{1}-1,\ldots ,\unicode[STIX]{x1D6FF}_{d}-1))$ . The established acyclicity of the left-hand side of (3.25.2) therefore implies the desired vanishing (3.25.1).◻

3.26 The $A_{\text{inf}}$ -algebras $A_{\text{cris}}^{(m)}$

For $m\in \mathbb{Z}_{{\geqslant}1}$ , we let $A_{\text{cris}}^{(m)}$ be the $p$ -adic completion of the $A_{\text{inf}}$ -subalgebra $A_{\text{cris}}^{0,\,(m)}$ of $A_{\text{inf}}[\frac{1}{p}]$ generated by the elements $\unicode[STIX]{x1D709}^{s}/s!$ with $s\leqslant m$ . In particular, $A_{\text{cris}}^{(m)}\cong A_{\text{inf}}$ for $m<p$ . In contrast, if $m\geqslant p$ , then, since $\unicode[STIX]{x1D707}^{p}/p!\in A_{\text{cris}}^{(m)}$ , the $p$ -adic and $(p,\unicode[STIX]{x1D707})$ -adic topologies of $A_{\text{cris}}^{(m)}$ agree. By its definition, $A_{\text{cris}}^{(m)}$ is $p$ -torsion free; in fact, although we will not use this, Proposition 5.36 below implies that $A_{\text{cris}}^{(m)}$ is even a domain. The map $\unicode[STIX]{x1D703}$ of (2.1.3) extends to  $A_{\text{cris}}^{(m)}$ :

(3.26.1) $$\begin{eqnarray}\unicode[STIX]{x1D703}:A_{\text{cris}}^{(m)}{\twoheadrightarrow}{\mathcal{O}}_{C}.\end{eqnarray}$$

Due to the ‘finite-type nature’ of the $A_{\text{inf}}$ -algebra $A_{\text{cris}}^{(m)}$ , more precisely, due to [Reference Bhatt, Morrow and ScholzeBMS18, 12.8(ii)], the systems of ideals

Equivalently,

(3.26.2) $$\begin{eqnarray}\text{for every}\quad n\geqslant 1,\quad \text{the map}\quad (A_{\text{cris}}^{(m)}/p^{n^{\prime }})[\unicode[STIX]{x1D707}]\rightarrow A_{\text{cris}}^{(m)}/p^{n}\quad \text{vanishes for large}\quad n^{\prime }>n.\end{eqnarray}$$

Therefore, by taking the inverse limit over $n$ of the sequences

(3.26.3) $$\begin{eqnarray}0\rightarrow (A_{\text{cris}}^{(m)}/p^{n})[\unicode[STIX]{x1D707}]\rightarrow A_{\text{cris}}^{(m)}/p^{n}\xrightarrow[{}]{\unicode[STIX]{x1D707}}A_{\text{cris}}^{(m)}/p^{n}\rightarrow A_{\text{cris}}^{(m)}/(\unicode[STIX]{x1D707},p^{n})\rightarrow 0,\end{eqnarray}$$

we conclude that

(3.26.4)

The Frobenius automorphism of $A_{\text{inf}}$ preserves the subring $A_{\text{cris}}^{0,\,(m)}\subset A_{\text{inf}}[\frac{1}{p}]$ : indeed, for $m\geqslant p$ , since $\unicode[STIX]{x1D709}=\sum _{i=0}^{p-1}[\unicode[STIX]{x1D716}^{i/p}]$ and $\unicode[STIX]{x1D709}^{p}\in pA_{\text{cris}}^{0,\,(m)}$ , we have $\unicode[STIX]{x1D711}(\unicode[STIX]{x1D709})=\sum _{i=0}^{p-1}[\unicode[STIX]{x1D716}^{i}]$ and $\unicode[STIX]{x1D711}(\unicode[STIX]{x1D709})\in pA_{\text{cris}}^{0,\,(m)}$ . Thus, the Frobenius induces a ring endomorphism

$$\begin{eqnarray}\unicode[STIX]{x1D711}:A_{\text{cris}}^{(m)}\rightarrow A_{\text{cris}}^{(m)},\end{eqnarray}$$

which, via the map $\unicode[STIX]{x1D703}$ , intertwines the absolute Frobenius of ${\mathcal{O}}_{C}/p$ (compare with (2.1.3)).

3.27 The $A(R)$ -algebras $A_{\text{cris}}^{(m)}(R)$

The ‘relative version’ of $A_{\text{cris}}^{(m)}$ (respectively, a ‘highly ramified cover’ of this relative version) is the $A(R)$ -algebra (respectively,  $\mathbb{A}_{\text{inf}}(R_{\infty })$ -algebra)

$$\begin{eqnarray}A_{\text{cris}}^{(m)}(R):=A(R)\,\widehat{\otimes }_{A_{\text{inf}}}\,A_{\text{cris}}^{(m)}\quad (\text{respectively,}\quad \mathbb{A}_{\text{cris}}^{(m)}(R_{\infty }):=\mathbb{A}_{\text{inf}}(R_{\infty })\,\widehat{\otimes }_{A_{\text{inf}}}\,A_{\text{cris}}^{(m)}),\end{eqnarray}$$

where the completion is $(p,\unicode[STIX]{x1D707})$ -adic (equivalently, $p$ -adic if $m\geqslant p$ ). In the case $m<p$ , due to Lemma 3.13 and §3.26, we have $A_{\text{cris}}^{(m)}(R)\cong A(R)$ and $\mathbb{A}_{\text{cris}}^{(m)}(R_{\infty })\cong \mathbb{A}_{\text{inf}}(R_{\infty })$ .

Due to the decomposition (3.14.5), the subring $A_{\text{cris}}^{(m)}(R)\subset \mathbb{A}_{\text{cris}}^{(m)}(R_{\infty })$ is an $A_{\text{cris}}^{(m)}(R)$ -module direct summand. Explicitly, the decomposition of $\mathbb{A}_{\text{inf}}(R_{\infty }^{\Box })$ described in §3.14 gives the decomposition

(3.27.1)

where the completion is $(p,\unicode[STIX]{x1D707})$ -adic (equivalently, $p$ -adic if $m\geqslant p$ ), and $\mathbb{A}_{\text{cris}}^{(m)}(R_{\infty })$ is $(p,\unicode[STIX]{x1D707})$ -adically formally étale over $\mathbb{A}_{\text{cris}}^{(m)}(R_{\infty }^{\Box })$ (see §3.14). In particular, (3.26.2) holds with $A_{\text{cris}}^{(m)}$ replaced by $\mathbb{A}_{\text{cris}}^{(m)}(R_{\infty }^{\Box })$ , and hence also by $\mathbb{A}_{\text{cris}}^{(m)}(R_{\infty })$ . Consequently, the generalization of (3.26.4) holds too:

(3.27.2)

In addition, by (3.27.1) and the formal étaleness, each $\mathbb{A}_{\text{cris}}^{(m)}(R_{\infty })$ is $p$ -torsion free. By §§3.14 and 3.26, the rings $A_{\text{cris}}^{(m)}(R)$ and $\mathbb{A}_{\text{cris}}^{(m)}(R_{\infty })$ come equipped with compatible $A_{\text{cris}}^{(m)}$ -semilinear Frobenius endomorphisms that are compatible as $m$ varies.

The group $\unicode[STIX]{x1D6E5}$ acts continuously, Frobenius-equivariantly, and $A_{\text{cris}}^{(m)}$ -linearly on $A_{\text{cris}}^{(m)}(R)$ and $\mathbb{A}_{\text{cris}}^{(m)}(R_{\infty })$ . For each $\unicode[STIX]{x1D6FF}\in \unicode[STIX]{x1D6E5}$ , the $A_{\text{inf}}$ -module endomorphism $\frac{\unicode[STIX]{x1D6FF}-1}{\unicode[STIX]{x1D707}}$ of $A(R)$ induces an $A_{\text{cris}}^{(m)}$ -module endomorphism $\frac{\unicode[STIX]{x1D6FF}-1}{\unicode[STIX]{x1D707}}$ of $A_{\text{cris}}^{(m)}(R)$ that satisfies $\unicode[STIX]{x1D6FF}=1+\unicode[STIX]{x1D707}\,\cdot \,\frac{\unicode[STIX]{x1D6FF}-1}{\unicode[STIX]{x1D707}}$ . In particular, $\unicode[STIX]{x1D6E5}$ acts trivially on $A_{\text{cris}}^{(m)}(R)/\unicode[STIX]{x1D707}$ .

3.28 The $A_{\text{cris}}^{(m)}$ -base change of the edge map

Since $A_{\text{cris}}^{(m)}\cong A_{\text{inf}}$ for $m<p$ , for the sake of analyzing the map $e\,\widehat{\otimes }_{A_{\text{inf}}}^{\mathbb{L}}\,A_{\text{cris}}^{(m)}$ , where $e$ is as in (3.15.1), we suppose that $m\geqslant p$ . Then, for each $n>0$ , we have $A_{\text{cris}}^{(m)}/p^{n}\cong A_{\text{cris}}^{(m)}/(p^{n},\unicode[STIX]{x1D707}^{n^{\prime }})$ for every large enough $n^{\prime }>0$ (see §3.26). Consequently, since $(p^{n},\unicode[STIX]{x1D707}^{n^{\prime }})$ is an $\mathbb{A}_{\text{inf}}(R_{\infty })$ -regular sequence with $\mathbb{A}_{\text{inf}}(R_{\infty })/(p^{n},\unicode[STIX]{x1D707}^{n^{\prime }})$ flat over $A_{\text{inf}}/(p^{n},\unicode[STIX]{x1D707}^{n^{\prime }})$ (see Lemma 3.13), the projection formula [SP, 0944] and Lemma 3.7 imply that

$$\begin{eqnarray}\textstyle R\unicode[STIX]{x1D6E4}_{\text{cont}}(\unicode[STIX]{x1D6E5},\mathbb{A}_{\text{inf}}(R_{\infty }))\,\widehat{\otimes }_{A_{\text{inf}}}^{\mathbb{L}}\,A_{\text{cris}}^{(m)}\cong R\unicode[STIX]{x1D6E4}_{\text{cont}}(\unicode[STIX]{x1D6E5},\mathbb{A}_{\text{cris}}^{(m)}(R_{\infty })).\end{eqnarray}$$

Consequently, the edge map $e$ defined in (3.15.1) gives rise to the map

(3.28.1) $$\begin{eqnarray}e\,\widehat{\otimes }_{A_{\text{inf}}}^{\mathbb{L}}\,A_{\text{cris}}^{(m)}:R\unicode[STIX]{x1D6E4}_{\text{cont}}(\unicode[STIX]{x1D6E5},\mathbb{A}_{\text{cris}}^{(m)}(R_{\infty }))\rightarrow R\unicode[STIX]{x1D6E4}_{\text{pro}\acute{\text{e}}\text{t}}(\mathfrak{X}_{C}^{\text{ad}},\mathbb{A}_{\text{inf}})\,\widehat{\otimes }_{A_{\text{inf}}}^{\mathbb{L}}\,A_{\text{cris}}^{(m)}.\end{eqnarray}$$

Since $[\mathfrak{m}^{\flat }]$ kills each $H^{i}(\operatorname{Cone}(e))$ (see §3.15) and $[\mathfrak{m}^{\flat }]^{2}=[\mathfrak{m}^{\flat }]$ , by using a free $A_{\text{inf}}$ -module resolution of $A_{\text{cris}}^{(m)}/p^{n}$ and the definition [SP, 064M], we see that $[\mathfrak{m}^{\flat }]$ also kills each $H^{i}(\operatorname{Cone}(e)\otimes _{A_{\text{inf}}}^{\mathbb{L}}A_{\text{cris}}^{(m)}/p^{n})$ . Consequently, by [SP, 0D6K], the ideal $[\mathfrak{m}^{\flat }]A_{\text{inf}}$ kills each $H^{i}(\operatorname{Cone}(e)\,\widehat{\otimes }_{A_{\text{inf}}}^{\mathbb{L}}\,A_{\text{cris}}^{(m)})$ , to the effect that, by Lemma 3.17 (and [Reference Bhatt and ScholzeBS15, 3.4.4 and 3.4.14]), so does $W(\mathfrak{m}^{\flat })$ . In conclusion,

(3.28.2) $$\begin{eqnarray}W(\mathfrak{m}^{\flat })\quad \text{kills the cohomology modules of}\quad \operatorname{Cone}(e\,\widehat{\otimes }_{A_{\text{inf}}}^{\mathbb{L}}\,A_{\text{cris}}^{(m)})\simeq \operatorname{Cone}(e)\,\widehat{\otimes }_{A_{\text{inf}}}^{\mathbb{L}}\,A_{\text{cris}}^{(m)}.\end{eqnarray}$$

By applying Lemma 3.18, we will show in Theorem 3.34 that $L\unicode[STIX]{x1D702}_{(\unicode[STIX]{x1D707})}(e\,\widehat{\otimes }_{A_{\text{inf}}}^{\mathbb{L}}\,A_{\text{cris}}^{(m)})$ is an isomorphism. Thus, we need to know that the $A_{\text{inf}}$ -modules $H_{\text{cont}}^{i}(\unicode[STIX]{x1D6E5},\mathbb{A}_{\text{cris}}^{(m)}(R_{\infty })/\unicode[STIX]{x1D707})$ have no nonzero $W(\mathfrak{m}^{\flat })$ -torsion (compare with Proposition 3.19 for $\mathbb{A}_{\text{inf}}(R_{\infty })/\unicode[STIX]{x1D707}$ ). The following result is a step in that direction.

Proof. By the $p$ -adic completeness of $\mathbb{A}_{\text{cris}}^{(m)}(R_{\infty })/\unicode[STIX]{x1D707}$ (see (3.27.2)), we may focus on the rings $\mathbb{A}_{\text{cris}}^{(m)}(R_{\infty })/(\unicode[STIX]{x1D707},p^{n})$ . The argument for the latter is similar to that of [Reference Bhatt, Morrow and ScholzeBMS18, 12.8(iii)] and uses approximation by Noetherian rings. Namely, by the $(p,\unicode[STIX]{x1D711}^{-1}(\unicode[STIX]{x1D707}))$ -adic completeness of $A_{\text{inf}}$ , the assignment

(3.29.1)

By [Reference Bhatt, Morrow and ScholzeBMS18, 4.31], this makes $A_{\text{inf}}$ a faithfully flat $\mathbb{Z}_{p}\unicode[STIX]{x27E6}T\unicode[STIX]{x27E7}$ -algebra. Thus, letting $M$ be the mod $((T+1)^{p}-1,p^{n})$ reduction of the $\mathbb{Z}_{p}\unicode[STIX]{x27E6}T\unicode[STIX]{x27E7}$ -subalgebra of $\mathbb{Z}_{p}\unicode[STIX]{x27E6}T\unicode[STIX]{x27E7}[\frac{1}{p}]$ generated by the $\frac{1}{s!}(\sum _{i=0}^{p-1}(T+1)^{i})^{s}$ with $s\leqslant m$ , we have the identification

$$\begin{eqnarray}\mathbb{A}_{\text{cris}}^{(m)}(R_{\infty })/(\unicode[STIX]{x1D707},p^{n})\cong M\otimes _{\mathbb{ Z}_{p}\unicode[STIX]{x27E6}T\unicode[STIX]{x27E7}/((T+1)^{p}-1,\,p^{n})}\mathbb{A}_{\text{inf}}(R_{\infty })/(\unicode[STIX]{x1D707},p^{n}).\end{eqnarray}$$

The $(\mathbb{Z}_{p}\unicode[STIX]{x27E6}T\unicode[STIX]{x27E7}/((T+1)^{p}-1,p^{n}))$ -flatness of $\mathbb{A}_{\text{inf}}(R_{\infty })/(\unicode[STIX]{x1D707},p^{n})$ ensures that the $\unicode[STIX]{x1D711}^{-1}(\unicode[STIX]{x1D707})$ -torsion submodule of $\mathbb{A}_{\text{cris}}^{(m)}(R_{\infty })/(\unicode[STIX]{x1D707},p^{n})$ is the base change of the $T$ -torsion submodule $M[T]\subset M$ . Consequently, since $\unicode[STIX]{x1D711}^{-1}(\unicode[STIX]{x1D707})\in W(\mathfrak{m}^{\flat })$ , the consideration of the $p$ -adic filtration of $M[T]$ reduces us to proving that

which follows from Lemma 3.12. ◻

To relate $H_{\text{cont}}^{i}(\unicode[STIX]{x1D6E5},\mathbb{A}_{\text{cris}}^{(m)}(R_{\infty })/\unicode[STIX]{x1D707})$ to $H_{\text{cont}}^{i}(\unicode[STIX]{x1D6E5},\mathbb{A}_{\text{inf}}(R_{\infty })/\unicode[STIX]{x1D707})$ in Proposition 3.33, we will use the following general result about exactness properties of $p$ -adically completed tensor products. For concreteness, we state it for $A_{\text{inf}}$ and its algebra $A_{\text{cris}}^{(m)}$ , but the proof is not specific to these choices.

Lemma 3.30. For a fixed $m\geqslant p$ , consider the following condition on an $A_{\text{inf}}$ -module $L$ :

(⋆)

which means that for every $j,n$ , the map $\operatorname{Tor}_{j}^{A_{\text{inf}}}(L,A_{\text{cris}}^{(m)}/p^{n^{\prime }})\rightarrow \operatorname{Tor}_{j}^{A_{\text{inf}}}(L,A_{\text{cris}}^{(m)}/p^{n})$ vanishes for some $n^{\prime }>n$ . For a bounded complex

$$\begin{eqnarray}M^{\bullet }=\ldots \rightarrow M^{i}\xrightarrow[{}]{d^{i}}M^{i+1}\rightarrow \ldots\end{eqnarray}$$

of $A_{\text{inf}}$ -modules, if each $M^{i}$ and each $H^{i}(M^{\bullet })$ satisfy ( $\star$ ), then, for every $i$ , we have

(3.30.1) $$\begin{eqnarray}\textstyle H^{i}(M^{\bullet }\,\widehat{\otimes }_{A_{\text{inf}}}\,A_{\text{cris}}^{(m)})\cong \mathop{\varprojlim }\nolimits_{n}(H^{i}(M^{\bullet }\otimes _{A_{\text{inf}}}A_{\text{cris}}^{(m)}/p^{n}))\cong H^{i}(M^{\bullet })\,\widehat{\otimes }_{A_{\text{inf}}}\,A_{\text{cris}}^{(m)}.\end{eqnarray}$$

Proof. For an inverse system $\{0\rightarrow I_{n}^{\prime }\rightarrow I_{n}\rightarrow I_{n}^{\prime \prime }\rightarrow 0\}\text{}_{n>0}$ of short exact sequences of abelian groups, $\{I_{n}\}\text{}_{n>0}$ is Mittag–Leffler with vanishing eventual images if and only if so are both $\{I_{n}^{\prime }\}\text{}_{n>0}$ and $\{I_{n}^{\prime \prime }\}\text{}_{n>0}$ . Therefore, the short exact sequences

(3.30.2) $$\begin{eqnarray}0\rightarrow \operatorname{Ker}(d^{i})\rightarrow M^{i}\rightarrow \operatorname{Im}(d^{i})\rightarrow 0\quad \text{and}\quad 0\rightarrow \operatorname{Im}(d^{i-1})\rightarrow \operatorname{Ker}(d^{i})\rightarrow H^{i}(M^{\bullet })\rightarrow 0\end{eqnarray}$$

imply, by descending induction on $i$ , that each $\operatorname{Ker}(d^{i})$ and each $\operatorname{Im}(d^{i})$ satisfy ( $\star$ ). Consequently, these sequences remain short exact after applying $-\,\widehat{\otimes }_{A_{\text{inf}}}\,A_{\text{cris}}^{(m)}$ , to the effect that the flanking terms of (3.30.1) get identified. By construction, this identification is compatible with the canonical maps to $\mathop{\varprojlim }\nolimits_{n}(H^{i}(M^{\bullet }\otimes _{A_{\text{inf}}}A_{\text{cris}}^{(m)}/p^{n}))$ , so it remains to establish the second identification in (3.30.1).

By [SP, 0662 and 0130], the spectral sequences associated to a double complex give the following spectral sequences that converge to $H^{i+j}(M^{\bullet }\otimes _{A_{\text{inf}}}^{\mathbb{L}}A_{\text{cris}}^{(m)}/p^{n})$ :

$$\begin{eqnarray}\vphantom{E}^{(n)}E_{2}^{ij}=H^{i}(H^{j}(M^{\bullet })\otimes _{A_{\text{inf}}}^{\mathbb{L}}A_{\text{cris}}^{(m)}/p^{n})\quad \text{and}\quad \vphantom{E}^{(n)\prime }E_{1}^{ij}=H^{j}(M^{i}\otimes _{A_{\text{inf}}}^{\mathbb{L}}A_{\text{cris}}^{(m)}/p^{n}),\end{eqnarray}$$

where the differential on the $\vphantom{E}^{(n)\prime }E_{1}$ -page is $H^{j}(d^{i}\otimes _{A_{\text{inf}}}^{\mathbb{L}}A_{\text{cris}}^{(m)}/p^{n})$ . As $n$ varies, both families of spectral sequences form inverse systems. Moreover, by assumption, the systems $\{\vphantom{E}^{(n)}E_{2}^{ij}\}\text{}_{n>0}$ with $i\neq 0$ and $\{\vphantom{E}^{(n)\prime }E_{1}^{ij}\}\text{}_{n>0}$ with $j\neq 0$ are Mittag–Leffler with vanishing eventual images. This persists to the subsequent pages: namely, by the first sentence of the proof, to the systems $\{\vphantom{E}^{(n)}E_{s}^{ij}\}\text{}_{n>0}$ with $i\neq 0$ and $\{\vphantom{E}^{(n)\prime }E_{s}^{ij}\}\text{}_{n>0}$ with $j\neq 0$ for any $s\leqslant \infty$ . Consequently, the edge maps

$$\begin{eqnarray}H^{i}(M^{\bullet })\otimes A_{\text{cris}}^{(m)}/p^{n}\rightarrow H^{i}(M^{\bullet }\otimes ^{\mathbb{L}}A_{\text{cris}}^{(m)}/p^{n})\quad \text{and}\quad H^{i}(M^{\bullet }\otimes ^{\mathbb{L}}A_{\text{cris}}^{(m)}/p^{n})\rightarrow H^{i}(M^{\bullet }\otimes A_{\text{cris}}^{(m)}/p^{n})\end{eqnarray}$$

become isomorphisms after applying the functor $\mathop{\varprojlim }\nolimits_{n}$ . It remains to note that then so does their composition, which is the canonical map $H^{i}(M^{\bullet })\otimes _{A_{\text{inf}}}A_{\text{cris}}^{(m)}/p^{n}\rightarrow H^{i}(M^{\bullet }\otimes _{A_{\text{inf}}}A_{\text{cris}}^{(m)}/p^{n})$ .◻