Proof.

1. (1) Firstly $Y_{ {\mathrm {pro\acute {e}t}}}$ admits finite projective limits due to [Reference ScholzeSch13, Lemma 3.10(vii)]. By [Reference HuberHub96, §3.5], there is an equivalence of sites $\mathfrak {Y}_{ {\mathrm {\acute {e}t}}} \cong \mathfrak {Y}_{\text {red}, {\mathrm {\acute {e}t}}}$ (idem. for $\mathfrak {X}_{ {\mathrm {\acute {e}t}}}$ ), where $\mathfrak {Y}_{\text {red}} = (\mathfrak {Y}, \mathcal {O}_{\mathfrak {Y}}/\mathscr {I})$ is the reduced scheme underlying $\mathfrak {Y}$ (here $\mathscr {I}$ is the vanishing sheaf of ideals of $\mathcal {O}_{\mathfrak {Y}}$ ). Now $\mathfrak {Y}_{\text {red}, {\mathrm {\acute {e}t}}}$ has a final object (namely $\mathfrak {Y}_{\text {red}}$ ) and fiber products exist. Thus by [Sta18, Tag 04AS, Lemma 4.18.4(1) $\kern2.00em \iff \kern2.00em$ (3)] $\mathfrak {Y}_{ {\mathrm {\acute {e}t}}}$ (idem. for $\mathfrak {X}_{ {\mathrm {\acute {e}t}}}$ ) admits finite projective limits.

2. (2) Both functors preserve final objects and commute with fiber products. Therefore they also commute with finite projective limits. With the identification $\mathfrak {Y}_{ {\mathrm {\acute {e}t}}} \cong \mathfrak {Y}_{\text {red}, {\mathrm {\acute {e}t}}}$ (idem. for $\mathfrak {X}_{ {\mathrm {\acute {e}t}}}$ ) from (1), we get a morphism of sites $\mathfrak {X}_{ {\mathrm {\acute {e}t}}} \to \mathfrak {Y}_{ {\mathrm {\acute {e}t}}}$ by [Sta18, Tag 04I0]. There is a natural morphism of sites $\nu \colon Y_{ {\mathrm {pro\acute {e}t}}} \to Y_{ {\mathrm {\acute {e}t}}}$ , so it suffices to show that the natural functor $\mathfrak {Y}_{ {\mathrm {\acute {e}t}}} \to Y_{ {\mathrm {\acute {e}t}}}$ induces a morphism of sites $Y_{ {\mathrm {\acute {e}t}}} \to \mathfrak {Y}_{ {\mathrm {\acute {e}t}}}$ . This is [Reference HuberHub96, Lemma 3.5.1(ii)].

Notation. We will refer to coverings of the above by ‘type (a)’, ‘type (b)’, ‘type (c)’ and ‘type (d)’, respectively.

Proof.

1. (1) Choose any object $T = (W \to \mathfrak {U} \leftarrow \mathfrak {V})$ in D and fix an affine covering $\{\mathfrak {U}_i \}$ of $\mathfrak {U}$ in $\mathfrak {U}_{ {\mathrm {\acute {e}t}}}$ . Then $\{\mathfrak {U}_i \times _{\mathfrak {U}} \mathfrak {V} \}$ is a covering of $\mathfrak {V}$ in $\mathfrak {V}_{ {\mathrm {\acute {e}t}}}$ . Therefore

   $$\begin{align*}\{(W \to \mathfrak{U} \leftarrow \mathfrak{U}_i \times_{\mathfrak{U}} \mathfrak{V}) \} \end{align*}$$

   is a covering of T of type (b). Then by using coverings of type (d):

   

   where U and $U_i$ are the adic generic fibers of $\mathfrak {U}$ and $\mathfrak {U}_i$ , respectively, we can assume $\mathfrak {U}$ is affine. Using coverings of type (b) again, we can assume $\mathfrak {V}$ is affine. Finally, by using coverings of type (a) and [Reference ScholzeSch13, Lemma 3.12(ii)], proves that the family is generating. For the last part, note that taking the adic generic fiber commutes with fiber products of formal schemes. Therefore fiber products in D are calculated component-wise and now it is clear that the family is stable under fiber products.

2. (2) By part (1), the objects appearing there are generating, so we can consider the site whose objects are the objects appearing in (1) together with the coverings generated by types (a)-(d). This is a site of definition for $Y_{ {\mathrm {pro\acute {e}t}}}\times _{\mathfrak {Y}_{ {\mathrm {\acute {e}t}}}}\mathfrak {X}_{ {\mathrm {\acute {e}t}}}$ . By [Reference ScholzeSch13, Lemma 3.12(i)], $\varprojlim _{i \in I} {\mathrm {Spa}}(S_{i}, S_{i}^{+})$ is a quasi-compact object in $Y_{ {\mathrm {pro\acute {e}t}}}$ , and $ {\mathrm {Spf}}(R)$ is quasi-compact in $\mathfrak {X}_{ {\mathrm {\acute {e}t}}}$ . It follows that all covering families of type (a) or (b) of Z have a finite refinement.

3. (3) By parts (1) and (2), and the criterion of [SGA4, Exposé VI, Proposition 2.1], all objects as in (1) are coherent. We check that (i ter) of Proposition 2.2 in loc. cit. is satisfied. For this we further restrict ourselves to objects of the form

   $$\begin{align*}Z':= (\varprojlim_{i \in I} {\mathrm{Spa}}(S_{i}, S_{i}^{+}) \to {\mathrm{Spf}}(S) \leftarrow {\mathrm{Spf}}(R)) \end{align*}$$

   such that $ {\mathrm {Spf}}(S) \to \mathfrak {Y}$ (resp. $ {\mathrm {Spf}}(R) \to \mathfrak {X}$ ) factors over an affine open subset $\mathfrak {Y}' \subset \mathfrak {Y}$ (resp. $\mathfrak {X}' \subset \mathfrak {X}$ ). This is still a generating family for D and

   $$\begin{align*}Z' \times_{(Y \to \mathfrak{Y} \leftarrow \mathfrak{X})} Z' = Z'\times_{(Y ' \to \mathfrak{Y}' \leftarrow \mathfrak{X}')} Z', \end{align*}$$

   where $Y'$ is the (affinoid) adic generic fiber of $\mathfrak {Y}'$ .

4. (4) By part (3) the topos $Y_{ {\mathrm {pro\acute {e}t}}}\times _{\mathfrak {Y}_{ {\mathrm {\acute {e}t}}}}\mathfrak {X}_{ {\mathrm {\acute {e}t}}}$ is algebraic. Moreover by [Reference ScholzeSch13, Proposition 3.12(iii)], $Y_{ {\mathrm {pro\acute {e}t}}}$ is an algebraic topos. We use the criterion in [SGA4, Exposé VI, Proposition 3.2] to check that p is a coherent morphism. By [Reference ScholzeSch13, Proposition 3.12(ii)-(iii)], objects of the form $\varprojlim _{i \in I} {\mathrm {Spa}}(S_{i}, S_{i}^{+})$ are coherent and are a generating family for $Y_{ {\mathrm {pro\acute {e}t}}}$ . We can further restrict to the class $\varprojlim _{i \in I} {\mathrm {Spa}}(S_{i}, S_{i}^{+})$ that have the additional property that $\varprojlim _{i \in I} {\mathrm {Spa}}(S_{i}, S_{i}^{+}) \to Y$ factors over an open affinoid $Y' \subset Y$ where $Y'$ is the adic generic fiber of some open affine $\mathfrak {Y}' \subset \mathfrak {Y}$ . This is still a generating family for $Y_{ {\mathrm {pro\acute {e}t}}}$ . It suffices to check that

   (2.5) $$ \begin{align} p^{-1}(\varprojlim_{i \in I} {\mathrm{Spa}}(S_{i}, S_{i}^{+})) = (\varprojlim_{i \in I} {\mathrm{Spa}}(S_{i}, S_{i}^{+}) \to \mathfrak{Y} \leftarrow \mathfrak{X}) \end{align} $$

   is a coherent object of $Y_{ {\mathrm {pro\acute {e}t}}}\times _{\mathfrak {Y}_{ {\mathrm {\acute {e}t}}}}\mathfrak {X}_{ {\mathrm {\acute {e}t}}}$ . For this we will use the criterion in [SGA4, Exposé VI, Corollaire 1.17]. We have a covering of type (c):

   

   Since the morphism $\mathfrak {X} \to \mathfrak {Y}$ is quasi-compact and quasi-separated, it follows that $\mathfrak {X}' \subset \mathfrak {X}$ is quasi-compact and quasi-separated. Thus by using coverings of type (b), we obtain a finite covering of (2.5)

   $$\begin{align*}\{ Z_j:= (\varprojlim_{i \in I} {\mathrm{Spa}}(S_{i}, S_{i}^{+}) \to \mathfrak{Y}' \leftarrow \mathfrak{X}_j) \to (\varprojlim_{i \in I} {\mathrm{Spa}}(S_{i}, S_{i}^{+}) \to \mathfrak{Y} \leftarrow \mathfrak{X}) \}_{j \in J} \end{align*}$$

   where each $\mathfrak {X}_j \subset \mathfrak {X}'$ is affine open (in particular by part (3) each $Z_j$ is coherent). Moreover for $j, k \in J$

   (2.6) $$ \begin{align} Z_j \times_{(\varprojlim_{i \in I} {\mathrm{Spa}}(S_{i}, S_{i}^{+}) \to \mathfrak{Y} \leftarrow \mathfrak{X})} Z_k = (\varprojlim_{i \in I} {\mathrm{Spa}}(S_{i}, S_{i}^{+}) \to \mathfrak{Y}' \leftarrow \mathfrak{X}_j \cap \mathfrak{X}_k). \end{align} $$

   Since $\mathfrak {X}'$ is quasi-separated, $\mathfrak {X}_j \cap \mathfrak {X}_k$ is a finite union of affine opens. By using coverings of type (b) and [SGA4, Exposé VI, Proposition 1.3], it follows that the object in (2.6) is quasi-compact.

Proof. This is proved in the same way as the first part of Proposition 2.7(1), except one uses that affinoid perfectoids form a basis for the pro-étale topology (cf. [Reference ScholzeSch13, Proposition 4.8]).

Proof. This is essentially proven in [Reference ScholzeSch12, §5]. Recall that $(-)^a$ denotes the functor sending an $\mathcal {O}$ -module to the corresponding almost module, and $(-)_*$ is its right adjoint. Lemma 5.3(ii) in loc. cit. gives $T^{\circ } \subset (T^{\circ a})_{*} \subset T$ , which implies

(2.7) $$ \begin{align} (T^{\circ a})_{*}[\tfrac{1}{p}] = T. \end{align} $$

By Proposition 5.5 in loc. cit., $T^{\circ a}$ is a perfectoid $\mathcal {O}^{a}$ -algebra and so by Lemma 5.6 in loc. cit.

(2.8) $$ \begin{align} (T^{\circ a})_{*}[\tfrac{1}{p}]^{\circ} = (T^{\circ a})_{*}. \end{align} $$

Therefore (2.7)-(2.8) give $T^{\circ } = (T^{\circ a})_{*}$ . Therefore by Lemma 5.3(iii) in loc. cit.

$$\begin{align*}T^{\circ}/x = (T^{\circ a})_{*}/x \subset (T^{\circ a}/x)_{*}. \end{align*}$$

Finally by Proposition 4.4 in loc. cit. $(T^{\circ a}/x)_{*} = {\mathrm {Hom}}_{\mathcal {O}^{a}}(\mathcal {O}^{a}, T^{\circ a}/x)$ has no non-zero $\mathfrak {m}$ -torsion.

Proof. Since Y is locally of finite type over $ {\mathrm {Spa}} (C, \mathcal {O})$ , it follows that $ {\mathrm {Spa}}(S_i, S_i^{+})$ is topologically of finite type over $ {\mathrm {Spa}} (C, \mathcal {O})$ and so by [Reference HuberHub94, Lemma 4.4] $S_i^{+} = S_i^{\circ }$ . The idea of the proof is to modify the terms in the colimit $\varinjlim _{i \in I} S_i^{\circ }$ , so that the projection morphisms $S_i^{\circ } \to S_{\infty }$ become well behaved. A clue is that $S_{\infty }$ is reduced (as it is a perfectoid ring, cf.[Reference Bhatt, Morrow and ScholzeBMS18, p. 255]).

For each $i \in I$ , let $S_{i, \text {red}}$ be the quotient of $S_i$ by its nilradical. The quotient morphism $S_i \to S_{i, \text {red}}$ induces a morphism of adic spaces $ {\mathrm {Spa}}(S_{i, \text {red}}, S_{i, \text {red}}^\circ ) \to {\mathrm {Spa}}(S_i, S_i^{\circ })$ which is an isomorphism on the underlying topological spaces (note that $S_{i, \text {red}}^{\circ }$ is the integral closure of $S_i^{\circ }$ in $S_{i, \text {red}}$ ). Furthermore for sufficiently large $j \geq i$ we claim that the diagram

is cartesian; indeed we may assume the ring morphism $S_i \to S_j$ is finite étale (and so $S_{j, \text {red}} = S_{i, \text {red}} \otimes _{S_i} S_j$ ) and $ {\mathrm {Spa}} (S_j, S_j^{\circ })\to {\mathrm {Spa}} (S_i, S_i^{\circ })$ is surjective. Moreover, again by Lemma 4.4 in loc. cit., $S_{j, \text {red}}^{\circ }$ is the integral closure of $S_{i, \text {red}}^{\circ } \otimes _{S_i^{\circ }} S_j^{\circ }$ in $S_{j, \text {red}}$ . In particular the morphism $ {\mathrm {Spa}}(S_{j, \text {red}}, S_{j, \text {red}}^\circ ) \to {\mathrm {Spa}}(S_{i, \text {red}}, S_{i, \text {red}}^\circ )$ is surjective, as surjectivity is preserved by base change. This implies that for each $n \geq 1$ , $S_{i, \text {red}}^{\circ }/p^n \to S_{j, \text {red}}^{\circ }/p^n$ is injectiveFootnote ⁷ and therefore after taking limits, the same is true for $S_{i, \text {red}}^{\circ } \to S_{j, \text {red}}^{\circ }$ (for each $i \in I$ , $S_{i, \text {red}}^{\circ }$ is reduced and so by [Reference Bosch, Güntzer and RemmertBGR84, §6.2.4, Theorem 1], $S_{i, \text {red}}^{\circ }$ is bounded and in particular a ring of definition for $S_{i, \text {red}}$ equipped with the p-adic topology).

The morphisms $S_i \to S_{\infty }[\tfrac {1}{p}]$ factor through $S_{i, \text {red}}$ (as $S_{\infty }[\tfrac {1}{p}]$ is a perfectoid C-algebra and in particular reduced [Reference ScholzeSch12, Proposition 5.8]). Therefore $S_i^{\circ } \to S_{\infty }$ factors through $S_{i, \text {red}}^{\circ }$ . We will need to compare $\varinjlim _{i \in I} S_i^{\circ }$ and $\varinjlim _{i \in I} S_{i, \text {red}}^{\circ }$ . For this we show:

By Lemma 2.14, it follows that $S_{\infty }$ coincides with the p-adic completion of $\varinjlim _{i \in I} S_{i, \text {red}}^{\circ }$ : indeed we obtain a surjection $\varinjlim _{i \in I} S_i^{\circ } \to \varinjlim _{i \in I} S_{i, \text {red}}^{\circ }$ which induces a surjection on the p-adic completions (cf. [Sta18, Tag 0315]). Finally the composition $(\varinjlim _{i \in I} S_i^{\circ })^{\wedge } \to (\varinjlim _{i \in I} S_{i, \text {red}}^{\circ })^{\wedge } \to S_{\infty }$ is an isomorphism.

Since $S_{i, \text {red}}^{\circ }$ is a subring of $S_{i, \text {red}}$ (a C-algebra), it follows that $S_{i, \text {red}}^{\circ }$ is torsionfree as an $\mathcal {O}$ -module and hence flat. Moreover by the finiteness theorem of Grauert-Remmert (cf. [Reference Bosch, Güntzer and RemmertBGR84, §6.4.1, Corollary 5]), it follows that $S_{i, \text {red}}^{\circ }$ is topologically of finite type over $\mathcal {O}$ (cf. [Reference ScholzeSch13, Theorem 9.4]). By a flattening theorem of Raynaud-Gruson, flatness together with topologically of finite type implies topologically of finite presentation (cf. [Reference BoschBos14, §7.3, Corollary 5]) and so $S_{i, \text {red}}^{\circ }$ is topologically of finite presentation over $\mathcal {O}$ .

Therefore, by [Reference Bhatt, Morrow and ScholzeBMS18, Lemma 8.10]Footnote ⁸ , $S_{i, \text {red}}^{\circ }$ is the p-adic completion of a free $\mathcal {O}$ -module. In particular for each $0 \not = x \in \mathfrak {m}$ , the $\mathcal {O}$ -modules $S_{i, \text {red}}^{\circ }/x$ have no nonzero $\mathfrak {m}$ -torsion. The surjectivity of $ {\mathrm {Spa}}(S_{j, \text {red}}, S_{j, \text {red}}^\circ ) \to {\mathrm {Spa}}(S_{i, \text {red}}, S_{i, \text {red}}^\circ )$ implies $S_{i, \text {red}}^{\circ }/x \to S_{j, \text {red}}^{\circ }/x$ is injectiveFootnote ⁹ . It follows that the direct limit $\varinjlim _{i \in I} S_{i, \text {red}}^{\circ }/x$ has no nonzero $\mathfrak {m}$ -torsion. Therefore $S_{\infty }/x$ has no nonzero $\mathfrak {m}$ -torsion. The result now follows from Remark 2.12.

Proof. This is essentially the same argument as [Reference BhattBha18, Lemma 4.9] (in the case $S = \mathcal {O}$ the argument in loc. cit. shows that it can be arranged so that the first morphism $ {\mathrm {Spf}}(R) \to {\mathrm {Spf}}(S \{t_1^{\pm 1}, \ldots , t_d^{\pm 1} \})$ is finite étale).

Proof. The first statement follows from Proposition 2.8 together with Lemma 2.15 (by using coverings of type (b)). Finally, the last statement follows from Lemma 2.13.

Proof. Note that $Rs_{*}\mathbb {A}_{ {\mathrm {inf}},X}^{W} = \mathbb {A}_{ {\mathrm {inf}},X}$ (as p-adic derived completion commutes with forgetting W) and since $\nu _f \circ s = r \circ v_{f}^{W}$ we obtain $R\nu _{f*}\mathbb {A}_{ {\mathrm {inf}},X} = Rr_{*}R\nu _{f*}^{W}\mathbb {A}_{ {\mathrm {inf}},X}^{W}$ . Therefore $A\Omega _{\mathfrak {X}/\mathfrak {Y}} = L\eta _{\mu }Rr_{*}R\nu _{f*}^{W}\mathbb {A}_{ {\mathrm {inf}},X}^{W}$ . Pick a $p^{-1}((\mu ))$ -torsion-free resolution $C^{\bullet }$ of $R\nu _{f*}^{W}\mathbb {A}_{ {\mathrm {inf}},X}^{W}$ in the sense of [Reference Bhatt, Morrow and ScholzeBMS18, pg. 288]. Since $r_*$ is an exact functor (because it is identity at the level of topoi), it follows that $r_{*}C^{\bullet }$ is a $(\mu )$ -torsion free complex representing $Rr_{*}C^{\bullet }$ . Thus $L\eta _{\mu }Rr_{*}C^{\bullet }$ is represented by the complex $\eta _{\mu }r_{*}C^{\bullet }$ . Finally note that $\eta _{\mu }r_{*} = r_{*}\eta _{p^{-1}((\mu ))}$ .

Proof. From the exact sequence

$$\begin{align*}0 \to W(\mathfrak{m}^\flat) \to A_{{\mathrm{inf}}} \to W(k) \to 0 \end{align*}$$

it suffices to show $\text {Tor}_{A_{ {\mathrm {inf}}}}^1(W(k), A_{ {\mathrm {inf}}}/\tilde {\xi })$ vanishes. The image of $\tilde {\xi }$ in $W(k)$ is p. Since $W(k)$ is p-torsion free, the result follows.

Proof. By [Sta18, Tag 0BLX], it suffices to show that for each affinoid perfectoid $U \in X_{ {\mathrm {pro\acute {e}t}}}$ , $R\Gamma (U, \mathbb {A}_{ {\mathrm {inf}}, X})$ is derived $(p, \mu )$ -adically complete. It is certainly derived p-adically complete by definition, so it suffices to show it is derived $\mu $ -adically complete. Since we are in the presheaf topos (it is replete), by [Reference Bhatt, Morrow and ScholzeBMS18, Lemma 6.15], it suffices to check each $H^{i}(U, \mathbb {A}_{ {\mathrm {inf}},X})$ is derived $\mu $ -adically complete. Now $\mu \in W(\mathfrak {m}^{\flat })$ and $H^{i}(U, \mathbb {A}_{ {\mathrm {inf}},X})$ is killed by $W(\mathfrak {m}^{\flat })$ for $i>0$ (cf. the statement just after [Reference Bhatt, Morrow and ScholzeBMS18, Lemma 5.6]). Therefore $H^{i}(U, \mathbb {A}_{ {\mathrm {inf}},X})$ is derived $\mu $ -adically complete for $i>0$ . For $i = 0$ , one has $H^{0}(U, \mathbb {A}_{ {\mathrm {inf}},X}) = W(R^{\flat +})$ where $ {\mathrm {Spa}}(R,R^{+})$ is the perfectoid space determined by U and this is even classically $\mu $ -adically complete.

Proof. Since $\phi ^{-1} \circ R\phi _{*} \cong {\mathrm {id}}$ , this is an immediate consequence of Lemma 2.27 (note that $\tilde {\xi } \in (p,\mu )$ ).

Notation. For $K \in D(X_{ {\mathrm {pro\acute {e}t}}}, \mathbb {Z}_p)$ we set $K_n := K \otimes ^{\mathbb {L}}_{\mathbb {Z}_p} \mathbb {Z}/p^n$ . We write $\underline {M}$ for the image of $M \in D(\mathbb {Z}_p)$ under the pullback $D(\mathbb {Z}_p) \to D(X_{ {\mathrm {pro\acute {e}t}}}, \mathbb {Z}_p)$ .

Proof.

1. (1) Since X is quasi-compact and pro-étale morphisms are open, it follows that we can find $a \leq b$ such that $K_1$ has tor amplitude in $[a,b]$ . For the remaining $K_n$ , the result follows from [Sta18, Tag 0942].

2. (2) The proof is the same as [Sta18, Tag 094I].

3. (3) Since K is p-adically derived complete, $K=R\varprojlim K_n$ . A priori, by part (2) each $K_n$ has finite locally constant cohomology sheaves on $X_{ {\mathrm {pro\acute {e}t}}}$ , but by Remark 3.2, they can be considered as finite locally constant sheaves on $X_{ {\mathrm {\acute {e}t}}}$ . The result now follows from [Reference ScholzeSch13, Theorem 4.9] and [Sta18, Tag 0A09].

4. (4) This follows from parts (1) and (3).

5. (5) This is an application of [Sta18, Tag 0D7U] to Remark 3.2.

Proof. Indeed by part (5) of Lemma 3.3, it suffices to verify the conditions of [Sta18, Tag 0D7U] (comparing finite locally constant sheaves on $X_{ {\mathrm {\acute {e}t}}}$ and $X_{\text {f} {\mathrm {\acute {e}t}}}$ ). This is achieved in [Reference ScholzeSch13, Theorem 4.9].

Proof. We will use the results of Lemma 3.3 without mention. The idea is to first trivialize $K_1$ for a finite étale covering $U_1 \to X$ and then construct a sequence of finite étale covers $U_{n+1} \to U_n$ trivializing $K_{n+1}$ . We proceed by induction on n.

Let us deal with $K_1$ first. We produce a finite étale cover $U_{1} \to X$ and a perfect complex of $\mathbb {Z}/p$ -modules $M_1$ such that $K_1 \lvert _{U_{1}} \cong \underline {M_1}\lvert _{U_{1}}$ . By Lemma 3.4, first note that $K_1$ comes (via pullback) of a complex in $D(X_{ {\mathrm {f\acute {e}t}}}, \mathbb {Z}/p)$ which has finite locally constant cohomology sheaves (in $X_{ {\mathrm {f\acute {e}t}}}$ ). By [Sta18, Tag 094G] we can choose a finite étale cover $U_{1} \to X$ (with $U_{1}$ affinoid connected) such that $K_1 \lvert _{U_{1}} \cong \underline {M_{1}}\lvert _{U_{1}}$ for some finite complex of finite $\mathbb {Z}/p$ -modules $M_{1}$ .

Suppose we have trivialized $K_n$ via surjective finite étale maps $U_{n} \to U_{1}$ with each $U_{n}$ affinoid connected. Then repeating the argument for $K_1$ with X replaced by $U_n$ , we produce a surjective finite étale map $U_{(n+1)} \to U_{n}$ trivializing $K_{n+1}$ with $U_{(n+1)}$ affinoid connected.

In summary we have produced a sequence of finite étale connected (affinoid) covers $U_{n+1} \to U_n$ , trivializing $K_{n+1}$ . The pro-object $U := \varprojlim U_n$ then gives a pro-(finite étale) cover trivializing all $K_n$ .

Proof. For the first part, note that the universal cover $X_{\infty }$ covers U (appearing in Lemma 3.6). Thus by Lemma 3.6, $K_n \lvert _{X_{\infty }} \cong \underline {M_n}$ for some perfect complex $M_n \in D(\mathbb {Z}/p^n)$ . Fix such a trivialization. Then by naturality it suffices to show that the canonical morphism

$$\begin{align*}\underline{R\Gamma(X_{\infty}, \underline{M_n})} \to \underline{M_n} \lvert_{X_{\infty}} \end{align*}$$

is a quasi-isomorphism. This follows because taking sections of constant sheaves of $\mathbb {Z}/p^n$ -modules over $X_{\infty }$ is exact (cf. proof of [Reference ScholzeSch12, Theorem 4.9]) and $X_{\infty }$ is connected (i.e. $H^{0}(X_{\infty }, \mathbb {F}_p) = \mathbb {F}_p$ ). Finally, perfectness of $M_n$ implies the perfectness of $R\Gamma (X_{\infty }, K_n)$ .

Proof. Working locally, we can assume that $X = {\mathrm {Spa}}(A, A^{+})$ is affinoid connected. Then by Corollary 3.7,

(3.2) $$ \begin{align} K \lvert_{X_{\infty}} = R\varprojlim K_n\vert_{X_{\infty}} \cong R\varprojlim \underline{R\Gamma(X_{\infty}, K_n)}. \end{align} $$

By [Sta18, Tag 09AW] we can find a finite complex of finite free $\mathbb {Z}_p$ -modules M and isomorphisms $M \otimes ^{\mathbb {L}}_{\mathbb {Z}_p} \mathbb {Z}/p^n \cong R\Gamma (X_{\infty }, K_n)$ compatible with the transition map $R\Gamma (X_{\infty }, K_{n+1}) \to R\Gamma (X_{\infty }, K_n)$ . Therefore (3.2) implies

$$\begin{align*}K \lvert_{X_{\infty}} \cong R\varprojlim (\underline{M}\lvert_{X_{\infty}} \otimes^{\mathbb{L}}_{\mathbb{Z}_p} \mathbb{Z}/p^n) \cong \widehat{\mathbb{Z}}_p \otimes_{\mathbb{Z}_p}^{\mathbb{L}} \underline{M}\lvert_{X_{\infty}}. \end{align*}$$

Proof. The localization morphism $A_{ {\mathrm {inf}}} \to A_{ {\mathrm {inf}}}[\tfrac {1}{\mu }]$ is flat, so the claim is a consequence of [Reference Bhatt, Morrow and ScholzeBMS18, Lemma 6.14].

Proof. This requires a slightly more delicate analysis compared to its uncompleted version in Lemma 3.10. Let $C^{\bullet }$ be the cone of the canonical morphism $A\Omega _{\mathfrak {X}/\mathfrak {Y}} \to R\nu _{f*}\mathbb {A}_{ {\mathrm {inf}}, X}$ given by [Reference Bhatt, Morrow and ScholzeBMS18, Lemma 6.10]. For each $m \geq 0$ , Lemma 6.9 in loc. cit. provides a map $(\mu )^{\otimes m} \otimes _{A_{ {\mathrm {inf}}}} \tau ^{[0,m]}R\nu _{f*}\mathbb {A}_{ {\mathrm {inf}}, X} \to \tau ^{[0,m]}A\Omega _{\mathfrak {X}/\mathfrak {Y}}$ which is part of the compositions

(3.6) $$ \begin{align} (\mu)^{\otimes m} \otimes_{A_{{\mathrm{inf}}}} \tau^{[0,m]}A\Omega_{\mathfrak{X}/\mathfrak{Y}} \to (\mu)^{\otimes m} \otimes_{A_{{\mathrm{inf}}}} \tau^{[0,m]}R\nu_{f*}\mathbb{A}_{{\mathrm{inf}}, X} \to \tau^{[0,m]}A\Omega_{\mathfrak{X}/\mathfrak{Y}} \end{align} $$

and

(3.7) $$ \begin{align} (\mu)^{\otimes m} \otimes_{A_{{\mathrm{inf}}}} \tau^{[0,m]}R\nu_{f*}\mathbb{A}_{{\mathrm{inf}}, X} \to \tau^{[0,m]}A\Omega_{\mathfrak{X}/\mathfrak{Y}} \to \tau^{[0,m]}R\nu_{f*}\mathbb{A}_{{\mathrm{inf}}, X} \end{align} $$

Moreover both (3.6)-(3.7) are just the identity tensored with the inclusion $(\mu )^{\otimes m} \hookrightarrow A_{ {\mathrm {inf}}}$ . Let $C_m$ be the cone of the truncated map $\tau ^{[0,m]}A\Omega _{\mathfrak {X}/\mathfrak {Y}} \to \tau ^{[0,m]}R\nu _{f*}\mathbb {A}_{ {\mathrm {inf}}, X}$ . Then by (3.6)-(3.7), $H^{i}(C_m)$ is killed by $\mu ^{2m}$ and so $H^{m}(C^{\bullet })$ is killed by $\mu ^{2m+2}$ . The Tor-spectral sequence then shows that $H^{m}(C^{\bullet } \otimes _{\mathbb {Z}}^{\mathbb {L}} \mathbb {Z}/p^n)$ is killed by $\mu ^{4m+6}$ for each $n \geq 1$ . By functoriality one obtains $R^{i}\varprojlim _{n}(H^{m}(C^{\bullet } \otimes _{\mathbb {Z}}^{\mathbb {L}} \mathbb {Z}/p^n))$ is also killed by $\mu ^{4m+6}$ for all $i \geq 0, n \geq 1$ . The spectral sequence

$$\begin{align*}R^{i}\varprojlim_{n}(H^{m}(C^{\bullet} \otimes_{\mathbb{Z}}^{\mathbb{L}} \mathbb{Z}/p^n)) \implies H^{i+m} (R\varprojlim_{n}(C^{\bullet} \otimes^{\mathbb{L}}_{\mathbb{Z}} \mathbb{Z}/p^n)) \end{align*}$$

then shows that $H^{m}(C^{\bullet \wedge })$ is killed by some power of $\mu $ , where $C^{\bullet \wedge }$ is the derived p-adic completion of $C^{\bullet }$ . Finally note that the cone of (3.5) is $C^{\bullet \wedge } \otimes ^{\mathbb {L}}_{A_{ {\mathrm {inf}}}} A_{ {\mathrm {inf}}}[\tfrac {1}{\mu }]$ which is thus quasi-isomorphic to the zero complex.

Proof. By Proposition 3.8, it suffices to show $Rf_{\eta , {\mathrm {pro\acute {e}t}}*}\widehat {\mathbb {Z}}_p \in D(Y_{ {\mathrm {pro\acute {e}t}}}, \mathbb {Z}_p)$ is lisse. We need to check that the conditions of Definition 3.1 are satisfied. By [Sta18, Tag 0A0G] it is p-adically derived complete. Moreover by the projection formula (cf. [Sta18, Tag 0944]) and [Reference ScholzeSch13, Corollary 3.17(ii)]

$$\begin{align*}Rf_{\eta, {\mathrm{pro\acute{e}t}}*}\widehat{\mathbb{Z}}_p \otimes^{\mathbb{L}}_{\mathbb{Z}_p} \mathbb{Z}/p \cong \nu^{*}Rf_{\eta, {\mathrm{\acute{e}t}}*}(\mathbb{Z}/p), \end{align*}$$

where $\nu \colon Y_{ {\mathrm {pro\acute {e}t}}} \to Y_{ {\mathrm {\acute {e}t}}}$ is the natural morphism of sites. By [Reference Scholze and WeinsteinSW20, Theorem 10.5.1], $Rf_{\eta , {\mathrm {\acute {e}t}}*}(\mathbb {Z}/p)$ has finite locally constant cohomology sheaves. Furthermore by base change (cf. [Reference HuberHub96, Proposition 2.6.1]) and vanishing of cohomology in large enough degrees (cf. [Reference ScholzeSch13, Theorem 5.1]), $Rf_{\eta , {\mathrm {\acute {e}t}}*}(\mathbb {Z}/p)$ has finite tor-dimension. This proves the lemma.

Proof. Let $\mathscr {C}$ be the cone. We will prove the statement in four steps.

1. Step 1: ‘ $[\mathfrak {m}^{\flat }]$ kills $H^{i}(\mathscr {C}^{\bullet }_n)(U)$ ’. Note that $H^{i}(\mathscr {C}^{\bullet }_n)$ sits in an exact sequence

   (3.9) $$ \begin{align} H^{i}(\mathscr{A}^{\bullet}_n) \to H^{i}(\mathscr{B}^{\bullet}_n) \to H^{i}(\mathscr{C}^{\bullet}_n) \to H^{i+1}(\mathscr{A}^{\bullet}_n) \to H^{i+1}(\mathscr{B}^{\bullet}_n). \end{align} $$

   The kernel and cokernel presheaves of the first and last arrows of (3.9) are killed by $[\mathfrak {m}^{\flat }]$ . Since sheafification preserves the property

   (3.10) $$ \begin{align} `\text{local sections killed by } [\mathfrak{m}^{\flat}]' \end{align} $$

   it follows that the cokernel of the first and last arrows is killed by $[\mathfrak {m}^{\flat }]$ . It is easy to check that (3.10) is also stable under extensions and this proves that $[\mathfrak {m}^{\flat }]$ kills $H^{i}(\mathscr {C}^{\bullet }_n)(U)$ .

2. Step 2: ‘ $[\mathfrak {m}^{\flat }]$ kills $H^{i}(U, \mathscr {C}^{\bullet }_n)$ ’. By the Leray spectral sequence (cf. [Sta18, Tag 0732])

   $$\begin{align*}H^{p}(U, H^q(\mathscr{C}^{\bullet}_n)) \implies H^{p+q}(U, \mathscr{C}^{\bullet}_n), \end{align*}$$

   it suffices to show that each $H^{p}(U, H^q(\mathscr {C}^{\bullet }_n))$ is killed by $[\mathfrak {m}^{\flat }]$ . Step 1 says that this is true for $p = 0$ . In other words for $m \in [\mathfrak {m}^{\flat }]$ , the multiplication map

   $$\begin{align*}m \colon H^q(\mathscr{C}^{\bullet}_n) \to H^q(\mathscr{C}^{\bullet}_n) \end{align*}$$

   is zero. By functoriality, this implies that the induced map

   $$\begin{align*}m \colon H^{p}(U, H^q(\mathscr{C}^{\bullet}_n)) \to H^{p}(U, H^q(\mathscr{C}^{\bullet}_n)) \end{align*}$$

   is also zero.

3. Step 3: ‘ $W(\mathfrak {m}^{\flat })$ kills $H^{i}(U, \mathscr {C}^{\bullet })$ ’. By [Sta18, Tag 0A0G] and [Reference Bhatt, Morrow and ScholzeBMS18, Lemma 6.15], $H^{i}(U, \mathscr {C}^{\bullet })$ is p-adically derived complete. Thus by [Reference Česnavičius and KoshikawaCK19, Lemma 3.17], it suffices to check that it is killed by $[\mathfrak {m}^{\flat }]$ . This follows from Step 2 and [Sta18, Tag 0D6K] (note that the $R^{1}\varprojlim $ of almost zero modules is almost zero).

4. Step 4: ‘ $W(\mathfrak {m}^{\flat })$ kills $H^{i}(\mathscr {C}^{\bullet })(U)$ ’. This follows from Step 3 and the fact that $H^{i}(\mathscr {C}^{\bullet })$ is the sheafification of $ U \mapsto H^{i}(U, \mathscr {C}^{\bullet })$ .

Proof. We check that the conditions of Lemma 3.13 are satisfied. By Lemma 3.12, $Rf_{\eta , {\mathrm {pro\acute {e}t}}*}\widehat {\mathbb {Z}}_p$ is an adic lisse complex on $Y_{ {\mathrm {pro\acute {e}t}}}$ . It follows that the LHS of (3.11) is p-adically derived complete. By [Sta18, Tag 0A0G] the RHS of (3.11), $Rf_{\eta , {\mathrm {pro\acute {e}t}}*}\mathbb {A}_{ {\mathrm {inf}},X}$ is p-adically derived complete.

Finally, after tensoring with $\mathbb {Z}/p^n$ and taking cohomology, (3.11) takes the shape

(3.12) $$ \begin{align} \nu^{*}R^{i}f_{\eta, {\mathrm{\acute{e}t}}*}\mathbb{Z}/p^n \otimes_{\widehat{\mathbb{Z}}_p} W(\widehat{\mathcal{O}}^{+, \flat}_{Y}) \to R^{i}f_{\eta, {\mathrm{pro\acute{e}t}}*}(W(\widehat{\mathcal{O}}^{+, \flat}_{X})/p^n). \end{align} $$

The proof of [Reference ScholzeSch13, Theorem 8.8] shows that (after taking sections) (3.12) is an almost isomorphism (w.r.t. $[\mathfrak {m}^{\flat }]$ ).

Proof. First we show the left-most quasi-isomorphism. Indeed Lemmas 3.10 and 3.11 imply that the canonical morphism

$$\begin{align*}A\Omega_{\mathfrak{X}/\mathfrak{Y}} \otimes_{A_{{\mathrm{inf}}}}^{\mathbb{L}} A_{{\mathrm{inf}}}[\tfrac{1}{\mu}] \to A\Omega_{\mathfrak{X}/\mathfrak{Y}}^{\wedge} \otimes_{A_{{\mathrm{inf}}}}^{\mathbb{L}} A_{{\mathrm{inf}}}[\tfrac{1}{\mu}] \end{align*}$$

is a quasi-isomorphism. The result now follows from [SGA4, Exposé VI, Théorème 5.1] (applied to Proposition 2.7(4)) and [Sta18, Tag 0A0G].

It remains to compare the two outer terms. Since $\mu \in W(\mathfrak {m}^{\flat })$ , this is a consequence of Lemma 3.10, Proposition 3.14 and [SGA4, Exposé VI, Théorème 5.1] (again applied to Proposition 2.7(4)).

Proof. Note first that each $S_i$ is a C-vector space and $S_i^{+} \subset S_i$ is a subring and therefore $S_i^{+}$ is a torsion-free $\mathcal {O}$ -module. The direct limit of torsion-free $\mathcal {O}$ -modules is again torsion free and so $\varinjlim _i S_i^{+}$ is a torsion-free $\mathcal {O}$ -module. Since $\mathcal {O}$ is a valuation ring by [Sta18, Tag 0539], this is equivalent to $\varinjlim _i S_i^{+}$ being a flat $\mathcal {O}$ -module. One then applies [Reference Gabber and RameroGR03, Lemma 7.1.6(i)] to conclude that the p-adic completion $S_{\infty }$ is a flat and therefore a torsion-free $\mathcal {O}$ -module.

Proof. By Lemma 4.1 (together with [Reference Gabber and RameroGR03, Lemma 7.1.6(i)]), it follows that $R_{\infty }^{\square }$ is a torsion free $\mathcal {O}$ -module. Since R is a p-completely étale $R^{\square }$ -algebra, it follows that R is the p-adic completion of an étale $R^{\square }$ -algebra $R'$ by [Sta18, Tag 04D1] and the topological invariance of étale maps. Therefore $R' \otimes _{R^{\square }} R_{\infty }^{\square }$ is a flat $R_{\infty }^{\square }$ -algebra. Hence, $R' \otimes _{R^{\square }} R_{\infty }^{\square }$ is a flat $\mathcal {O}$ -algebra. One then applies [Reference Gabber and RameroGR03, Lemma 7.1.6(i)] to conclude.

Proof. The first part follows the same idea as the proof of Corollary 4.2 and in fact we use the algebra $R'$ appearing there. Indeed $R' \otimes _{R^{\square }} R_{0}^{\square }$ is a flat $R_{0}^{\square }$ -algebra. Hence by [Reference Gabber and RameroGR03, Lemma 7.1.6(i)], so is $R \widehat {\otimes }_{R^{\square }}R^{\square }_0$ . As $R^{\square }_m$ is a finitely presented $R^{\square }_0$ -module, the result follows from [Reference Gabber and RameroGR03, Lemma 7.1.6(ii)]. For the second part note that by [Sta18, Tag 03GG] $S_{\infty }[t_1^{\pm 1/p^{m}}, \ldots , t_d^{\pm 1/p^{m}}]$ is integrally closed in $S_{\infty }[\tfrac {1}{p}][t_1^{\pm 1/p^{m}}, \ldots , t_d^{\pm 1/p^{m}}]$ and we conclude by [Reference BhattBha17, Lemma 5.1.2]. The last part follows from the second part and a similar argument.

Proof. We have $R_{\infty } \cong R \widehat {\otimes }_{R^{\square }} R_{\infty }^{\square }$ and the completed direct sum decomposition (4.2) shows that $R_{\infty }^{\square }$ has a p-adically completed direct sum decomposition into free $R^{\square }\widehat {\otimes }_{\mathcal {O}} S_{\infty }$ -modules. Therefore $R_{\infty }$ has a p-adically completed direct sum decomposition into free $R \widehat {\otimes }_{\mathcal {O}} S_{\infty }$ -modules. Since S is topologically of finite type over $\mathcal {O}$ , so is R. Moreover R is topologically of finite presentation by [Reference BoschBos14, §7.3, Corollary 5] and therefore R is the p-adic completion of a free $\mathcal {O}$ -module by [Reference Bhatt, Morrow and ScholzeBMS18, Lemma 8.10] (cf. footnote 8). The result now follows from Lemma 2.13.

Proof. Firstly by Lemma 2.11, it follows that $R_{\infty }/b$ has no nonzero $\mathfrak {m}$ -torsion if $(R_{\infty }[\tfrac {1}{p}])^{\circ } = R_{\infty }$ . With the same assumption, we will now show that $H^{i}_{ {\mathrm {cont}}}(\Delta , R_{\infty }/b)$ has no non-zero $\mathfrak {m}$ -torsion. As we are in the smooth case, we will have access to a decomposition of $M_{\infty }$ of which all of the terms will be $R \widehat {\otimes }_{R^{\square }} R_{0}^{\square }$ -modules of rank 1. This will allow us to treat each term one by one, analogous to the situation in [Reference Bhatt, Morrow and ScholzeBMS18, Proposition 8.9]. The semistable case handled in [Reference Česnavičius and KoshikawaCK19, Proposition 3.8] is slightly more delicate. With this in mind consider $S_{\infty } \cdot t_{1}^{a_1} \cdots t_d^{a_d}$ a summand of (4.2) and set $T := R_{0}^{\square } \cdot t_{1}^{a_1} \cdots t_d^{a_d}$ .

By [Reference Česnavičius and KoshikawaCK19, Lemma 3.7] (cf. [Reference Bhatt, Morrow and ScholzeBMS18, Lemma 7.3(ii)]) the $\mathcal {O}$ -module $H^{i}_{ {\mathrm {cont}}}(\Delta , T/b)$ is the i-th cohomology of the $R_{0}^{\square }/b$ -tensor product of d-complexes of the form $R_{0}^{\square }/b \xrightarrow {\zeta -1} R_{0}^{\square }/b$ for suitable p-power roots of unity $\zeta $ . By Lemma 4.1, one obtains

(4.7) $$ \begin{align} H^{i}_{{\mathrm{cont}}}(\Delta, T/b) = M \otimes_{D} R_{0}^{\square}, \end{align} $$

where M is a D-module for some discrete valuation ring $D \subset \mathcal {O}$ (the b is absorbed into the D-module M). Consider the decomposition

(4.8) $$ \begin{align} R_{\infty}/b \cong (R \widehat{\otimes}_S S_{\infty})/b \oplus \underbrace{\widehat{\bigoplus}_{T} ((R \widehat{\otimes}_{R^{\square}}R^{\square}_0) \otimes_{R^{\square}_0} T)/b}_{M_{\infty}/b}. \end{align} $$

where the direct sum runs over those T which contribute to $M_{\infty }$ . In order to show that $H^{i}_{ {\mathrm {cont}}}(\Delta , R_{\infty }/b)$ has no non-zero $\mathfrak {m}$ -torsion, we need to show that both $H^{i}_{ {\mathrm {cont}}}(\Delta , (R \widehat {\otimes }_S S_{\infty })/b)$ and $H^{i}_{ {\mathrm {cont}}}(\Delta , M_{\infty }/b)$ have no non-zero $\mathfrak {m}$ -torsion.

Since R is $R^{\square }$ -flat, [Reference Gabber and RameroGR03, Lemma 7.1.6(i)] implies $R \widehat {\otimes }_{R^{\square }}R^{\square }_0$ is $R^{\square }_0$ -flat. Moreover $(R \widehat {\otimes }_{R^{\square }}R^{\square }_0) \otimes _{R^{\square }_0} T$ is p-adically complete and so [Reference Česnavičius and KoshikawaCK19, Lemma 3.7] (cf. [Reference Bhatt, Morrow and ScholzeBMS18, Lemma 7.3(ii)]) gives

(4.9) $$ \begin{align} H^{i}_{{\mathrm{cont}}}(\Delta, ((R \widehat{\otimes}_{R^{\square}}R^{\square}_0) \otimes_{R^{\square}_0} T)/b) \cong (R \widehat{\otimes}_{R^{\square}}R^{\square}_0) \otimes_{R^{\square}_0} H^{i}_{{\mathrm{cont}}}(\Delta, T/b). \end{align} $$

Therefore by (4.7) and [Reference Česnavičius and KoshikawaCK19, Lemma 3.6], to show that $H^{i}_{ {\mathrm {cont}}}(\Delta , M_{\infty }/b)$ has no nonzero $\mathfrak {m}$ -torsion, it suffices to show that $(R \widehat {\otimes }_{R^{\square }}R^{\square }_0) \otimes _{R^{\square }_0} (M \otimes _{D} R_{0}^{\square })$ has no nonzero $\mathfrak {m}$ -torsion. This reduces to showing that $(R \widehat {\otimes }_{R^{\square }}R^{\square }_0) \otimes _{R^{\square }_0} R^{\square }_0/d \cong (R \widehat {\otimes }_{R^{\square }}R^{\square }_0)/d$ has no nonzero $\mathfrak {m}$ -torsion for some $d \in D$ . However

$$\begin{align*}(R \widehat{\otimes}_{R^{\square}}R^{\square}_0)/d \cong (R \widehat{\otimes}_S S_{\infty})\{t_1^{\pm 1}, \ldots, t_d^{\pm 1} \}/d \end{align*}$$

and since $(R \widehat {\otimes }_S S_{\infty })/d$ sits inside $R_{\infty }/d$ , it follows that $(R \widehat {\otimes }_{R^{\square }}R^{\square }_0)/d$ has no nonzero $\mathfrak {m}$ -torsion. Finally $\Delta $ acts trivially on $(R \widehat {\otimes }_S S_{\infty })/b$ and so $H^{i}_{ {\mathrm {cont}}}(\Delta ,(R \widehat {\otimes }_S S_{\infty })/b)$ has no nonzero $\mathfrak {m}$ -torsion.

It remains to show $\zeta _{p}-1$ kills the $\mathcal {O}$ -modules $H^{i}_{ {\mathrm {cont}}}(\Delta , M_{\infty })$ , without assuming $(R_{\infty }[\tfrac {1}{p}])^{\circ } = R_{\infty }$ ; we argue similarly and use the same notation. If T contributes to $M_{\infty }$ , then as in the proof of [Reference Česnavičius and KoshikawaCK19, Proposition 3.8], we conclude

(4.10) $$ \begin{align} \zeta_{p}-1 \text{ kills }H^{i}_{{\mathrm{cont}}}(\Delta, T). \end{align} $$

By (4.8)-(4.9) and [Reference Česnavičius and KoshikawaCK19, Lemma 3.6], we get the desired result.

Proof. Recall by §4.2, the edge map e is an almost isomorphism. Therefore $L\eta _{(\zeta _p - 1)}(e)$ is an almost isomorphism. By Proposition 4.5, i induces an isomorphism

$$\begin{align*}\frac{H^{i}_{{\mathrm{cont}}}(\Delta, R_{\infty})}{H^{i}_{{\mathrm{cont}}}(\Delta, R_{\infty})[\zeta_p -1]} \cong \frac{H^{i}_{{\mathrm{cont}}}(\Delta, R\widehat{\otimes}_S S_{\infty})}{H^{i}_{{\mathrm{cont}}}(\Delta, R\widehat{\otimes}_S S_{\infty})[\zeta_p -1]}. \end{align*}$$

Therefore $L\eta _{(\zeta _p - 1)}(R\Gamma _{ {\mathrm {cont}}}(i))$ is an isomorphism and hence $L\eta _{(\zeta _p - 1)}(e \circ R\Gamma _{ {\mathrm {cont}}}(i))$ is an almost isomorphism.

Suppose now $(R_{\infty }[\tfrac {1}{p}])^{\circ } = R_{\infty }$ . To show that $L\eta _{(\zeta _p - 1)}(e)$ is an isomorphism, it is enough to show that $L\eta _{(\zeta _p - 1)}(e \circ R\Gamma _{ {\mathrm {cont}}}(i))$ is an isomorphism. We will apply [Reference Bhatt, Morrow and ScholzeBMS18, Lemma 8.11(i)] (cf. [Reference Česnavičius and KoshikawaCK19, Lemma 3.4]). Proposition 4.5 ensures that the $\mathcal {O}$ -modules $H^{i}_{ {\mathrm {cont}}}(\Delta , R\widehat {\otimes }_S S_{\infty })$ have no nonzero $\mathfrak {m}$ -torsion, while [Reference Česnavičius and KoshikawaCK19, Lemma 3.7] (cf. [Reference Bhatt, Morrow and ScholzeBMS18, Lemma 7.3(ii)]) and the triviality of action of $\Delta $ show that the quotient $\tfrac {H^{i}_{ {\mathrm {cont}}}(\Delta , R\widehat {\otimes }_S S_{\infty })}{(\zeta _p -1)H^{i}_{ {\mathrm {cont}}}(\Delta , R\widehat {\otimes }_S S_{\infty })}$ is a direct sum of $\mathcal {O}$ -modules of the form $(R\widehat {\otimes }_S S_{\infty })/(\zeta _p -1)$ . These also have no nonzero $\mathfrak {m}$ -torsion. Therefore $L\eta _{(\zeta _p - 1)}(e \circ R\Gamma _{ {\mathrm {cont}}}(i))$ is an isomorphism.

Proof. By using Frobenius we can assume that $b \text { }\lvert \text { }p^{1/p^{\infty }}$ in $\mathcal {O}^{\flat }$ . Then Proposition 4.5 and the $\Delta $ -isomorphism $R_{\infty }^{\flat }/b \cong R_{\infty }/b^{\sharp }$ for some $b^{\sharp } \in \mathcal {O}$ gives the claim.

Proof. This follows from Lemma 4.7 (cf. [Reference Česnavičius and KoshikawaCK19, (3.14.1)]).