$q-\textnormal{CRYS}(\mathfrak{Y}/D)\xrightarrow{g_{1}}\mathfrak{Y}_{\operatorname{\acute{E}T}}^{\textnormal{Aff}}\xrightarrow{a}\mathfrak{Y}_{\operatorname{\acute{E}T}}\xleftarrow{g}\mathfrak{Y}_{\operatorname{\acute{e}t}}$

and so $(u^{q}_{\mathfrak{Y}})^{-1}=g_{1}^{s}a^{s}g_{s}$. Unravelling the
definitions, the LHS of (6.21) is the sheafification of the presheaf

$(E,J)\mapsto\varinjlim_{\operatorname{Spf}(E/J)\to
g(W)}((\mathcal{O}_{\mathfrak{X}_{\operatorname{\acute{e}t}}}/p^{n})(W\times_{\mathfrak{Y}}\mathfrak{X})\otimes_{\mathcal{O}_{\mathfrak{Y}_{\operatorname{\acute{e}t}}}(W)}E/[p]_{q}).$
(6.22)

Since $\mathfrak{X}$ and $\mathfrak{Y}$ are affine, the expression inside the
limit of (6.22) is in fact independent of $W$ (for $W$ affine) and so (6.22)
simplifies to the sheafification of

$(E,J)\mapsto(\mathcal{O}_{\mathfrak{X}_{\operatorname{\acute{e}t}}}/p^{n})(\mathfrak{X})\otimes_{\mathcal{O}_{\mathfrak{Y}_{\operatorname{\acute{e}t}}}(\mathfrak{Y})}E/[p]_{q}.$
(6.23)

We now deal with the RHS of (6.21). First note that
$p_{2}^{*}(\mathcal{O}_{\mathfrak{X}_{\operatorname{\acute{e}t}}}/p^{n})$ is
the sheaf

$Z\mapsto\overline{\mathcal{O}}^{(1)}(Z)/p^{n}=(E/[p]_{q}\otimes_{S}R)/p^{n}$

where $Z$ is of the form (6.9). Therefore
$p_{1*}p_{2}^{*}(\mathcal{O}_{\mathfrak{X}_{\operatorname{\acute{e}t}}}/p^{n})$
is the sheaf

$(E,J)\mapsto(\mathcal{O}_{\mathfrak{X}_{\operatorname{\acute{e}t}}}(\mathfrak{X})\otimes_{\mathcal{O}_{\mathfrak{Y}_{\operatorname{\acute{e}t}}}(\mathfrak{Y})}E/[p]_{q})/p^{n}.$
(6.24)

Clearly (6.23) coincides with (6.24). ∎

We will also need vanishing of cohomology.

######  Lemma 6.22.

Let $Z$ be of the form (6.9). The cohomology groups
$H^{i}(Z,\overline{\mathcal{O}}^{(1)}/p^{n})$ vanish for $i>0$ and $n\geq 1$.

###### Proof.

It suffices to observe that any covering of $Z$ induces an étale covering of
$\operatorname{Spf}(E/[p]_{q}\widehat{\otimes}_{S}R)$. The result now follows
by vanishing of high Čech cohomology for affine schemes. ∎

######  Proposition 6.23.

Suppose $f$ is proper and smooth, and $\mathscr{F}$ a locally free
$\mathcal{O}_{\mathfrak{X}_{\operatorname{\acute{e}t}}}/p^{n}$-module of
finite type for some $n\geq 1$. Then the canonical base change map induced by
(6.20)

$Lu_{\mathfrak{Y}}^{q*}Rf_{*}\mathscr{F}\to Rp_{1*}Lp_{2}^{*}\mathscr{F}$
(6.25)

is a quasi-isomorphism.

###### Proof.

The proof is very similar to the proof of the counterpart in the pro-étale
setting (cf. Proposition 5.24) and so we omit the details. In the current
situation however, we get a result on the nose because:

  1. (1)

We know the sections of $\overline{\mathcal{O}}^{(1)}/p^{n}$ (in the pro-étale
setting the sections of $\widehat{\mathcal{O}}^{+}_{D}/p^{n}$ were only known
up to almost ambiguity). This led us to a result on the nose of the affine
case (cf. Lemma 6.21).

  2. (2)

By the proof of Lemma 6.22 we know the vanishing of higher Čech cohomology of
$\overline{\mathcal{O}}^{(1)}/p^{n}$ for coverings of $Z$ of the form (6.9).
The analogous result in the pro-étale setting (cf. the proof of Lemma 5.5(i))
was only known up to almost ambiguity.

∎

###  6.5 $q$-crystalline vs pro-étale fibered products

In this section we specialize to the case
$(D,I)=(A_{\operatorname{inf}},(\xi))$ and assume that we have a
$\operatorname{Spf}(\mathcal{O})$-morphism
$f\colon\mathfrak{X}\to\mathfrak{Y}$ of $p$-adic formal schemes of type
(S)(b). In §6.1, we constructed the site

$q-\textnormal{CRYS}(\mathfrak{Y}/A_{\operatorname{inf}})\overleftarrow{\times}_{\mathfrak{Y}_{\operatorname{\acute{e}t}}}\mathfrak{X}_{\operatorname{\acute{e}t}}$
(6.26)

On the pro-étale side, in §2.1, we construced a site $D$. Let $D^{\prime}$ be
the subsite of $D$ consisting of objects appearing in Proposition 2.8. In this
case, $D$ and $D^{\prime}$ have the same associated topoi:
$Y_{\operatorname{pro\acute{e}t}}\times_{\mathfrak{Y}_{\operatorname{\acute{e}t}}}\mathfrak{X}_{\operatorname{\acute{e}t}}$.
We want to compare the $q$-crystalline complex
$q\Omega_{\mathfrak{X}/\mathfrak{Y}}$ (cf. Definition 6.15) and
$A\Omega_{\mathfrak{X}/\mathfrak{Y}}$. These two objects live in different
topoi
$(\mathfrak{Y}/D)_{q-\textnormal{CRYS}}\overleftarrow{\times}_{\mathfrak{Y}_{\operatorname{\acute{e}t}}}\mathfrak{X}_{\operatorname{\acute{e}t}}$
and
$Y_{\operatorname{pro\acute{e}t}}\times_{\mathfrak{Y}_{\operatorname{\acute{e}t}}}\mathfrak{X}_{\operatorname{\acute{e}t}}$,
respectively. So we need to first construct a morphism between the topoi.

######  Lemma 6.24.

The functor

$\displaystyle u\colon D^{\prime}$ $\displaystyle\to
q-\textnormal{CRYS}(\mathfrak{Y}/A_{\operatorname{inf}})\overleftarrow{\times}_{\mathfrak{Y}_{\operatorname{\acute{e}t}}}\mathfrak{X}_{\operatorname{\acute{e}t}}$
$\displaystyle(\varprojlim_{i\in
I}\operatorname{Spa}(S_{i},S_{i}^{+})\to\operatorname{Spf}(S)\leftarrow\operatorname{Spf}(R))$
$\displaystyle\mapsto((A_{\operatorname{inf}}(S_{\infty}),(\xi))\to\operatorname{Spf}(S)\leftarrow\operatorname{Spf}(R))$
(6.27)

is cocontinuous.

######  Remark 6.25.

We explain how the morphism
$\operatorname{Spf}(S_{\infty})\to\operatorname{Spf}(S)$ is defined so that
the RHS of (6.24) makes sense. Let $S^{+}$ be the integral closure of $S$ in
$S[\tfrac{1}{p}]$. Then the LHS of (6.24) gives a morphism $\varprojlim_{i\in
I}\operatorname{Spa}(S_{i},S_{i}^{+})\to\operatorname{Spa}(S[\frac{1}{p}],S^{+})$
in $Y_{\operatorname{pro\acute{e}t}}$. This gives a continuous morphism
$S^{+}\to\varinjlim S_{i}^{+}$. The composition $S\to S^{+}\to\varinjlim
S_{i}^{+}\to S_{\infty}$ gives the desired morphism.

###### Proof.

We show the cocontinuity for coverings of type (a) first. Let
$(E_{1},J_{1})\to(A_{\operatorname{inf}}(S_{\infty}),(\xi))$ be a covering in
$q-\textnormal{CRYS}(\mathfrak{Y}/(A_{\operatorname{inf}})$. We verify the
cocontinuity condition for this cover. By definition we have a $p$-completely
étale covering map
$\operatorname{Spf}(E_{1}/J_{1})\to\operatorname{Spf}(S_{\infty})$. So
$E_{1}/J_{1}$ is the $p$-adic derived completion of an étale
$S_{\infty}$-algebra
$\tfrac{S_{\infty}[x_{1},\dots,x_{n}]}{(f_{1},\ldots,f_{n})}$. Then

$E_{1}/(p,J_{1})=(S_{i_{1}}^{+}/p)[x_{1},\ldots,x_{n}]/(\overline{f}_{1},\ldots,\overline{f}_{n})\otimes_{S_{i_{1}}^{+}/p}\varinjlim_{i\in
I}S_{i}^{+}/p$

for some $i_{1}\in I$. Moreover by increasing $i_{1}$ if necessary, we can
assume
$\tfrac{(S_{i_{1}}^{+}/p)[x_{1},\dots,x_{n}]}{(\overline{f}_{1},\ldots,\overline{f}_{n})}$
is faithfully flat over $S_{i_{1}}^{+}/p$ (cf. [Sta18, Tag 07RR]). By [Sta18,
Tag 04D1] there exists some lift of
$\tfrac{(S_{i_{1}}^{+}/p)[x_{1},\dots,x_{n}]}{(\overline{f}_{1},\ldots,\overline{f}_{n})}$
to an étale $S_{i_{1}}^{+}$-algebra $T$. Then $E_{1}/J_{1}$ is the $p$-adic
derived completion of $T\otimes_{S_{i_{1}}^{+}}\varinjlim_{i\in I}S_{i}^{+}$.
We claim that

$\operatorname{Spa}(\widehat{T}[\tfrac{1}{p}],T^{\prime})\to\operatorname{Spa}(S_{i_{1}},S_{i_{1}}^{+}),$

where $\widehat{T}$ is the $p$-adic completion of $T$, and $T^{\prime}$ is the
integral closure of $\widehat{T}$ in $\widehat{T}[\tfrac{1}{p}]$, is an étale
covering map. Indeed it is étale by [Hub96, Corollary 1.7.3(iii)]. It is also
a covering because $T/p$ is faithfully flat over $S_{i_{1}}^{+}/p$. It follows
that the fiber product

$\operatorname{Spa}(\widehat{T}[\tfrac{1}{p}],T^{\prime})\times_{\operatorname{Spa}(S_{i_{1}},S_{i_{1}}^{+})}\varprojlim_{i\in
I}\operatorname{Spa}(S_{i},S_{i}^{+})$ (6.28)

exists in $Y_{\operatorname{pro\acute{e}t}}$ and is a covering of
$\varprojlim_{i\in I}\operatorname{Spa}(S_{i},S_{i}^{+})$. We can rewrite
(6.28) as

$\varprojlim_{i\in
I}\operatorname{Spa}(\widehat{T}[\tfrac{1}{p}]\widehat{\otimes}_{S_{i_{1}}}S_{i},{S^{\prime}_{i}}^{+})$

where ${S^{\prime}_{i}}^{+}$ is the integral closure of
$T^{\prime}\otimes_{S_{i_{1}}^{+}}S_{i}^{+}$ in
$\widehat{T}[\tfrac{1}{p}]\widehat{\otimes}_{S_{i_{1}}}S_{i}$. Therefore we
obtain a composition of maps

$T\otimes_{S_{i_{1}}^{+}}\varinjlim_{i\in I}S_{i}^{+}\to\varinjlim_{i\in
I}(T^{\prime}\otimes_{S_{i_{1}}^{+}}S_{i}^{+})\to\varinjlim{S^{\prime}_{i}}^{+}.$

This verifies the cocontinuty condition for coverings of type (a).

Coverings of type (b) are trivial to handle. We now deal with coverings of
type (c): let
$((A_{\operatorname{inf}}(S_{\infty}),(\xi))\to\mathfrak{Y}^{\prime}\leftarrow\mathfrak{X}^{\prime})$
be such a cover of the RHS of (6.24). By using the argument in the proof of
Lemma 6.12, we can refine this to a cover of the form
$((A_{\operatorname{inf}}(S^{\prime}_{\infty}),(\xi))\to\operatorname{Spf}(S^{\prime})\leftarrow\operatorname{Spf}(R^{\prime}))$
where
$\operatorname{Spf}(S^{\prime}_{\infty})\to\operatorname{Spf}(S_{\infty})$ is
an étale covering. This is a covering of type (a) followed by a covering of
type (c). Thus by the first part of this proof, we have reduced to checking
the cocontinuity condition for the case where we have a cover of the form
$((A_{\operatorname{inf}}(S_{\infty}),(\xi))\to\operatorname{Spf}(S^{\prime})\leftarrow\operatorname{Spf}(R^{\prime}))$
of type (c) of the RHS of (6.24). This means that
$\operatorname{Spf}(S_{\infty})\to\operatorname{Spf}(S)$ factors through
$\operatorname{Spf}(S^{\prime})\to\operatorname{Spf}(S)$. It suffices to show
that $\varprojlim_{i\in
I}\operatorname{Spa}(S_{i},S_{i}^{+})\to\operatorname{Spa}(S[1/p],S^{+})$ in
$Y_{\operatorname{pro\acute{e}t}}$ factors through
$\operatorname{Spa}(S^{\prime}[\tfrac{1}{p}],S^{\prime+})\to\operatorname{Spa}(S[\tfrac{1}{p}],S^{+})$,
where $S^{\prime+}$ is the integral closure of $S^{\prime}$ in
$S^{\prime}[\tfrac{1}{p}]$. Observe that a composition
$\operatorname{Spf}(S_{\infty})\to\operatorname{Spf}(S^{\prime})\to\operatorname{Spf}(S)$
is equivalent to a continuous $S$-morphism $S^{\prime}\to S_{\infty}$. Since
$S/p\to S^{\prime}/p$ is of finite presentation, there exists $i_{1}\in I$
such that $S^{\prime}/p\to\varinjlim_{i\in I}S^{+}_{i}/p$ factors through
$S^{+}_{i_{1}}/p$. Moreover by increasing $i_{1}$ if necessary, we can assume
$S\to\varinjlim_{i\in I}S^{+}_{i}$ factors through $S_{i_{1}}^{+}$. Then for
each $n\geq 1$ we have a commutative square

$\textstyle{S^{\prime}/p^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S^{+}_{i_{1}}/p}$$\textstyle{S/p^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S^{+}_{i_{1}}/p^{n},\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

where the left vertical arrow is étale and so there is a unique map
$S^{\prime}/p^{n}\to S_{i_{1}}^{+}/p^{n}$ rendering the diagram commutative.
Thus upon taking limits we obtain a continuous map $S^{\prime}\to
S^{+}_{i_{1}}$ compatible with the maps from $S$. The result now follows. ∎

Lemma 6.24 induces a morphism of topoi

$\mu\colon
Y_{\operatorname{pro\acute{e}t}}\times_{\mathfrak{Y}_{\operatorname{\acute{e}t}}}\mathfrak{X}_{\operatorname{\acute{e}t}}\to(\mathfrak{Y}/A_{\operatorname{inf}})_{q-\textnormal{CRYS}}\overleftarrow{\times}_{\mathfrak{Y}_{\operatorname{\acute{e}t}}}\mathfrak{X}_{\operatorname{\acute{e}t}}.$
(6.29)

###  6.6 Comparison with $q$-de Rham cohomology

In this short section we compare $q\Omega_{\mathfrak{X}/\mathfrak{Y}}$ with a
_relative_ $q$-de Rham complex (cf. [BS22, Theorem 16.22] and [Kos21, Theorem
7.17]). We work locally and suppose
$\operatorname{Spf}(R)\to\operatorname{Spf}(S)$ is a smooth
$\operatorname{Spf}(\mathcal{O})$-morphism. In addition we assume $R$ is _very
small_ over $S$, i.e. $R/p$ is generated over $S/p$ by units. This means that
there is a canonical functorial surjection $P\to
R\widehat{\otimes}_{S}S_{\infty}$ with kernel $J$, where
$P:=\mathbb{A}_{\operatorname{inf}}(S_{\infty})[\\{t_{s}\\}_{s\in
R^{\times}}]^{\wedge}$ and the completions are $(p,\tilde{\xi})$-adic. In this
context $q=[\epsilon]\in A_{\operatorname{inf}}$ and in order to define $q$-de
Rham cohomology in the case that we need, we verify the following result.

######  Lemma 6.26.

$\mathbb{A}_{\operatorname{inf}}(S_{\infty})$ is a flat $A$-algebra.

###### Proof.

By [BMS18, Remark 4.31], it suffices to show that $S_{\infty}^{\flat}$ is
$(\epsilon-1)$-torsion free. This follows from Lemma 4.1 and [CS19,
(2.1.2.2)]. ∎

We may regard $P$ as a
$\delta$-$\mathbb{A}_{\operatorname{inf}}(S_{\infty})$-algebra in a unique way
by setting $\delta(x_{s})=0$ for each $s\in R^{\times}$. Let $D_{J,q}(P)$ be
the $q$-PD envelope of $P\to R\widehat{\otimes}_{S}S_{\infty}$. The main
result is the following.

######  Theorem 6.27.

Let $Z$ be of the form the RHS of (6.24) such that in addition $R$ is very
small over $S$. Then there is a canonical functorial quasi-isomorphism
$R\Gamma(Z,q\Omega_{\mathfrak{X}/\mathfrak{Y}})\cong
q\Omega^{*,\square}_{D_{J,q}(P)/\mathbb{A}_{\operatorname{inf}}(S_{\infty})}$.

###### Proof.

As with the comparison with prismatic cohomology, the hard work is already
done in [BS22, §16]. Indeed by Lemma 6.14

$R\Gamma(Z,q\Omega_{\mathfrak{X}/\mathfrak{Y}})=R\Gamma((R\widehat{\otimes}_{S}S_{\infty})/\mathbb{A}_{\operatorname{inf}}(S_{\infty}))_{q-\textnormal{CRYS}},\mathcal{O}_{q-\textnormal{CRYS}}).$

The result now follows by Theorem 16.21 in loc.cit. and the comparison made in
Lemma 6.6. ∎

###  6.7 Comparison with $A\Omega_{\mathfrak{X}/\mathfrak{Y}}$

We specialize to the case $(D,I)=(A_{\operatorname{inf}},(\xi))$. Having
established the following ingredients:

  1. (1)

a way to pass from the pro-étale setting to the $q$-crystalline setting (cf.
(6.29)),

  2. (2)

the Hodge-Tate comparison for $q\Omega_{\mathfrak{X}/\mathfrak{Y}}$ (cf.
Corollary 6.19), and

  3. (3)

the Hodge-Tate comparison for $A\Omega_{\mathfrak{X}/\mathfrak{Y}}$ (cf.
Theorem 5.19),

in this section we are now ready to compare
$q\Omega_{\mathfrak{X}/\mathfrak{Y}}$ and
$A\Omega_{\mathfrak{X}/\mathfrak{Y}}$ (cf. [BS22, Theorem 17.2] and [Kos21,
Theorem 8.1]).

######  Theorem 6.28.

Assume that we have a smooth $\operatorname{Spf}(\mathcal{O})$-morphism
$f\colon\mathfrak{X}\to\mathfrak{Y}$ of $p$-adic formal schemes of type
(S)(b). Suppose $\mathfrak{X}$ and $\mathfrak{Y}$ are flat over $\mathcal{O}$.
There is a canonical functorial morphism

$\mu^{-1}q\Omega_{\mathfrak{X}/\mathfrak{Y}}\to
A\Omega_{\mathfrak{X}/\mathfrak{Y}}$ (6.30)

in $D(D,p^{-1}W(\widehat{\mathcal{O}}^{+,\flat}_{Y}))$ compatible with the
Frobenius such that the cohomology sheaves of the cone of (6.30) are killed by
$W(\mathfrak{m}^{\flat})\subset A_{\operatorname{inf}}$. If in addition
$\mathfrak{Y}$ is locally of finite type over $\mathcal{O}$, then (6.30) is an
isomorphism.

######  Remark 6.29.

We explain how to view the LHS and RHS of (6.30) as objects in
$D(D,p^{-1}W(\widehat{\mathcal{O}}^{+,\flat}_{Y}))$. For the RHS this is
already explained in Definition 2.18 together with Lemma 2.19. For the LHS,
first note that
$p^{-1}W(\widehat{\mathcal{O}}^{+,\flat}_{Y})=\mu^{-1}p_{1}^{-1}\mathcal{O}_{q-\textnormal{CRYS}}$.
Moreover, by temporarily regarding $\nu_{f}^{q}$ as the following morphism of
ringed topoi (we abuse notation and use $\mathcal{O}_{q-\textnormal{CRYS}}$ to
denote the structure sheaves on both $(\mathfrak{X}/D)_{q-\textnormal{CRYS}}$
and $(\mathfrak{Y}/D)_{q-\textnormal{CRYS}}$)

$(\nu_{f}^{q},\nu_{f}^{q\sharp})\colon((\mathfrak{X}/D)_{q-\textnormal{CRYS}},\mathcal{O}_{q-\textnormal{CRYS}})\to((\mathfrak{Y}/D)_{q-\textnormal{CRYS}}\overleftarrow{\times}_{\mathfrak{Y}_{\operatorname{\acute{e}t}}}\mathfrak{X}_{\operatorname{\acute{e}t}},p_{1}^{-1}\mathcal{O}_{q-\textnormal{CRYS}})$

naturally equips $q\Omega_{\mathfrak{X}/\mathfrak{Y}}$ as an object in
$D(q-\textnormal{CRYS}(\mathfrak{Y}/D)\overleftarrow{\times}_{\mathfrak{Y}_{\operatorname{\acute{e}t}}}\mathfrak{X}_{\operatorname{\acute{e}t}},p_{1}^{-1}\mathcal{O}_{q-\textnormal{CRYS}})$.

###### Proof.

We follow the main ideas of the proof of [BS22, Theorem 17.2]. Let $Z$ be of
the form the LHS of (6.24). One difference in the (current) relative case is
that the derived $Z$-sections of the presheaf
$A\Omega_{\mathfrak{X}/\mathfrak{Y}}^{\text{psh}}$ identify with the derived
$Z$-sections of $A\Omega_{\mathfrak{X}/\mathfrak{Y}}$ only up to almost
ambiguity (cf. (5.20)). For this reason (together with the fact that the
derived $Z$-sections of the presheaf version are explicit), we will work
directly with the presheaf.

We will use the notation developed in §5.1. In particular, addition to the
site $D$, the site $D^{\text{psh}}$ will be useful. In fact we will work with
the subsites $D_{\text{vs}}\subset D$ and $D^{\text{psh}}_{\text{vs}}\subset
D^{\text{psh}}$ consisting only of good objects $Z$ such that $R$ is very
small over $S$. Note that $D_{\text{vs}}$ and $D$ induce the same topoi. We
have
$\mu^{-1}q\Omega_{\mathfrak{X}/\mathfrak{Y}}=\phi^{-1}q\Omega_{\mathfrak{X}/\mathfrak{Y}}^{\text{psh}}$
where $q\Omega_{\mathfrak{X}/\mathfrak{Y}}^{\text{psh}}$ is the complex of
presheaves given by

$Z\mapsto R\Gamma(u(Z),q\Omega_{\mathfrak{X}/\mathfrak{Y}}),$

where $u$ is the functor in Lemma 6.24. By (5.2), it suffices to construct a
canonical functorial morphism

$q\Omega_{\mathfrak{X}/\mathfrak{Y}}^{\text{psh}}\to
A\Omega_{\mathfrak{X}/\mathfrak{Y}}^{\text{psh}}$ (6.31)

in
$D(D^{\text{psh}}_{\text{vs}},p^{-1}W(\widehat{\mathcal{O}}^{+,\flat}_{Y}))$
compatible with the Frobenius, whose cone satisfies the relevant conditions
(we abuse notation and denote $p^{-1}W(\widehat{\mathcal{O}}^{+,\flat}_{Y})$
for the presheaf $Z\mapsto\mathbb{A}_{\operatorname{inf}}(S_{\infty})$). We do
this in several steps. We assume henceforth that $R$ is very small over $S$.

_Step 1: Constructing explicit representatives for
$R\Gamma(Z,q\Omega_{\mathfrak{X}/\mathfrak{Y}}^{\text{psh}})$ and
$R\Gamma(Z,A\Omega_{\mathfrak{X}/\mathfrak{Y}}^{\text{psh}})$._ We rewrite
each term $R\Gamma(Z,q\Omega_{\mathfrak{X}/\mathfrak{Y}}^{\text{psh}})$ and
$R\Gamma(Z,A\Omega_{\mathfrak{X}/\mathfrak{Y}}^{\text{psh}})$ by explicit
representatives of Koszul complexes. Strictly speaking this will be only up to
almost ambiguity for the second term in general, and is one difference
compared to the cases of [BS22] and [Kos21]. For the term
$R\Gamma(Z,q\Omega_{\mathfrak{X}/\mathfrak{Y}}^{\text{psh}})$, this is made
possible by comparison with the $q$-de Rham complex (cf. Theorem 6.27). Indeed
$R\Gamma(u(Z),q\Omega_{\mathfrak{X}/\mathfrak{Y}})\cong
q\Omega^{*,\square}_{D_{J,q}(P)/\mathbb{A}_{\operatorname{inf}}(S_{\infty})}$
and by definition
$q\Omega^{*,\square}_{D_{J,q}(P)/\mathbb{A}_{\operatorname{inf}}(S_{\infty})}$
is the complex

$\operatorname{Kos}_{c}(D_{J,q}(P);\\{\nabla_{q,s}\\}_{s\in R^{\times}})$
(6.32)

defined as the term-wise $(p,[p]_{q})$-completed filtered colimit over all
finite subsets $\Sigma\subset R^{\times}$ of the corresponding cohomological
Koszul complexes for $\\{\nabla_{q,s}\\}_{s\in\Sigma}$. The treatment of the
term $R\Gamma(Z,A\Omega_{\mathfrak{X}/\mathfrak{Y}}^{\text{psh}})$ is slightly
more delicate. We use the notation from §4. For each finite subset
$\Sigma\subset R^{\times}$ containing $t_{1},\ldots,t_{d}$, there is a
corresponding profinite group $\Delta_{\Sigma}=\mathbb{Z}_{p}^{\Sigma}$ with
generators (over $\mathbb{Z}_{p}$) $\\{\sigma_{s}\\}_{s\in\Sigma}$ and a
pro-(finite étale) affinoid perfectoid $\Delta_{\Sigma}$-cover161616One takes
$R^{\square}_{\Sigma}:=S\\{\\{t_{s}^{\pm 1}\\}_{s\in\Sigma}\\}$ and defines
$R^{\square}_{\Sigma,m}$, and $R_{\Sigma,\infty}$ analogously as in (4.1) and
(4.3), respectively. For instance $R^{\square}_{\Sigma,m}$ contains a
$p^{m}$-th root $t_{s}^{1/p^{m}}$.

$\operatorname{Spa}(R_{\Sigma,\infty}[\tfrac{1}{p}],R_{\Sigma,\infty})\to
X\times_{Y}Y_{\infty}\hskip 5.69054pt\text{ that refines the
$\Delta=\Delta_{\\{t_{1},\ldots,t_{d}\\}}$-cover }\hskip
5.69054ptX_{\infty}\to X\times_{Y}Y_{\infty}$

compatible with the natural surjection
$\Delta_{\Sigma}\twoheadrightarrow\Delta$. Consider now the complex

$(\varinjlim_{\Sigma}\eta_{q-1}\operatorname{Kos}_{c}(\mathbb{A}_{\operatorname{inf}}(R_{\Sigma,\infty});\\{\sigma_{s}-1\\}_{s\in\Sigma}))^{\wedge}$

defined as the term-wise $(p,[p]_{q})$-completed filtered colimit over all
finite subsets $\Sigma\subset R^{\times}$ (containing $t_{1},\ldots,t_{d}$) of
the corresponding cohomological Koszul complexes. By Theorem 4.11, Remark 4.12
and (5.3), there is a morphism

$(\varinjlim_{\Sigma}\eta_{q-1}\operatorname{Kos}_{c}(\mathbb{A}_{\operatorname{inf}}(R_{\Sigma,\infty});\\{\sigma_{s}-1\\}_{s\in\Sigma}))^{\wedge}\to
R\Gamma(Z,A\Omega_{\mathfrak{X}/\mathfrak{Y}}^{\text{psh}})$ (6.33)

whose cone has its cohomology groups killed by $W(\mathfrak{m}^{\flat})$ (if
$\mathfrak{Y}$ is locally of finite type over $\mathcal{O}$, then it is a
quasi-isomorphism).

_Step 2: Constructing a map
$R\Gamma(Z,q\Omega_{\mathfrak{X}/\mathfrak{Y}}^{\text{psh}})\to
R\Gamma(Z,A\Omega_{\mathfrak{X}/\mathfrak{Y}}^{\text{psh}})$._ By
(6.32)-(6.33), it is enough to construct a functorial comparison map

$\operatorname{Kos}_{c}(D_{J,q}(P);\\{\nabla_{q,s}\\}_{s\in
R^{\times}})\to(\varinjlim_{\Sigma}\eta_{q-1}\operatorname{Kos}_{c}(\mathbb{A}_{\operatorname{inf}}(R_{\Sigma,\infty});\\{\sigma_{s}-1\\}_{s\in\Sigma}))^{\wedge}.$
(6.34)

By [BMS18, Lemma 7.9], there is a canonical isomorphism

$\operatorname{Kos}_{c}(D_{J,q}(P);\\{\nabla_{q,s}\\}_{s\in
R^{\times}})\cong\eta_{q-1}\operatorname{Kos}_{c}(D_{J,q}(P);\\{\gamma_{s}-1\\}_{s\in
R^{\times}})$

and so to construct (6.34), it suffices to construct a canonical map

$D_{J,q}(P)\to\mathbb{A}_{\operatorname{inf}}((\varinjlim_{\Sigma}R_{\Sigma,\infty})^{\wedge})$
(6.35)

of $\mathbb{A}_{\operatorname{inf}}(S_{\infty})$-algebras that intertwines
$\sigma_{s}$ and $\gamma_{s}$ (the completion appearing on the RHS is
$p$-adic). We have a commutative diagram

$\textstyle{P\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{A}_{\operatorname{inf}}((\varinjlim_{\Sigma}R_{\Sigma,\infty})^{\wedge})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{R\widehat{\otimes}_{S}S_{\infty}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(\varinjlim_{\Sigma}R_{\Sigma,\infty})^{\wedge},}$

where the top map is induced by mapping $t_{s}$ to
$[(\ldots,\tilde{t}_{s}^{1/p},\tilde{t}_{s})]$ (where
$\tilde{t}_{s}^{1/p^{m}}$ denotes the image of $t_{s}^{1/p^{m}}\in
R^{\square}_{\Sigma,m}$ in $(\varinjlim_{\Sigma}R_{\Sigma,\infty})^{\wedge}$).
The action of $\sigma_{s}$ on
$\mathbb{A}_{\operatorname{inf}}((\varinjlim_{\Sigma}R_{\Sigma,\infty})^{\wedge})$
can be computed as in the last paragraph of §4.2 (where it is done in the case
$s=t_{i}$ on $\mathbb{A}_{\operatorname{inf}}(R_{\infty})$). This action
clearly intertwines with the action of $\gamma_{s}$ on $P$. Moreover
$P\to\mathbb{A}_{\operatorname{inf}}((\varinjlim_{\Sigma}R_{\Sigma,\infty})^{\wedge})$
is a $\delta$-map and so extends uniquely to a $\delta$-map
$D_{J,q}(P)\to\mathbb{A}_{\operatorname{inf}}((\varinjlim_{\Sigma}R_{\Sigma,\infty})^{\wedge})$.
The uniqueness also implies that it intertwines $\sigma_{s}$ and $\gamma_{s}$
(to see this one considers the compositions
$P\xrightarrow{\gamma_{s}}P\to\mathbb{A}_{\operatorname{inf}}(R_{\infty})$ and
$P\to\mathbb{A}_{\operatorname{inf}}(R_{\infty})\xrightarrow{\sigma_{s}}\mathbb{A}_{\operatorname{inf}}(R_{\infty})$).
In summary we have constructed the desired map (6.35).

_Step 3: Showing the cone of map
$R\Gamma(Z,q\Omega_{\mathfrak{X}/\mathfrak{Y}}^{\text{psh}})\to
R\Gamma(Z,A\Omega_{\mathfrak{X}/\mathfrak{Y}}^{\text{psh}})$ from Step 2
satisfies the relevant conditions._ It suffices to check that the map (6.34)
has cone whose cohomology groups are killed by $W(\mathfrak{m}^{\flat})$ (and
a quasi-isomorphism if $\mathfrak{Y}$ is locally of finite type over
$\mathcal{O}$). We use (a variant of) the criterion given in [BS22, Lemma
17.4]. Let $K_{1}$ and $K_{2}$ be the source and target of (6.34). By the
Hodge-Tate comparison for $q\Omega_{\mathfrak{X}/\mathfrak{Y}}$ (cf. Corollary
6.19) there is an isomorphism

$\Omega_{R/S}^{\bullet}\widehat{\otimes}_{S}\mathbb{A}_{\operatorname{inf}}(S_{\infty})/[p]_{q}\xrightarrow{\sim}H^{\bullet}(K_{1}/[p]_{q})\\{\bullet\\}$

of differential graded
$\mathbb{A}_{\operatorname{inf}}(S_{\infty})/[p]_{q}$-algebras. Let

$\eta_{K_{1}}\colon
R\widehat{\otimes}_{S}\mathbb{A}_{\operatorname{inf}}(S_{\infty})/[p]_{q}\to
H^{0}(K_{1}/[p]_{q})$ (6.36)

be the morphism in degree 0. In the proof of the de Rham specialization (cf.
Theorem 5.20) we constructed an isomorphism

$\Omega_{R/S}^{\bullet}\widehat{\otimes}_{S}\mathbb{A}_{\operatorname{inf}}(S_{\infty})/[p]_{q}=(q^{*}\Omega^{\bullet}_{\mathfrak{X}/\mathfrak{Y}})(Z)\to
C^{\bullet}_{Z}.$ (6.37)

Note that the twist appearing in
$\Omega_{R/S}^{\bullet}\widehat{\otimes}_{S}\mathbb{A}_{\operatorname{inf}}(S_{\infty})/[p]_{q}$
was implicit and is coming from (5.7). Moreover by Remark 4.12, there is a
natural morphism

$C^{\bullet}_{Z}\to H^{\bullet}(K_{2}/[p]_{q})\\{\bullet\\}$ (6.38)

whose cone has its cohomology groups killed by $W(\mathfrak{m}^{\flat})$ (if
$\mathfrak{Y}$ is locally of finite type over $\mathcal{O}$ then it is an
isomorphism). Composing (6.37)-(6.38) gives

$\Omega_{R/S}^{\bullet}\widehat{\otimes}_{S}\mathbb{A}_{\operatorname{inf}}(S_{\infty})/[p]_{q}\to
H^{\bullet}(K_{2}/[p]_{q})\\{\bullet\\}$ (6.39)

whose cone has cohomology groups killed by $W(\mathfrak{m}^{\flat})$ in
general, and if $\mathfrak{Y}$ is locally of finite type over $\mathcal{O}$,
then it is an isomorphism. Let

$\eta_{K_{2}}\colon
R\widehat{\otimes}_{S}\mathbb{A}_{\operatorname{inf}}(S_{\infty})/[p]_{q}\to
H^{0}(K_{2}/[p]_{q})$ (6.40)

be the morphism in degree 0 of (6.39). We remark that $\eta_{K_{2}}$ is
$S_{\infty}$-linear. To see this, note that (6.38) is clearly
$S_{\infty}$-linear in each degree and $\widehat{\mathcal{O}}^{+}_{D}(Z)\to
C^{0}_{Z}$ is $S_{\infty}$-linear as mentioned in the proof of Theorem 5.20.
One then checks that the composition induced by (6.34) and (6.36)

$R\widehat{\otimes}_{S}\mathbb{A}_{\operatorname{inf}}(S_{\infty})/[p]_{q}\xrightarrow{\eta_{K_{1}}}H^{0}(K_{1}/[p]_{q})\to
H^{0}(K_{2}/[p]_{q})$

identifies with $\eta_{K_{2}}$. This can be done by reducing to the case
$R=S\\{t_{1}^{\pm 1}\\}$ where one uses that the morphisms are
$S_{\infty}$-linear and keeps track of where the coordinate $t_{1}$ is mapped
to. Finally by a similar argument to the proof of [BS22, Lemma 17.4] (where in
place of derived Nakayama, one uses the _almost_ version, cf. Lemma 6.30), one
obtains that (6.34) has cone whose cohomology groups are killed by
$W(\mathfrak{m}^{\flat})$ (and is a quasi-isomorphism if $\mathfrak{Y}$ is
locally of finite type over $\mathcal{O}$ by loc.cit.).

In summary _Steps 1-3_ show that the morphism (6.34) constructed is functorial
in $Z$ and satisfies the relevant conditions. Similarly by fixing an injective
resolution $\mathbb{A}_{\operatorname{inf},X}\to I^{\bullet}$ in
$X_{\operatorname{pro\acute{e}t}}$, the edge map (4.15) comes from a
functorial morphism of actual complexes. This implies that (6.33) globalizes.
This produces the map (6.31) as promised. ∎

Almost in the next statement is w.r.t. $[\mathfrak{m}^{\flat}]\subset
A_{\operatorname{inf}}$.

######  Lemma 6.30 (almost derived Nakayama).

Let $(\mathscr{C},A_{\operatorname{inf}})$ be a ringed site and suppose that
$\mathscr{A}\in D(\mathscr{C},A_{\operatorname{inf}})$ is almost derived
$[p]_{q}$-adically complete. If
$\mathscr{A}\otimes^{\mathbb{L}}_{A_{\operatorname{inf}}}A_{\operatorname{inf}}/[p]_{q}$
is almost zero, then so is $\mathscr{A}$.

###### Proof.

By induction on $n\geq 1$, one obtains that
$\mathscr{A}\otimes^{\mathbb{L}}_{A_{\operatorname{inf}}}A_{\operatorname{inf}}/[p]_{q}^{n}$
is almost zero. Hence by taking the derived limit, we get that
$R\varprojlim_{n}(\mathscr{A}\otimes^{\mathbb{L}}_{A_{\operatorname{inf}}}A_{\operatorname{inf}}/[p]_{q}^{n})$
is almost zero. But by assumption the canonical morphism $\mathscr{A}\to
R\varprojlim_{n}(\mathscr{A}\otimes^{\mathbb{L}}_{A_{\operatorname{inf}}}A_{\operatorname{inf}}/[p]_{q}^{n})$
is an almost quasi-isomorphism and so the result follows. ∎

###  6.8 Comparison with
$R\Gamma_{A_{\operatorname{inf}}}(\mathfrak{X}/\mathfrak{Y})$

In this section we again specialize to the case
$(D,I)=(A_{\operatorname{inf}},(\xi))$ and unless otherwise stated, assume
that we have a $\operatorname{Spf}(\mathcal{O})$-morphism
$f\colon\mathfrak{X}\to\mathfrak{Y}$ of $p$-adic formal schemes of type
(S)(b). In addition we suppose $\mathfrak{X}$ and $\mathfrak{Y}$ are flat over
$\mathcal{O}$. We now move to the comparison between (relative)
$q$-crystalline cohomology
($R\Gamma_{q-\textnormal{CRYS}}(\mathfrak{X}/\mathfrak{Y})$, cf. Definition
6.15) and $A_{\operatorname{inf}}$-cohomology
($R\Gamma_{A_{\operatorname{inf}}}(\mathfrak{X}/\mathfrak{Y})$). Let
$Y^{\prime}_{\operatorname{pro\acute{e}t}}\subset
Y_{\operatorname{pro\acute{e}t}}$ be the subsite consisting of affinoid
perfectoid objects. By the proof of Lemma 6.24, the functor of sites

$\displaystyle u_{\operatorname{pro\acute{e}t}}\colon
Y^{\prime}_{\operatorname{pro\acute{e}t}}$ $\displaystyle\to
q-\textnormal{CRYS}(\mathfrak{Y}/A_{\operatorname{inf}})$
$\displaystyle\varprojlim_{i\in I}\operatorname{Spa}(S_{i},S_{i}^{+})$
$\displaystyle\mapsto(\mathbb{A}_{\operatorname{inf}}(S_{\infty}),(\xi))$

is cocontinuous and so induces a morphism of topoi

$\mu_{\operatorname{pro\acute{e}t}}\colon
Y_{\operatorname{pro\acute{e}t}}\to(\mathfrak{Y}/A_{\operatorname{inf}})_{q-\textnormal{CRYS}}.$

We now have a priori, two different morphisms of topoi
$u^{q}_{\mathfrak{Y}}\circ\mu_{\operatorname{pro\acute{e}t}}$ and $\nu_{Y}$
from $Y_{\operatorname{pro\acute{e}t}}$ to
$\mathfrak{Y}_{\operatorname{\acute{e}t}}$. As a reality check, we verify that
they coincide.

######  Lemma 6.31.

The two morphisms of topoi
$u^{q}_{\mathfrak{Y}}\circ\mu_{\operatorname{pro\acute{e}t}}\colon
Y_{\operatorname{pro\acute{e}t}}\to\mathfrak{Y}_{\operatorname{\acute{e}t}}$
and $\nu_{Y}\colon
Y_{\operatorname{pro\acute{e}t}}\to\mathfrak{Y}_{\operatorname{\acute{e}t}}$
are the same.

###### Proof.

Indeed for a sheaf $\mathscr{F}$ on $Y_{\operatorname{pro\acute{e}t}}$ and
$\mathfrak{V}$ an affine object in $\mathfrak{Y}_{\operatorname{\acute{e}t}}$

$(u^{q}_{\mathfrak{Y}*}\mu_{\operatorname{pro\acute{e}t},*}\mathscr{F})(\mathfrak{V})=\varprojlim_{\operatorname{Spf}(E/J)\to\mathfrak{V}}\varprojlim_{(A_{\operatorname{inf}}(S_{\infty}),(\xi))\to(E,J)}\mathscr{F}(\varprojlim_{i\in
I}\operatorname{Spa}(S_{i},S_{i}^{+}))$ (6.41)

where the left-most limit is taking place along
$\mathfrak{Y}_{\operatorname{\acute{E}T}}^{\textnormal{Aff}}$ and the middle
limit along $q-\textnormal{CRYS}(\mathfrak{Y}/A_{\operatorname{inf}})$.
Combining these two limits implies the RHS of (6.41) can be rewritten as

$\varprojlim_{\operatorname{Spf}(S_{\infty})\to\mathfrak{V}}\mathscr{F}(\varprojlim_{i\in
I}\operatorname{Spa}(S_{i},S_{i}^{+}))$

with the left-most limit taking place along
$\mathfrak{Y}_{\operatorname{\acute{E}T}}^{\textnormal{Aff}}$. The morphism
$\operatorname{Spf}(S_{\infty})\to\mathfrak{V}$ is equivalent to giving a
morphism on the generic fibers
$\operatorname{Spa}(S_{\infty}[\tfrac{1}{p}],S_{\infty})\to V$ (in the
category of adic spacs over $Y$) because $S_{\infty}$ is integrally closed in
$S_{\infty}[\tfrac{1}{p}]$. We claim that giving a morphism
$\operatorname{Spa}(S_{\infty}[\tfrac{1}{p}],S_{\infty})\to V$ (in the
category of adic spaces over $Y$) is equivalent to giving a morphism
$\varprojlim_{i\in I}\operatorname{Spa}(S_{i},S_{i}^{+})\to V$ in
$Y_{\operatorname{pro\acute{e}t}}$. This is similar to the proof of the last
part in Lemma 6.24 and we omit the details. ∎

######  Lemma 6.32.

Assume that we have a proper smooth $\operatorname{Spf}(\mathcal{O})$-morphism
$f\colon\mathfrak{X}\to\mathfrak{Y}$ of $p$-adic formal schemes of type (S)(b)
with $\mathfrak{X}$ and $\mathfrak{Y}$ flat over $\mathcal{O}$. Then the
cohomology sheaves of the cone of the canonical morphism

$\mu_{\operatorname{pro\acute{e}t}}^{-1}R\Gamma_{q-\textnormal{CRYS}}(\mathfrak{X}/\mathfrak{Y})\to(\mu_{\operatorname{pro\acute{e}t}}^{-1}R\Gamma_{q-\textnormal{CRYS}}(\mathfrak{X}/\mathfrak{Y}))^{\wedge}$

are killed by $[\mathfrak{m}^{\flat}]\subset A_{\operatorname{inf}}$ (the
completion on the RHS is derived $[p]_{q}$-adic).

###### Proof.

Take a covering $\\{U_{j}\to Y\\}_{j}$ by affinoid perfectoid objects of $Y$
and write $U_{j}=\varprojlim_{i\in
I_{j}}\operatorname{Spa}(S_{ij},S_{ij}^{+})$. Then

$\mu_{\operatorname{pro\acute{e}t}}^{-1}R\Gamma_{q-\textnormal{CRYS}}(\mathfrak{X}/\mathfrak{Y})\lvert_{U_{j}}=\mu_{\operatorname{pro\acute{e}t}}^{-1}(R\Gamma_{q-\textnormal{CRYS}}(\mathfrak{X}/\mathfrak{Y})\lvert_{(\mathbb{A}_{\inf}(S_{j,\infty}),(\xi))}).$

By Proposition 6.17(ii),
$(R\Gamma_{q-\textnormal{CRYS}}(\mathfrak{X}/\mathfrak{Y})\lvert_{(\mathbb{A}_{\inf}(S_{j,\infty}),(\xi))}$
is represented by a strictly perfect object in
$D((\mathfrak{Y}/A_{\operatorname{inf}})_{q-\textnormal{CRYS}}/(\mathbb{A}_{\inf}(S_{j,\infty}),(\xi)),\mathcal{O}_{q-\textnormal{CRYS}}\lvert_{(\mathbb{A}_{\inf}(S_{j,\infty}),(\xi))})$.
Now an easy calculation gives
$\mu_{\operatorname{pro\acute{e}t}}^{-1}\mathcal{O}_{q-\textnormal{CRYS}}=W(\widehat{\mathcal{O}}^{+,\flat}_{Y})$.
Moreover the cone of the canonical morphism

$W(\widehat{\mathcal{O}}^{+,\flat}_{Y})\to\mathbb{A}_{\operatorname{inf},Y}$

has its cohomology sheaves killed by $[\mathfrak{m}^{\flat}]$ (cf. [BMS18,
Lemma 5.6]). Since $\mathbb{A}_{\operatorname{inf},Y}$ is derived
$[p]_{q}$-complete by Corollary 2.28, the result follows. ∎

######  Theorem 6.33.

Assume that we have a proper smooth $\operatorname{Spf}(\mathcal{O})$-morphism
$f\colon\mathfrak{X}\to\mathfrak{Y}$ of $p$-adic formal schemes of type (S)(b)
with $\mathfrak{X}$ and $\mathfrak{Y}$ flat over $\mathcal{O}$. There is a
canonical functorial morphism

$\mu_{\operatorname{pro\acute{e}t}}^{-1}R\Gamma_{q-\textnormal{CRYS}}(\mathfrak{X}/\mathfrak{Y})\to
R\Gamma_{A_{\operatorname{inf}}}(\mathfrak{X}/\mathfrak{Y})$ (6.42)

in
$D(Y_{\operatorname{pro\acute{e}t}},W(\widehat{\mathcal{O}}^{+,\flat}_{Y}))$
compatible with the Frobenius such that the cohomology sheaves of the cone of
(6.42) are killed by $[\mathfrak{m}^{\flat}]\subset A_{\operatorname{inf}}$.

######  Remark 6.34.

We explain how to view the LHS and RHS of (6.42) as objects in
$D(Y_{\operatorname{pro\acute{e}t}},W(\widehat{\mathcal{O}}^{+,\flat}_{Y}))$.
For the RHS this is already explained in Definition 2.18 together with Lemma
2.19. For the LHS, first note
$R\Gamma_{q-\textnormal{CRYS}}(\mathfrak{X}/\mathfrak{Y})$ is naturally an
object of
$D(q-\textnormal{CRYS}(\mathfrak{Y}/A_{\operatorname{inf}}),\mathcal{O}_{q-\textnormal{CRYS}})$.
Since
$\mu_{\operatorname{pro\acute{e}t}}^{-1}\mathcal{O}_{q-\textnormal{CRYS}}=W(\widehat{\mathcal{O}}^{+,\flat}_{Y})$,
we get that the LHS can be viewed as an object in
$D(Y_{\operatorname{pro\acute{e}t}},W(\widehat{\mathcal{O}}^{+,\flat}_{Y}))$.

###### Proof.

The strategy is to work modulo $[p]_{q}$ and to gradually massage the LHS
towards the RHS of (6.42). The following schematic diagram summarises this
“massage process”:

$\textstyle{\mu_{\operatorname{pro\acute{e}t}}^{-1}R\Gamma_{q-\textnormal{CRYS}}(\mathfrak{X}/\mathfrak{Y})/[p]_{q}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\textnormal{I})}$$\textstyle{\Omega_{\mathfrak{X}/\mathfrak{Y}}^{\bullet}\lvert_{(\mathfrak{Y}/A_{\operatorname{inf}})_{q-\textnormal{CRYS}}\overleftarrow{\times}_{\mathfrak{Y}_{\operatorname{\acute{e}t}}}\mathfrak{X}_{\operatorname{\acute{e}t}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\textnormal{II})}$$\textstyle{R\Gamma_{A_{\operatorname{inf}}}(\mathfrak{X}/\mathfrak{Y})/[p]_{q}}$$\textstyle{\Omega_{\mathfrak{X}/\mathfrak{Y}}^{\bullet}\lvert_{Y_{\operatorname{pro\acute{e}t}}\times_{\mathfrak{Y}_{\operatorname{\acute{e}t}}}\mathfrak{X}_{\operatorname{\acute{e}t}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces,}$$\scriptstyle{(\textnormal{III})}$

where steps (I) and (III) are combinations of the relevant Hodge-Tate
comparisons, and step (II) is a combination of the base changes from the pro-
étale and $q$-crystalline side. Finally after working modulo $[p]_{q}$, one
can lift the results as both sides of (6.42) are (almost) derived
$[p]_{q}$-adically complete. We now begin the proof.

We define (6.42) as the composition

$\mu_{\operatorname{pro\acute{e}t}}^{-1}R\Gamma_{q-\textnormal{CRYS}}(\mathfrak{X}/\mathfrak{Y})\to
Rp_{*}\mu^{-1}q\Omega_{\mathfrak{X}/\mathfrak{Y}}\xrightarrow{Rp_{*}(\eqref{eq:compmorAOMeqOMe})}R\Gamma_{A_{\operatorname{inf}}}(\mathfrak{X}/\mathfrak{Y})$

where the first map is induced by the canonical base change map for the
commutative171717Commutativity can be checked at the level of sites: by
definition $\mu$ and $\mu_{\operatorname{pro\acute{e}t}}$ are induced by
cocontinuous functors and as mentioned in the proof of Lemma 6.9, so are $p$
and $p_{1}$. diagram of ringed topoi

$\textstyle{(Y_{\operatorname{pro\acute{e}t}}\times_{\mathfrak{Y}_{\operatorname{\acute{e}t}}}\mathfrak{X}_{\operatorname{\acute{e}t}},p^{-1}W(\widehat{\mathcal{O}}^{+,\flat}_{Y}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu}$$\scriptstyle{p}$$\textstyle{((\mathfrak{Y}/D)_{q-\textnormal{CRYS}}\overleftarrow{\times}_{\mathfrak{Y}_{\operatorname{\acute{e}t}}}\mathfrak{X}_{\operatorname{\acute{e}t}},p_{1}^{-1}\mathcal{O}_{q-\textnormal{CRYS}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{1}}$$\textstyle{(Y_{\operatorname{pro\acute{e}t}},W(\widehat{\mathcal{O}}^{+,\flat}_{Y}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{\operatorname{pro\acute{e}t}}}$$\textstyle{((\mathfrak{Y}/D)_{q-\textnormal{CRYS}},\mathcal{O}_{q-\textnormal{CRYS}}).}$
(6.43)

Note that since the morphism (6.30) is constructed in
$D(D,p^{-1}W(\widehat{\mathcal{O}}^{+,\flat}_{Y}))$, the morphism (6.42) is in
$D(Y_{\operatorname{pro\acute{e}t}},W(\widehat{\mathcal{O}}^{+,\flat}_{Y}))$.

By Corollary 5.13, $A\Omega_{\mathfrak{X}/\mathfrak{Y}}$ is almost derived
$[p]_{q}$-adically complete and so the same is true for
$R\Gamma_{A_{\operatorname{inf}}}(\mathfrak{X}/\mathfrak{Y})$. Similarly by
Lemma 6.32,
$\mu_{\operatorname{pro\acute{e}t}}^{-1}R\Gamma_{q-\textnormal{CRYS}}(\mathfrak{X}/\mathfrak{Y})$
is almost derived $[p]_{q}$-adically complete. Therefore by _almost derived
Nakayama_ (cf. Lemma 6.30), it suffices to show that the induced morphism

$\mu_{\operatorname{pro\acute{e}t}}^{-1}R\Gamma_{q-\textnormal{CRYS}}(\mathfrak{X}/\mathfrak{Y})\otimes^{\mathbb{L}}_{A_{\operatorname{inf}}}A_{\operatorname{inf}}/[p]_{q}\to
R\Gamma_{A_{\operatorname{inf}}}(\mathfrak{X}/\mathfrak{Y})\otimes^{\mathbb{L}}_{A_{\operatorname{inf}}}A_{\operatorname{inf}}/[p]_{q}.$
(6.44)

is an almost quasi-isomorphism. The LHS of (6.44) can be rewritten as

$\mu_{\operatorname{pro\acute{e}t}}^{-1}Rp_{1*}(q\Omega_{\mathfrak{X}/\mathfrak{Y}}\otimes^{\mathbb{L}}_{A_{\operatorname{inf}}}A_{\operatorname{inf}}/[p]_{q}).$

We have a spectral sequence

$R^{i}p_{1*}H^{j}(q\Omega_{\mathfrak{X}/\mathfrak{Y}}\otimes^{\mathbb{L}}_{A_{\operatorname{inf}}}A_{\operatorname{inf}}/[p]_{q})\implies
R^{i+j}p_{1*}(q\Omega_{\mathfrak{X}/\mathfrak{Y}}\otimes^{\mathbb{L}}_{A_{\operatorname{inf}}}A_{\operatorname{inf}}/[p]_{q})$

whose terms on the second page satisfy

$\displaystyle
R^{i}p_{1*}H^{j}(q\Omega_{\mathfrak{X}/\mathfrak{Y}}\otimes^{\mathbb{L}}_{A_{\operatorname{inf}}}A_{\operatorname{inf}}/[p]_{q})$
$\displaystyle\overset{(a)}{\cong}R^{i}p_{1*}p_{2}^{*}\Omega^{j}_{\mathfrak{X}/\mathfrak{Y}}\\{-j\\}$
$\displaystyle\overset{(b)}{\cong}H^{i}(Lu^{q*}_{\mathfrak{Y}}Rf_{*}\Omega^{j}_{\mathfrak{X}/\mathfrak{Y}}\\{-j\\})$
$\displaystyle\overset{(c)}{\cong}H^{i}((u^{q}_{\mathfrak{Y}})^{-1}Rf_{*}\Omega^{j}_{\mathfrak{X}/\mathfrak{Y}}\\{-j\\}\otimes^{\mathbb{L}}_{(u^{q}_{\mathfrak{Y}})^{-1}\mathcal{O}_{\mathfrak{Y}_{\operatorname{\acute{e}t}}}}\overline{\mathcal{O}}_{q-\textnormal{CRYS}}^{(1)}),$

where (a) follows from the (global) Hodge-Tate comparison (cf. Corollary
6.20), (b) follows from base change (cf. Proposition 6.23), noting that
$\Omega^{j}_{\mathfrak{X}/\mathfrak{Y}}$ is a locally free
$\mathcal{O}_{\mathfrak{X}_{\operatorname{\acute{e}t}}}$-module, and (c) is by
definition of $u^{q*}_{\mathfrak{Y}}$. We compute further

$\displaystyle\mu^{-1}_{\operatorname{pro\acute{e}t}}R^{i}p_{1*}H^{j}(q\Omega_{\mathfrak{X}/\mathfrak{Y}}\otimes^{\mathbb{L}}_{A_{\operatorname{inf}}}A_{\operatorname{inf}}/[p]_{q})$
$\displaystyle\overset{(1)}{\cong}H^{i}(\nu_{Y}^{-1}Rf_{*}\Omega^{j}_{\mathfrak{X}/\mathfrak{Y}}\\{-j\\}\otimes^{\mathbb{L}}_{\nu_{Y}^{-1}\mathcal{O}_{\mathfrak{Y}_{\operatorname{\acute{e}t}}}}\mu^{-1}_{\operatorname{pro\acute{e}t}}\overline{\mathcal{O}}_{q-\textnormal{CRYS}}^{(1)})$
$\displaystyle\overset{(2)}{\cong}H^{i}(\nu_{Y}^{-1}Rf_{*}\Omega^{j}_{\mathfrak{X}/\mathfrak{Y}}\\{-j\\}\otimes^{\mathbb{L}}_{\nu_{Y}^{-1}\mathcal{O}_{\mathfrak{Y}_{\operatorname{\acute{e}t}}}}\widehat{\mathcal{O}}^{+}_{Y})\otimes_{A_{\operatorname{inf}}/\xi,\varphi}A_{\operatorname{inf}}/[p]_{q}$
$\displaystyle\overset{(3)}{\cong}H^{i}(L\nu_{Y}^{*}Rf_{*}\Omega^{j}_{\mathfrak{X}/\mathfrak{Y}}\\{-j\\})\otimes_{A_{\operatorname{inf}}/\xi,\varphi}A_{\operatorname{inf}}/[p]_{q}$

where (1) follows from Lemma 6.31, (2) follows from the fact that
$\mu^{-1}_{\operatorname{pro\acute{e}t}}\overline{\mathcal{O}}_{q-\textnormal{CRYS}}^{(1)}$
is the sheaf $\varprojlim_{i\in
I}\operatorname{Spa}(S_{i},S_{i}^{+})\mapsto\mathbb{A}_{\operatorname{inf}}(S_{\infty})/[p]_{q}$,
and (3) follows by definition upon viewing $\nu_{Y}$ as a morphism of ringed
topoi
$(Y_{\operatorname{pro\acute{e}t}},\widehat{\mathcal{O}}^{+}_{Y})\to(\mathfrak{Y}_{\operatorname{\acute{e}t}},\mathcal{O}_{\mathfrak{Y}_{\operatorname{\acute{e}t}}})$.

By Corollary 5.27, the base change morphism

$L\nu_{Y}^{*}Rf_{*}\Omega^{j}_{\mathfrak{X}/\mathfrak{Y}}\\{-j\\}\to
Rp_{*}Lq^{*}\Omega^{j}_{\mathfrak{X}/\mathfrak{Y}}\\{-j\\}$

is an almost isomorphism (via the canonical map $A_{\operatorname{inf}}\to
A_{\operatorname{inf}}/\xi$, $[\mathfrak{m}^{\flat}]$ is sent to
$\mathfrak{m}$). Therefore the expression on the RHS of (3) is almost
isomorphic to

$R^{i}p_{*}q^{*}\Omega^{j}_{\mathfrak{X}/\mathfrak{Y}}\\{-j\\}\otimes_{A_{\operatorname{inf}}/\xi,\varphi}A_{\operatorname{inf}}/[p]_{q}.$

Moreover by the Hodge-Tate comparison (cf. Theorem 5.19), there is an almost
isomorphism

$R^{i}p_{*}q^{*}\Omega^{j}_{\mathfrak{X}/\mathfrak{Y}}\\{-j\\}\otimes_{A_{\operatorname{inf}}/\xi,\varphi}A_{\operatorname{inf}}/[p]_{q}\to
R^{i}p_{*}H^{j}(\widetilde{\Omega}_{\mathfrak{X}/\mathfrak{Y}})\otimes_{A_{\operatorname{inf}}/\xi,\varphi}A_{\operatorname{inf}}/[p]_{q}.$

There is a spectral sequence (with second page terms)

$R^{i}p_{*}H^{j}(\widetilde{\Omega}_{\mathfrak{X}/\mathfrak{Y}})\otimes_{A_{\operatorname{inf}}/\xi,\varphi}A_{\operatorname{inf}}/[p]_{q}\implies
Rp_{*}(\widetilde{\Omega}_{\mathfrak{X}/\mathfrak{Y}}\otimes_{A_{\operatorname{inf}}/\xi,\varphi}A_{\operatorname{inf}}/[p]_{q})$

By construction the morphism (6.30) was shown to be compatible with the Hodge-
Tate comparisons on the $q$-crystalline side and the pro-étale side, and so we
conclude that there is an almost quasi-isomorphism

$\mu_{\operatorname{pro\acute{e}t}}^{-1}Rp_{1*}(q\Omega_{\mathfrak{X}/\mathfrak{Y}}\otimes^{\mathbb{L}}_{A_{\operatorname{inf}}}A_{\operatorname{inf}}/[p]_{q})\to
Rp_{*}(\widetilde{\Omega}_{\mathfrak{X}/\mathfrak{Y}}\otimes_{A_{\operatorname{inf}}/\xi,\varphi}A_{\operatorname{inf}}/[p]_{q}).$
(6.45)

compatible with (6.44). Finally by (5.4) the map

$R\Gamma_{A_{\operatorname{inf}}}(\mathfrak{X}/\mathfrak{Y})\otimes^{\mathbb{L}}_{A_{\operatorname{inf}}}A_{\operatorname{inf}}/[p]_{q}\to
Rp_{*}(\widetilde{\Omega}_{\mathfrak{X}/\mathfrak{Y}}\otimes_{A_{\operatorname{inf}}/\xi,\varphi}A_{\operatorname{inf}}/[p]_{q})$
(6.46)

is an almost quasi-isomorphism (with the twist by $\varphi$ being implicit
there). Combining (6.45)-(6.46), gives that (6.44) is an almost quasi-
isomorphism. ∎

###  6.9 Perfectness of
$R\Gamma_{A_{\operatorname{inf}}}(\mathfrak{X}/\mathfrak{Y})$

In this short section we record (almost) perfectness of
$R\Gamma_{A_{\operatorname{inf}}}(\mathfrak{X}/\mathfrak{Y})$ together with
base change (cf. Proposition 6.35 for the precise result). We will rely on the
comparison with $q$-crystalline cohomology (cf. Theorem 6.33), however similar
results can be obtained (albeit via a longer argument) using only the de Rham
specialization of $A\Omega_{\mathfrak{X}/\mathfrak{Y}}$ (cf. Theorem 5.20).
Almost in the next proposition is with respect to
$[\mathfrak{m}^{\flat}]\subset A_{\operatorname{inf}}$.

######  Proposition 6.35.

Assume that we have a proper smooth $\operatorname{Spf}(\mathcal{O})$-morphism
$f\colon\mathfrak{X}\to\mathfrak{Y}$ of $p$-adic formal schemes with
$\mathfrak{X}$ and $\mathfrak{Y}$ flat over $\mathcal{O}$. Let $U\in
Y_{\operatorname{pro\acute{e}t}}$ be an affinoid perfectoid object determining
the perfectoid space $\operatorname{Spa}(R,R^{+})$. Then

  1. (i)

$R\Gamma(U,R\Gamma_{A_{\operatorname{inf}}}(\mathfrak{X}/\mathfrak{Y}))$ is
almost quasi-isomorphic to a perfect object in $D(W(R^{\flat+}))$.

  2. (ii)

$R\Gamma_{A_{\operatorname{inf}}}(\mathfrak{X}/\mathfrak{Y})\lvert_{U}$ is
almost quasi-isomorphic to a strictly perfect object in
$D(Y_{\operatorname{pro\acute{e}t}}/U,W(\widehat{\mathcal{O}}^{+,\flat}_{Y}\lvert_{U}))$.

  3. (iii)

$R\Gamma_{A_{\operatorname{inf}}}(\mathfrak{X}/\mathfrak{Y})$ is almost quasi-
isomorphic to a perfect object in
$D(Y_{\operatorname{pro\acute{e}t}},W(\widehat{\mathcal{O}}^{+,\flat}_{Y}))$.

  4. (iv)

For a morphism of affinoid perfectoids $U^{\prime}\to U$ (in
$Y_{\operatorname{pro\acute{e}t}}$) with $U^{\prime}$ determining the
perfectoid space $\operatorname{Spa}(R^{\prime},R^{\prime+})$, the natural
morphism

$R\Gamma(U,R\Gamma_{A_{\operatorname{inf}}}(\mathfrak{X}/\mathfrak{Y}))\otimes^{\mathbb{L}}_{W(R^{\flat+})}W(R^{\prime\flat+})\to
R\Gamma(U^{\prime},R\Gamma_{A_{\operatorname{inf}}}(\mathfrak{X}/\mathfrak{Y}))$

has the cohomology groups of its cone killed by $[\mathfrak{m}^{\flat}]$.

###### Proof.

Parts (i)-(iii) are an immediate consequence of

  1. (1)

the analogous perfectness results of
$R\Gamma_{q-\textnormal{CRYS}}(\mathfrak{X}/\mathfrak{Y})$ (cf. Proposition
6.17(i)-(iii)) and

  2. (2)

the comparison of
$R\Gamma_{A_{\operatorname{inf}}}(\mathfrak{X}/\mathfrak{Y})$ with
$R\Gamma_{q-\textnormal{CRYS}}(\mathfrak{X}/\mathfrak{Y})$ (cf. Theorem 6.33).

Finally part (iv) follows from part (ii). ∎

###  6.10 Comparison with relative $p$-adic étale cohomology

Assume that we have a proper smooth $\operatorname{Spf}(\mathcal{O})$-morphism
$f\colon\mathfrak{X}\to\mathfrak{Y}$ of $p$-adic formal schemes locally of
finite type and flat over $\mathcal{O}$. In this section we compare
$R\Gamma_{q-\textnormal{CRYS}}(\mathfrak{X}/\mathfrak{Y})$ with $p$-adic étale
cohomology (i.e.
$Rf_{\eta,\operatorname{pro\acute{e}t}*}\widehat{\mathbb{Z}}_{p}$). This will
involve considering the derived $p$-adic completions as
$\mu\notin[\mathfrak{m}^{\flat}]$.

######  Theorem 6.36.

We have an identification

$(\mu_{\operatorname{pro\acute{e}t}}^{-1}R\Gamma_{q-\textnormal{CRYS}}(\mathfrak{X}/\mathfrak{Y}))^{\wedge}\otimes_{A_{\operatorname{inf}}}^{\mathbb{L}}A_{\operatorname{inf}}[\tfrac{1}{\mu}]\cong
Rf_{\eta,\operatorname{pro\acute{e}t}*}\widehat{\mathbb{Z}}_{p}\otimes_{\widehat{\mathbb{Z}}_{p}}^{\mathbb{L}}\mathbb{A}_{\operatorname{inf},Y}[\tfrac{1}{\mu}],$

where the completion on the LHS is derived $p$-adic.

###### Proof.

Denote by
$R\Gamma_{A_{\operatorname{inf}}}(\mathfrak{X}/\mathfrak{Y})^{\wedge}$ the
derived $p$-adic completion of
$R\Gamma_{A_{\operatorname{inf}}}(\mathfrak{X}/\mathfrak{Y})$. Then Theorem
6.33 and [CK19, Lemma 3.17] induces a quasi-isomorphism

$(\mu_{\operatorname{pro\acute{e}t}}^{-1}R\Gamma_{q-\textnormal{CRYS}}(\mathfrak{X}/\mathfrak{Y}))^{\wedge}\otimes_{A_{\operatorname{inf}}}^{\mathbb{L}}A_{\operatorname{inf}}[\tfrac{1}{\mu}]\cong
R\Gamma_{A_{\operatorname{inf}}}(\mathfrak{X}/\mathfrak{Y})^{\wedge}\otimes_{A_{\operatorname{inf}}}^{\mathbb{L}}A_{\operatorname{inf}}[\tfrac{1}{\mu}]$

as $\mu\in W(\mathfrak{m}^{\flat})$. The result now follows from Theorem 3.15.
∎

##  7 Applications

###  7.1 Relative Hodge-Tate spectral sequence

In this section we record a relative Hodge-Tate spectral sequence in Theorem
7.2 and is independent of the $q$-crystalline section. This only uses the
results developed in §5. In the context of a discretely valued complete
nonarchimedean extension of $\mathbb{Q}_{p}$ with perfect residue field, such
a sequence already exists by [CS17, Corollary 2.2.4] (cf. [AG22, Remarque
1.4.7]). In the context of schemes, results have been obtained in [AG22]. It
is an interesting question to understand the compatibility of these spectral
sequences. Our approach to establishing the relative Hodge-Tate spectral
sequence mirrors that of loc.cit. (we both pass through a fiber product of
topoi) of which the following table summarises the similarities of the two
approaches.

Our approach |  Approach of Abbes-Gros
---|---
Use of pro-étale site |  Use of Faltings’ site
Hodge-Tate specialization of $A\Omega_{\mathfrak{X}/\mathfrak{Y}}$ (cf. Theorems 5.2 and 5.19) |  Hodge-Tate comparison of the _structure sheaf_ of the relative Faltings’ topos (cf. [AG22, Théorème 6.6.4])
Base change for fiber product of topoi (cf. Proposition 5.24) |  Base change for the relative Faltings’ site (cf. [AG22, Théorème 6.5.31])

Let us mention one difference between the two approaches is that we get rid of
the “junk torsion” (via the décalage functor) while Abbes-Gros do not.

We begin by comparing the sheaf of Kähler differentials on the special and
generic fibers.

######  Lemma 7.1.

Suppose $f\colon\mathfrak{X}\to\mathfrak{Y}$ is a proper smooth morphism of
$p$-adic formal schemes of type (S)(b) over $\mathcal{O}$. Suppose
$\mathfrak{X}$ and $\mathfrak{Y}$ are flat over
$\operatorname{Spf}(\mathcal{O})$ and $X$ and $Y$ are locally of finite type
over $\operatorname{Spa}(C,\mathcal{O})$. Consider the commutative diagram of
ringed topoi

$\textstyle{(X,\mathcal{O}_{X})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\text{sp}_{\mathfrak{X}}}$$\scriptstyle{f_{\eta}}$$\textstyle{(\mathfrak{X},\mathcal{O}_{\mathfrak{X}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{\operatorname{Zar}}}$$\textstyle{(Y,\mathcal{O}_{Y})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\text{sp}_{\mathfrak{Y}}}$$\textstyle{(\mathfrak{Y},\mathcal{O}_{\mathfrak{Y}}),}$

and $\mathscr{F}$ a locally free $\mathcal{O}_{\mathfrak{X}}$-module. Then the
canonical base change morphism

$L\text{sp}_{\mathfrak{Y}}^{*}Rf_{\operatorname{Zar}*}\mathscr{F}\to
Rf_{\eta*}L\text{sp}_{\mathfrak{X}}^{*}\mathscr{F}$

is a quasi-isomorphism.

###### Proof.

First note that by [Sta18, Tag 0A1H],
$Rf_{\operatorname{Zar}*}\mathscr{F}\otimes^{\mathbb{L}}_{\mathcal{O}_{\mathfrak{Y}}}\mathcal{O}_{\mathfrak{Y}}/p^{n}$
is a perfect object of $D(\mathcal{O}_{\mathfrak{Y}}/p^{n})$ and its formation
commutes with base change. Therefore by taking the derived limit we obtain
$Rf_{\operatorname{Zar}*}\mathscr{F}$ itself is a perfect object of
$D(\mathcal{O}_{\mathfrak{Y}})$. Therefore
$L\text{sp}_{\mathfrak{Y}}^{*}Rf_{\operatorname{Zar}*}\mathscr{F}$ is a
perfect object of $D(\mathcal{O}_{Y})$.

The claim is local for the Zariski topology on $\mathfrak{Y}$, so we can
assume it is affine, say $\mathfrak{Y}=\operatorname{Spf}(S)$. Let $U\subseteq
Y$ be any open affinoid of $Y$ such that
$L\text{sp}_{\mathfrak{Y}}^{*}Rf_{\operatorname{Zar}*}\mathscr{F}\lvert_{U}$
is strictly perfect. Then for any rational subset $U^{\prime}\subseteq U$, by
Tate acyclicity

$R\Gamma(U,L\text{sp}_{\mathfrak{Y}}^{*}Rf_{\operatorname{Zar}*}\mathscr{F})\otimes_{\mathcal{O}_{Y}(U)}^{\mathbb{L}}\mathcal{O}_{Y}(U^{\prime})\cong
R\Gamma(U^{\prime},L\text{sp}_{\mathfrak{Y}}^{*}Rf_{\operatorname{Zar}*}\mathscr{F}).$
(7.1)

Moreover by Kiehl’s finiteness theorem [Kie67, Theorem 3.3]

$(R^{i}f_{\eta*}\text{sp}_{\mathfrak{X}}^{*}\mathscr{F})(U)\otimes_{O_{Y}(U)}\mathcal{O}_{Y}(U^{\prime})=(R^{i}f_{\eta*}\text{sp}_{\mathfrak{X}}^{*}\mathscr{F})(U^{\prime}).$
(7.2)

Thus by Tate acyclicity and flatness of $\mathcal{O}_{Y}(U^{\prime})$ over
$\mathcal{O}_{Y}(U)$ (cf. [Hub94, pg. 530]), (7.2) implies

$R\Gamma(U,Rf_{\eta*}L\text{sp}_{\mathfrak{X}}^{*}\mathscr{F})\otimes_{\mathcal{O}_{Y}(U)}^{\mathbb{L}}\mathcal{O}_{Y}(U^{\prime})\cong
R\Gamma(U^{\prime},Rf_{\eta*}L\text{sp}_{\mathfrak{X}}^{*}\mathscr{F}).$ (7.3)

Therefore (7.1) and (7.3) implies that it suffices to check that the induced
morphism

$R\Gamma(U,L\text{sp}_{\mathfrak{Y}}^{*}Rf_{\operatorname{Zar}*}\mathscr{F})\to
R\Gamma(U,Rf_{\eta*}L\text{sp}_{\mathfrak{X}}^{*}\mathscr{F})$

is a quasi-isomorphism where in addition $U$ comes from some affine open
$\mathfrak{U}\subseteq\mathfrak{Y}$.

We can assume without loss of generality that $\mathfrak{U}=\mathfrak{Y}$ and
$Rf_{\operatorname{Zar}*}\mathscr{F}$ is represented by a strictly perfect
complex $\mathscr{E}^{\bullet}$. In that case
$R\Gamma(U,L\text{sp}_{\mathfrak{Y}}^{*}Rf_{\operatorname{Zar}*}\mathscr{F})$
is represented by the complex
$\mathscr{E}^{\bullet}(\mathfrak{Y})\otimes_{S}S[\tfrac{1}{p}]$, which in turn
is just
$R\Gamma(\mathfrak{Y},Rf_{\operatorname{Zar}*}\mathscr{F})\otimes_{S}^{\mathbb{L}}S[\tfrac{1}{p}]$.
Choose some finite open affine cover of $\mathfrak{X}$. Then
$R\Gamma(\mathfrak{Y},Rf_{\operatorname{Zar}*}\mathscr{F})$ is computed via
the associated Čech complex $C^{\bullet}$ and
$R\Gamma(U,Rf_{\eta*}L\text{sp}_{\mathfrak{X}}^{*}\mathscr{F})$ is computed by
$C^{\bullet}\otimes_{S}S[\tfrac{1}{p}]$. Since $S[\tfrac{1}{p}]$ is flat over
$S$, the result now follows. ∎

######  Theorem 7.2.

Suppose $f\colon\mathfrak{X}\to\mathfrak{Y}$ is a proper smooth morphism of
$p$-adic formal schemes of type (S)(b) over $\mathcal{O}$. Assume that
$\mathfrak{X}$ and $\mathfrak{Y}$ are flat over $\mathcal{O}$. There is an
$E_{2}$-spectral sequence

$E_{2}^{i,j}=H^{i}(L\nu_{Y}^{*}Rf_{*}\Omega^{j}_{\mathfrak{X}/\mathfrak{Y}}\\{-j\\})\implies
H^{i+j}(R\Gamma_{A_{\operatorname{inf}}}(\mathfrak{X}/\mathfrak{Y})\otimes^{\mathbb{L}}_{A_{\operatorname{inf}},\theta\circ\varphi^{-1}}\mathcal{O})$
(7.4)

in $D(Y_{\operatorname{pro\acute{e}t}},\mathcal{O}^{a})$. Moreover, if both
$\mathfrak{X}$ and $\mathfrak{Y}$ are locally of finite type over
$\mathcal{O}$ then upon inverting $p$, (7.4) takes the shape

$E_{2}^{i,j}=H^{i}(Rf_{\eta*}\Omega^{j}_{X/Y}\otimes^{\mathbb{L}}_{\mathcal{O}_{Y}}\widehat{\mathcal{O}}_{Y}(-j))\implies
R^{i+j}f_{\eta,\operatorname{pro\acute{e}t}*}\widehat{\mathbb{Z}}_{p}\otimes_{\widehat{\mathbb{Z}}_{p}}\widehat{\mathcal{O}}_{Y}$
(7.5)

in $D(Y_{\operatorname{pro\acute{e}t}},C)$ (here the $f_{\eta*}$ is
pushforward between analytic topologies). In addition, if for any $i,j\geq 0$,
$R^{i}f_{\eta*}\Omega^{j}_{X/Y}$ is locally free of finite type over
$\mathcal{O}_{Y}$, then (7.5) takes the shape

$E_{2}^{i,j}=R^{i}f_{\eta*}\Omega^{j}_{X/Y}\otimes_{\mathcal{O}_{Y}}\widehat{\mathcal{O}}_{Y}(-j)\implies
R^{i+j}f_{\eta,\operatorname{pro\acute{e}t}*}\widehat{\mathbb{Z}}_{p}\otimes_{\widehat{\mathbb{Z}}_{p}}\widehat{\mathcal{O}}_{Y}$
(7.6)

and it degenerates.

######  Remark 7.3.

Let $K$ be a discretely valued complete nonarchimedean extension of
$\mathbb{Q}_{p}$ with perfect residue field and ring of integers
$\mathcal{O}_{K}$. Then if $f_{\eta}$ comes from base change of a proper
smooth morphism of smooth adic spaces over
$\operatorname{Spa}(K,\mathcal{O}_{K})$, then by [Sch13, Theorem 8.8(ii)] and
base change, for any $i,j\geq 0$, $R^{i}f_{\eta*}\Omega^{j}_{X/Y}$ is locally
free of finite type over $\mathcal{O}_{Y}$.

###### Proof.

By Theorem 5.2 and the projection formula (cf. [Sta18, Tag 0944]) we get a
morphism in $D(Y_{\operatorname{pro\acute{e}t}},\mathcal{O})$

$R\Gamma_{A_{\operatorname{inf}}}(\mathfrak{X}/\mathfrak{Y})\otimes^{\mathbb{L}}_{A_{\operatorname{inf}},\theta\circ\varphi^{-1}}\mathcal{O}\to
Rp_{*}\widetilde{\Omega}_{\mathfrak{X}/\mathfrak{Y}}$

whose image in $D(Y_{\operatorname{pro\acute{e}t}},\mathcal{O}^{a})$ is an
almost isomorphism. Therefore we obtain an $E_{2}$-spectral sequence (the
Cartan-Leray spectral sequence)

$R^{i}p_{*}H^{j}(\widetilde{\Omega}_{\mathfrak{X}/\mathfrak{Y}})\implies
H^{i+j}(R\Gamma_{A_{\operatorname{inf}}}(\mathfrak{X}/\mathfrak{Y})\otimes^{\mathbb{L}}_{A_{\operatorname{inf}},\theta\circ\varphi^{-1}}\mathcal{O})$

in $D(Y_{\operatorname{pro\acute{e}t}},\mathcal{O}^{a})$. By Theorem 5.19, we
get a morphism
$R^{i}p_{*}q^{*}\Omega^{j}_{\mathfrak{X}/\mathfrak{Y}}\\{-j\\}\to
R^{i}p_{*}H^{j}(\widetilde{\Omega}_{\mathfrak{X}/\mathfrak{Y}})$ in
$D(Y_{\operatorname{pro\acute{e}t}},\mathcal{O})$, whose image in
$D(Y_{\operatorname{pro\acute{e}t}},\mathcal{O}^{a})$ is an almost
isomorphism. By Corollary 5.27, the morphism

$H^{i}(L\nu_{Y}^{*}Rf_{*}\Omega^{j}_{\mathfrak{X}/\mathfrak{Y}}\\{-j\\})\to
R^{i}p_{*}q^{*}\Omega^{j}_{\mathfrak{X}/\mathfrak{Y}}\\{-j\\}$

is an almost isomorphism. This proves (7.4).

Suppose now both $X$ and $Y$ are locally of finite type over
$\operatorname{Spa}(C,\mathcal{O})$. We now invert $p$ and perform some yoga
to obtain (7.5). Indeed

$\displaystyle
H^{i}(L\nu_{Y}^{*}Rf_{*}\Omega^{j}_{\mathfrak{X}/\mathfrak{Y}}\\{-j\\})\otimes_{\mathcal{O}}C$
$\displaystyle\overset{(i)}{\cong}H^{i}(L\nu_{Y}^{*}Rf_{*}\Omega^{j}_{\mathfrak{X}/\mathfrak{Y}}\\{-j\\}\otimes_{\mathcal{O}}^{\mathbb{L}}C)$
$\displaystyle\overset{(ii)}{\cong}H^{i}(Rf_{*}\Omega^{j}_{\mathfrak{X}/\mathfrak{Y}}\otimes_{\mathcal{O}_{\mathfrak{Y}_{\operatorname{\acute{e}t}}}}^{\mathbb{L}}\widehat{\mathcal{O}}_{Y}(-j))$
$\displaystyle\overset{(iii)}{\cong}H^{i}(Rf_{\operatorname{Zar},*}\Omega^{j}_{\mathfrak{X}/\mathfrak{Y}}\otimes^{\mathbb{L}}_{\mathcal{O}_{\mathfrak{Y}}}\widehat{\mathcal{O}}_{Y}(-j))$
$\displaystyle\overset{(iv)}{\cong}H^{i}(Rf_{\eta*}\Omega^{j}_{X/Y}\otimes^{\mathbb{L}}_{\mathcal{O}_{Y}}\widehat{\mathcal{O}}_{Y}(-j))$

where (i) follows from localization being flat, (ii) is by definition of
$L\nu_{Y}^{*}$ (we have dropped the $\nu_{Y}^{-1}$) and the fact that the
cokernel of $\mathcal{O}(1)\hookrightarrow\mathcal{O}\\{1\\}$ is killed by
$p^{1/(p-1)}$, (iii) from the fact that coherent cohomology can be computed in
the étale or Zariski site, and (iv) is a consequence of Lemma 7.1. The
identification

$H^{i+j}(R\Gamma_{A_{\operatorname{inf}}}(\mathfrak{X}/\mathfrak{Y})\otimes^{\mathbb{L}}_{A_{\operatorname{inf}},\theta\circ\varphi^{-1}}\mathcal{O})[\tfrac{1}{p}]\cong
R^{i+j}f_{\eta,\operatorname{pro\acute{e}t}*}\widehat{\mathbb{Z}}_{p}\otimes_{\widehat{\mathbb{Z}}_{p}}\widehat{\mathcal{O}}_{Y}$

follows from Theorem 3.15 (note that $\mu$ is invertible in
$A_{\operatorname{inf}}/\tilde{\xi}$). This proves (7.5) of which (7.6) is a
direct consequence.

We now prove degeneration of (7.6). By Lemma 3.12 the complex
$Rf_{\eta,\operatorname{pro\acute{e}t}*}\widehat{\mathbb{Z}}_{p}$ is adic
lisse and in particular it is locally a perfect complex of
$\widehat{\mathbb{Z}}_{p}$-modules. Thus

$R^{i+j}f_{\eta,\operatorname{pro\acute{e}t}*}\widehat{\mathbb{Z}}_{p}\otimes_{\widehat{\mathbb{Z}}_{p}}\widehat{\mathcal{O}}_{Y}=R^{i+j}f_{\eta,\operatorname{pro\acute{e}t}*}\widehat{\mathbb{Z}}_{p}\otimes_{\widehat{\mathbb{Z}}_{p}}\widehat{\mathbb{Z}}_{p}[\tfrac{1}{p}]\otimes_{\widehat{\mathbb{Z}}_{p}[\tfrac{1}{p}]}\widehat{\mathcal{O}}_{Y}$

is a locally free $\widehat{\mathcal{O}}_{Y}$-module. Recall that locally free
sheaves of finite rank on $Y_{\operatorname{pro\acute{e}t}}$ correspond to
locally free sheaves of finite rank on the analytic topology on $Y$ by [Sch13,
Lemma 7.3]). So the rank (locally) of such objects is determined by the rank
at geometric points of rank 1. Fix an integer $m\geq 0$. By [Hub96,
Proposition 2.6.1] applied181818It is enough to look at
$R^{i+j}f_{\eta,\operatorname{\acute{e}t}*}(\mathbb{Z}/p^{n})$ for each $n$ as
$Rf_{\eta,\operatorname{pro\acute{e}t}*}\widehat{\mathbb{Z}}_{p}$ is adic
lisse. to $R^{i+j}f_{\eta,\operatorname{\acute{e}t}*}(\mathbb{Z}/p^{n})$ for
each $n$, commutation with arbitrary base change for the terms
$R^{i}f_{\eta*}\Omega^{j}_{X/Y}$ (by locally freeness assumption) and the
degeneration of the Hodge-Tate spectral sequence (cf. [BMS18, Theorem
13.3(ii)]), we get a covering $\\{U_{k}\to Y\\}_{k}$ in
$Y_{\operatorname{pro\acute{e}t}}$ such that restricted to $U_{k}$ each term
in the quadrant $i+j\leq m$ of the second page of (7.6) is free (as well as
the abutment
$R^{i+j}f_{\eta,\operatorname{pro\acute{e}t}*}\widehat{\mathbb{Z}}_{p}\otimes_{\widehat{\mathbb{Z}}_{p}}\widehat{\mathcal{O}}_{Y}$)
and

$\sum_{i+j=l}\textrm{rank}(R^{i}f_{\eta*}\Omega^{j}_{X/Y}\otimes_{\mathcal{O}_{Y}}\widehat{\mathcal{O}}_{Y}(-j)\lvert_{U_{k}})=\textrm{rank}(R^{l}f_{\eta,\operatorname{pro\acute{e}t}*}\widehat{\mathbb{Z}}_{p}\otimes_{\widehat{\mathbb{Z}}_{p}}\widehat{\mathcal{O}}_{Y}\lvert_{U_{k}})$
(7.7)

for all $k$ and any $0\leq l\leq m$. Since affinoid perfectoids form a basis
for $Y_{\operatorname{pro\acute{e}t}}$, we can assume further that each
$U_{k}$ is affinoid perfectoid and determines a perfectoid space
$\operatorname{Spa}(S_{k},S_{k}^{+})$. Then by vanishing of higher cohomology
(cf. [Sch13, Lemma 4.10(v)]), taking the $U_{k}$-sections of (7.6) gives a
spectral sequence consisting of finite free $S_{k}$-modules. It suffices to
show that the resulting spectral sequence degenerates. Consider a differential
$d_{2}^{i,j}\colon E_{2}^{i,j}(U_{k})=S_{k}^{a}\to
E_{2}^{i+2,j-1}(U_{k})=S_{k}^{b}$ for any $i+j\leq m-1$. Then for each minimal
prime ideal $\mathfrak{p}$ of $S_{k}$, the localization $S_{k,\mathfrak{p}}$
is a field as $S_{k}$ is reduced (cf. [Sta18, Tag 00EU]). Thus by (7.7),
$d_{2}^{i,j}$ localized at $\mathfrak{p}$ is the zero morphism. Thus the
induced morphism

$\prod_{\mathfrak{p}\text{
minimal}}(S_{k,\mathfrak{p}})^{a}\to\prod_{\mathfrak{p}\text{
minimal}}(S_{k,\mathfrak{p}})^{b},$ (7.8)

where the products run over all minimal prime ideals of $S_{k}$, is the zero
morphism. Moreover $S_{k}\to\prod_{\mathfrak{p}\text{
minimal}}S_{k,\mathfrak{p}}$ is an embedding (cf. [Sta18, Tag 00EW]).
Therefore $d_{2}^{i,j}$ is the zero morphism. This proves degeneration. ∎

###  7.2 Torsion in étale pushforward

The result in this section makes use of the $q$-crystalline theory from §6.
Although one could avoid this by using the de Rham specialization for
$A\Omega_{\mathfrak{X}/\mathfrak{Y}}$ (cf. Theorem 5.20), one would still need
to prove a perfectness result such as Proposition 6.35.

######  Theorem 7.4.

Suppose $f\colon\mathfrak{X}\to\mathfrak{Y}$ is a proper smooth morphism of
$p$-adic formal schemes locally of finite type and flat over $\mathcal{O}$.
Fix some $j\geq 0$. Suppose that
$R^{i}f_{\operatorname{dR}*}\mathcal{O}_{\mathfrak{X}}$ is a finite locally
free $\mathcal{O}_{\mathfrak{Y}}$-module for all $i\geq j$. Then for all
$i\geq j$

  1. (1)

$R^{i}f_{\eta,\operatorname{pro\acute{e}t}*}\widehat{\mathbb{Z}}_{p}$ is
$p$-torsion free,

  2. (2)

if $\varprojlim_{i\in I}\operatorname{Spa}(S_{i},S_{i}^{+})$ is an affinoid
perfectoid of $Y_{\operatorname{pro\acute{e}t}}$ determining the perfectoid
space $\operatorname{Spa}(S_{\infty}[\tfrac{1}{p}],S_{\infty})$, then
$H^{i}(R\Gamma((\mathbb{A}_{\operatorname{inf}}(S_{\infty}),(\xi)),R\Gamma_{q-\textnormal{CRYS}}(\mathfrak{X}/\mathfrak{Y})))$
is a finite locally free $\mathbb{A}_{\operatorname{inf}}(S_{\infty})$-module
and

  3. (3)

if $\mathfrak{Y}$ is in addition smooth and $(E,J)$ is any object in
$(\mathfrak{Y}/A_{\operatorname{inf}})_{q-\textnormal{CRYS}}$, then

$H^{i}(R\Gamma((E,J),R\Gamma_{q-\textnormal{CRYS}}(\mathfrak{X}/\mathfrak{Y})))$
is a finite locally free $E$-module.

######  Remark 7.5.

A slight variant of part (2) is also proved in [MT20, Lemma 1.38] by another
method. The key point in the argument of loc.cit. is to take a judicious base
change to a rank 1 perfectoid valuation ring, at which point the result of
[BMS18, Theorem 1.8] can be applied. We take a different approach and instead
rely on the comparison with relative $p$-adic étale cohomology (cf. Theorem
6.36). T. Tsuji also pointed out that (1) immediately follows from the
analagous result in [BMS18, Theorem 14.5(ii)] (cf. Remark 14.4 in loc.cit.)
together with [Hub96, Proposition 2.6.1] and Lemma 3.12.

###### Proof.

We first show part (2) and we use the notation from §6. Let $(E,J)$ be an
object in $(\mathfrak{Y}/A_{\operatorname{inf}})_{q-\textnormal{CRYS}}$. Then
by the de Rham comparison with $q$-crystalline/prismatic cohomology (cf.
[BS22, Corollary 15.4]) and (6.13)

$R\Gamma_{\operatorname{dR}}((\operatorname{Spf}(E/[p]_{q})\times_{\mathfrak{Y}}\mathfrak{X})/\operatorname{Spf}(E/[p]_{q}))\cong
R\Gamma((E,J),R\Gamma_{q-\textnormal{CRYS}}(\mathfrak{X}/\mathfrak{Y}))\otimes^{\mathbb{L}}_{A_{\operatorname{inf}},\theta}\mathcal{O}.$
(7.9)

Note that by Proposition 6.17(i), the RHS of (7.9) is already ($p$-adically
derived) complete. Since
$R^{i}f_{\operatorname{dR}*}\mathcal{O}_{\mathfrak{X}}$ is a finite locally
free $\mathcal{O}_{\mathfrak{Y}}$-module for all $i\geq j$ and
$Rf_{\operatorname{dR}*}\mathcal{O}_{\mathfrak{X}}$ is a perfect complex and
commutes with arbitrary base change,
$R^{i}f_{\operatorname{dR}*}\mathcal{O}_{\mathfrak{X}}$ commutes with
arbitrary base change and so
$R^{i}f^{\prime}_{\operatorname{dR}*}\mathcal{O}_{\operatorname{Spf}(E/[p]_{q})\times_{\mathfrak{Y}}\mathfrak{X}}$
is a finite locally free $\mathcal{O}_{\operatorname{Spf}(E/[p]_{q})}$-module
(here
$f^{\prime}\colon\operatorname{Spf}(E/[p]_{q})\times_{\mathfrak{Y}}\mathfrak{X}\to\operatorname{Spf}(E/[p]_{q})$
is obtained from base change) for all $i\geq j$. By affine vanishing of higher
cohomology, it follows that $H^{i}$ of the LHS of (7.9) is a finite locally
free $E/[p]_{q}$-module for all $i\geq j$. Therefore

$H^{i}(R\Gamma((E,J),R\Gamma_{q-\textnormal{CRYS}}(\mathfrak{X}/\mathfrak{Y}))\otimes^{\mathbb{L}}_{A_{\operatorname{inf}},\theta}\mathcal{O})$
(7.10)

is a finite locally free $E/[p]_{q}$-module for all $i\geq j$.

Let $\varprojlim_{i\in I}\operatorname{Spa}(S_{i},S_{i}^{+})$ be an affinoid
perfectoid of $Y_{\operatorname{pro\acute{e}t}}$ determining the perfectoid
space $\operatorname{Spa}(S_{\infty}[\tfrac{1}{p}],S_{\infty})$ and suppose
$(E,J)=(\mathbb{A}_{\operatorname{inf}}(S_{\infty}),(\xi))$. We now proceed by
descending induction on $i\geq j$ to prove that

$H^{i}:=H^{i}(R\Gamma((E,J),R\Gamma_{q-\textnormal{CRYS}}(\mathfrak{X}/\mathfrak{Y})))$

is a finite locally free $E$-module. Since
$R\Gamma((E,J),R\Gamma_{q-\textnormal{CRYS}}(\mathfrak{X}/\mathfrak{Y}))$ is a
perfect complex in $D(E)$ (cf. Proposition 6.17(i)), it is quasi-isomorphic to
a complex $E^{\bullet}$ of finite projective $E$-modules with $E^{k}=0$ for
$k\geq m+1$ for some $m\geq i$. Then $H^{m}$ is a finite $E$-module and
$H^{m}\otimes_{A_{\operatorname{inf}},\theta}\mathcal{O}$ coincides with
(7.10) for $i=m$. Assume without loss of generality
$H^{m}\otimes_{A_{\operatorname{inf}},\theta}\mathcal{O}$ is a finite free
$E/[p]_{q}$-module and choose a basis $\overline{B}$ (this does not affect the
form of $(E,J)$, as it is enough to take a Zariski cover of $\mathfrak{Y}$).
By Nakayama’s lemma, the basis $\overline{B}$ lifts to a generating set $B$ of
$H^{m}$. Moreover by Proposition 6.17(ii), [Sch13, Theorem 6.5(ii)] (in
particular the sentence immediately following [BMS18, Lemma 5.6]) and the
equality
$\mu_{\operatorname{pro\acute{e}t}}^{-1}\mathcal{O}_{q-\textnormal{CRYS}}=W(\widehat{\mathcal{O}}^{+,\flat}_{Y})$
we get

$H^{m}[\tfrac{1}{\mu}]=H^{m}(R\Gamma(\varprojlim_{i\in
I}\operatorname{Spa}(S_{i},S_{i}^{+}),(\mu^{-1}_{\operatorname{pro\acute{e}t}}R\Gamma_{q-\textnormal{CRYS}}(\mathfrak{X}/\mathfrak{Y}))^{\wedge}))[\tfrac{1}{\mu}].$
(7.11)

By Theorem 6.36, (7.11) implies $H^{m}[\tfrac{1}{\mu},\tfrac{1}{p}]$ is the
$E[\tfrac{1}{\mu},\tfrac{1}{p}]$-module

$(R^{m}f_{\eta,\operatorname{pro\acute{e}t}*}\widehat{\mathbb{Z}}_{p})(\varprojlim_{i\in
I}\operatorname{Spa}(S_{i},S_{i}^{+}))\otimes_{(\widehat{\mathbb{Z}}_{p})(\varprojlim_{i\in
I}\operatorname{Spa}(S_{i},S_{i}^{+}))}E[\tfrac{1}{\mu},\tfrac{1}{p}].$ (7.12)

Since $Rf_{\eta,\operatorname{pro\acute{e}t}*}\widehat{\mathbb{Z}}_{p}$ is
adic lisse we can trivialize it (locally on
$Y_{\operatorname{pro\acute{e}t}}$) and after applying base change as in
footnote 18, (7.12) base changed to
$\mathbb{A}_{\operatorname{inf}}(S_{\infty}^{\prime})[\tfrac{1}{\mu},\tfrac{1}{p}]$
(where $\varprojlim_{i^{\prime}\in
I^{\prime}}\operatorname{Spa}(S^{\prime}_{i^{\prime}},{S^{\prime}_{i^{\prime}}}^{+})\to\varprojlim_{i\in
I}\operatorname{Spa}(S_{i},S_{i}^{+})$ is a pro-étale map of affinoid
perfectoids and $\varprojlim_{i^{\prime}\in
I^{\prime}}\operatorname{Spa}(S^{\prime}_{i^{\prime}},{S^{\prime}_{i^{\prime}}}^{+})$
determines the perfectoid space
$\operatorname{Spa}(S_{\infty}^{\prime}[\tfrac{1}{p}],S_{\infty}^{\prime})$)
is a free
$\mathbb{A}_{\operatorname{inf}}(S_{\infty}^{\prime})[\tfrac{1}{\mu},\tfrac{1}{p}]$-module
of rank

$\textrm{rank}_{\mathbb{Z}_{p}}((H^{m}_{\operatorname{\acute{e}t}}(X\times_{Y}\operatorname{Spa}(K,K^{\circ}),\mathbb{Z}_{p})))$
(7.13)

for any rank $1$ geometric point $\operatorname{Spa}(K,K^{\circ})$ of $Y$
which lies above $\varprojlim_{i^{\prime}\in
I^{\prime}}\operatorname{Spa}(S^{\prime}_{i^{\prime}},{S^{\prime}_{i^{\prime}}}^{+})$.
By the comparison of $B^{+}_{\operatorname{dR}}$-cohomology with étale
cohomology (cf. [BMS18, Theorem 13.1]) and the fact that
$B^{+}_{\operatorname{dR}}$-cohomology specializes to de Rham cohomology (cf.
the expression immediately after [BMS18, Definition 13.18]), (7.13) is
equal191919One also uses Lemma 7.1 to rewrite
$R^{i}f^{\prime}_{\operatorname{dR}*}\mathcal{O}_{\operatorname{Spf}(E/[p]_{q})\times_{\mathfrak{Y}}\mathfrak{X}}$
as the de Rham pushforward along the generic fiber. to $|\overline{B}|$. Now

$\varprojlim_{i\in
J}\operatorname{Spa}(T_{i},T_{i}^{+})\mapsto(R^{m}f_{\eta,\operatorname{pro\acute{e}t}*}\widehat{\mathbb{Z}}_{p})(\varprojlim_{i\in
J}\operatorname{Spa}(T_{i},T_{i}^{+}))\otimes_{(\widehat{\mathbb{Z}}_{p})(\varprojlim_{i\in
J}\operatorname{Spa}(T_{i},T_{i}^{+}))}\mathbb{A}_{\operatorname{inf}}(T_{\infty})[\tfrac{1}{\mu},\tfrac{1}{p}]$
(7.14)

is a sheaf on affinoid perfectoids in $Y_{\operatorname{pro\acute{e}t}}$.
Since (7.13) equals $|\overline{B}|=|B|$, the generating set $B$ is a basis
(pro-étale locally on $\varprojlim_{i\in
I}\operatorname{Spa}(S_{i},S_{i}^{+})$) for the sections of (7.14). Therefore,
it must be a basis for (7.12). Hence, as
$\mathbb{A}_{\operatorname{inf}}(S_{\infty})$ is $\mu$-torsion (and
$p$-torsion) free, it is a basis for $H^{m}$. This proves the base case of the
induction. For the inductive step, one repeats the argument noting that the
locally finite freeness of $H^{i}$ for $i>i^{\prime}\geq j$ implies
$H^{i^{\prime}}\otimes_{A_{\operatorname{inf}},\theta}\mathcal{O}$ coincides
with (7.10) for $i=i^{\prime}$. Moreover since
$R\Gamma((E,J),R\Gamma_{q-\textnormal{CRYS}}(\mathfrak{X}/\mathfrak{Y}))$ is a
perfect complex in $D(E)$, the inductive step also shows $H^{i^{\prime}}$ is a
finite $E$-module. This proves (2).

Part (1) then follows from Theorem 6.36: indeed

$(R^{i}f_{\eta,\operatorname{pro\acute{e}t}*}\widehat{\mathbb{Z}}_{p}\otimes_{\widehat{\mathbb{Z}}_{p}}^{\mathbb{L}}\mathbb{A}_{\operatorname{inf},Y}[\tfrac{1}{\mu}])(\varprojlim_{i\in
I}\operatorname{Spa}(S_{i},S_{i}^{+}))$

is a finite locally free
$\mathbb{A}_{\operatorname{inf}}(S_{\infty})[\tfrac{1}{\mu}]$-module for all
$i\geq j$ and so
$(R^{i}f_{\eta,\operatorname{pro\acute{e}t}*}\widehat{\mathbb{Z}}_{p})(\varprojlim_{i\in
I}\operatorname{Spa}(S_{i},S_{i}^{+}))$ is $p$-torsion free.

We now show part (3) by descent to the case of part (2). First we show the map
$S\to S_{\infty}$ (cf. Remark 6.25) is quasisyntomic (for small enough Zariski
neighbourhood $\operatorname{Spf}(S)$ of $\mathfrak{Y}$ and a choice of
$S_{\infty}$). Since $\mathfrak{Y}$ is assumed to be smooth, we can take $S$
to be a small $p$-completely smooth $\mathcal{O}$-algebra (recall this means
that there is an étale map
$\operatorname{Spf}(S)\to\operatorname{Spf}(\mathcal{O}\\{t_{1}^{\pm
1},\ldots,t_{n}^{\pm 1}\\})$). Above $\operatorname{Spa}(C\\{t_{1}^{\pm
1},\ldots,t_{n}^{\pm 1}\\},\mathcal{O}\\{t_{1}^{\pm 1},\ldots,t_{n}^{\pm
1}\\})$, we have the affinoid perfectoid covering

$\varprojlim_{m\geq 0}\operatorname{Spa}(C\\{t_{1}^{\pm
1/p^{m}},\ldots,t_{n}^{\pm 1/p^{m}}\\},\mathcal{O}\\{t_{1}^{\pm
1/p^{m}},\ldots,t_{n}^{\pm 1/p^{m}}\\}).$

The induced map $\mathcal{O}\\{t_{1}^{\pm 1},\ldots,t_{n}^{\pm
1}\\}\to\mathcal{O}\\{t_{1}^{\pm 1/p^{\infty}},\ldots,t_{n}^{\pm
1/p^{\infty}}\\}$ is quasisyntomic. The map $S\to S_{\infty}$, induced by
($p$-completed) base change, is again quasisyntomic by [BMS19, Lemma 4.16(2)].
Now suppose $\operatorname{Spf}(E/J)\to\mathfrak{Y}$ factors through
$\operatorname{Spf}(S)$. Then again by loc.cit. the induced map
$E/\tilde{\xi}\to S_{\infty}\widehat{\otimes}_{S^{(1)}}E/\tilde{\xi}$ is
quasisyntomic (the completion is $p$-adic and
$S^{(1)}:=S\otimes_{A_{\operatorname{inf}}/\xi,\varphi}A_{\operatorname{inf}}/\tilde{\xi}$).
By [BS22, Proposition 7.11], there exists an object

$(E^{\prime}\to E^{\prime}/\tilde{\xi}\leftarrow
S_{\infty}\widehat{\otimes}_{S^{(1)}}E/\tilde{\xi})\in((S_{\infty}\widehat{\otimes}_{S^{(1)}}E/\tilde{\xi})/E)_{{{\mathbbl{\Delta}}}},$

and moreover the map $S_{\infty}\widehat{\otimes}_{S^{(1)}}E/\tilde{\xi})\to
E^{\prime}/\varphi(\xi)$ is $p$-completely faithfully flat. Therefore the
composition $E/\tilde{\xi}\to E^{\prime}/\tilde{\xi}$ is also $p$-completely
faithfully flat. By universality (cf. Lemma 4.7 in loc.cit.), the induced map
$S_{\infty}\to E^{\prime}/\tilde{\xi}$ lifts uniquely to a map
$(\mathbb{A}_{\operatorname{inf}}(S_{\infty}),\tilde{\xi})\to(E^{\prime},\tilde{\xi})$
of prisms. Therefore by part (2)

$H^{i}(R\Gamma((\mathbb{A}_{\operatorname{inf}}(S_{\infty}),(\xi)),R\Gamma_{q-\textnormal{CRYS}}(\mathfrak{X}/\mathfrak{Y})))\otimes_{\mathbb{A}_{\operatorname{inf}}(S_{\infty})}E^{\prime}$
(7.15)

is a finite locally free $E^{\prime}$-module. The expression (7.15) coincides
with

$H^{i}(R\Gamma((E,J),R\Gamma_{q-\textnormal{CRYS}}(\mathfrak{X}/\mathfrak{Y})))\widehat{\otimes}_{E}E^{\prime},$

where the completion is $(p,\tilde{\xi})$-adic (to see this, identify
$q$-crystalline cohomology with prismatic cohomology (6.13)-(6.14) and apply
[BS22, Theorem 1.8(5)]). By a similar argument as the proof of [ALB20, Lemma
A.12] (in particular using Lemma A.8 of loc.cit.), one concludes that

$H^{i}(R\Gamma((E,J),R\Gamma_{q-\textnormal{CRYS}}(\mathfrak{X}/\mathfrak{Y})))$

is a finite locally free $E$-module. In general (in the case when the morphism
$\operatorname{Spf}(E/J)\to\mathfrak{Y}$ does not necessarily factor through
$\operatorname{Spf}(S)$), there is a covering
$(E^{\prime},J^{\prime})\to(E,J)$ (i.e.
$\operatorname{Spf}(E^{\prime}/J^{\prime})\to\operatorname{Spf}(E/J)$ is a
$p$-completely étale covering), such that
$H^{i}(R\Gamma((E^{\prime},J^{\prime}),R\Gamma_{q-\textnormal{CRYS}}(\mathfrak{X}/\mathfrak{Y})))$
is a finite locally free $E^{\prime}$-module. One then repeats the argument as
the proof of [ALB20, Lemma A.12], as before. ∎

######  Remark 7.6.

In the setting of Theorem 7.4, there is also a Frobenius structure on

$H^{i}(R\Gamma((\mathbb{A}_{\operatorname{inf}}(S_{\infty}),(\xi)),R\Gamma_{q-\textnormal{CRYS}}(\mathfrak{X}/\mathfrak{Y})))$
(7.16)

and by the isogeny theorem (cf. [BS22, Theorem 1.8(6)]), the linearization
morphism (induced by Frobenius) is an isomorphism after inverting
$\tilde{\xi}$. Together with part (2) of Theorem 7.4, the expression (7.16)
gives rise to a _relative Breuil-Kisin-Fargues module_. This is made precise
in [MT20], where various categories (e.g. $\mu$-small generalised
representations over $\mathbb{A}_{\operatorname{inf}}(S_{\infty})$ for a
particular $S_{\infty}$ and relative Breuil-Kisin-Fargues modules with
Frobenius structure) are shown to be equivalent. In the language of loc.cit.
(7.16) can be upgraded to a generalised representation over
$\mathbb{A}_{\operatorname{inf}}(S_{\infty})$ (with the
$\mathbb{Z}_{p}(1)^{d}$-action202020In this context, one takes a small Zariski
neighbourhood $\operatorname{Spf}(S)$ of $Y$ which can be described using
$d$-tori-coordinates. The $S$-algebra $S_{\infty}$ is then obtained from $S$
by adjoining all $p$-th roots of the tori-coordinates and this gives the usual
$\mathbb{Z}_{p}(1)^{d}$-action on $\mathbb{A}_{\inf}(S_{\infty})$, which in
turn induces a $\mathbb{Z}_{p}(1)^{d}$-action on (7.16). satisfying a certain
triviality condition modulo $\mu$) with Frobenius structure.

###  7.3 The relative absolute crystalline comparison

Let $f\colon\mathfrak{X}\to\mathfrak{Y}$ be a smooth
$\operatorname{Spf}(\mathcal{O})$-morphism of $p$-adic formal schemes of type
(S)(b). Assume that $\mathfrak{X}$ and $\mathfrak{Y}$ are flat over
$\mathcal{O}$. In this section we prove a relative version of the absolute
crystalline comparison isomorphism (cf. [BMS18, Theorem 12.1]).

As usual let
$X:=\mathfrak{X}\times_{\operatorname{Spf}(\mathcal{O})}\operatorname{Spec}(\mathcal{O}/p)$
and
$Y:=\mathfrak{Y}\times_{\operatorname{Spf}(\mathcal{O})}\operatorname{Spec}(\mathcal{O}/p)$
be the fibers modulo $p$ of $\mathfrak{X}$ and $\mathfrak{Y}$, respectively.
In addition let $\textnormal{CRYS}(X/A_{\textnormal{crys}})$ and
$\textnormal{CRYS}(Y/A_{\textnormal{crys}})$ be the absolute (big) crystalline
sites of $X$ and $Y$, respectively. We denote the corresponding topoi by
$(X/A_{\textnormal{crys}})_{\textnormal{CRYS}}$ (resp.
$(Y/A_{\textnormal{crys}})_{\textnormal{CRYS}}$). Let
$\mathcal{O}^{\textnormal{CRYS}}_{X/\mathbb{Z}_{p}}$ be the structure sheaf on
$(X/A_{\textnormal{crys}})_{\textnormal{CRYS}}$. The morphism $f$ induces a
functor of topoi

$f^{\textnormal{CRYS}}\colon(X/A_{\textnormal{crys}})_{\textnormal{CRYS}}\to(Y/A_{\textnormal{crys}})_{\textnormal{CRYS}}.$

Let $Y^{\prime}_{\operatorname{pro\acute{e}t}}\subset
Y_{\operatorname{pro\acute{e}t}}$ be the subsite consisting of affinoid
perfectoid objects. By killing $\mu=q-1$ in the target of
$u_{\operatorname{pro\acute{e}t}}$ in §6.8, we obtain a cocontinuous functor
of sites

$\displaystyle u_{\operatorname{pro\acute{e}t}}^{\textnormal{CRYS}}\colon
Y^{\prime}_{\operatorname{pro\acute{e}t}}$
$\displaystyle\to\textnormal{CRYS}(Y/A_{\textnormal{crys}})$
$\displaystyle\varprojlim_{i\in I}\operatorname{Spa}(S_{i},S_{i}^{+})$
$\displaystyle\mapsto(\mathbb{A}_{\operatorname{inf}}(S_{\infty})/\mu,(\xi/\mu))$

which induces a morphism of topoi

$\mu_{\operatorname{pro\acute{e}t}}^{\textnormal{CRYS}}\colon
Y_{\operatorname{pro\acute{e}t}}\to(Y/A_{\textnormal{crys}})_{\textnormal{CRYS}}.$

We can now state the absolute crystalline comparison in the current context:

######  Theorem 7.7.

Assume that we have a smooth $\operatorname{Spf}(\mathcal{O})$-morphism
$f\colon\mathfrak{X}\to\mathfrak{Y}$ of $p$-adic formal schemes of type
(S)(b). Suppose in addition that $\mathfrak{X}$ and $\mathfrak{Y}$ are flat
over $\mathcal{O}$. There is a canonical functorial morphism (the completion
on the target is derived $p$-adic)

$(\mu^{\textnormal{CRYS}}_{\operatorname{pro\acute{e}t}})^{-1}Rf_{*}^{\textnormal{CRYS}}\mathcal{O}^{\textnormal{CRYS}}_{X/\mathbb{Z}_{p}}\to
R\Gamma_{A_{\operatorname{inf}}}(\mathfrak{X}/\mathfrak{Y})\widehat{\otimes}_{A_{\operatorname{inf}}}A_{\textnormal{crys}}$
(7.17)

in $D(D,W(\widehat{\mathcal{O}}^{+,\flat}_{Y}))$ compatible with the Frobenius
such that the cohomology sheaves of the cone of (7.17) are killed by
$[\mathfrak{m}^{\flat}]\subset A_{\operatorname{inf}}$.

###### Proof.

Let
$r\colon(\mathfrak{Y}/A_{\operatorname{inf}})_{q-\textnormal{CRYS}}\to(Y/A_{\textnormal{crys}})_{\textnormal{CRYS}}$
be the natural morphism of topoi such that
$\mu_{\operatorname{pro\acute{e}t}}^{\textnormal{CRYS}}=r\circ\mu_{\operatorname{pro\acute{e}t}}$.
Then by [BS22, Theorem 16.14]
$r^{-1}Rf_{*}^{\textnormal{CRYS}}\mathcal{O}^{\textnormal{CRYS}}_{X/\mathbb{Z}_{p}}\cong
R\Gamma_{q-\textnormal{CRYS}}(\mathfrak{X}/\mathfrak{Y})\widehat{\otimes}_{A_{\operatorname{inf}}}A_{\textnormal{crys}}$.
The result now follows from Theorem 6.33. ∎

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