# Overconvergent prismatic cohomology

Andreas Langer  Harrison Building, University of Exeter, Exeter, EX4 4QF,
Devon, UK<EMAIL_ADDRESS>
(Date: August 06, 2023
This research was supported by EPSRC grant EP/T005351/1)

###### Abstract.

In this note I define an overconvergent version of prisms and prismatic
cohomology as introduced by Bhatt and Scholze and show that overconvergent
prismatic cohomology specialises to $p$-adic cohomologies, like Monsky-
Washnitzer resp. rigid cohomology for smooth varieties over a perfect field,
the de Rham cohomology of smooth weak formal schemes over a perfectoid ring
and the étale cohomology of its generic fibre. Besides, I give an
overconvergent version of the complex $A\Omega$ of Bhatt-Morrow-Scholze and
relate it to overconvergent prismatic cohomology.

##  0\. Introduction

In their fundamental work [BS22] Bhatt and Scholze introduced a new $p$-adic
cohomology theory, called prismatic cohomology, which has a universal
character and is based on their notion of prisms and the prismatic site. Fix a
prime number $p$. A prism is a pair $(A,I)$ that consists of a $\delta$-ring
$A$ and an ideal $I\subset A$ satisfying some properties ([BS22, Definition
1.1]) such that $A$ is $(p,I)$-adically complete. The $\delta$-structure on
$A$ induces a Frobenius lift $\varphi:A\rightarrow A$. In all cases that we
consider $I$ is a principal ideal generated by a non-zero divisor $d$. The
main examples in [BS22] are the perfect prisms (1) and (2) and the non-perfect
prism (3):

  1. (1)

$(W(k),(p))$ for a perfect field $k$ of characteristic $p$. Then $\varphi$ is
the Witt vector Frobenius and $d=p$. (It is called the crystalline prism).

  2. (2)

Let $\mathbb{C}_{p}$ be the completion of an algebraic closure of
$\mathbb{Q}_{p}$ and $\mathcal{O}=\mathcal{O}_{\mathbb{C}_{p}}$ its ring of
integers with tilt $\mathcal{O}^{\flat}=\displaystyle\varprojlim_{x\mapsto
x^{p}}\mathcal{O}/p$. There is a natural surjection

$\theta:A_{\mathrm{inf}}(\mathcal{O}):=W(\mathcal{O}^{\flat})\rightarrow\mathcal{O}$

with $\ker\theta$ a principal ideal, and let $\varphi$ be the Witt vector
Frobenius. Then $(A_{\mathrm{inf}}(\mathcal{O}),\ker\theta)$ is a perfect
prism.

An important example of a non-perfect prism is the Breuil-Kisin prism:

  1. (3)

Let $K$ be a finite totally ramified extension of $W(k)[1/p]$. Fix a
uniformiser $\pi$ and let $E(u)\in W(k)[u]$ denote the Eisenstein polynomial
such that $E(\pi)=0$. Then $(W(k)\llbracket u\rrbracket,(E(u)))$ is a prism
with $\varphi$ sending $u$ to $u^{p}$.

Now let $(A,I)$ be a prism which is bounded, i.e.
$(A/I)[p^{\infty}]\cong(A/I)[p^{n}]$ for some $n$. For a smooth $p$-adic
formal scheme $X$ over $\mathrm{Spf}(A/I)$ Bhatt and Scholze introduced the
prismatic site $(X/A)_{{{\mathbbl{\Delta}}}}$ with structure sheaf
$\mathcal{O}_{{{\mathbbl{\Delta}}}}$. An object of the prismatic site is a
bounded prism $(B,I)$ over $(A,I)$ with a map $\mathrm{Spf}(B/IB)\rightarrow
X$ satisfying a compatibility condition. Then the prismatic cohomology is
defined as

$R\Gamma_{{{\mathbbl{\Delta}}}}(X/A):=R\Gamma((X/A)_{{{\mathbbl{\Delta}}}},\mathcal{O}_{{{\mathbbl{\Delta}}}})\,.$

The main comparison results, demonstrating the universal character of
prismatic cohomology, are stated in [BS22, Theorem 1.8]. In particular, Bhatt
and Scholze show that if $X$ is smooth and proper, in the above examples
(1)-(3) $R\Gamma_{{{\mathbbl{\Delta}}}}(X/A)$ is a perfect $(p,d)$-complete
complex in $D(A)$, equipped with a $\varphi_{A}$-linear operator $\varphi$,
and

  * -

a Frobenius descent of crystalline cohomology
$R\Gamma_{\mathrm{cris}}(X/W(k))$ in case (1)

  * -

a Frobenius descent of the $A_{\mathrm{inf}}$-cohomology
$R\Gamma_{\mathrm{inf}}(X)$ defined in [BMS18] in case (2)

  * -

is isomorphic to Breuil-Kisin cohomology $R\Gamma_{\mathrm{BK}}(X)$ as defined
in [BMS19] in case (3).

Then Koshikawa and Yao [Kos22], [KY23] introduced a logarithmic version of
prismatic cohomology, by extending the notion of $\delta$-ring to the log
context, leading to the definition of logarithmic prisms and the logarithmic
prismatic site. Then they establish, in the log-smooth case, analogous
comparison results for logarithmic prismatic cohomology (log-crystalline, log-
de Rham, étale, Hodge-Tate) see [KY23, Theorem 2]. They also recover the
$A_{\mathrm{inf}}$-cohomology in the semistable case constructed by
Česnavičius and Koshikawa [ČK19] and construct Breuil-Kisin cohomology in the
semistable case by Breuil-Kisin descent along the canonical map of prisms

$(W(k)\llbracket
u\rrbracket,(E(u)))\rightarrow(A_{\mathrm{inf}}(\mathcal{O}),\ker\theta);\ \
u\mapsto[\pi^{\flat}]$

for a compatible system $\pi,\pi^{1/p},\ldots$ of $p$-power roots of $\pi$ in
$\mathcal{O}$.

The original motivation in our work was to impose an overconvergence condition
on the prismatic site such that the hypercohomology of the corresponding
overconvergent structure sheaf rationally recovers – when the base prism is
the crystalline prism – the rigid cohomology of smooth varieties over a
perfect field $k$. This naturally led to the notion of dagger prisms defined
below. Although we only treat the smooth case it is probably straightforward
to extend the comparison with rigid cohomology to the semistable case as well,
by introducing dagger versions of the logarithmic prisms and logarithmic
prismatic site of Koshikawa-Yao, which will lead to a more general definition
of overconvergent logarithmic prismatic cohomology. If one replaces
$p$-adically formal $\mathrm{Spf}\,\mathcal{O}$-schemes by weak formal schemes
over $\mathrm{Spf}\,\mathcal{O}$, then our construction of the overconvergent
prismatic site leads to dagger versions of the de Rham and étale comparison
results as stated in [BS22, Theorem 1.8], see Theorem 0.11 and Theorem 0.12
below. Using the pro-étale site on affinoid dagger varieties we will construct
a dagger version of $A_{\mathrm{inf}}$-cohomology and compare it with
overconvergent prismatic cohomology. The proof relies on an almost purity
property for overconvergent Witt vectors on perfectoid dagger algebras.

I consider this work as an another endeavour to introduce the notion of
overconvergence into the constructions of relative $p$-adic Hodge theory. It
was also inspired by the recent progress on the study of prismatic
$F$-crystals and their close connection to $p$-adic Galois representations. To
explain this, let $K$ be a $p$-adic field, $K_{\infty}$ the completion of the
infinite cyclotomic extension $K(\mu_{p^{\infty}})$ with $K_{\infty}^{\flat}$
its tilt. Let $\Gamma_{K}=\mathrm{Gal}(K_{\infty}/K)$. Fontaine established an
equivalence between the category of étale $(\varphi,\Gamma_{K})$-modules
$\mathrm{Mod}^{\varphi,\Gamma_{K},\mathrm{\acute{e}t}}_{W(K_{\infty}^{\flat})}$
and the category $\mathrm{Rep}_{\mathbb{Z}_{p}}(G_{K})$ of finite free
$\mathbb{Z}_{p}$-representations of the absolute Galois group. Wu [Wu21] gave
a prismatic approach to this equivalence by comparing both categories with the
category of prismatic $F$-crystals in
$\mathcal{O}_{{{\mathbbl{\Delta}}}}[1/I]_{p}^{\wedge}$-modules over the
absolute prismatic site $(\mathcal{O}_{K})_{{{\mathbbl{\Delta}}}}$. Then using
results of Bhatt-Scholze [BS23], Marks [Mar23] generalised the result of Wu
and gave a geometric relativisation by replacing
$\mathrm{Rep}_{\mathbb{Z}_{p}}(G_{K})$ by $\mathcal{O}_{K}$-local systems on
the generic fibre of a formal scheme over $\mathrm{Spf}\,\mathcal{O}_{L}$ ($L$
a finite extension of $\mathbb{Q}_{p}$) and relating this to the category of
Laurent $F$-crystals on the absolute prismatic site, i.e. vector bundles over
a certain structure sheaf
$\mathcal{O}_{{{\mathbbl{\Delta}}}}[1/I]_{\pi}^{\wedge}$, equipped with an
isomorphism $\varphi^{\ast}\mathcal{M}\cong\mathcal{M}$, for a fixed
uniformiser $\pi\in\mathcal{O}_{L}$. To $\pi$ one can associate a Lubin-Tate
formal group $\mathcal{G}$ over $\mathcal{O}_{L}$. Then, for a $p$-adic field
$K\supset L$, let $K_{\infty}$ be the $p$-adic completion of
$K(\mathcal{G}[\pi^{\infty}])$, $K_{\infty}^{\flat}$ its tilt and
$\Gamma_{K}=\mathrm{Gal}(K_{\infty}/K)$. Then Kisin and Ren [KR09] construct a
period ring $A_{K}\subset
W(K_{\infty}^{\flat})\otimes_{W(\mathbb{F}_{q})}\mathcal{O}_{L}$ and extend
Fontaine’s theory to Lubin-Tate $(\varphi_{q},\Gamma_{K})$-modules to obtain
an equivalence between the category of étale
$(\varphi_{q},\Gamma_{K})$-modules
$\mathrm{Mod}_{A_{K}}^{\varphi_{q},\Gamma_{K},\mathrm{\acute{e}t}}$ and the
category $\mathrm{Rep}_{\mathcal{O}_{K}}(G_{L})$ of finite free
$\mathcal{O}_{K}$-representations of $G_{L}$. Marks shows that the result of
Kisin-Ren is a special case of his result on Laurent $F$-crystals on the
absolute prismatic site ([Mar23, Theorem 1.6]). Following ideas of Cherbonnier
and Colmez [CC98], Forquaux and Xie [FX13] introduced the notion of
overconvergence to $K$-linear representations of $G_{L}$ and established an
equivalence between the category of overconvergent $K$-representations of
$G_{L}$ and étale $(\varphi_{q},\Gamma_{K})$-modules over the Robba ring
([FX13, Proposition 1.5]). It is therefore natural to search for a category of
overconvergent Laurent $F$-crystals on the (overconvergent) prismatic site of
weak formal schemes and to relate it to étale local systems on the generic
fibre in a way that recovers the result of Forquaux-Xie. This will be a future
project.

In the following, we give the basic definitions of dagger prisms and
overconvergent prismatic cohomology and will state the main results of the
paper.

We fix a perfect base prism $(A,I$) [BS22, Definition 1.1 and Example 1.3] and
assume that $I=(d)$ is principal. Our main examples are $(A,I)=(W(k),(p))$,
$k$ a perfect field of characteristic $p>0$, and
$(A,I)=(A_{\mathrm{inf}}(\mathcal{O})=W(\mathcal{O}^{\flat}),I=(d))$ where
$\mathcal{O}=\mathcal{O}_{\mathbb{C}_{p}}$, $\mathcal{O}^{\flat}$ its tilt and
$d$ a generator of the kernel of the ghost map
$\theta:W(\mathcal{O}^{\flat})\rightarrow\mathcal{O}$ (for example, let
$\epsilon=(1,\zeta_{p},\zeta_{p^{2}},\ldots)\in\mathcal{O}^{\flat}$ and
$d=1+[\epsilon^{1/p}]+\cdots+[\epsilon^{1/p}]^{p-1}$).

######  Definition 0.1.

A dagger prism of finite type over $A$ is a pair $(S,\varphi)$ where $S$ is an
$A$-algebra of finite type and $\varphi:S\rightarrow S$ is a Frobenius lift
such that the following properties hold:

  1. i)

$S$ is $J=(p,d)$-adically separated.

  2. ii)

The $J=(p,d)$-adic completion $\widehat{S}$ of $S$, together with the induced
lifting $\widehat{\varphi}$ of the Frobenius, is a bounded prism and
$A\rightarrow\widehat{S}$ is a map of prisms.

  3. iii)

There exists finitely many elements $x_{1},\ldots,x_{r}\in S$ such that for
any $s\in\widehat{S}$ we have:

     * -

$s=\sum_{\kappa\in\mathbb{N}^{r}}a_{\kappa}\underline{x}^{\kappa}$

and for $m\in\mathbb{N}$, $a_{\kappa}\in J^{m}$ up to finitely many $\kappa$.

     * -

$s\in S$ if and only if there exists $\epsilon>0$ and $C\in\mathbb{R}$ such
that

$\gamma_{\epsilon}(s):=\inf_{\kappa\in\mathbb{N}^{r}}(v_{J}(a_{\kappa})-\epsilon|\kappa|)>C\,.$

Here $v_{J}(a_{\kappa}):=n$ if $a_{\kappa}\in J^{n}$ and $a_{\kappa}\notin
J^{n+1}$.

######  Remark 0.2.

  1. i)

$S$ is weakly complete with respect to the ideal $J=(p,d)$ in the sense of
[MW68, Definition 1.2]

  2. ii)

If $\overline{S}=S/J$ is smooth over $\overline{A}=A/J$ and $S$ is a flat
$A$-algebra, then $S$ is a weak formalisation of $\overline{S}$ in the sense
of [MW68, Definition 3.2].

  3. iii)

Assume $\overline{S}$ is smooth over $\overline{A}$ or a complete transversal
intersection and $S$ is a weak formalisation. Then $S$ is a very smooth
lifting of $\overline{S}$ to $(A,I)$ in the sense of [MW68, Definition 2.5].
In particular, given a diagram

${S}$${S}$${S/J}$${S/J}$$\pi$$\overline{\varphi}$

where $\overline{\varphi}$ is the Frobenius, there exists a lifting of
Frobenius $\varphi$ on $S$ and $(\widehat{S},\varphi)$ is a prism [MW68,
Theorem 3.6 and Definition 2.4].

  4. iv)

$S/dS$ is a weakly complete $A/d$-algebra with respect to the $p$-adic
topology.

######  Definition 0.3.

An $A$-algebra is called a dagger prism if it is a direct limit $\varinjlim
S_{I}$ of dagger prisms of finite type $S_{I}$ with generators
$I=(x_{1},\ldots,x_{r})$.

######  Example 0.4.

  * -

Let $(A,(d))=(W(k),(p))$ and $\overline{S}$ a smooth $k$-algebra. Then a
Monsky-Washnitzer lift $S^{\dagger}$ of $\overline{S}$ defines a dagger prism
over $(W(k),(p))$.

  * -

Let $S$ be a dagger prism of finite type. The perfection
$\varinjlim_{\varphi}S=\varinjlim(S\xrightarrow{\varphi}S\xrightarrow{\varphi}S\xrightarrow{\varphi}\cdots)$
is a dagger prism. A variant of these perfect prisms which we will call dagger
perfections will be very important. For a dagger prism $S$ of finite type, we
can, by Lemma 1.1 below, write $S=\varinjlim_{\epsilon}S_{\epsilon}$ where
$S_{\epsilon}$ is a $(p,d)$-adically complete subring of $S$. Let
$\widehat{S}$ be the associated prism. Let
$\widehat{(S_{\epsilon}/p)^{\mathrm{perf}}}$ be the completion of the colimit
perfection of $S_{\epsilon}/p$. Define

$W^{\dagger}(\widehat{(S_{\epsilon}/p)^{\mathrm{perf}}})=\\{(z_{0},z_{1},\ldots)\in
W(\widehat{(S_{\epsilon}/p)^{\mathrm{perf}}})\,:\,\varinjlim_{i}(i+\gamma_{\epsilon}(z_{i})/p^{i})=\infty\\}\,.$

######  Definition 0.5.

With the above notation,
$S^{\dagger,\mathrm{perf}}:=\varinjlim_{\epsilon}W^{\dagger}(\widehat{(S_{\epsilon}/p)^{\mathrm{perf}}})$
is called the dagger perfection of $S$.

######  Remark 0.6.

Let $\widehat{(\widehat{S}^{\mathrm{perf}})}$ be the $(p,d)$-completed
perfection of the prism $\widehat{S}$. By [BS22, Lemma 3.9 and Theorem 3.10],
$\widehat{(\widehat{S}^{\mathrm{perf}})}=W\left(\widehat{(\widehat{S}/p)^{\mathrm{perf}}}\right)$
and we have a commutative diagram

${S}$${S^{\dagger,\mathrm{perf}}}$${\widehat{S}}$${W\left(\widehat{(\widehat{S}/p)^{\mathrm{perf}}}\right)}$

Since $S^{\dagger,\mathrm{perf}}$ is weakly complete with respect to the
$(p,d)$-adic topology and perfect, the map $S\rightarrow\widehat{S}\rightarrow
W\left(\widehat{(\widehat{S}/p)^{\mathrm{perf}}}\right)$ factors through
$S^{\dagger,\mathrm{perf}}$ and $S^{\dagger,\mathrm{perf}}$ contains the
perfection $\varinjlim_{\varphi}S$ of $S$.

In analogy to the prismatic site [BS22, §4], we now define the dagger
prismatic site.

######  Definition 0.7.

Let $(A,(d))$ be a perfect prism and $X$ a $p$-adic weak formal
$\mathrm{Spf}(A/d)$-scheme. The dagger prismatic site
$(X/A)_{{{\mathbbl{\Delta}}}}^{\dagger}$ is the opposite category of the
category of dagger prisms $(B,(d))$ over $(A,(d))$ with a map
$\mathrm{Spf}^{\dagger}(B/d)\rightarrow X$, where
$\mathrm{Spf}^{\dagger}(B/d)$ denotes the affine weak formal $p$-adic scheme
associated to the weakly complete $A/d$-algebra $B/d$. We endow an object
$(B,(d))$ with faithfully flat covers $(B,(d))\rightarrow(C,(d))$ and write an
object in $(X/A)_{{{\mathbbl{\Delta}}}}^{\dagger}$ as

$\mathrm{Spf}^{\dagger}B\leftarrow\mathrm{Spf}^{\dagger}(B/d)\rightarrow X\,.$

Let
$(X/A)_{{{\mathbbl{\Delta}}}}^{\dagger,\mathrm{perf}}\subset(X/A)_{{{\mathbbl{\Delta}}}}^{\dagger}$
be the full subcategory of objects
$\mathrm{Spf}^{\dagger}B\leftarrow\mathrm{Spf}^{\dagger}(B/d)\rightarrow X$
for which $B$ is a dagger perfection, as in Definition 0.5.

Note that the weak formal scheme $\mathrm{Spf}^{\dagger}\,B$ is well-defined:
The dagger prism $B$ defines a presheaf on the underlying formal scheme
$\mathrm{Spf}\,\widehat{B}$ in the obvious way and since the ideal of
definition is finitely generated (generated by $p$ and $d$) one can follow
Meredith’s proof [Mer72] to show that the presheaf is in fact a sheaf,
defining $\mathrm{Spf}^{\dagger}\,B$.

Our main example is as follows:

######  Example 0.8.

Let $A=W(\mathcal{O}^{\flat})$, $S=A^{\dagger}\langle U_{1}^{\pm
1},\ldots,U_{d}^{\pm 1}\rangle$. Then define $A^{\dagger}\langle U_{1}^{\pm
1/p^{\infty}},\ldots,U_{d}^{\pm 1/p^{\infty}}\rangle$ as
$S^{\dagger\mathrm{perf}}$. Note that there is a canonical isomorphism of the
$(p,d)$-completed $A$-algebra $A\langle\underline{U}^{\pm
1/p^{\infty}}\rangle\xrightarrow{\sim}W(\mathcal{O}\langle\underline{T}^{\pm
1/p^{\infty}}\rangle^{\flat})$ given by
$U_{i}^{1/p^{r}}\mapsto[(T_{i}^{1/p^{r}},T_{i}^{1/p^{r+1}},\ldots,)]$ [BMS18,
p. 70]. We define $W^{\dagger}(\mathcal{O}^{\dagger}\langle T_{1}^{\pm
1/p^{\infty}},\ldots,T_{d}^{\pm 1/p^{\infty}}\rangle^{\flat})$ as the image of
$A^{\dagger}\langle U_{1}^{\pm 1/p^{\infty}},\ldots,U_{d}^{\pm
1/p^{\infty}}\rangle$ under this isomorphism. It is a perfect $(p,d)$-weakly
complete subalgebra of $W(\mathcal{O}\langle\underline{T}^{\pm
1/p^{\infty}}\rangle^{\flat})$.

In the following we define structure sheaves on the dagger prismatic site. We
first consider the affine case, so let $X=\mathrm{Spf}^{\dagger}R$ be a weak
formal $\mathcal{O}$-scheme. We define two presheaves on
$(R/A)^{\dagger}_{{{\mathbbl{\Delta}}}}$ by

$\mathcal{O}^{\dagger}_{{{\mathbbl{\Delta}}}}(R\rightarrow B/dB\leftarrow
B)=B$

and

$\overline{\mathcal{O}}^{\dagger}_{{{\mathbbl{\Delta}}}}(R\rightarrow
B/dB\leftarrow B)=B/d\,.$

Both presheaves are equipped with an action of $\varphi$. We will show that
$\mathcal{O}^{\dagger}_{{{\mathbbl{\Delta}}}}$ and
$\overline{\mathcal{O}}^{\dagger}_{{{\mathbbl{\Delta}}}}$ are sheaves, and we
point out that in the affine case presheaf cohomology computes the correct
cohomology (see for example [Bha18a, Lecture V]).

######  Definition 0.9.

  1. a)

We define overconvergent prismatic cohomology by

${{\mathbbl{\Delta}}}_{R/A}^{\dagger}:=R\Gamma((R/A)_{{{\mathbbl{\Delta}}}}^{\dagger},\mathcal{O}_{{{\mathbbl{\Delta}}}}^{\dagger})\,.$

It is represented by a complex of dagger prisms over $A$ (this follows from
the discussion after Lemma 1.5 below).

  2. b)

Dagger-Hodge-Tate cohomology is defined as

$\overline{{{\mathbbl{\Delta}}}}_{R/A}^{\dagger}:=R\Gamma((R/A)_{{{\mathbbl{\Delta}}}}^{\dagger},\overline{\mathcal{O}}_{{{\mathbbl{\Delta}}}}^{\dagger})\,.$

This is represented by a complex of $p$-adically weakly complete
$\mathcal{O}$-algebras.

Due to the sheaf property the definition extends to weak formal
$\mathcal{O}$-schemes $X$ to get complexes
$R\Gamma((X/A)_{{{\mathbbl{\Delta}}}}^{\dagger},\mathcal{O}_{{{\mathbbl{\Delta}}}}^{\dagger})$
and likewise
$R\Gamma((X/A)_{{{\mathbbl{\Delta}}}}^{\dagger},\overline{\mathcal{O}}_{{{\mathbbl{\Delta}}}}^{\dagger})$.

Then we have the following main results.

######  Theorem 0.10.

(= Theorem 2.2, Corollary 2.3)

  1. a)

Let $\overline{R}$ be a smooth $k$-algebra with Monsky-Washnitzer lift
$R^{\dagger}$. Then we have a canonical isomorphism

${{\mathbbl{\Delta}}}^{\dagger}_{\overline{R}/W(k)}\otimes\mathbb{Q}\simeq\Omega^{\bullet}_{R^{\dagger}/W(k)}\otimes\mathbb{Q}\,.$

  2. b)

Let $X$ be a smooth $k$-scheme. Then we have an isomorphism

$R\Gamma((X/A)_{{{\mathbbl{\Delta}}}}^{\dagger},\mathcal{O}_{{{\mathbbl{\Delta}}}}^{\dagger})\otimes\mathbb{Q}\simeq
R\Gamma_{\mathrm{rig}}(X/W(k)\otimes\mathbb{Q})\,.$

######  Theorem 0.11.

(= Theorem 3.1) Let $(A,d)$ be a perfect prism, $A/d=\mathcal{O}$ and
$X/\mathcal{O}$ a smooth weak formal scheme with generic fibre
$X_{\eta}=X\times_{\mathrm{Spec}\,\mathcal{O}}\mathrm{Spec}\,\mathcal{O}[1/p]$.
Let $\mu:(X_{\eta})_{\mathrm{\acute{e}t}}\rightarrow X_{\mathrm{\acute{e}t}}$.
Then we have a canonical isomorphism

$R\mu_{\ast}\mathbb{Z}/p^{n}\simeq({{\mathbbl{\Delta}}}^{\dagger}_{X/A}[1/d]/p^{n})^{\varphi=1}\,.$

If $X=\mathrm{Spf}^{\dagger}S$ is an affine weak formal scheme with generic
fibre $\mathrm{Spec}\,S[1/p]$, then we have

$R\Gamma(\mathrm{Spec}\,S[1/p],\mathbb{Z}/p^{n})\simeq({{\mathbbl{\Delta}}}^{\dagger}_{S/A}[1/d]/p^{n})^{\varphi=1}\,.$

######  Theorem 0.12.

(= Theorem 5.14) Let $A=A_{\mathrm{inf}}(\mathcal{O})$, $X/\mathcal{O}$ a
smooth weak formal scheme. Then we have an isomorphism

$\varphi^{\ast}R\Gamma((X/A)_{{{\mathbbl{\Delta}}}}^{\dagger},\mathcal{O}_{{{\mathbbl{\Delta}}}}^{\dagger})\otimes^{L}A/[p]_{q}A\cong\Omega^{\dagger\bullet}_{X/\mathcal{O}}\,.$

######  Remark 0.13.

Thus theorems 0.11 and 0.12 are the dagger analogues of [BS22, Theorem 9.1]
and [BS22, Theorem 1.8(3)].

Finally we define an overconvergent version
$A^{\dagger}\Omega_{\mathcal{X}/\mathcal{O}}$ of the complex
$A\Omega_{\mathcal{X}/\mathcal{O}}$ of Bhatt-Morrow-Scholze, for a smooth weak
formal $\mathcal{O}$-scheme $\mathcal{X}$ (for the definition see §5) and
prove the following comparison result.

######  Theorem 0.14.

(= Theorem 5.3) Let $\mathcal{X}$ be a smooth weak formal scheme over
$\mathrm{Spf}\,\mathcal{O}$. Then we have a $\varphi$-equivariant quasi-
isomorphism

$R\Gamma(\mathcal{X},A^{\dagger}\Omega_{\mathcal{X}/\mathcal{O}})\otimes^{L\dagger}_{A_{\inf}}B_{\mathrm{cris}}^{+}\cong\varphi^{\ast}\left(R\Gamma((\mathcal{X}/A_{\inf})_{{{\mathbbl{\Delta}}}},\mathcal{O}_{{{\mathbbl{\Delta}}}}^{\dagger})\right)\otimes^{L\dagger}_{A_{\inf}}B_{\mathrm{cris}}^{+}\,.$

In the last section we define an overconvergent version
$W^{\dagger}\Omega^{\bullet}_{\mathcal{X}/\mathcal{O}}$ of the relative de
Rham-Witt complex $W\Omega^{\bullet}_{\mathcal{X}/\mathcal{O}}$, generalising
the construction in [DLZ11], and compare it with
$A\Omega^{\dagger}_{\mathcal{X}/\mathcal{O}}$.

######  Theorem 0.15.

(=Theorem 6.5) Let $\mathcal{X}$ be a weak formal smooth $\mathcal{O}$-scheme.
Then

$A\Omega^{\dagger}_{\mathcal{X}/\mathcal{O}}\otimes^{L}_{A_{\inf}}W(\mathcal{O})\cong
W^{\dagger}\Omega^{\bullet}_{\mathcal{X}/\mathcal{O}}\,.$

##  1\. Perfect dagger prisms, perfectoid dagger algebras and the structure
sheaf of the overconvergent prismatic site

In this section we prove some properties of dagger prisms and in particular
give a dagger version of an equivalence of categories due to Bhatt-Scholze
between perfect prisms and perfectoid algebras. Finally we prove the sheaf
property of the structure (pre-)sheaves
$\mathcal{O}_{{{\mathbbl{\Delta}}}}^{\dagger}$ and
$\overline{\mathcal{O}}_{{{\mathbbl{\Delta}}}}^{\dagger}$.

######  Lemma 1.1.

Let $S$ be a dagger prism of finite type with generators $x_{1},\ldots,x_{r}$.
Then $S=\varinjlim_{\epsilon}S_{\epsilon}$ where $S_{\epsilon}$ is a
$(p,d)$-adically complete subring of $S$.

###### Proof.

Define $A^{\dagger}\langle T_{1},\ldots,T_{r}\rangle$ to be the weak
completion of $A[T_{1},\ldots,T_{r}]$ with respect to the $(p,d)$-adic
topology. Then

$\lambda\,:\,A^{\dagger}\langle T_{1},\ldots,T_{r}\rangle\rightarrow
S\,;\,T_{i}\mapsto x_{i}$

is surjective. Let

$A_{\epsilon}\langle T_{1},\ldots,T_{r}\rangle=\\{f\in A\langle
T_{1},\ldots,T_{r}\rangle\,|\,f\text{ has radius of convergence
}p^{\epsilon},\epsilon>0,\text{ in }A[1/p]\langle
T_{1},\ldots,T_{r}\rangle\\}\,.$

This is a $(p,d)$-adically complete subring of $A\langle
T_{1},\ldots,T_{r}\rangle$ and we have $f\in A_{\epsilon}\langle
T_{1},\ldots,T_{r}\rangle$ for $f=\sum_{I}a_{I}\underline{T}^{I}$, ${}_{I}\in
A$ if and only if $\varinjlim_{I}(v_{J}(a_{I})-\epsilon|I|)=\infty$.
$A_{\epsilon}\langle T_{1},\ldots,T_{r}\rangle=\\{f\in A\langle
T_{1},\ldots,T_{r}\rangle$ is equipped with the Gauss norm
$\gamma_{\epsilon}(f)=\min_{I}\\{v_{J}(a_{I})-\epsilon|I|\\}>-\infty$ and
$\,A^{\dagger}\langle
T_{1},\ldots,T_{r}\rangle=\varinjlim_{\epsilon}\,A_{\epsilon}\langle
T_{1},\ldots,T_{r}\rangle$. Then $S_{\epsilon}=\lambda(A_{\epsilon}\langle
T_{1},\ldots,T_{r}\rangle)$ is a $(p,d)$-adically complete subring of $S$ and
$S=\varinjlim_{\epsilon}S_{\epsilon}$. ∎

######  Remark.

Note that $S_{\epsilon}$ is not a prism because it does not admit a lifting of
Frobenius (in general).

Note that the perfection of dagger prisms are already weakly complete with
respect to the $(p,d)$-adic topology. Indeed, let
$S=\varinjlim_{\epsilon}S_{\epsilon}$ as in Lemma 1.1. Let
$\varphi(x_{i})=x_{i}^{p}+p\delta(x_{i})$ for $i=1,\ldots,r$. Choose
$\epsilon$ such that $\gamma_{\epsilon}(p\delta(x_{i}))>0$ for all $i$. This
is possible since $\gamma_{\epsilon}(p)=1$. Then for any $s\in S_{\epsilon}$
we have $\varphi(s)\in S_{\epsilon/p}$ and hence

$\varinjlim_{\varphi}S=\varinjlim_{\epsilon}\varinjlim(S_{\epsilon}\xrightarrow{\varphi}S_{\epsilon/p}\xrightarrow{\varphi}S_{\epsilon/p^{2}}\xrightarrow{\varphi}\cdots)$

is weakly complete.

######  Definition 1.2.

An $\mathcal{O}$-algebra $R$ is a perfectoid dagger algebra if it is weakly
complete with respect to the $\pi$-adic topology for some element $\pi\in R$
with $p\in\pi^{p}R$, the Frobenius $\varphi:R/p\rightarrow R/p$ is surjective
and the $\pi$-adic completion $\widehat{R}$ is perfectoid. Moreover, we
require that $\widehat{R}$ is equipped with a family of Gauss norms
$\gamma_{\epsilon}$ such that
$R_{\epsilon}:=\\{r\in\widehat{R}\,:\,\gamma_{\epsilon}(r)\text{ is
finite}\\}$ is a perfectoid $\mathcal{O}$-algebra and such that
$R=\varinjlim_{\epsilon}R_{\epsilon}$.

Then we have a dagger version of [BS22, Theorem 3.10]:

######  Proposition 1.3.

There is an equivalence of categories between dagger perfect prisms $B$ and
perfectoid dagger $\mathcal{O}$-algebras $R$. The two functors are given as

$G\,:\,B\mapsto B/d$

and

$H\,:\,R\mapsto A_{\mathrm{inf}}^{\dagger}(R)=W^{\dagger}(R^{\flat})$

where the definition of $W^{\dagger}(R^{\flat})$ will be given below. It is a
generalisation of $W^{\dagger}(\mathcal{O}^{\dagger}\langle\underline{T}^{\pm
1/p^{\infty}}\rangle^{\flat})=H(R)$ for
$R=\mathcal{O}^{\dagger}\langle\underline{U}^{\pm 1/p^{\infty}}\rangle$.

###### Proof.

We assume that a perfectoid dagger $\mathcal{O}$-algebra $R$ can be written as
a direct limit $R=\varinjlim_{\epsilon}R_{\epsilon}$ where $R_{\epsilon}$ is a
perfectoid $\mathcal{O}$-algebra. We omit here the exact description of
$R_{\epsilon}$ which will become clear when we define an equivalence of
categories between perfectoid dagger $K$-algebras and perfectoid dagger
$K^{\flat}$-algebras for a perfectoid field $K$ and its tilt $K^{\flat}$.

Let $R_{\epsilon}^{\flat}$ be the tilt of $R_{\epsilon}$, so
$R^{\flat}_{\epsilon}=\varprojlim R_{\epsilon}/p$, and define

$B=H(R)=\varinjlim
W^{\dagger}(R_{\epsilon}^{\flat})=:A_{\mathrm{inf}}^{\dagger}(R)\,.$

It is the smallest $(p,d)$-weakly complete subring of $W(\widehat{R}^{\flat})$
containing $[x]$ for $x\in
R^{\flat}:=\varinjlim_{\epsilon}R^{\flat}_{\epsilon}$.

Conversely, let $B$ be a dagger perfect prism obtained as
$B=S^{\dagger\mathrm{perf}}$ for a dagger prism $S$. Then let
$(S/d)^{\mathrm{perfd}}=\varinjlim_{\epsilon}(S_{\epsilon}/d)^{\mathrm{perfd}}$
be the dagger perfectoidisation of $S/d$ and
$((S/d)^{\mathrm{perfd}})^{\flat}$ its tilt. Then we have

$(S/p)^{\mathrm{perfd}}=\varinjlim_{\epsilon}\widehat{(S_{\epsilon}/p)^{\mathrm{perf}}}=((S/d)^{\mathrm{perfd}})^{\flat}$

and $\varinjlim_{\epsilon}\widehat{S_{\epsilon}^{\mathrm{perf}}}$ is the
smallest $(p,d)$-weakly complete perfect $A$-subalgebra of
$W((\widehat{(S/d)^{\mathrm{perfd}}})^{\flat})$ containing $S$. By definition,
this consists of elements $\sum p^{i}[x_{i}]^{1/p^{i}}$,
$x_{i}\in(S/p)^{\mathrm{perfd}}$ such that there exists $\epsilon>0$ with
$\inf_{i}\\{i+\frac{1}{p^{i}}\gamma_{\epsilon}(x_{i})\\}>-\infty$, where
$\gamma_{\epsilon}$ is a Gauss norm with respect to the $d$-adic topology. We
define

$W^{\dagger}(((S/d)^{\mathrm{perfd}})^{\flat}):=W^{\dagger}((S/p)^{\mathrm{perfd}})=\varinjlim_{\epsilon}\widehat{S_{\epsilon}^{\mathrm{perf}}}=B$

and $G(B)=B/d=R$. This is $p$-adically weakly complete. Let
$\pi\in\widehat{R}$ be the element satisfying $\pi^{p}\neq p$ in the
perfectoid ring $\widehat{R}$. It can be chosen to being in $R$: Let
$d=a_{0}\mod p$ for $a_{0}\in R^{\flat}$. Define $\pi$ to be the image of
$[a_{0}^{1/p}]$ in $R$. Then $\pi^{p}$ divides $p$ in $R$. Evidently
$\varphi:R/p\rightarrow R/p$ is surjective and hence $R$ is dagger perfectoid.

After taking $(p,d)$-, resp. $\pi$-, adic completion, the functors $G\circ H$
and $H\circ G$ are identities. Since $G$ maps perfect dagger prisms to
perfectoid dagger $\mathcal{O}$-algebras and $H$ maps perfectoid dagger
$\mathcal{O}$-algebras to perfect dagger prisms, $G\circ H$ and $H\circ G$ are
again identities as functors. The proposition follows. ∎

We will compute the (pre-)sheaf cohomology by Čech-Alexander complexes,
explained in the next section. Now we show that
$\mathcal{O}_{{{\mathbbl{\Delta}}}}^{\dagger}$ and
$\overline{\mathcal{O}}_{{{\mathbbl{\Delta}}}}^{\dagger}$ are in fact sheaves
on $(R/A)_{{{\mathbbl{\Delta}}}}^{\dagger}$ with respect to the Grothendieck
topology generated by faithfully flat covers.

Let $(A,d)$ be a prism, and $S$ an overconvergent prism over $(A,d)$. As in
Lemma 1.1, write $S=\varinjlim_{\epsilon}S_{\epsilon}$ where $S_{\epsilon}$ is
a $(p,d)$-complete $A$-algebra. We consider the site of all $(p,d)$-complete
$A$-algebras with the topology where covers are faithfully flat
$(p,d)$-complete coverings. Now let $S\rightarrow B$ be a faithfully flat
cover of dagger prisms. Writing $B=\displaystyle\varinjlim_{\delta\rightarrow
0}B_{\delta}$ as in Lemma 1.1, we can assume that there exists
$\delta=\phi(\epsilon)$ for a continuous function
$\phi:]0,\epsilon_{0}[\rightarrow]0,\phi(\epsilon_{0})[$ with
$\displaystyle\varinjlim_{\epsilon\rightarrow 0}\phi(\epsilon)=0$, such that
$B_{\phi(\epsilon)}$ is faithfully flat over $S_{\epsilon}$. After renaming
indices $B_{\epsilon}$ is faithfully flat over $S_{\epsilon}$. By [BS22,
Corollary 3.12] the functor that sends $(S,d)\rightarrow S$ (resp. $S/d$)
forms a sheaf with vanishing higher cohomology. Indeed, consider the
$(p,d)$-complete Čech-nerve
$B^{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptscriptstyle\bullet$}}}}}}_{\epsilon}$
of $S_{\epsilon}\rightarrow B_{\epsilon}$. By the proof of [BS22, Corollary
3.12] the corresponding total complex is exact and the sheaf property holds.
Since
$B^{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptscriptstyle\bullet$}}}}}}=\displaystyle\varinjlim_{\epsilon\rightarrow
0}B_{\epsilon}^{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptscriptstyle\bullet$}}}}}}$
is the weakly completed (with respect to the $(p,d)$-adic topology) Čech-nerve
of $S\rightarrow B$, it satisfies faithfully flat descent and vanishing higher
cohomology as well, by exactness of the direct limit. The proof for $S/d$ is
similar.

Then we have

######  Proposition 1.4.

$\mathcal{O}_{{{\mathbbl{\Delta}}}}^{\dagger}$ and
$\overline{\mathcal{O}}_{{{\mathbbl{\Delta}}}}^{\dagger}$ define sheaves on
$(R/A)_{{{\mathbbl{\Delta}}}}^{\dagger}$.

The proof is straightforward from the previous arguments, using weakly
completed Čech-nerves of faithfully flat covers of dagger prisms.

We have a natural map

$v:\mathrm{Shv}(R/A)_{{{\mathbbl{\Delta}}}}^{\dagger}\rightarrow\mathrm{Shv}(\mathrm{Spf}^{\dagger}R)_{\mathrm{\acute{e}t}}$

which gives a complex of étale sheaves

${{\mathbbl{\Delta}}}^{\dagger}:=Rv_{\ast}\mathcal{O}_{{{\mathbbl{\Delta}}}}^{\dagger}\,.$

Again we can define dagger prismatic cohomology as
$R\Gamma(\mathrm{Spf}^{\dagger}R,{{\mathbbl{\Delta}}}^{\dagger})$.

By the obvious gluing process we obtain sheaves
$\mathcal{O}_{{{\mathbbl{\Delta}}}}^{\dagger}$ and
$\overline{\mathcal{O}}_{{{\mathbbl{\Delta}}}}^{\dagger}$ for any weak formal
scheme $X$ and can define overconvergent prismatic cohomology as
$R\Gamma(X,{{\mathbbl{\Delta}}}^{\dagger})$.

In analogy to prismatic cohomology we describe now the computation of
overconvergent prismatic cohomology for smooth weak formal affine schemes
$X=\mathrm{Spf}^{\dagger}R$ over $\mathrm{Spf}\,A/d$ using the Čech-Alexander
complex. We recall the following lemma from [Stacks, Tag 07JM] (see also
[Bha18a, Lecture V, Lemma 4.3]).

######  Lemma 1.5.

Let $\mathscr{C}$ be a small category admitting finite non-empty products. Let
$\mathcal{F}$ be an abelian presheaf. Assume there exists a weakly final
object $X\in\mathscr{C}$, i.e. $\mathrm{Hom}(X,Y)\neq 0$ for all
$Y\in\mathscr{C}$. Then $R\Gamma(\mathscr{C},\mathcal{F})$ is computed by the
chain complex (the totalisation of) attached to

$\mathcal{F}(X)\rightarrow\mathcal{F}(X\times
X)\rightrightarrows\mathcal{F}(X\times X\times
X)\mathrel{\begin{subarray}{c}\textstyle\rightarrow\\\\[-1.72218pt]
\textstyle\rightarrow\\\\[-1.72218pt]
\textstyle\rightarrow\end{subarray}}\cdots$

by applying $\mathcal{F}$ to the Čech-nerve of $X$.

As in [Bha18a, Lecture V, Corollary 5.2] it is proved that
$(R/A)_{{{\mathbbl{\Delta}}}}^{\dagger}$ admits finite non-empty coproducts.

There exists a free $A$-algebra (a polynomial algebra) $F_{0}$ with a
surjection $F_{0}\rightarrow R$, let $J_{0}=\ker(F_{0}\rightarrow R)$ and let
$F_{\delta}$ be the weakly completed free $\delta$-$A$-algebra on $F_{0}$,
following the construction in [BS22, 4.17]. So if
$F_{0}=A[T_{1},\ldots,T_{s}]$ then $F_{\delta}=A^{\dagger}\langle
T_{i},\delta(T_{i}),\delta^{2}(T_{i}),\ldots\rangle_{i=1}^{s}$. Let
$J_{0}=J_{0}F_{\delta}$ be the ideal in $F_{\delta}$ generated by $J_{0}$.
Then we construct the dagger prism $(F,dF)$ by taking
$F_{\delta}^{\dagger}\\{\frac{J_{0}}{d}\\}$ where $\dagger$ means weak
completion with respect to the $(p,d)$-adic topology. It has the obvious
universal property. We get the following commutative diagram

${A}$${F_{\delta}}$${F}$${A/d}$${F_{\delta}/J_{0}F_{\delta}}$${F/dF}$

where the maps on the top are $\delta$-maps. We obtain an object $X$ in
$(R/A)_{{{\mathbbl{\Delta}}}}^{\dagger}$

$X:=(R\rightarrow F/dF\leftarrow F)\,.$

Then $X$ is a weakly initial object. Indeed, for any $(R\rightarrow
B/dB\leftarrow B)\in(R/A)_{{{\mathbbl{\Delta}}}}^{\dagger}$ there exists a map
$F_{0}\rightarrow B$ of $A$-algebras compatible with $R\rightarrow B/dB$. It
extends to a map of $\delta$-$A$-algebras $F_{\delta}\rightarrow B$ and
finally to a map of dagger prisms $F\rightarrow B$ by the universal property
of $F$.

By Lemma 1.5 ${{\mathbbl{\Delta}}}_{R/A}^{\dagger}$ is computed by the
cosimplicial $\delta$-$A$-algebra

$F^{0}\rightarrow F^{1}\rightrightarrows
F^{2}\mathrel{\begin{subarray}{c}\textstyle\rightarrow\\\\[-1.72218pt]
\textstyle\rightarrow\\\\[-1.72218pt]
\textstyle\rightarrow\end{subarray}}\cdots$

where $F^{n}=\mathcal{O}_{{{\mathbbl{\Delta}}}}^{\dagger}(X^{\times(n+1)})$,
so $F=F^{0}$ and $F^{n}$ is a $d$-torsion-free $(p,d)$-weakly complete
$\delta$-ring.

It can be constructed as follows: Let
$F_{0}^{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptscriptstyle\bullet$}}}}}}$
be the $(p,d)$-weakly completed Čech-nerve of
$A\rightarrow(F_{0})^{\dagger}=F^{0}_{0}$ and let
$J^{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptscriptstyle\bullet$}}}}}}$
be the kernel of the augmentation
$F_{0}^{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptscriptstyle\bullet$}}}}}}\rightarrow
F_{0}^{0}\rightarrow R$. To each
$F_{0}^{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptscriptstyle\bullet$}}}}}}$
we apply the above construction of a weakly completed $\delta$-$A$-algebra
$F_{\delta}^{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptscriptstyle\bullet$}}}}}}$
on
$F_{0}^{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptscriptstyle\bullet$}}}}}}$
with ideal
$J^{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptscriptstyle\bullet$}}}}}}F_{\delta}^{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptscriptstyle\bullet$}}}}}}$
and take the associated prism
$F^{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptscriptstyle\bullet$}}}}}}=F_{\delta}^{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptscriptstyle\bullet$}}}}}\dagger}\\{\frac{J^{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptscriptstyle\bullet$}}}}}}}{d}\\}$
to obtain a cosimplicial object
$(F^{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptscriptstyle\bullet$}}}}}}\rightarrow
F^{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptscriptstyle\bullet$}}}}}}/dF^{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptscriptstyle\bullet$}}}}}}\leftarrow
R)$ in $(R/A)_{{{\mathbbl{\Delta}}}}^{\dagger}$ which is the Čech-nerve of
$(F^{0}\rightarrow F^{0}/dF^{0}\leftarrow R)$, the latter being the weakly
final object of the topos
$\mathrm{Shv}(R/A)_{{{\mathbbl{\Delta}}}}^{\dagger}$. Hence
${{\mathbbl{\Delta}}}_{R/A}^{\dagger}\cong\text{totalisation of
}F^{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptscriptstyle\bullet$}}}}}}$.

In the next section we derive our first comparison theorem, namely for the
base prism $(A=W(k),I=(p))$ we can compare (rational) overconvergent prismatic
cohomology of $(X/A)_{{{\mathbbl{\Delta}}}}^{\dagger}$, for a smooth
$k$-scheme $X$, with rigid cohomology.

##  2\. The comparison with Monsky-Washnitzer, respectively rigid, cohomology

Let
$\overline{A}=k[T_{1},\ldots,T_{d}]/(\overline{f}_{1},\ldots,\overline{f}_{r})$
be a smooth $k$-algebra with Monsky-Washnitzer lift
$A^{\dagger}=W(k)^{\dagger}[T_{1},\ldots,T_{d}]/(f_{1},\ldots,f_{r})$.
Consider the following frame over $W(k)$:

$B=W(k)[T_{1},\ldots,T_{d}][Y_{1},\ldots,Y_{r}]/(f_{1}-pY_{1},\ldots,f_{r}-pY_{r})\,.$

Then $B$ is an Elkik lift of
$\overline{B}=B/p=\overline{A}[Y_{1},\ldots,Y_{r}]$. Let $B^{\dagger}$ be the
Monsky-Washnitzer (= weak) completion of $B$, which is a Monsky-Washnitzer
lift of $\overline{B}$. We have

$\Omega^{\bullet}_{B^{\dagger}/W(k)}\otimes\mathbb{Q}\cong\Omega^{\bullet}_{A^{\dagger}/W(k)}\otimes\mathbb{Q}$

by [MW68, Theorem 5.4]. Now we use the ideas in [BS22, Construction 4.17,
4.18] to compute overconvergent prismatic cohomology using a Čech-Alexander
complex.

Let
$M^{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptscriptstyle\bullet$}}}}}}$
be the weak completed Čech-nerve of $W(k)\rightarrow
M^{0}=W(k)[T_{1},\ldots,T_{d}]$, so
$M^{n}=(W(k)[T_{1},\ldots,T_{d}])^{\otimes(n+1)})^{\dagger}$. Let
$J^{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptscriptstyle\bullet$}}}}}}$
be the kernel of
$M^{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptscriptstyle\bullet$}}}}}}\rightarrow
M^{0}\rightarrow\overline{A}$. Take for each
$M^{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptscriptstyle\bullet$}}}}}}$
the associated weakly completed free $\delta$-$W(k)$-algebra
$F_{\delta}^{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptscriptstyle\bullet$}}}}}}$
on
$M^{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptscriptstyle\bullet$}}}}}}$
with ideal
$J^{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptscriptstyle\bullet$}}}}}}F_{\delta}^{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptscriptstyle\bullet$}}}}}}$
and let
$B^{\dagger\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptscriptstyle\bullet$}}}}}}=F^{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptscriptstyle\bullet$}}}}}\dagger}_{\delta}\langle\frac{J^{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptscriptstyle\bullet$}}}}}}}{p}\rangle$,
which is the Čech-nerve of $B^{\dagger
0}=F_{\delta}^{0\dagger}\langle\frac{J^{0}}{p}\rangle$. Then
$(B^{\dagger\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptscriptstyle\bullet$}}}}}}\rightarrow
B^{\dagger\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptscriptstyle\bullet$}}}}}}/p\leftarrow\overline{A})$
is the Čech-nerve as cosimplicial object of $(B^{\dagger 0}\rightarrow
B^{\dagger 0}/p\leftarrow\overline{A})$ in
$(\overline{A}/W(k))_{{{\mathbbl{\Delta}}}}^{\dagger}$. Then it follows from
Lemma 1.5 and the subsequent construction that any object in
$(\overline{A}/W(k))_{{{\mathbbl{\Delta}}}}^{\dagger}$ receives a map from
$(B^{\dagger 0}\rightarrow B^{\dagger 0}/p\leftarrow\overline{A})$ hence the
latter object is a weakly initial object in
$(\overline{A}/W(k))_{{{\mathbbl{\Delta}}}}^{\dagger}$. This implies that
${{\mathbbl{\Delta}}}^{\dagger}_{\overline{A}/W(k)}$ is computed by
$B^{\dagger\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptscriptstyle\bullet$}}}}}}$.
Let $f_{1},\ldots,f_{r(n)}$ be generators of the ideal $J^{n}$. Then

$B^{\dagger n}=F^{n\dagger}_{\delta}\langle
Y_{1},\ldots,Y_{r(n)}\rangle/(f_{1}-pY_{1},\ldots,f_{r(n)}-pY_{r(n)})\,.$

Then $B^{\dagger n}\otimes_{W(k)}k=\overline{A}[Z_{1},Z_{2},\ldots]$ is a
polynomial algebra over $\overline{A}$ in infinitely many variables. It is
then clear that $B^{\dagger n}$ is a direct limit of Monsky-Washnitzer lifts
of polynomial algebras over $\overline{A}$ in finitely many variables.
Applying the argument [MW68, Theorem 5.4] as at the beginning of this section
we conclude that

$\Omega^{\bullet}_{B^{\dagger
n}/W(k)}\otimes\mathbb{Q}\cong\Omega^{\bullet}_{A^{\dagger}/W(k)}\otimes\mathbb{Q}$

for all $n$. We will see that an argument analogous to [BdJ11] implies that

${{\mathbbl{\Delta}}}_{\overline{A}/W(k)}^{\dagger}\otimes\mathbb{Q}\cong\Omega^{\bullet}_{A^{\dagger}/W(k)}\otimes\mathbb{Q}\,.$

Indeed, let $M^{r,s}=\Omega^{r}_{B^{\dagger s}/W(k)}\otimes\mathbb{Q}$, and
consider $M^{\bullet,\bullet}$ as a first quadrant double complex.

######  Lemma 2.1.

For $i>0$ the cosimplicial module
$\Omega^{i}_{B^{\dagger\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptscriptstyle\bullet$}}}}}}/W(k)}\otimes\mathbb{Q}$
is homotopy equivalent to zero.

###### Proof.

This holds for the cosimplicial module
$\Omega^{i}_{M^{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptscriptstyle\bullet$}}}}}}/W(k)}$
by taking the weak completion of
$\Omega^{i}_{P^{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptscriptstyle\bullet$}}}}}}/W(k)}$
where
$P^{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptscriptstyle\bullet$}}}}}}$
is the Čech-nerve of $W(k)\rightarrow W(k)[T_{1},\ldots,T_{d}]$ and using
[BdJ11, Lemma 2.15, Lemma 2.17]. The same argument implies the statement for
$\Omega^{i}_{F_{\delta}^{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptscriptstyle\bullet$}}}}}}/W(k)}$
and then one has to tensor the cosimplicial module
$\Omega^{i}_{F_{\delta}^{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptscriptstyle\bullet$}}}}}}/W(k)}$
with the
$F_{\delta}^{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptscriptstyle\bullet$}}}}}}$-module
$B^{\dagger\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptscriptstyle\bullet$}}}}}}$
to prove the lemma. ∎

Each column complex $M^{\bullet,s}$ is quasi-isomorphic to
$M^{\bullet,0}=\Omega^{\bullet}_{A^{\dagger}/W(k)}\otimes\mathbb{Q}$. The
complex $M^{0,\bullet}$ computes
${{\mathbbl{\Delta}}}_{\overline{A}/W(k)}^{\dagger}\otimes\mathbb{Q}$. The
total complex $\mathrm{Tot}(M^{\bullet,\bullet})$ computes the cohomology of
the de Rham complex $\Omega^{\bullet}_{A^{\dagger}/W(k)}\otimes\mathbb{Q}$ by
the first spectral sequence associated to the double complex
$M^{\bullet,\bullet}$. On the other hand, by Lemma 2.1,
$\mathrm{Tot}(M^{\bullet,\bullet})$ also computes the cohomology of
$M^{0,\bullet}$ by the second spectral sequence. Hence we have proven:

######  Theorem 2.2.

Let $\overline{R}$ be a smooth $k$-algebra with Monsky-Washnitzer lift
$R^{\dagger}$. Then we have a canonical isomorphism

${{\mathbbl{\Delta}}}_{\overline{R}/W(k)}^{\dagger}\otimes\mathbb{Q}\cong\Omega^{\bullet}_{R^{\dagger}/W(k)}\otimes\mathbb{Q}\,.$

Now we globalise the above comparison: Let $\\{U_{i}\\}$ be an affine
covering, $U_{i}=\mathrm{Spec}\,A_{i}\subset X$, of $X$, a smooth $k$-scheme.
$R\Gamma(X,{{\mathbbl{\Delta}}}_{X/W(k)}^{\dagger})$ is the total complex of

$\prod_{i}{{\mathbbl{\Delta}}}_{U_{i}/W(k)}^{\dagger}\rightrightarrows\prod_{i,j}{{\mathbbl{\Delta}}}_{U_{i}\cap
U_{j}/W(k)}^{\dagger}\mathrel{\begin{subarray}{c}\textstyle\rightarrow\\\\[-1.72218pt]
\textstyle\rightarrow\\\\[-1.72218pt]
\textstyle\rightarrow\end{subarray}}\cdots\,.$

On the other hand, we have a commutative diagram

$\displaystyle\prod_{i}{{\mathbbl{\Delta}}}_{U_{i}/W(k)}^{\dagger}\otimes\mathbb{Q}$$\displaystyle\prod_{i,j}{{\mathbbl{\Delta}}}_{U_{i}\cap
U_{j}/W(k)}^{\dagger}\otimes\mathbb{Q}$$\displaystyle\prod_{i}W^{\dagger}\Omega^{\bullet}_{U_{i}/k}\otimes\mathbb{Q}$$\displaystyle\prod_{i,j}W^{\dagger}\Omega^{\bullet}_{U_{i}\cap
U_{j}/k}\otimes\mathbb{Q}$$\cdots$$\cdots$$\simeq$$\simeq$

where the isomorphisms come from [DLZ11, Corollary 3.25]: we have isomorphisms

${{\mathbbl{\Delta}}}_{\mathrm{Spec}\,\overline{B}/W(k)}^{\dagger}\otimes\mathbb{Q}\simeq\Omega^{\bullet}_{B^{\dagger}/W(k)}\otimes\mathbb{Q}\simeq
W^{\dagger}\Omega^{\bullet}_{\overline{B}/k}\otimes\mathbb{Q}$

for any smooth $k$-algebra $\overline{B}$ with Monsky-Washnitzer lift
$B^{\dagger}$. This implies the existence of a map

$R\Gamma(X,{{\mathbbl{\Delta}}}^{\dagger}_{X/W(k)}\otimes\mathbb{Q})=$$\mathrm{Tot}\biggl{(}\displaystyle\prod_{i}{{\mathbbl{\Delta}}}_{U_{i}/W(k)}^{\dagger}\otimes\mathbb{Q}$$\displaystyle\prod_{i,j}{{\mathbbl{\Delta}}}_{U_{i}\cap
U_{j}/W(k)}^{\dagger}\otimes\mathbb{Q}$$\cdots\biggr{)}$$\mathrm{Tot}\biggl{(}\displaystyle\prod_{i}W^{\dagger}\Omega^{\bullet}_{U_{i}/k}\otimes\mathbb{Q}$$\displaystyle\prod_{i,j}W^{\dagger}\Omega^{\bullet}_{U_{i}\cap
U_{j}/k}\otimes\mathbb{Q}$$\cdots\biggr{)}$$R\Gamma(X,W^{\dagger}\Omega^{\bullet}_{X/k}\otimes\mathbb{Q})=$$\simeq$$\simeq$

due to the sheaf properties of ${{\mathbbl{\Delta}}}^{\dagger}_{X/W(k)}$ and
$W^{\dagger}\Omega^{\bullet}_{X/k}$, which by construction is an isomorphism.

######  Corollary 2.3.

Let $X$ be a smooth $k$-scheme. Then we have an isomorphism

$R\Gamma(X,{{\mathbbl{\Delta}}}^{\dagger}_{X/W(k)}\otimes\mathbb{Q})\cong
R\Gamma_{\mathrm{rig}}(X/W(k)\otimes\mathbb{Q})$

between rational overconvergent prismatic and rigid cohomology.

###### Proof.

See [DLZ11, Theorem 4.40] for the case that $X$ is quasiprojective, and
[Law18] for the general case. ∎

##  3\. Étale comparison of overconvergent prismatic cohomology

Let $(A,d)$ be a perfect prism, $A/d=\mathcal{O}$, and $X/\mathcal{O}$ a
smooth weak formal scheme with generic fibre
$X_{\eta}=X\times_{\mathrm{Spec}\,\mathcal{O}}\mathrm{Spec}\,\mathcal{O}[1/p]$.
Then we have

######  Theorem 3.1.

Let
$\mu:(X_{\eta})_{\mathrm{\acute{e}t}}\rightarrow(X)_{\mathrm{\acute{e}t}}$.
Then we have a canonical isomorphism

$R\mu_{\ast}\mathbb{Z}/p^{n}\cong\left({{\mathbbl{\Delta}}}_{X/A}^{\dagger}[1/d]/p^{n}\right)^{\varphi=1}\,.$

If $X=\mathrm{Spf}^{\dagger}S$ is affine with generic fibre
$\mathrm{Spec}\,S[1/p]$ an affinoid dagger variety, then we have

$R\Gamma(\mathrm{Spec}\,S[1/p],\mathbb{Z}/p^{n})\cong\left({{\mathbbl{\Delta}}}_{S/A}^{\dagger}[1/d]/p^{n}\right)^{\varphi=1}\,.$

######  Remark 3.2.

  * -

Theorem 3.1 is the dagger analogue of [BS22, Theorem 9.1].

  * •

Let $\widehat{X}=\mathrm{Spf}\,\widehat{S}$ be the formal smooth scheme
associated to $\mathrm{Spf}^{\dagger}S$ with generic fibre
$\mathrm{Spec}\,\widehat{S}[1/p]$, a rigid variety. Then

$R\Gamma(\mathrm{Spec}\,S[1/p],\mathbb{Z}/p^{n})\cong
R\Gamma(\mathrm{Spec}\,\widehat{S}[1/p],\mathbb{Z}/p^{n})\,.$

The étale cohomology of an affinoid dagger variety coincides with the étale
cohomology of its associated affinoid variety [CN20, Propositon 3.5]. Hence
the theorem implies that
$\left({{\mathbbl{\Delta}}}_{S/A}^{\dagger}[1/d]/p^{n}\right)^{\varphi=1}$ is
isomorphic to
$\left({{\mathbbl{\Delta}}}_{\widehat{S}/A}[1/d]/p^{n}\right)^{\varphi=1}$.

###### Proof.

Let
$C^{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptscriptstyle\bullet$}}}}}}$
be the Čech-Alexander complex that computes
${{\mathbbl{\Delta}}}_{S/A}^{\dagger}$. We have a simplicial object
$(C^{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptscriptstyle\bullet$}}}}}}\rightarrow
C^{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptscriptstyle\bullet$}}}}}}/dC^{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptscriptstyle\bullet$}}}}}}\leftarrow
S)$ in $(S/A)^{\dagger}_{{{\mathbbl{\Delta}}}}$. Each $C^{i}$ is a dagger
prism and is weakly complete with respect to the $(p,d)$-adic topology:
$C^{i}=(B_{\delta}^{i})^{\dagger}\\{\frac{J^{i}}{d}\\}$ where $B_{\delta}^{i}$
is the $(p,d)$-weakly completed $\delta$-$B^{i}$-algebra associated to $B^{i}$
and where
$B^{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptscriptstyle\bullet$}}}}}}$
is the Čech-nerve of $A\rightarrow B^{0}$, a $(p,d)$-weakly complete free
$A$-algebra with kernel $J^{0}=\ker(B^{0}\rightarrow S)$ and
$J^{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptscriptstyle\bullet$}}}}}}=\ker(B^{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptscriptstyle\bullet$}}}}}}\rightarrow
S)$. Then $C^{i}=\displaystyle\varinjlim_{\epsilon\rightarrow
0}C^{i}_{\epsilon}$ (as in Lemma 1.1). The family of Gauss norms
$\gamma_{\epsilon}$ defining $C^{i}_{\epsilon}$ can be chosen in a compatible
way on
$C^{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptscriptstyle\bullet$}}}}}}$
to get simplicial complexes
$C^{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptscriptstyle\bullet$}}}}}}_{\epsilon}$
of $(p,d)$-complete $A$-algebras with
$\varinjlim_{\epsilon}C_{\epsilon}^{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptscriptstyle\bullet$}}}}}}=C^{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptscriptstyle\bullet$}}}}}}$.
Then
$C_{\epsilon}^{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptscriptstyle\bullet$}}}}}}/p$
is $d$-adically complete. Using [BS22, Lemma 9.2], we see that
$C_{\epsilon}^{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptscriptstyle\bullet$}}}}}}/p\in
D_{\mathrm{comp}}(\mathcal{O}^{\flat}[F])$ and taking
$\varinjlim_{\epsilon}C_{\epsilon}^{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptscriptstyle\bullet$}}}}}}/p$,
$\epsilon>0$, commutes with the functors $M\mapsto M^{\varphi=1}$ and
$M\mapsto M[1/d]^{\varphi=1}$, hence

$({{\mathbbl{\Delta}}}^{\dagger}_{S/A}[1/d]/p)^{\varphi=1}=\varinjlim_{\epsilon\rightarrow
0}(C_{\epsilon}^{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptscriptstyle\bullet$}}}}}}/p)[1/d]^{\varphi=1}\,.$

Define
${{\mathbbl{\Delta}}}_{S/A,\mathrm{perf}}^{\dagger}:=C^{\bullet\dagger,\mathrm{perf}}$
using Definition 0.5. We have

$({{\mathbbl{\Delta}}}_{S/A}^{\dagger}[1/d]/p)^{\varphi=1}\xrightarrow[\varphi]{\sim}({{\mathbbl{\Delta}}}_{S/A}^{\dagger}/p[1/d])^{\varphi=1}$

hence we get

$({{\mathbbl{\Delta}}}_{S/A}^{\dagger}[1/d]/p^{n})^{\varphi=1}\xrightarrow{\sim}({{\mathbbl{\Delta}}}_{S/A,\mathrm{perf}}^{\dagger}[1/d]/p^{n})^{\varphi=1}\,.$

Indeed, without loss of generality $n=1$. Then

$({{\mathbbl{\Delta}}}_{S/A}^{\dagger}[1/d]/p)^{\varphi=1}=\varinjlim_{\epsilon\rightarrow
0}(C_{\epsilon}/p[1/d])^{\varphi=1}$

and

${{\mathbbl{\Delta}}}^{\dagger}_{S/A,\mathrm{perf}}[1/d]/p)^{\varphi=1}=\displaystyle\varinjlim_{\epsilon\rightarrow
0}((C_{\epsilon}^{\bullet}/p)^{\mathrm{perfd}}[1/d])^{\varphi=1}$

where $(C_{\epsilon}^{\bullet}/p)^{\mathrm{perfd}}$ is the $d$-completed
filtered colimit of
$C_{\epsilon}^{\bullet}/p\xrightarrow{\varphi}C_{\epsilon}^{\bullet}/p\xrightarrow{\varphi}\cdots$.
As each map acts trivially on applying $((-)[1/d])^{\varphi=1}$, [BS22, Lemma
9.2] implies that

$\displaystyle({{\mathbbl{\Delta}}}_{S/A,\mathrm{perf}}^{\dagger}[1/d]/p)^{\varphi=1}$
$\displaystyle=\varinjlim_{\epsilon\rightarrow
0}(C^{\bullet}_{\epsilon}/p[1/d])^{\varphi=1}$
$\displaystyle=({{\mathbbl{\Delta}}}^{\dagger}_{S/A}[1/d]/p)^{\varphi=1}$

as claimed.

Let $S$ be a dagger perfectoid with $p$-adic completion $\widehat{S}$.
Consider the composite map

$\alpha\,:\,{{\mathbbl{\Delta}}}_{S/A}^{\dagger}\rightarrow{{\mathbbl{\Delta}}}_{\widehat{S}/A}\cong
W(\widehat{S}^{\flat})$

where the isomorphism is derived from [BS22, Lemma 4.7]. Since
$(W^{\dagger}(S^{\flat})\rightarrow W^{\dagger}(S^{\flat})/d=S)$ is an object
in $(\mathrm{Spf}^{\dagger}S/A)_{{{\mathbbl{\Delta}}}}$, there exists a unique
map ${{\mathbbl{\Delta}}}_{S/A}^{\dagger}\rightarrow W^{\dagger}(S^{\flat})$
compatible with ${{\mathbbl{\Delta}}}_{S/A}\xrightarrow{\sim}W(S^{\flat})$.
Then the analogue of [BS22, Lemma 4.7] also holds: Let $S\rightarrow B/d$ be a
map, for a dagger prism $B$ with completion $\widehat{B}$ we have a
commutative diagram

${W^{\dagger}(S^{\flat})}$${B}$${W(S^{\flat})}$${\widehat{B}}$

where the lower horizontal arrow is induced by [BS22, Lemma 4.7] and the upper
horizontal arrow is induced by the canonical map
${{\mathbbl{\Delta}}}^{\dagger}_{S/A}\rightarrow B$ which exists by the
universal property of ${{\mathbbl{\Delta}}}^{\dagger}_{S/A}$. Hence
${{\mathbbl{\Delta}}}^{\dagger}_{S/A}\cong W^{\dagger}(S^{\flat})$.

In the following we reduce the proof of Theorem 3.1 to the case that $S$ is
dagger perfectoid.

For an affine weak formal scheme $\mathrm{Spf}\,S$ with formal completion
$\mathrm{Spf}\,\widehat{S}$ we have an isomorphism (Remark 3.2)

$F(S):=R\Gamma(\mathrm{Spec}\,S[1/p],\mathbb{Z}/p^{n})\cong
R\Gamma(\mathrm{Spec}\,\widehat{S}[1/p],\mathbb{Z}/p^{n})=:F(\widehat{S})$

and we write $S=\varinjlim_{\epsilon}S_{\epsilon}$ where $S_{\epsilon}$ is
$p$-adically complete and $\\{S_{\epsilon}\\}_{\epsilon}$ defines the dagger
structure on $\mathrm{Spf}\,\widehat{S}=\mathrm{Spf}\,S$. Then

$F(S)=\varinjlim_{\epsilon}R\Gamma(\mathrm{Spec}\,S_{\epsilon}[1/p],\mathbb{Z}/p^{n})=\varinjlim_{\epsilon}F(S_{\epsilon})\,.$

In the proof of [BS22, Theorem 9.1] a comparison map

$F(-)\rightarrow G(-):=({{\mathbbl{\Delta}}}_{-/A}[1/d]/p^{n})^{\varphi=1}$

of $\mathrm{arc}_{p}$-sheaves is constructed on
$\mathrm{fSch}_{/\mathrm{Spf}\,\mathcal{O}}$ and shown to be an isomorphism.

######  Definition 3.3.

A homomorphism of weakly complete $\mathcal{O}$-algebras $S\rightarrow T$ is
called an $\mathrm{arc}_{p}$-cover if $T=\varinjlim_{\epsilon}T_{\epsilon}$
has a dagger presentation with $p$-complete $\mathcal{O}$-algebras
$T_{\epsilon}$ such that $S_{\epsilon}\rightarrow T_{\epsilon}$ is an
$\mathrm{arc}_{p}$-cover in the sense of [BM21, Definition 6.14].

Note that $F$ is an $\mathrm{arc}_{p}$-sheaf by [BM21, Corollary 6.17] on
$\mathrm{Spf}\,S_{\epsilon}$, hence
$F(S)=\varinjlim_{\epsilon}F(S_{\epsilon})$ is an $\mathrm{arc}_{p}$-sheaf on
$\mathrm{Spf}\,S$.

Now let again $C^{\bullet}$ be the Čech-Alexander complex computing
${{\mathbbl{\Delta}}}^{\dagger}_{S/A}$ and
${{\mathbbl{\Delta}}}_{S/A,\mathrm{perf}}^{\dagger}=C^{\bullet\dagger,\mathrm{perf}}$
as above. Then we have

$\displaystyle{{\mathbbl{\Delta}}}^{\dagger}_{S/A,\mathrm{perf}}/p^{n}$
$\displaystyle=C^{\bullet\dagger,\mathrm{perf}}/p^{n}$
$\displaystyle=\varinjlim_{\epsilon\rightarrow
0}W^{\dagger}((C^{\bullet}_{\epsilon}/p)^{\mathrm{perfd}})/p^{n}$
$\displaystyle=\varinjlim_{\epsilon\rightarrow
0}W_{n}((C^{\bullet}_{\epsilon}/p)^{\mathrm{perfd}})$
$\displaystyle=\varinjlim_{\epsilon\rightarrow
0}{{\mathbbl{\Delta}}}_{S_{\epsilon}/A,\mathrm{perf}}/p^{n}\,.$

This implies that

$({{\mathbbl{\Delta}}}^{\dagger}_{S/A,\mathrm{perf}}[1/d]/p^{n})^{\varphi=1}=\varinjlim_{\epsilon\rightarrow
0}({{\mathbbl{\Delta}}}_{S_{\epsilon}/A,\mathrm{perf}}[1/d]/p^{n})^{\varphi=1}\,.$

Since
$({{\mathbbl{\Delta}}}_{S_{\epsilon}/A,\mathrm{perf}}[1/d]/p^{n})^{\varphi=1}$
is an $\mathrm{arc}_{p}$-sheaf on $\mathrm{Spf}\,S_{\epsilon}$,
$({{\mathbbl{\Delta}}}^{\dagger}_{S/A,\mathrm{perf}}[1/d]/p^{n})^{\varphi=1}$
is an $\mathrm{arc}_{p}$-sheaf on $\mathrm{Spf}\,S$.

One defines the $\mathrm{arc}$-topology for weak formal $p$-adic schemes by
using $p$-adic weakly complete valuation rings of rank $1$ instead, and shows
that there exists an $\mathrm{arc}$-cover $S\rightarrow T$ with $T$ dagger
perfectoid. The proof is entirely similar to [BS22, §8] for $p$-adic formal
schemes. The proof of [BS22, Proposition 8.10] transfers to affine perfectoid
dagger schemes $\mathrm{Spf}\,T$; that is,
$H^{i}_{\mathrm{arc}}(\mathrm{Spf}\,T,\mathcal{O})=0$ and the structure
presheaf on $\mathrm{Spf}\,T$ is a sheaf.

In order to show that the map

$F(S)=\varinjlim_{\epsilon}F(S_{\epsilon})\rightarrow({{\mathbbl{\Delta}}}^{\dagger}_{S/A,\mathrm{perf}}[1/d]/p^{n})^{\varphi=1}\simeq\varinjlim_{\epsilon\rightarrow
0}({{\mathbbl{\Delta}}}_{S_{\epsilon}/A,\mathrm{perf}}[1/d]/p^{n})^{\varphi=1}$

is an isomorphism, we can work $\mathrm{arc}$-locally and hence from now on we
may assume that $S$ is dagger perfectoid. We claim that we have an isomorphism

(3.3.1) $R\Gamma(\mathrm{Spec}\,S[1/p],\mathbb{Z}/p^{n})\cong
R\Gamma(\mathrm{Spec}\,S^{\flat}[1/d],\mathbb{Z}/p^{n})\,.$

Indeed, we have

$\displaystyle R\Gamma(\mathrm{Spec}\,S^{\flat}[1/d],\mathbb{Z}/p^{n})$
$\displaystyle=R\Gamma(\mathrm{Spa}\,S^{\flat}[1/d],\mathbb{Z}/p^{n})$
$\displaystyle=R\Gamma(\mathrm{Spa}\,\widehat{S}^{\flat}[1/d],\mathbb{Z}/p^{n})$
$\displaystyle=R\Gamma(\mathrm{Spec}\,\widehat{S}^{\flat}[1/d],\mathbb{Z}/p^{n})$

and likewise for $S$. This follows from the comparison theorem of
Huber/Scholze for étale cohomology of rigid/adic spaces. The isomorphism

$R\Gamma(\mathrm{Spa}\,\widehat{S}[1/p],\mathbb{Z}/p^{n})\cong
R\Gamma(\mathrm{Spa}\,\widehat{S}^{\flat}[1/d],\mathbb{Z}/p^{n})$

follows from [Sch12, Theorem 1.11]. This shows that we have (3.3.1).

It remains to check

$\displaystyle R\Gamma(\mathrm{Spec}\,S^{\flat}[1/d],\mathbb{Z}/p^{n})$
$\displaystyle=({{\mathbbl{\Delta}}}^{\dagger}_{S/A}[1/d]/p^{n})^{\varphi=1}$
$\displaystyle=(W_{n}(S^{\flat})[1/d])^{\varphi=1}\,.$

By Artin-Schreier-Witt we have

$\mathbb{Z}/p^{n}\cong(W_{n}(S^{\flat})[1/d])^{\varphi=1}$

and since by the dagger analogue of [Sch13, Theorem 4.9, Lemma 4.10] we have

$W_{n}(S^{\flat})[1/d]\xrightarrow{\varphi-1}W_{n}(S^{\flat})[1/d]$

is surjective, we conclude that
$H^{i}(\mathrm{Spa}\,S^{\flat}[1/d],\mathbb{Z}/p^{n})$ vanishes for $i>0$.
This concludes the proof of Theorem 3.1. ∎

##  4\. The tilting correspondence for perfectoid dagger algebras

Let $K$ be a perfectoid field, with absolute value $|\,\,\,|$, ring of
integers $\mathcal{O}$ in $K$, and $w$ an element with $|w|=1/p$. Consider the
ring $R=K\langle T_{1}^{1/p^{\infty}},\ldots,T_{d}^{1/p^{\infty}}\rangle$ of
$w$-adically converging power series, so
$z=\sum_{I\in\mathbb{N}[1/p]^{d}}a_{I}\underline{T}^{I}$, $a_{I}\in K$ is in
$K\langle T_{1}^{1/p^{\infty}},\ldots,T_{d}^{1/p^{\infty}}\rangle$ if and only
if $|a_{I}|\rightarrow 0$. Then $K\langle
T_{1}^{1/p^{\infty}},\ldots,T_{d}^{1/p^{\infty}}\rangle$ is the
perfectoidisation of the Tate algebra $K\langle T_{1},\ldots,T_{d}\rangle$,
equipped with the Gauss norm $\|z\|=\max_{I}\\{|a_{I}|\\}$. Let
$R^{\circ}\subset R$ be the subring with $a_{I}\in\mathcal{O}$ and let
$\lambda\in\mathcal{O}$, $|\lambda|<1$, we write $\lambda=w^{\epsilon}$,
$\epsilon>0$. Then

$\left\\{z=\sum_{I\in\mathbb{N}[1/p]^{d}}a_{I}\underline{T}^{I}\,:\,\varinjlim_{I}|a_{I}||\lambda|^{-|I|}=0\right\\}$

is for every $\lambda=w^{\epsilon}$ a Banach algebra $R_{\epsilon}$ with Gauss
norm $\max\\{|a_{I}|w^{\epsilon}|^{-|I|}\\}$. We write

$R_{\epsilon}=K\langle(\lambda T_{1})^{1/p^{\infty}},\ldots,(\lambda
T_{d})^{1/p^{\infty}}\rangle\,,$

so we have convergence for $|\lambda T_{i}|\leq 1$. We have an isomorphism

$\displaystyle R_{\epsilon}\rightarrow K\langle
S_{1}^{1/p^{\infty}},\ldots,S_{d}^{1/p^{\infty}}\rangle$
$\displaystyle\,\,T_{i}\mapsto\lambda^{-1}S_{i},\text{ (so }S_{i}=\lambda
T_{i})\,.$

Let $|\lambda|<|\lambda^{\prime}|<1$, $\lambda=w^{\epsilon}$,
$\lambda^{\prime}=w^{\epsilon^{\prime}}$. Then we have an inclusion

${R_{\epsilon}}$${R_{\epsilon^{\prime}}}$${K\langle
S_{1}^{1/p^{\infty}},\ldots,S_{d}^{1/p^{\infty}}\rangle}$${K\langle{S^{\prime}}_{1}^{1/p^{\infty}},\ldots,{S^{\prime}}_{d}^{1/p^{\infty}}\rangle}$${S_{i}}$${\frac{\lambda}{\lambda^{\prime}}S^{\prime}_{i}}$$\wr$$\wr$

and we obtain a map of power bounded elements

$\mathcal{O}\langle
S_{1}^{1/p^{\infty}},\ldots,S_{d}^{1/p^{\infty}}\rangle\rightarrow\mathcal{O}\langle{S^{\prime}}_{1}^{1/p^{\infty}},\ldots,{S^{\prime}}_{d}^{1/p^{\infty}}\rangle$

inducing $R^{\circ}_{\epsilon}\rightarrow R_{\epsilon^{\prime}}^{\circ}$.
Define $S=\varinjlim_{\epsilon}R_{\epsilon}=\bigcup_{\epsilon}R_{\epsilon}$.
We define

$S^{\circ}=\\{f\in
S\,:\,\forall\alpha\in\mathcal{O},|\alpha|<1,\exists\lambda=w^{\epsilon}\text{
such that }f\in R_{\epsilon}\text{ and }\alpha f\in
R^{\circ}_{\epsilon}\\}\,.$

Now let $K$ be a perfectoid field of characteristic $0$, $K^{\flat}$ its tilt
(of characteristic $p>0$). Let $\mathcal{O}$, resp. $\mathcal{O}^{\flat}$, be
the ring of integers in $K$, resp. in $K^{\flat}$. We construct perfectoid
dagger algebras over $K$, resp. over $K^{\flat}$, and derive a tilting
correspondence. Let $A/K^{\flat}$ be a reduced affinoid dagger algebra with
integral elements $A^{+}/\mathcal{O}^{\flat}$. We fix a presentation
$(K^{\flat})^{\dagger}\langle T_{1},\ldots,T_{d}\rangle\rightarrow A$ (and
$(\mathcal{O}^{\flat})^{+}\langle T_{1},\ldots,T_{d}\rangle\rightarrow
A^{+}$). Define Gauss norms $\gamma_{\epsilon}$ on $\widehat{A}$ via the
presentation and let $A_{\epsilon}=\\{a\in
A\,:\,\gamma_{\epsilon}(a)>-\infty\\}$. $A_{\epsilon}$ is a $\varpi$-adically
complete Tate algebra over $K^{\flat}$, with integral elements
$A^{+}_{\epsilon}$ over $\mathcal{O}^{\flat}$. Let
$\widehat{A}_{\epsilon}^{\mathrm{perf}}$ be the $\varpi$-adically completed
perfection, with integral elements $\widehat{A}_{\epsilon}^{+\mathrm{perf}}$.
This is a perfectoid $K^{\flat}$-algebra with

$\mathrm{Spa}\,(\widehat{A}_{\epsilon}^{\mathrm{perf}},\widehat{A}_{\epsilon}^{+\mathrm{perf}})=\mathrm{Spa}\,(\widehat{A}_{\epsilon},\widehat{A}_{\epsilon}^{+})$

(see [Sch12, Proposition 6.11]). Define

$(A^{\mathrm{perf}},A^{+\mathrm{perf}})=\varinjlim_{\epsilon}(\widehat{A}^{\mathrm{perf}}_{\epsilon},\widehat{A}_{\epsilon}^{+\mathrm{perf}})\,.$

This is a perfectoid dagger algebra over $K^{\flat}$ with

$\mathrm{Spa}\,(A^{\mathrm{perf}},A^{+\mathrm{perf}})=\mathrm{Spa}\,(\widehat{A},\widehat{A}^{+})\,.$

The chosen presentation extends to a presentation
$R_{\epsilon}\twoheadrightarrow\widehat{A}_{\mathrm{\epsilon}}^{\mathrm{perf}}$,
$R^{\circ}_{\epsilon}\twoheadrightarrow\widehat{A}_{\mathrm{\epsilon}}^{\mathrm{+perf}}$,
and hence $S=\varinjlim_{\epsilon}R_{\epsilon}\twoheadrightarrow
A^{\mathrm{perf}}$, $S^{\circ}\twoheadrightarrow A^{+\mathrm{perf}}$. Let
$(B_{\epsilon},B_{\epsilon}^{+})$ be the untilt of
$(\widehat{A}^{\mathrm{perf}}_{\epsilon},\widehat{A}_{\epsilon}^{+\mathrm{perf}})$
under Scholze’s tilting correspondence. Then
$\mathrm{Spa}\,(B_{\epsilon},B_{\epsilon}^{+})$ is homeomorphic to its tilt
$\mathrm{Spa}\,(B_{\epsilon},B_{\epsilon}^{+})^{\flat}=\mathrm{Spa}\,(\widehat{A}_{\epsilon}^{\mathrm{perf}},\widehat{A}_{\epsilon}^{+\mathrm{perf}})$
[Sch12, Theorem 6.3].

Define $(B,B^{+})=\varinjlim_{\epsilon}(B_{\epsilon},B_{\epsilon}^{+})$. This
is a perfectoid dagger algebra over $K$. Then we have the analogue of [Sch12,
Proposition 6.17]:

######  Theorem 4.1.

There is an equivalence of perfectoid dagger spaces over $K$ and perfectoid
dagger spaces over $K^{\flat}$, by associating to a perfectoid dagger space
$X^{\dagger}/K$ its tilt $(X^{\flat})^{\dagger}/K^{\flat}$.

###### Proof.

The reduction to the case of an affinoid dagger space is shown in the same way
as in [Sch12, Proposition 6.17]. For any affinoid perfectoid dagger algebra
$(B,B^{+})$ with tilt $(A^{\mathrm{perf}},A^{+\mathrm{perf}})$, the associated
presheaves $\mathcal{O}_{X}$ over $X=\mathrm{Spa}(B,B^{+})$ and
$\mathcal{O}_{X^{\flat}}$ on $X^{\flat}=\mathrm{Spa}(B,B^{+})^{\flat}$,
defined by taking direct limits of the restrictions of
$\mathcal{O}_{X_{\epsilon}}$ on
$X_{\epsilon}=\mathrm{Spa}(B_{\epsilon},B_{\epsilon}^{+})$ and
$\mathcal{O}_{X_{\epsilon}^{\flat}}$ on
$X^{\flat}_{\epsilon}=\mathrm{Spa}(B_{\epsilon},B_{\epsilon}^{+})^{\flat}$ to
$X$ resp. $X^{\flat}$ are sheaves by [Sch12, Proposition 6.14], and [Sch12,
Theorem 6.3] holds verbatim for affinoid perfectoid dagger spaces. In
particular, we have a homeomorphism

$\mathrm{Spa}(B,B^{+})\cong\mathrm{Spa}(B,B^{+})^{\flat}\
(=\mathrm{Spa}(\widehat{B},\widehat{B}^{+})=\mathrm{Spa}(\widehat{B},\widehat{B}^{+})^{\flat})$

which commutes with the homeomorphism between the associated affinoid
perfectoid spaces. For an affinoid perfectoid space the equivalence follows
from the above construction, namely

$\mathrm{Spa}\,(B,B^{+})^{\flat}=\mathrm{Spa}\,(A^{\mathrm{perf}},A^{+\mathrm{perf}})\,.$

∎

The main example is

$\mathrm{Spa}\,(K^{\dagger}\langle
T_{1}^{1/p^{\infty}},\ldots,T_{d}^{1/p^{\infty}}\rangle,\mathcal{O}^{\dagger}\langle
T_{1}^{1/p^{\infty}},\ldots,T_{d}^{1/p^{\infty}}\rangle)$

and its tilt

$\mathrm{Spa}\,(K^{\flat\dagger}\langle
T_{1}^{1/p^{\infty}},\ldots,T_{d}^{1/p^{\infty}}\rangle,\mathcal{O}^{\flat\dagger}\langle
T_{1}^{1/p^{\infty}},\ldots,T_{d}^{1/p^{\infty}}\rangle)$

where $\dagger$ means $\varpi$-adic weak completion.

##  5\. A dagger version of $A\Omega$

Let $(A,I)=(A_{\mathrm{inf}}(\mathcal{O}),I=(d))$. Let
$\mathcal{X}/\mathrm{Spf}\,\mathcal{O}$ be a weak formal smooth scheme and let
$X$ be its generic fibre, a dagger variety. Let $X_{\mathrm{pro\acute{e}t}}$
be the pro-étale site on $X$, given locally by

$(U_{i+1}\xrightarrow{\mathrm{f\acute{e}t}}U_{i}\xrightarrow{\mathrm{f\acute{e}t}}\cdots\xrightarrow{\mathrm{f\acute{e}t}}U_{1}\xrightarrow{\mathrm{\acute{e}t}}X)$

where $U_{i}$ are affinoid dagger varieties, $U_{i+1}\rightarrow U_{i}$ is
finite étale for $i\geq 1$ and $U_{1}\rightarrow X$ is étale. Let
$U=\varprojlim_{i}\mathrm{Spec}\,U_{i}$. Let
$T_{i}=\varinjlim_{\epsilon}T_{\epsilon,i}$ be the affinoid dagger algebra
corresponding to $U_{i}$, with the notation of §4. Let
$T_{\epsilon,i}^{\mathrm{perf}}$ be the perfectoidisation arising from the
$T_{\epsilon,i}$, with integral elements
$T^{\circ,\mathrm{perf}}_{\epsilon,i}$. We have the structure sheaf
$\mathcal{O}(U)=\varinjlim_{\epsilon}T_{\epsilon}^{\mathrm{perf}}$, and
$\mathcal{O}^{+}(U)=\varinjlim_{\epsilon}T^{\circ,\mathrm{perf}}_{\epsilon}$.
These algebras are weakly complete with respect to the $p$-adic topology. Then
$(\mathcal{O}(U),\mathcal{O}^{+}(U))$ is a perfectoid dagger algebra over
$\mathbb{C}_{p}$. Let $Z=\mathrm{Spa}\,(\mathcal{O}(U),\mathcal{O}^{+}(U))$
and let
$Z^{\flat}=\mathrm{Spa}\,(\mathcal{O}(U^{\flat}),\mathcal{O}^{\flat+}(U))$ be
its tilt. Let
$v:X_{\mathrm{pro\acute{e}t}}\rightarrow\mathcal{X}_{\mathrm{Zar}}$ be the
canonical map of topoi.

We want to define a dagger version of $A\Omega_{\widehat{X}/\mathcal{O}}$
where $\widehat{X}$ is the associated formal smooth scheme and
$A\Omega_{\widehat{X}/\mathcal{O}}$ is defined as in [BMS18]. So we first need
to give a reasonable definition of
$A_{\inf,X}^{\dagger}=W^{\dagger}(\mathcal{O}_{X}^{+\flat})$. Let us first
consider the case $\mathcal{X}=\mathrm{Spf}^{\dagger}R$ for
$R=\mathcal{O}^{\dagger}\langle T_{1}^{\pm 1},\ldots,T_{d}^{\pm 1}\rangle$. We
have an isomorphism

$A_{\inf}\langle\underline{U}^{\pm 1/p^{\infty}}\rangle\cong
W(\mathcal{O}\langle\underline{T}^{\pm 1/p^{\infty}}\rangle^{\flat})$

of complete $A_{\inf}$-algebras [BMS18, p. 70] with respect to the
$(p,d)$-adic topology. Define the $(p,d)$-weakly complete analogue

(5.0.1) $\displaystyle A^{\dagger}_{\inf}\langle\underline{U}^{\pm
1/p^{\infty}}\rangle$ $\displaystyle=\varinjlim_{\epsilon\rightarrow
0}A_{\inf,\epsilon}\langle\underline{U}^{\pm 1/p^{\infty}}\rangle$
$\displaystyle=\varinjlim_{\epsilon\rightarrow
0}A_{\inf}\langle(p^{\epsilon}\underline{U})^{\pm 1/p^{\infty}}\rangle$
$\displaystyle=\varinjlim_{\epsilon\rightarrow
0}W^{\dagger}(\mathcal{O}\langle(p^{\epsilon}\underline{T})^{\pm
1/p^{\infty}}\rangle^{\flat})$
$\displaystyle=:W^{\dagger}(\mathcal{O}^{\dagger}\langle\underline{T}^{\pm
1/p^{\infty}}\rangle^{\flat})\,.$

To be precise, $A_{\inf}\langle(p^{\epsilon}\underline{U})^{\pm
1/p^{\infty}}\rangle$ are the power series in $\underline{U}^{\pm
1/p^{\infty}}$ with radius of convergence $p^{\epsilon}$ in
$A_{\inf}[1/p]\langle\underline{U}^{\pm 1/p^{\infty}}\rangle$. Then
$W^{\dagger}(\mathcal{O}^{\dagger}\langle\underline{T}^{\pm
1/p^{\infty}}\rangle^{\flat})$ consists of Witt vectors $(w_{0},w_{1},\ldots)$
such that there exists $\epsilon>0$ with
$\inf_{i}\\{i+\gamma_{\epsilon}(w_{i})/p^{i}\\}>-\infty$. In particular,
$w_{i}\in\mathcal{O}\langle(p^{\epsilon}\underline{T})^{\pm
1/p^{\infty}}\rangle^{\flat}$.

For $R$ étale over $\mathcal{O}^{\dagger}\langle T_{1}^{\pm
1},\ldots,T_{d}^{\pm 1}\rangle$ we have, using a lifting $A^{\dagger}(R)$ of
$R$ over $A_{\inf}$ under the map $\theta:A_{\inf}\rightarrow\mathcal{O}$,

(5.0.2) $A^{\dagger}(R)\otimes_{A_{\inf}^{\dagger}\langle\underline{U}^{\pm
1}\rangle}A_{\inf}^{\dagger}\langle\underline{U}^{\pm
1/p^{\infty}}\rangle=A_{\inf}^{\dagger}(R_{\infty})=:W^{\dagger}((R_{\infty})^{\flat})$

where
$(R_{\infty})^{\flat}=(R\otimes_{\mathcal{O}^{\dagger}\langle\underline{T}^{\pm
1}\rangle}\mathcal{O}^{\dagger}\langle\underline{T}^{\pm
1/p^{\infty}}\rangle)^{\flat}$. (This is a dagger version of [BMS18, p. 70]).

This construction sheafifies (see also the proof of Proposition 5.4 below),
namely we have

$W^{\dagger}(\mathcal{O}_{X}^{+\flat})=\displaystyle\varinjlim_{\epsilon\rightarrow
0}W^{\dagger}(\mathcal{O}_{X_{\epsilon}}^{+\flat})$

for $X_{\epsilon}=\mathrm{Spf}\,R_{\epsilon}$ (such that $R_{\epsilon}$ is
étale over $\mathcal{O}\langle p^{\epsilon}\underline{T}^{\pm 1}\rangle$ and
$R=\varinjlim_{\epsilon}R_{\epsilon}$) and global sections equal to
$W^{\dagger}((R_{\infty})^{\flat})$. We define for any $X=\mathrm{Spec}\,S$
the sheaf $A^{\dagger}_{\mathrm{inf},X}$ on $X_{\mathrm{pro\acute{e}t}}$

$A^{\dagger}_{\mathrm{inf},X}:=W^{\dagger}(\mathcal{O}_{X}^{+\flat})$

by local conditions using a covering by small affines and using that
$A_{\inf,X}$ is a sheaf on $\widehat{X}_{\mathrm{pro\acute{e}t}}$.

######  Definition 5.1.

We define a dagger version of $A\Omega$ as follows: Let $\mathcal{X}$ be a
weak formal smooth scheme over $\mathcal{O}$. Then

$A^{\dagger}\Omega_{\mathcal{X}/\mathcal{O}}=L\eta_{\mu}(Rv_{\ast}A^{\dagger}_{\inf,X})$

where $L\eta_{\mu}$ is the décalage functor and $\mu=[\epsilon]-1\in
W(\mathcal{O}^{\flat})$.

Now let $\mathcal{X}=\mathrm{Spf}^{\dagger}R$ be small affine as above. We
have a local-to-global map

$A^{\dagger}\Omega^{\mathrm{pro\acute{e}t}}_{\mathcal{X}/\mathcal{O}}:=L\eta_{\mu}R\Gamma_{\mathrm{pro\acute{e}t}}(\mathcal{X},A^{\dagger}_{\inf,X})\rightarrow
R\Gamma(\mathcal{X},A^{\dagger}\Omega_{\mathcal{X}/\mathcal{O}})\,.$

######  Lemma 5.2.

The local-to-global map is a quasi-isomorphism.

###### Proof.

Since $L\eta_{\mu}$ commutes with direct limits, we have:

$\displaystyle
A^{\dagger}\Omega^{\mathrm{pro\acute{e}t}}_{\mathcal{X}/\mathcal{O}}$
$\displaystyle\cong\varinjlim_{\epsilon}L\eta_{\mu}R\Gamma_{\mathrm{pro\acute{e}t}}(\mathcal{X}_{\epsilon},A^{\dagger}_{\inf,X_{\epsilon}})$
$\displaystyle\cong\varinjlim_{\epsilon}R\Gamma(\mathcal{X}_{\epsilon},A^{\dagger}\Omega_{\mathcal{X}_{\epsilon}/\mathcal{O}})$
$\displaystyle\cong
R\Gamma(\mathcal{X},A^{\dagger}\Omega_{\mathcal{X}/\mathcal{O}})$

where the middle quasi-isomorphism follows from [BMS18, Proposition 9.14]. ∎

Our main theorem for comparing overconvergent prismatic cohomology with
$A^{\dagger}\Omega$ can be formulated as follows:

######  Theorem 5.3.

Let $\mathcal{X}$ be weak formal smooth over $\mathrm{Spf}\,\mathcal{O}$. Then
we have a $\varphi$-equivariant quasi-isomorphism

$R\Gamma(\mathcal{X},A^{\dagger}\Omega_{\mathcal{X}/\mathcal{O}})\otimes^{L\dagger}_{A_{\inf}}B_{\mathrm{cris}}^{+}\cong\varphi^{\ast}\left(R\Gamma((\mathcal{X}/A_{\inf})_{{{\mathbbl{\Delta}}}},\mathcal{O}_{{{\mathbbl{\Delta}}}}^{\dagger})\right)\otimes^{L\dagger}_{A_{\inf}}B_{\mathrm{cris}}^{+}$

where $B_{\mathrm{cris}}^{+}$ is Fontaine’s ring.

###### Proof.

It suffices to show the analogous statement for
$A^{\dagger}\Omega^{\mathrm{pro\acute{e}t}}_{\mathcal{X}/\mathcal{O}}$ where
$\mathcal{X}=\mathrm{Spf}^{\dagger}R$ is small affine. We view
$A_{\inf}(\mathcal{O})=W(\mathcal{O}^{\flat})$ as a perfect
$\mathbb{Z}\llbracket q-1\rrbracket$-algebra, for $q=[\epsilon]$,
$\epsilon=(1,\zeta_{p},\zeta_{p^{2}},\ldots)\in\mathcal{O}^{\flat}$. The
outline of the proof is as follows: We relate
$A^{\dagger}\Omega^{\mathrm{pro\acute{e}t}}_{\mathcal{X}/\mathcal{O}}$ to
group cohomology, where the group is $\Gamma=\mathbb{Z}_{p}(1)^{d}$, the
Galois group of the pro-étale extension $\mathcal{U}/U$. The group cohomology
can be computed by a dagger Koszul complex, which can be related to a dagger
$q$-de Rham complex that computes overconvergent prismatic cohomology.

To be precise, let $R$ be small étale over $\mathcal{O}^{\dagger}\langle
T_{1}^{\pm 1},\ldots,T_{d}^{\pm 1}\rangle$ and
$R_{i}=R\otimes_{\mathcal{O}^{\dagger}\langle\underline{T}^{\pm
1}\rangle}\mathcal{O}^{\dagger}\langle\underline{T}^{\pm 1/p^{i}}\rangle$.
Then $\mathcal{U}=``\varprojlim"\mathrm{Spec}\,R_{i}[1/p]$ is a Galois cover
of $U=\mathrm{Spec}\,R[1/p]$. We have an action of $\mu_{p^{i}}^{d}$ on
$R_{i}[1/p]$ in the usual way:
$\underline{\xi}=(\xi_{1},\ldots,\xi_{d})\in\mu_{p^{i}}^{d}$ acts via
$\underline{\xi}T_{1}^{i_{1}/p^{i}}\cdots
T_{d}^{i_{d}/p^{i}}=\xi_{1}^{i_{1}}\cdots\xi_{d}^{i_{d}}T_{1}^{i_{1}/p^{i}}\cdots
T_{d}^{i_{d}/p^{i}}$. We get a Cartan-Leray spectral sequence

$H^{n}_{\mathrm{cont}}(\mu_{p^{i}}^{d},H^{l}_{\mathrm{pro\acute{e}t}}(U_{i},W^{\dagger}(\mathcal{O}^{+\flat}_{X}))\Rightarrow
H^{n+l}_{\mathrm{pro\acute{e}t}}(U,W^{\dagger}(\mathcal{O}^{+\flat}_{X}))$

for $U_{i}=\mathrm{Spa}\,R[1/p]$ and after taking limits over $i$ yielding the
derived version

$R\Gamma_{\mathrm{gp}}(\mathbb{Z}_{p}(1)^{d},R\Gamma_{\mathrm{pro\acute{e}t}}(\mathcal{U},W^{\dagger}(\mathcal{O}^{+\flat}_{X}))\simeq
R\Gamma_{\mathrm{pro\acute{e}t}}(U,W^{\dagger}(\mathcal{O}^{+\flat}_{X}))\,.$

We have the following proposition, which is an almost purity version for
overconvergent Witt vectors:

######  Proposition 5.4.

  * -

$H^{i}_{\mathrm{pro\acute{e}t}}(\mathcal{U},W^{\dagger}(\mathcal{O}^{+\flat}_{X}))$
is almost zero, killed by $W(\mathfrak{m}^{\flat})$, for $i>0$.

  * -

$H^{0}_{\mathrm{pro\acute{e}t}}(\mathcal{U},W^{\dagger}(\mathcal{O}^{+\flat}_{X}))=W^{\dagger}((R_{\infty})^{\flat})$
where $(R_{\infty})^{\flat}$ is the dagger tilt of the perfectoid dagger
algebra $R_{\infty}$.

###### Proof.

As before, let
$W^{\dagger}(\mathcal{O}_{X}^{+\flat})=\displaystyle\varinjlim_{\epsilon\rightarrow
0}W^{\dagger}(\mathcal{O}_{X_{\epsilon}}^{+\flat})$ for
$\mathcal{X}_{\epsilon}=\mathrm{Spf}\,R_{\epsilon}$. By [Sch13, 4.10, 5.11] we
have
$H^{i}_{\mathrm{pro\acute{e}t}}(\mathcal{U},W(\mathcal{O}^{+\flat}_{X_{\epsilon}}))$
is almost zero for $i>0$ and
$H^{0}_{\mathrm{pro\acute{e}t}}(\mathcal{U},W(\mathcal{O}^{+\flat}_{X_{\epsilon}}))=W(R_{\infty,\epsilon}^{\flat})$.
We need the same assertions for the overconvergent Witt vectors. Now let
$S^{\circ}=R_{\infty,\epsilon}^{\flat}$ be as above, which is a perfectoid
algebra over $\mathcal{O}_{K^{\flat}}$ and
$S=R_{\infty,\epsilon}[1/p]^{\flat}$ which is a perfectoid
$K^{\flat}$-algebra. Let $S^{\prime}/S$ be finite étale of degree $n$, then
$S^{\prime}$ is a perfectoid $K^{\flat}$-algebra and $S^{\prime\circ}$, the
integral closure of $S^{\circ}$ in $S^{\prime}$, is almost finite étale over
$S^{\circ}$ by [Sch12, Proposition 5.23]. By [DLZ12, Corollary 2.46]
$W^{\dagger}(S^{\prime})$ is finite étale over $W^{\dagger}(S)$ of degree $n$.
The proof in [DLZ12], which is given for a finite étale extension of a
finitely generated algebra over a perfect field, easily transfers to
perfectoid algebras. Under different assumptions the result is also proved in
[DK15].

Let
$X_{\epsilon}^{\flat}=\mathrm{Spa}(\mathcal{O}_{X_{\epsilon}}^{\flat},\mathcal{O}_{X_{\epsilon}}^{+\flat})$.
We will show that $W^{\dagger}(\mathcal{O}_{X_{\epsilon}}^{\flat})$ is a sheaf
on $X^{\flat}_{\epsilon}$ for the topology generated by rational subdomains.
Moreover, for any étale covering $\\{U_{\epsilon,i}\\}$ of
$X_{\epsilon}^{\flat}$, where $U_{\epsilon,i}$ is a rational subdomain of a
finite étale map to a rational subdomain of $X_{\epsilon}^{\flat}$, using
étale acyclicity of the sheaf $\mathcal{O}_{X_{\epsilon}}^{\flat}$ ([Sch12,
Proposition 7.13]), the total complex associated to the simplicial complex

(5.4.1)
$0\rightarrow\varinjlim_{\epsilon}W^{\dagger}(\mathcal{O}(X_{\epsilon}^{\flat}))\rightarrow\prod_{i}\varinjlim_{\epsilon}W^{\dagger}(\mathcal{O}(U_{\epsilon,i}))\rightrightarrows\prod_{i,j}\varinjlim_{\epsilon}W^{\dagger}(\mathcal{O}(U_{\epsilon,i}\times
U_{\epsilon,j}))\mathrel{\begin{subarray}{c}\textstyle\rightarrow\\\\[-1.72218pt]
\textstyle\rightarrow\\\\[-1.72218pt]
\textstyle\rightarrow\end{subarray}}\cdots$

is exact. Indeed, let $S_{\epsilon}=(R_{\infty,\epsilon}[1/p])^{\flat}$, a
perfectoid $K^{\flat}$-algebra, and let $S^{\prime}_{\epsilon}$ be finite
étale, again a perfectoid $K^{\flat}$-algebra. We may assume that there are
surjections (with the notation of §4)

$\rho_{\epsilon}:K^{\flat}\langle
w^{\epsilon}T_{1}^{1/p^{\infty}},\ldots,w^{\epsilon}T_{d}^{1/p^{\infty}}\rangle\twoheadrightarrow
S_{\epsilon}$

and likewise for $S^{\prime}_{\epsilon}$. Define finite Gauss norms
$\gamma_{S_{\epsilon}}$ on $S_{\epsilon}$, resp.
$\gamma_{S^{\prime}_{\epsilon}}$ on $S^{\prime}_{\epsilon}$, as follows: let
$w^{\epsilon}t_{i}$ be the image of $w^{\epsilon}T_{i}$ in $S_{\epsilon}$ and
define
$\gamma^{\rho_{\epsilon}}(z)=\inf_{I}(v(a_{I})-\epsilon|k_{I}|)>-\infty$ for
$z=\sum_{k_{I}\in\mathbb{N}[1/p]^{d}}a_{I}(w^{\epsilon}t_{I})^{k_{I}}\in
S_{\epsilon}$ and $|k_{I}|=\sum_{i}k_{i}$, and then
$\gamma_{S_{\epsilon}}(z):=\sup_{\rho_{\epsilon}}\gamma^{\rho_{\epsilon}}(z)$,
and likewise for $S^{\prime}_{\epsilon}$. Now let $U_{\epsilon}$ be a rational
subdomain of $\mathrm{Spa}\,S^{\prime}_{\epsilon}$, hence

$\displaystyle U_{\epsilon}$
$\displaystyle=\mathrm{Spa}\,S^{\prime}_{\epsilon}\langle(w^{\epsilon}Y_{1})^{1/p^{\infty}},\ldots,(w^{\epsilon}Y_{s})^{1/p^{\infty}}\rangle/\langle
f_{i}-gY_{i}\rangle$
$\displaystyle=\mathrm{Spa}\,S^{\prime}_{\epsilon}\left\langle\frac{(w^{\epsilon}f_{1})^{1/p^{\infty}},\ldots,(w^{\epsilon}f_{s})^{1/p^{\infty}}}{g^{1/p^{\infty}}}\right\rangle$
$\displaystyle=\mathrm{Spa}\,S^{\prime}_{U_{\epsilon}}\,.$

Define the Gauss norms $\gamma_{S^{\prime}_{U_{\epsilon}}}$ by adding a degree
valuation for each new variable $Y_{i}$. It then follows from the techniques
in [DLZ12, §1] that the restriction of $\gamma_{S^{\prime}_{U_{\epsilon}}}$ to
$S_{\epsilon}$ is linearly equivalent to $\gamma_{S_{\epsilon}}$, that is,
there exists $c>0$ such that

$\gamma_{S^{\prime}_{U_{\epsilon}}}(z)\geq\gamma_{S_{\epsilon}}(z)\geq
c\gamma_{S^{\prime}_{U_{\epsilon}}}(z)$

for all $z\in S_{\epsilon}$. We may assume that
$\gamma_{S^{\prime}_{U_{\epsilon}}}(z)$ is negative, so that $c>1$. Then
$\frac{1}{c}\gamma_{S_{\epsilon}}(z)\geq\gamma_{S^{\prime}_{U_{\epsilon}}}(z)$,
but from the definitions we have the inequality
$\gamma_{S_{\epsilon/c}}(z)\geq\frac{1}{c}\gamma_{S_{\epsilon}}(z)$ and hence
$\gamma_{S_{\epsilon/c}}(z)\geq\gamma_{S^{\prime}_{U_{\epsilon}}}(z)$ for all
$z\in S_{\epsilon}$. Now let $(z_{0},z_{1},\ldots)\in W(S_{\epsilon})$ be a
Witt vector such that its image lies in
$W^{\dagger}(S^{\prime}_{U_{\epsilon}})$. Then
$\varinjlim_{i}\left(i+\frac{\gamma_{S^{\prime}_{U_{\epsilon}}}(z_{i})}{p^{i}}\right)=\infty$,
hence $(z_{0},z_{1},\ldots)\in W^{\dagger}(S_{\epsilon/c})$. To show
acyclicity in higher degrees it suffices to consider the two cases

  1. i)

$U_{\epsilon}=\mathrm{Spa}\,S^{\prime}_{\epsilon}$ finite étale faithfully
flat over $\mathrm{Spa}\,S_{\epsilon}$. Then the complex (5.4.1) associated to
the acyclic complex

$0\rightarrow S_{\epsilon}\rightarrow S^{\prime}_{\epsilon}\rightrightarrows
S^{\prime}_{\epsilon}\otimes_{S_{\epsilon}}S^{\prime}_{\epsilon}\mathrel{\begin{subarray}{c}\textstyle\rightarrow\\\\[-1.72218pt]
\textstyle\rightarrow\\\\[-1.72218pt]
\textstyle\rightarrow\end{subarray}}S^{\prime}_{\epsilon}\otimes_{S_{\epsilon}}S^{\prime}_{\epsilon}\otimes_{S_{\epsilon}}S^{\prime}_{\epsilon}\mathrel{\begin{subarray}{c}\textstyle\rightarrow\\\\[-1.72218pt]
\textstyle\rightarrow\\\\[-1.72218pt] \textstyle\rightarrow\\\\[-1.72218pt]
\textstyle\rightarrow\end{subarray}}\cdots$

is acyclic because $W^{\dagger}(S^{\prime}_{\epsilon})$ is finite étale and
faithfully flat over $W^{\dagger}(S_{\epsilon})$.

  2. ii)

$\\{U_{\epsilon,i}\\}$ is a covering by rational subdomains of
$\mathrm{Spa}\,S_{\epsilon}$. Then by using the same reduction argument as in
[BGR84, Chapter 8.2], it suffices to consider Laurent coverings and then by an
induction argument the case that $\mathrm{Spa}\,S_{\epsilon}$ is covered by
$\mathrm{Spa}\,S_{\epsilon}\langle w^{\epsilon}f\rangle$ and
$\mathrm{Spa}\,S_{\epsilon}\langle w^{\epsilon}f^{-1}\rangle$ for some $f\in
S_{\epsilon}$. In this case (5.4.1) becomes

(5.4.2) $\displaystyle 0\rightarrow\varinjlim_{\epsilon\rightarrow
0}W^{\dagger}(S_{\epsilon}))\rightarrow$
$\displaystyle\varinjlim_{\epsilon}(W^{\dagger}(S_{\epsilon}\langle(w^{\epsilon}f)^{1/p^{\infty}}\rangle\times
W^{\dagger}(S_{\epsilon}\langle(w^{\epsilon}f^{-1})^{1/p^{\infty}}\rangle)$
$\displaystyle\rightarrow\varinjlim_{\epsilon}W^{\dagger}(S_{\epsilon}\langle(w^{\epsilon}f)^{1/p^{\infty}},(w^{\epsilon}f^{-1})^{1/p^{\infty}}\rangle)\rightarrow
0\,.$

Let $S^{\dagger}\langle
f^{1/p^{\infty}}\rangle=\varinjlim_{\epsilon}S_{\epsilon}\langle(w^{\epsilon}f)^{1/p^{\infty}}\rangle$
be the corresponding perfectoid dagger algebra. Then

$\displaystyle
W^{\dagger}(S_{\epsilon}\langle(w^{\epsilon}f)^{1/p^{\infty}}\rangle)=\\{z=(z_{0},z_{1},\ldots)$
$\displaystyle\in W(S^{\dagger}\langle f^{1/p^{\infty}}\rangle)\,|\,\forall
i:\,z_{i}\in S_{\epsilon}\langle(w^{\epsilon}f)^{1/p^{\infty}}$
$\displaystyle\text{ and
}\varinjlim_{i}\left(\frac{\gamma_{S_{\epsilon}\langle(w^{\epsilon
f)^{1/p^{\infty}}}}(z_{i})}{p^{i}}+i\right)=\infty\\}\,.$

Likewise,
$W^{\dagger}(S_{\epsilon}\langle(w^{\epsilon}f^{-1})^{1/p^{\infty}}\rangle)$
and
$W^{\dagger}(S_{\epsilon}\langle(w^{\epsilon}f)^{1/p^{\infty}},(w^{\epsilon}f^{-1})^{1/p^{\infty}}\rangle)$
are defined.

Let $\widehat{S}\langle f^{1/{p^{\infty}}}\rangle$ be the $w$-adic completion
of $S^{\dagger}\langle f^{1/{p^{\infty}}}\rangle$. Under the isomorphism

$W(\widehat{S}\langle
f^{1/{p^{\infty}}}\rangle)=W(\widehat{S})\langle[f]^{1/p^{\infty}}\rangle$

(where $\langle\,\cdot\,\rangle$ denotes $p$-$d$-adic completion), let

$R(f):=W^{\dagger}(S_{\epsilon})\langle[w^{\epsilon}f]^{1/p^{\infty}}\rangle$

be the image of
$W^{\dagger}(S_{\epsilon}\langle[w^{\epsilon}f]^{1/p^{\infty}}\rangle)$.
Likewise, for a variable $\eta$, under the isomorphism

$W(\widehat{S}\langle\eta^{1/{p^{\infty}}}\rangle)=W(\widehat{S})\langle[\eta]^{1/p^{\infty}}\rangle$

let

$R(\eta):=W^{\dagger}(S_{\epsilon})\langle[w^{\epsilon}\eta]^{1/p^{\infty}}\rangle$

be the image of
$W^{\dagger}(S_{\epsilon}\langle[w^{\epsilon}\eta]^{1/p^{\infty}}\rangle)$.
Similarly for variables $\xi,\eta$

$R(\xi,\eta):=W^{\dagger}(S_{\epsilon})\langle[w^{\epsilon}\xi]^{1/p^{\infty}},[w^{\epsilon}\eta]^{1/p^{\infty}}\rangle$

and

$R(\xi,\xi^{-1}):=W^{\dagger}(S_{\epsilon})\langle[w^{\epsilon}\xi]^{1/p^{\infty}},[w^{\epsilon}\xi^{-1}]^{1/p^{\infty}}\rangle$

are defined.

Now consider the following commutative diagram

(5.4.3) ${0}$${0}$ ${I(\xi)\times I(\eta)}$${I(\xi)_{R(\xi,\xi^{-1})}}$
${0}$${W^{\dagger}(S_{\epsilon})}$${R(\xi)\times
R(\eta)}$${R(\xi,\xi^{-1})}$${0}$${0}$${W^{\dagger}(S_{\epsilon})}$${R(f)\times
R(f^{-1})}$${R(f,f^{-1})}$${0}$ ${0}$${0}$ $\lambda^{\prime}$$\lambda$$=$

$\lambda$ is the map
$(h_{1}(w^{\epsilon}\xi),h_{2}(w^{\epsilon}\eta))\mapsto(h_{1}(w^{\epsilon}\xi)-h_{2}(w^{\epsilon}\xi^{-1}))$,
$\lambda^{\prime}$ is induced by $\lambda$, the vertical maps are the
canonical ones given by $[w^{\epsilon}\xi]^{k}\mapsto[w^{\epsilon}f]^{k}$ and
$[w^{\epsilon}\eta]^{k}\mapsto[w^{\epsilon}f^{-1}]^{k}$ for
$k\in\mathbb{Z}_{\geq 0}[1/p]$. $I(\xi)$ denotes the $(p,d)$-completed ideal
in $R(\xi)$ generated by $[w^{\epsilon}\xi]^{k}-[w^{\epsilon}f]^{k}$,
$k\in\mathbb{Z}_{\geq 0}[1/p]$ and, likewise, $I(\eta)$ is the
$(p,d)$-completed ideal in $R(\eta)$ generated by $1-[f]^{k}[\eta]^{k}$,
$k\in\mathbb{Z}_{\geq 0}[1/p]$. It is then clear that the second column is
exact.

Now

$R(f,f^{-1})=R(\xi,\eta)/\langle([w^{\epsilon}\xi]^{k})-[w^{\epsilon}f]^{k}),(1-[f]^{k}[\eta]^{k})\rangle_{k\in\mathbb{Z}_{\geq
0}[1/p]}$

and since the ideal

$\langle([w^{\epsilon}\xi]^{k})-[w^{\epsilon}f]^{k}),(1-[f]^{k}[\eta]^{k})\rangle_{k\in\mathbb{Z}_{\geq
0}[1/p]}$

coincides with the ideal

$\langle([w^{\epsilon}\xi]^{k})-[w^{\epsilon}f]^{k}),(1-[\xi]^{k}[\eta]^{k})\rangle_{k\in\mathbb{Z}_{\geq
0}[1/p]}$

we obtain

$R(f,f^{-1})=R(\xi,\xi^{-1})/I(\xi)_{R(\xi,\xi^{-1})}$

and hence the third column is exact too. (Note that $I(\xi)_{R(\xi,\xi^{-1})}$
is the $(p,d)$-completed ideal in $R(\xi,\xi^{-1})$ generated by
$[w^{\epsilon}\xi]^{k}-[w^{\epsilon}f]^{k}$ for $k\in\mathbb{Z}_{\geq
0}[1/p]$.

The equations

(5.4.4) $R(\xi,\xi^{-1})=R(\xi)+\langle
w^{\epsilon},\xi^{-1}\rangle^{1/p^{\infty}}R(\xi^{-1})$

and

$I(\xi)_{R(\xi,\xi^{-1})}=I(\xi)_{R(\xi)}+\langle
1-[f]^{k}[\xi^{-1}]^{k}\rangle_{k\in\mathbb{Z}_{\geq 0}[1/p]}R(\xi^{-1})$

show the surjectivity of $\lambda$ and $\lambda^{\prime}$, and the exactness
of the first row in (5.4.3). The second row in (5.4.3) is easily seen to be
exact too.

We already know that after taking $\varinjlim_{\epsilon}$ the last row is
exact in the middle. But a simple diagram chase in (5.4.3) yields that the
last row is already exact before taking $\varinjlim_{\epsilon}$. Hence the
complex (5.4.1) is acyclic. Hence
$H^{i}_{\mathrm{\acute{e}t}}(X^{\flat},W^{\dagger}(\mathcal{O}^{\flat}_{X}))=H^{i}_{\mathrm{pro\acute{e}t}}(\mathcal{U},W^{\dagger}(\mathcal{O}^{\flat}_{X}))=0$
for $i>0$.

To obtain an almost vanishing result for the cohomology of the sheaves
$W^{\dagger}(\mathcal{O}_{X_{\epsilon}}^{+\flat})$ we need an appropriate
definition of almost finite étaleness for $W(\mathcal{O}^{\flat})$-modules.

######  Definition 5.5.

  * a)

Let $R$ be a $W(\mathcal{O}^{\flat})$-algebra and $N$ an $R$-module. Then $N$
is uniformly finitely generated if there exists some integer $n$ such that for
all $\epsilon\in W(\mathfrak{m}^{\flat})$ there exists an $R$-module
$N_{\epsilon}$, finitely generated by $n$ elements, and a map
$f_{\epsilon}:N_{\epsilon}\rightarrow N$ such that the kernel and cokernel of
$f_{\epsilon}$ are annihilated by $\epsilon$.

  * b)

Let $R$ be a $W(\mathcal{O}^{\flat})$-algebra and $N$ an $R$-module. Then $N$
is almost projective over $R$ if for all $R$-modules $X$ and all $i>0$,
$\mathrm{Ext}_{R}^{i}(N,X)$ is almost zero (i.e. annihilated by
$W(\mathfrak{m}^{\flat})$).

  * c)

A morphism of $W(\mathcal{O}^{\flat})$-modules $A\rightarrow B$ is almost
unramified if $A\otimes_{W(\mathcal{O}^{\flat})}W(K^{\flat})\rightarrow
B\otimes_{W(\mathcal{O}^{\flat})}W(K^{\flat})$ is unramified and the
corresponding idempotent
$e\in(B\otimes_{W(\mathcal{O}^{\flat})}W(K^{\flat}))\otimes_{A\otimes_{W(\mathcal{O}^{\flat})}W(K^{\flat})}(B\otimes_{W(\mathcal{O}^{\flat})}W(K^{\flat}))$
defines an almost element in $B\otimes_{A}B$ : that is $e$ lies in
$\mathrm{Hom}(W(\mathfrak{m}^{\flat}),B\otimes_{A}B)$ under the map
$\epsilon\mapsto\epsilon e$.

  * d)

A morphism $A\rightarrow B$ of $W(\mathcal{O}^{\flat})$-modules is almost
finite étale if it is almost unramified, almost projective and uniformly
almost finitely presented.

######  Proposition 5.6.

The assumptions are as above, so $S^{\prime}/S$ is finite étale of degree $n$,
$S^{\prime\circ}/S^{\circ}$ is almost finite étale. Then
$W^{\dagger}(S^{\prime\circ})$ is almost finite étale over
$W^{\dagger}(S^{\circ})$.

###### Proof.

Let $\overline{e}\in S^{\prime}\otimes_{S}S^{\prime}$ be the idempotent
showing that $S^{\prime}$ is unramified over $S$. Then for all
$\overline{\epsilon}\in\mathfrak{m}^{\flat}$
$\overline{\epsilon}\cdot\overline{e}\in
S^{\prime\circ}\otimes_{S^{\circ}}S^{\prime\circ}$ (see [Sch12, Proposition
5.23]). Then $e:=[\overline{e}]\in
W^{\dagger}(S^{\prime}\otimes_{S}S^{\prime})=W^{\dagger}(S^{\prime})\otimes_{W^{\dagger}(S)}W^{\dagger}(S^{\prime})$
is an idempotent showing unramifiedness of $W^{\dagger}(S^{\prime})$ over
$W^{\dagger}(S)$. Then for all $\epsilon\in W(\mathfrak{m}^{\flat})$,
$\epsilon=(\epsilon_{0},\epsilon_{1},\epsilon_{2},\ldots)$ we have
$\epsilon\cdot
e=\epsilon[\overline{e}]=(\epsilon_{0}\overline{e},\epsilon_{1}\overline{e},\epsilon_{2}\overline{e},\ldots)$
is an element in
$W^{\dagger}(S^{\prime\circ}\otimes_{S^{\circ}}S^{\prime\circ})$. Write

$e=\left[\displaystyle\sum_{\stackrel{{\scriptstyle
i=1}}{{j=1}}}^{n}\lambda_{ij}(x_{i}\otimes x_{j})\right]$

for an $S$-basis $x_{1},\ldots,x_{n}$ of $S^{\prime}$. Choose
$u\in\mathfrak{m}^{\flat}$ such that $ux_{j}=y_{j}\in S^{\prime\circ}$ for all
$j$. Then

$e=\left[\displaystyle\sum_{\stackrel{{\scriptstyle
i=1}}{{j=1}}}^{n}\lambda_{ij}(x^{\prime}_{i}\otimes y_{j})\right]$

with $x^{\prime}_{i}=x_{i}/u$. Using [LZ04, Lemma A.9] we can also write - for
each $z\in S^{\prime}\otimes_{S}S^{\prime}$ and $l\in\mathbb{N}$:

$z=\sum_{i=1}^{n}\xi_{ij}^{(l)}(x_{i}\otimes x_{j}^{p^{l}})$

and hence

$z=\sum_{i=1}^{n}\xi_{ij}^{(l)}x^{\prime}_{i,l}\otimes y_{j}^{p^{l}}$

for $x^{\prime}_{i,l}=x_{i}/u^{p^{l}}$. We have $y_{j}^{p^{l}}\in
S^{\prime\circ}$ for all $l$, by construction then, in
$W^{\dagger}(S^{\prime})\otimes_{W^{\dagger}(S)}W^{\dagger}(S^{\prime})$, we
can write $e$ as follows:

(5.6.1)
$e=\sum_{i,j}[\lambda_{ij}(x_{i}^{\prime})]\otimes[y_{j}]-\sum_{i,j}[0,m_{ij}^{(1)},m_{ij}^{(2)},\ldots]$

where $m_{ij}^{(l)}=\xi_{ij}^{(l)}[x_{i,l}^{\prime}\otimes y_{j}^{p^{l}}]$
with uniquely determined $\xi_{ij}^{(l)}\in S$. Hence we can write

(5.6.2)
$e=\sum_{i,j}[\lambda_{ij}(x_{i}^{\prime})]\otimes[y_{j}]-\sum_{i,j}[0,\xi_{ij}^{(1)}x^{\prime}_{i,1},\xi_{ij}^{(2)}x^{\prime}_{i,2},\ldots]\otimes[y_{j}]$

as an element in
$W^{\dagger}(S^{\prime})\otimes_{W^{\dagger}(S)}W^{\dagger}(S^{\prime})$.

For $\epsilon_{0}\in\mathfrak{m}^{\flat}$, $[\epsilon_{0}]e\in
W^{\dagger}(S^{\prime\circ}\otimes_{S^{\circ}}S^{\prime\circ})$ as above. The
uniqueness of the representation (5.6.2) of $e$ implies that
$\epsilon_{0}\lambda_{ij}x^{\prime}_{i}\in S^{\prime\circ}$ for all $i,j$ and
likewise
$[\epsilon_{0}][0,\xi_{ij}^{(1)}x^{\prime}_{i,1},\xi_{ij}^{(2)}x^{\prime}_{i,2},\ldots]\in
W^{\dagger}(S^{\prime\circ})$ and therefore $[\epsilon_{0}]e\in
W^{\dagger}(S^{\prime\circ})\otimes_{W^{\dagger}(S^{\circ})}W^{\dagger}(S^{\prime\circ})$.
We also see that $V^{s}[\epsilon_{s}]e=p^{s}[\epsilon_{s}]^{1/p^{s}}e\in
W^{\dagger}(S^{\prime\circ})\otimes_{W^{\dagger}(S^{\circ})}W^{\dagger}(S^{\prime\circ})$
for all $\epsilon_{s}\in\mathfrak{m}^{\flat}$, and therefore for all
$\epsilon\in W(\mathfrak{m}^{\flat})$ we have $\epsilon\cdot e\in
W^{\dagger}(S^{\prime\circ})\otimes_{W^{\dagger}(S^{\circ})}W^{\dagger}(S^{\prime\circ})$
and therefore $W^{\dagger}(S^{\prime\circ})$ is almost unramified over
$W^{\dagger}(S^{\circ})$.

Let $\epsilon\in W(\mathfrak{m}^{\flat})$ and $\epsilon\cdot
e=\sum_{i=1}^{r}a_{i}\otimes b_{i}$ for $a_{i},b_{i}\in
W^{\dagger}(S^{\prime\circ})$ (so in the following $\epsilon$ is fixed).
Consider the map

$W^{\dagger}(S^{\prime\circ})\rightarrow
W^{\dagger}(S^{\circ})^{n},s\mapsto(\mathrm{Tr}_{W^{\dagger}(S^{\prime})/W^{\dagger}(S)}(s,b_{1}),\ldots,\mathrm{Tr}_{W^{\dagger}(S^{\prime})/W^{\dagger}(S)}(s,b_{n}))$

and the map $W^{\dagger}(S^{\circ})^{n}\rightarrow
W^{\dagger}(S^{\prime\circ})$ defined by
$(r_{1},\ldots,r_{n})\mapsto\sum_{i=1}^{n}a_{i}r_{i}$. As in the proof of
[Sch12, Proposition 5.23] one easily checks that the composite map
$W^{\dagger}(S^{\prime\circ})\rightarrow W^{\dagger}(S^{\circ})^{n}\rightarrow
W^{\dagger}(S^{\prime\circ})$ is multiplication by $\epsilon$. This shows that
$W^{\dagger}(S^{\prime\circ})$ is a uniformly almost finitely presented almost
projective $W^{\dagger}(S^{\circ})$-module and proves Proposition 5.6. ∎

Now consider the sequence (5.4.1) at the “integral level”, namely the total
complex associated to the simplicial complex using the presheaves
$\mathcal{O}^{+\flat}_{X_{\epsilon}}$, resp.
$W^{\dagger}(\mathcal{O}^{+\flat}_{X_{\epsilon}})$. We have

######  Lemma 5.7.

The total complex associated to the simplicial complex

(5.7.1)
$0\rightarrow\varinjlim_{\epsilon}W^{\dagger}(\mathcal{O}_{X_{\epsilon}}^{+\flat})(X_{\epsilon}^{\flat})\rightarrow\prod_{i}\varinjlim_{\epsilon}W^{\dagger}(\mathcal{O}_{X_{\epsilon}}^{+\flat})(U_{i})\rightrightarrows\prod_{i,j}\varinjlim_{\epsilon}W^{\dagger}(\mathcal{O}_{X_{\epsilon}}^{+\flat})(U_{i}\times
U_{j})\mathrel{\begin{subarray}{c}\textstyle\rightarrow\\\\[-1.72218pt]
\textstyle\rightarrow\\\\[-1.72218pt]
\textstyle\rightarrow\end{subarray}}\cdots$

has cohomology killed by $W(\mathfrak{m}^{\flat})$.

###### Proof.

Again we consider the two cases

  1. i)

$U_{\epsilon}=\mathrm{Spa}\,S^{\prime}_{\epsilon}$ finite étale faithfully
flat over $\mathrm{Spec}\,\S_{\epsilon}$. Since
$W^{\dagger}(S^{\prime}_{\epsilon})$ is finite étale over
$W^{\dagger}(S_{\epsilon})$, the canonical map

$W^{\dagger}(S^{\prime}_{\epsilon})\otimes_{W^{\dagger}(S_{\epsilon})}W^{\dagger}(S^{\prime}_{\epsilon})\rightarrow
W^{\dagger}(S^{\prime}_{\epsilon}\otimes_{S_{\epsilon}}S^{\prime}_{\epsilon})$

defined by

$(\sum_{i\geq 0}p^{i}[x_{i}]^{1/p^{i}})\otimes(\sum_{j\geq
0}p^{j}[x_{j}]^{1/p^{j}})\mapsto\sum_{i,j\geq 0}p^{i+j}[x_{i}^{1/p^{i}}\otimes
x_{j}^{1/p^{j}}]$

for $x_{i},x_{j}\in S^{\prime}_{\epsilon}$, is an isomorphism. Proposition 5.6
implies that the same map, defined at integral level,

$W^{\dagger}(S^{\prime\circ}_{\epsilon})\otimes_{W^{\dagger}(S^{\circ}_{\epsilon})}W^{\dagger}(S^{\prime\circ}_{\epsilon})\rightarrow
W^{\dagger}(S^{\prime\circ}_{\epsilon}\otimes_{S^{\circ}_{\epsilon}}S^{\prime\circ}_{\epsilon})$

has kernel and cokernel killed by $W(\mathfrak{m}^{\flat})$. Moreover,
Proposition 5.6 implies that the complex (5.7.1) associated to the acyclic
complex

$0\rightarrow S_{\epsilon}\rightarrow S^{\prime}_{\epsilon}\rightrightarrows
S^{\prime}_{\epsilon}\otimes_{S_{\epsilon}}S^{\prime}_{\epsilon}\mathrel{\begin{subarray}{c}\textstyle\rightarrow\\\\[-1.72218pt]
\textstyle\rightarrow\\\\[-1.72218pt]
\textstyle\rightarrow\end{subarray}}\cdots\,,$

namely

$0\rightarrow W^{\dagger}(S^{\circ}_{\epsilon})\rightarrow
W^{\dagger}(S^{\prime\circ}_{\epsilon})\rightrightarrows
W^{\dagger}(S^{\prime\circ}_{\epsilon}\otimes_{S^{\circ}_{\epsilon}}S^{\prime\circ}_{\epsilon})\mathrel{\begin{subarray}{c}\textstyle\rightarrow\\\\[-1.72218pt]
\textstyle\rightarrow\\\\[-1.72218pt]
\textstyle\rightarrow\end{subarray}}\cdots$

has cohomology killed by $W(\mathfrak{m}^{\flat})$. Indeed, for $\kappa\in
W(\mathfrak{m}^{\flat})$, let
$\lambda_{\kappa}:W^{\dagger}(S^{\circ}_{\epsilon})^{n}\rightarrow
W^{\dagger}(S^{\prime\circ}_{\epsilon})$ be the map considered in the proof of
Proposition 5.6 such that the cohomology of the cone of $\lambda_{\kappa}$ is
killed by $\kappa$. The corresponding simplicial complex for
$M:=W^{\dagger}(S^{\circ}_{\epsilon})^{n}$

$0\rightarrow W^{\dagger}(S^{\circ}_{\epsilon})\rightarrow M\rightrightarrows
M\otimes_{W^{\dagger}(S^{\circ}_{\epsilon})}M\mathrel{\begin{subarray}{c}\textstyle\rightarrow\\\\[-1.72218pt]
\textstyle\rightarrow\\\\[-1.72218pt]
\textstyle\rightarrow\end{subarray}}\cdots$

is obviously acyclic, hence the cohomology of the complex (5.7.1) is killed by
$\kappa$. Since this holds for all $\kappa$, the claim follows.

  2. ii)

Now let $\\{U_{\epsilon,i}\\}$ be a covering by rational subdomains of
$\mathrm{Spa}\,S_{\epsilon}$. Using again the same reduction argument as in
[BGR84, §8.2] it suffices to consider Laurent coverings and then by induction
the case $\mathrm{Spa}\,S_{\epsilon}=\mathrm{Spa}\,S_{\epsilon}\langle
w^{\epsilon}f\rangle\cup\mathrm{Spa}\,S_{\epsilon}\langle
w^{\epsilon}f^{-1}\rangle$ for some $f\in S_{\epsilon}$.

In the limit we have the covering
$\mathrm{Spa}\,S^{\dagger}=\mathrm{Spa}\,S^{\dagger}\langle
f\rangle\cup\mathrm{Spa}\,S^{\dagger}\langle 1/f\rangle$ where $S^{\dagger}$
is a perfectoid dagger algebra. Since we consider integral elements we can use
the argument in [Sch12, Lemma 6.4] to assume that $f\in S^{\dagger,\circ}$ at
the expense of writing
$\mathrm{Spa}\,S^{\dagger}=\mathrm{Spa}\,S^{\dagger}\langle
f\rangle\cup\mathrm{Spa}\,S^{\dagger}\langle w^{N}/f\rangle$ for some $N$. The
Laurent covering does not change. Then, to show Lemma 5.7, we need to show
that the sequence

(5.7.2) $\displaystyle 0\rightarrow\varinjlim_{\epsilon\rightarrow
0}W^{\dagger}(S^{\circ}_{\epsilon})\rightarrow$
$\displaystyle\varinjlim_{\epsilon\rightarrow
0}W^{\dagger}(S^{\circ}_{\epsilon}\langle(w^{\epsilon}f)^{1/p^{\infty}}\rangle)\times
W^{\dagger}(S^{\circ}_{\epsilon}\langle(w^{\epsilon}w^{N}/f)^{1/p^{\infty}}\rangle)$
$\displaystyle\rightarrow\varinjlim_{\epsilon\rightarrow
0}W^{\dagger}(S^{\circ}_{\epsilon}\langle(w^{\epsilon}f)^{1/p^{\infty}},(w^{\epsilon}w^{N}/f)^{1/p^{\infty}}\rangle)\rightarrow
0$

has cohomology killed by $W(\mathfrak{m}^{\flat})$.

Define, in analogy to the ‘generic’ case in diagram (5.4.3),

$\displaystyle
R^{\circ}(f)=W^{\dagger}(S^{\circ}_{\epsilon}\langle(w^{\epsilon}f)^{1/p^{\infty}}\rangle)$
$\displaystyle
R^{\circ}(\eta)=W^{\dagger}(S^{\circ}_{\epsilon}\langle(w^{\epsilon}\eta)^{1/p^{\infty}}\rangle)\text{
for a variable }\eta\text{ and likewise for }\xi$ $\displaystyle
R^{\circ}(w^{N}/f)=W^{\dagger}(S^{\circ}_{\epsilon}\langle(w^{\epsilon}w^{N}f^{-1})^{1/p^{\infty}}\rangle)$
$\displaystyle
R^{\circ}(\xi,w^{N}/\xi)=W^{\dagger}(S^{\circ}_{\epsilon}\langle(w^{\epsilon}\xi)^{1/p^{\infty}},(w^{\epsilon}w^{N}\xi^{-1})^{1/p^{\infty}}\rangle)\text{
and likewise for }R^{\circ}(f,w^{N}/f)\,.$

In analogy to diagram (5.4.3) we consider the following commutative diagram at
“integral” level:

(5.7.3) ${0}$${0}$ ${I^{\circ}(\xi)\times
I^{\circ}(\eta)}$${I^{\circ}(\xi)_{R^{\circ}(\xi,w^{N}/\xi)}}$
${0}$${W^{\dagger}(S^{\circ}_{\epsilon})}$${R^{\circ}(\xi)\times
R^{\circ}(\eta)}$${R^{\circ}(\xi,w^{N}/\xi)}$${0}$${0}$${W^{\dagger}(S^{\circ}_{\epsilon})}$${R^{\circ}(f)\times
R^{\circ}(w^{N}/f)}$${R^{\circ}(f,w^{N}/f)}$${0}$ ${0}$${0}$
$\lambda^{\prime}$$\lambda$$=$

$\lambda$ is the map

$(\sum_{i}p^{i}[h_{1}^{(i)}(w^{\epsilon}\xi)]^{1/p^{i}},\sum_{i}p^{i}[h_{2}^{(i)}(w^{\epsilon}\eta)]^{1/p^{i}})\mapsto\sum_{i}p^{i}[h_{1}^{(i)}(w^{\epsilon}\xi)]-h_{2}^{(i)}(w^{\epsilon}w^{N}\eta^{-1})]^{1/p^{i}}$
$I^{\circ}(\xi)=\\{\sum_{i}p^{i}[w^{\epsilon}\xi-w^{\epsilon}f]^{r_{i}}[\theta_{i}]\,|\,r_{i}\in\mathbb{Z}_{\geq
0}[1/p],\theta_{i}\in S^{\circ}_{\epsilon}\langle
w^{\epsilon}\xi\rangle^{1/p^{\infty}}\\}$

where we only consider elements that still lie in
$W^{\dagger}(S^{\circ}_{\epsilon}\langle
w^{\epsilon}\xi\rangle^{1/p^{\infty}})$, i.e. which are overconvergent.

$I^{\circ}(\eta)=\\{\sum_{i}p^{i}[w^{\epsilon}w^{N}-w^{\epsilon}\eta
f]^{r_{i}}[\theta_{i}]\,|\,r_{i}\in\mathbb{Z}_{\geq 0}[1/p],\theta_{i}\in
S^{\circ}_{\epsilon}\langle w^{\epsilon}\eta\rangle^{1/p^{\infty}}\\}$

again with the condition that these elements are overconvergent, hence lie in
$W^{\dagger}(S_{\epsilon}^{\circ}(w^{\epsilon}\eta)^{1/p^{\infty}})$.
$I^{\circ}(\xi)_{R^{\circ}(\xi,w^{N}/\xi)}$ is defined analogously, where we
assume that $r_{i}\in
S^{\circ}_{\epsilon}((w^{\epsilon}\xi)^{1/p^{\infty}},(w^{\epsilon}w^{N}\xi^{-1})^{1/p^{\infty}})$.
The map $\lambda^{\prime}$ in diagram (5.7.3) is then induced by $\lambda$.
The vertical maps are the obvious ones given by
$([h_{1}[w^{\epsilon}\xi],[h_{2}(w^{\epsilon}\eta)])\mapsto(([h_{1}[w^{\epsilon}f],[h_{2}(w^{\epsilon}w^{N}/f)])$.
Since

$[w^{\epsilon}\xi-w^{\epsilon}f]^{k}\cdot[w^{N}\xi^{-1}]^{k}=[w^{\epsilon}w^{N}-w^{\epsilon}w^{N}f\xi^{-1}]^{k}$

for $k\in\mathbb{Z}_{\geq 0}[1/p]$, we see that $\lambda^{\prime}$ is
bijective. As in the generic case (5.4.3) it is easy to see that the middle
row in (5.7.3) is exact. Now consider, in particular, the map
$\mu:R^{\circ}(\eta)\rightarrow R^{\circ}(w^{N}/f)$ in the second column in
(5.7.3). Then, modulo $p$, it follows from the proof of [Sch12, Lemma 6.4]
that

$S^{\circ}_{\epsilon}\langle(w^{\epsilon}\eta)^{1/p^{\infty}}\rangle/(I^{\circ}(\eta)\mod
p)\rightarrow
S^{\circ}_{\epsilon}\langle(w^{\epsilon}w^{N}f^{-1})^{1/p^{\infty}}\rangle$

is an almost isomorphism, the kernel is killed by $\mathfrak{m}^{\flat}$. This
implies that the map

$\mu:R^{\circ}(\eta)/I^{\circ}(\eta)\rightarrow R^{\circ}(w^{N}/f)$

is an almost isomorphism with respect to the ideal
$[\mathfrak{m}^{\flat}]=\langle[x]\,|\,x\in\mathfrak{m}^{\flat}\rangle$, hence
$[\mathfrak{m}^{\flat}]\ker\mu\subset I^{\circ}(\eta)$.

Now consider the corresponding rings/ideals for the full ring of Witt vectors:
$\widehat{R}^{\circ}(f)=W(S^{\circ}_{\epsilon}\langle(w^{\epsilon}f)^{1/p^{\infty}}\rangle)$,
likewise $\widehat{R}^{\circ}(\eta)$, $\widehat{R}^{\circ}(w^{N}/f)$,
$\widehat{R}^{\circ}(\xi,w^{N}/\xi)$, $\widehat{R}^{\circ}(f,w^{N}/f)$,
obtained by taking $p$-adic completions, also $\widehat{I}^{\circ}(\eta)$, the
$p$-adic completion of $I^{\circ}(\eta)$. We have the induced map

$\widehat{\mu}:\widehat{R}^{\circ}(\eta)\rightarrow\widehat{R}^{\circ}(w^{N}/f)\,.$

Since $\widehat{I}^{\circ}(\eta)$ is $p$-adically complete and
$W(\mathfrak{m}^{\flat})$ is the $p$-adic completion of
$[\mathfrak{m}^{\flat}]$, we conclude that
$W(\mathfrak{m}^{\flat})\ker\widehat{\mu}\subset\widehat{I}^{\circ}(\eta)$, so
$W(\mathfrak{m}^{\flat})$ kills $\ker\widehat{\mu}/\widehat{I}^{\circ}(\eta)$.

On the other hand, returning to overconvergent elements, we see that
multiplication by $\lambda\in W(\mathfrak{m}^{\flat})$ maps $\ker\mu$ to
overconvergent elements in $R^{\circ}(\eta)$, hence to $I^{\circ}(\eta)$,
hence $W(\mathfrak{m}^{\flat})$ kills $\ker\mu/I^{\circ}(\eta)$. A diagram
chase in (5.7.3) yields then that the cohomology of the bottom row in (5.7.3)
is killed by $W(\mathfrak{m}^{\flat})$.

This finishes the proof of Lemma 5.7. ∎

This finishes the proof of Proposition 5.4. ∎

######  Corollary 5.8.

We have an almost quasi-isomorphism

$R\Gamma_{\mathrm{gp}}(\mathbb{Z}_{p}(1)^{d},W^{\dagger}(R_{\infty}^{\flat}))\simeq
R\Gamma_{\mathrm{pro\acute{e}t}}(U,W^{\dagger}(\mathcal{O}^{+\flat}_{X}))\,.$

(This means that the cohomology of the cone is killed by
$W(\mathfrak{m}^{\flat})$).

The décalage functor $L\eta_{\mu}$ applied to
$R\Gamma_{\mathrm{pro\acute{e}t}}(U,W^{\dagger}(\mathcal{O}^{+\flat}_{X}))$
yields, by definition,
$A^{\dagger}\Omega_{\mathcal{X}/\mathcal{O}}^{\mathrm{pro\acute{e}t}}$. We
have the following lemma:

######  Lemma 5.9.

$L\eta_{\mu}$ transforms the almost quasi-isomorphism in Corollary 5.8 into a
quasi-isomorphism.

###### Proof.

Since $W(\mathfrak{m}^{\flat})\neq W(\mathfrak{m}^{\flat})^{2}$ we cannot
apply the usual technology of almost mathematics. Instead we use the following
proposition of Bhatt [Bha18b, Lemma 6.14]:

######  Proposition 5.10.

Let $\mathfrak{m}$ be an ideal of a ring $A$ and $f$ a non-zero divisor in
$\mathfrak{m}$. Let $\sigma:C\rightarrow D$ be a homomorphism of complexes of
$A$-modules such that

  1. i)

the cone of $\sigma$ is killed by $\mathfrak{m}$.

  2. ii)

all cohomology groups of $C\otimes^{L}A/fA$ contain no non-zero elements
killed by $\mathfrak{m}^{2}$.

Then $L\eta_{f}C\rightarrow L\eta_{f}D$ is a quasi-isomorphism.

To continue the proof of Lemma 5.9, property i) in Proposition 5.10 is
satisfied since our map is an almost quasi-isomorphism. For ii) we use
Morrow’s arguments in his notes [Mor20, p. 42]. We need to show that the
cohomology of
$R\Gamma_{\mathrm{gp}}(\Gamma,W^{\dagger}((R^{\dagger}_{\infty})^{\flat}/\mu)))$
has no non-zero elements killed by $W(\mathfrak{m}^{\flat})^{2}$. Now,
$R\Gamma_{\mathrm{gp}}(\Gamma,W^{\dagger}((R^{\dagger}_{\infty})^{\flat}/\mu)))$
is quasi-isomorphic to the weak $p$-adic completion of

$\bigoplus_{k_{1},\ldots,k_{d}\in\mathbb{Z}[1/p]}K_{A_{\inf}/\mu
A_{\inf}}([\epsilon]^{k_{1}}-1,\ldots,[\epsilon]^{k_{d}}-1)$

which is a dagger completion of Koszul complexes. By [BMS18, Lemma 7.10], the
cohomology of each of these complexes is the weak $p$-adic completion of a
finite direct sum of copies of $A_{\inf}/\mu A_{\inf}([\epsilon]^{k}-1)$ and
$A_{\inf}/([\epsilon]^{k}-1)A_{\inf}$ for varying $k\in\mathbb{Z}[1/p]$. It is
shown in [Mor20, p. 42] that these $p$-torsion-free modules contain no non-
zero elements killed by $W(\mathfrak{m}^{\flat})^{2}$. ∎

Now we are going to relate the group cohomology
$L\eta_{\mu}R\Gamma_{\mathrm{gp}}(\Gamma,W^{\dagger}((R_{\infty})^{\flat})))$
to a dagger version of the $q$-de Rham complex. In analogy to the
$\Gamma$-equivariant decomposition

$A_{\inf}(\widehat{R}_{\infty})=A(\widehat{R})^{\square}\oplus
A_{\inf}(\widehat{R}_{\infty})^{\mathrm{nonint}}$

(see below [BMS18, Lemma 9.6]) we have the dagger version

$A_{\inf}^{\dagger}(R_{\infty})=W^{\dagger}((R_{\infty})^{\flat})=A(R)^{\square}\oplus
A^{\dagger}_{\inf}(R_{\infty})^{\mathrm{nonint}}\,.$

Here $A(R)^{\square}$ is a lifting of $R$ over $A_{\inf}$, it is étale over
$A_{\inf}^{\dagger}(\underline{U}^{\pm 1})$. Following the proof of [BMS18,
Lemma 9.6], we conclude that
$H^{i}_{\mathrm{cont}}(\Gamma,A_{\inf}^{\dagger}(R_{\infty})^{\mathrm{nonint}})$
is killed by $\mu$ and hence
$L\eta_{\mu}R\Gamma_{\mathrm{cont}}(\Gamma,A_{\inf}^{\dagger}(R_{\infty})^{\mathrm{nonint}})$
vanishes.

On the other hand, we have the $q$-derivation

$\frac{\partial_{q}}{\partial_{q}\log(U_{i})}=\frac{\gamma_{i}-1}{[\epsilon]-1}$

action on $A(R)^{\square}$ (see [BMS18, §9.2]). It is well-known that we can
compute group cohomology via Koszul complexes. The differentials in the
complex

$R\Gamma_{\mathrm{cont}}(\Gamma,A(R)^{\square})=K_{A(R)^{\square}}(\gamma_{1}-1,\ldots,\gamma_{d}-1)$

are divisible by $\mu=[\epsilon]-1$ and we get by [BMS18, Lemma 7.9]

$\eta_{\mu}R\Gamma_{\mathrm{cont}}(\Gamma,A(R)^{\square})=K_{A(R)^{\square}}\left(\frac{\gamma_{1}-1}{[\epsilon]-1},\ldots,\frac{\gamma_{d}-1}{[\epsilon]-1}\right)$

and we define the $q$dagger de Rham complex of $A(R)^{\square}$ as

$q\Omega^{\dagger\bullet}_{A(R)^{\square}/A_{\inf}}:=K_{A(R)^{\square}}\left(\frac{\gamma_{1}-1}{[\epsilon]-1},\ldots,\frac{\gamma_{d}-1}{[\epsilon]-1}\right)\,.$

Hence, by definition, we have a quasi-isomorphism

$L\eta_{\mu}R\Gamma_{\mathrm{cont}}(\Gamma,W^{\dagger}(R_{\infty})^{\flat}))=q\Omega^{\dagger\bullet}_{A(R)^{\square}/A_{\inf}}\,.$

We will show that the $q$-dagger de Rham complex computes the overconvergent
prismatic cohomology after applying
$\otimes^{\dagger}_{A_{\inf}}B_{\mathrm{cris}}^{+}$.

Let $A=A_{\inf}$ and let $R/\mathcal{O}$ ($\mathcal{O}=A/d$) be weakly
complete, smooth. So $d=1+[\epsilon^{1/p}]+\cdots+[\epsilon^{1/p}]^{p-1}$
which is a generator of $\ker(A\rightarrow\mathcal{O})$. Then

$\varphi(d):=[p]_{q}=\frac{q^{p}-1}{q-1}$

for $q=[\epsilon]$ which is the $q$-analogue of $p$. Let
$\varphi^{\ast}R=R^{(1)}/(A/[p]_{q})$ be the base change of $R$ along
$A\xrightarrow{\varphi}A$. Consider $P_{0}$, a free $A$-algebra, weakly
$(p,d)$-completed with an action of $\varphi$ and a surjection
$P_{0}\twoheadrightarrow R$. Let
$P_{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptscriptstyle\bullet$}}}}}}:=P_{0}\rightrightarrows
P_{1}\mathrel{\begin{subarray}{c}\textstyle\rightarrow\\\\[-1.72218pt]
\textstyle\rightarrow\\\\[-1.72218pt]
\textstyle\rightarrow\end{subarray}}\cdots$ be the $(p,d)$-weakly completed
Čech-nerve of $A\rightarrow P_{0}$, and let
$J_{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptscriptstyle\bullet$}}}}}}=\ker(P_{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptscriptstyle\bullet$}}}}}}\rightarrow
R)$ be the kernel of the augmentation and
$P_{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptscriptstyle\bullet$}}}}},\delta}$
the associated
$\delta$-$P_{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptscriptstyle\bullet$}}}}}}$-algebra.
Then $C_{i}:=P_{i,\delta}^{\dagger}\\{\frac{J_{i}}{d}\\}$ is a dagger prism
($\dagger$ = $(p,d)$-weakly completed). We know that
$C_{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptscriptstyle\bullet$}}}}}}$,
the $(p,d)$-weakly completed Čech-nerve of $A\rightarrow C_{0}$ computes
${{\mathbbl{\Delta}}}_{R/A_{\inf}}^{\dagger}$ equipped with its
$\varphi$-action. Then

$\displaystyle\varphi^{\ast}(C_{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptscriptstyle\bullet$}}}}}})$
$\displaystyle=\varphi_{A}^{\ast}(C_{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptscriptstyle\bullet$}}}}}})=\varphi_{P_{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptscriptstyle\bullet$}}}}},\delta}}^{\ast}(C_{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptscriptstyle\bullet$}}}}}})$
$\displaystyle=\varphi_{P_{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptscriptstyle\bullet$}}}}},\delta}}^{\ast}\left(P_{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptscriptstyle\bullet$}}}}},\delta}^{\dagger}\left\\{\frac{J_{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptscriptstyle\bullet$}}}}}}}{d}\right\\}\right)=P_{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptscriptstyle\bullet$}}}}},\delta}^{\dagger}\left\\{\frac{\varphi(J_{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptscriptstyle\bullet$}}}}}})}{[p]_{q}}\right\\}$

where we have used that $A\rightarrow
P_{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptscriptstyle\bullet$}}}}},\delta}$
is a cosimplicial homotopy equivalence. Then we have a ($(p,[q]_{p})$-weakly
completed) $[p]_{q}$-PD version of Berthelot’s Poincaré lemma [Ber74, Lemma V
2.1.2]:

######  Proposition 5.11.

$\varphi^{\ast}(C_{n})=P_{n,\delta}^{\dagger}\left\\{\frac{\varphi(J_{n})}{[p]_{q}}\right\\}$
is a $(p,[q]_{p})$-weakly completed $[p]_{q}$-PD-polynomial algebra over
$P_{0,\delta}^{\dagger}\left\\{\frac{\varphi(J_{0})}{[p]_{q}}\right\\}$, so
for $N:=P_{0,\delta}^{\dagger}\left\\{\frac{\varphi(J_{0})}{[p]_{q}}\right\\}$
we have

$\varphi^{\ast}(C_{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptscriptstyle\bullet$}}}}}})\cong
N^{\dagger}\langle
T_{1},\ldots,T_{s}\rangle\left\langle\frac{T_{1}^{p}}{[p]_{q}},\ldots,\frac{T_{s}^{p}}{[p]_{q}}\right\rangle\,.$

######  Lemma 5.12.

Let $S$ be a $(p,[p]_{q})$-weakly complete $\mathbb{Z}_{q}\llbracket
q-1\rrbracket$-algebra. Then the $q$-dagger de Rham complex

$q\Omega^{\dagger\bullet}_{S\langle
T_{1},\ldots,T_{s}\rangle\left\langle\frac{T_{1}^{p}}{[p]_{q}},\ldots,\frac{T_{s}^{p}}{[p]_{q}}\right\rangle}$

is acyclic, where the derivative is given by the $q$-derivation
$\nabla_{q,n}:T^{n}\mapsto[n]_{q}T^{n-1}$.

###### Proof.

Before taking the $(p,[p]_{q})$ weak completion, the proof is analogous to the
proof that the usual de Rham complex of a PD-polynomial algebra is acyclic.
Taking $(p,[p]_{q})$ weak completions preserves acyclicity. ∎

######  Corollary 5.13.

$q\Omega^{\dagger\bullet}_{\varphi^{\ast}(C_{n})/A}$ is quasi-isomorphic to
$q\Omega^{\dagger\bullet}_{\varphi^{\ast}(C_{0})/A}$.

We observe that $q\Omega^{\dagger\bullet}_{\varphi^{\ast}(C_{0})/A}$ is the
$q$-dagger de Rham complex over the $(p,[p]_{q})$-weakly completed $q$-PD-
envelope $D^{\dagger}_{J_{0},q}(P_{0,\delta})$, which is the Koszul complex
$K_{D^{\dagger}_{J_{0},q}(P_{0,\delta})}(\nabla_{q,1},\ldots,\nabla_{q,s})$.
But since we have a smooth lifting $A(R)^{\square}$ of $R$ over $A$, we have
also considered the $q$-de Rham complex
$q\Omega^{\dagger\bullet}_{A(R)^{\square}/A_{\inf}}=K_{A(R)^{\square}}(\nabla_{q,1},\ldots,\nabla_{q,s})$.
We explain the relationship between these complexes.

Let again
$C_{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptscriptstyle\bullet$}}}}}}$
be the Čech-Alexander complex that computes
${{\mathbbl{\Delta}}}_{R/A}^{\dagger}$ and let
$\varphi^{\ast}C_{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptscriptstyle\bullet$}}}}}}=P_{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptscriptstyle\bullet$}}}}},\delta}^{\dagger}\left\\{\frac{\varphi(J_{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptscriptstyle\bullet$}}}}}})}{[p]_{q}}\right\\}={{\mathbbl{\Delta}}}_{R^{(1)}/A}^{\dagger}$.
By the argument in [BdJ11, Theorem 2.12], using Corollary 5.13, the total
complex of the simplicial complex
$\varphi^{\ast}C_{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptscriptstyle\bullet$}}}}}}$
is quasi-isomorphic to the $q$-dagger de Rham complex

$q\Omega^{\dagger\bullet}_{\varphi^{\ast}(C_{0})/A}=K_{D^{\dagger}_{J_{0},q}(P_{0})}(\nabla_{q,1},\ldots,\nabla_{q,s})\,.$

Now
$\varphi^{\ast}(C_{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptscriptstyle\bullet$}}}}}})=A_{\varphi}\otimes_{A}C_{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptscriptstyle\bullet$}}}}}}\cong
C_{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.65}{$\scriptscriptstyle\bullet$}}}}}}$
with

$\displaystyle C_{i}$
$\displaystyle=A^{\dagger}[X_{1},\ldots,X_{d},T_{1},\ldots,T_{s}]/\langle
g_{1}-T_{1}d,\ldots,g_{s}-T_{s}d\rangle$