$\textstyle{\pi^{-1}R\pi_{*}\mathcal{F}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi^{-1}f}$$\textstyle{\pi^{-1}R\pi_{*}\mathcal{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{F}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widetilde{f}}$$\textstyle{\mathcal{G}.}$

∎

Now we are able to prove the comparison theorem.

######  Theorem 5.5.5.

Let $X$ be a rigid space over ${\mathrm{B^{+}_{dR,e}}}$. Then there exists a
natural filtered morphism from the analytic derived de Rham complex to the
Hodge-filtered infinitesimal cohomology sheaf as below

${\widehat{\mathrm{dR}}^{\mathrm{an}}}_{X/\Sigma_{e}}\longrightarrow(Ru_{X/\Sigma_{e}*}\mathcal{O}_{X/\Sigma_{e}},~{}Ru_{X/\Sigma_{e}*}\mathcal{J}_{X/\Sigma_{e}},~{}Ru_{X/\Sigma_{e}*}\mathcal{J}_{X/\Sigma_{e}}^{2},~{}\ldots),$

Moroever, the induced map between their underlying complexes is an isomorphism

$\underline{\widehat{\mathrm{dR}}}_{X/\Sigma_{e}}^{\mathrm{an}}\longrightarrow
Ru_{X/\Sigma_{e}*}\mathcal{O}_{X/\Sigma_{e}}.$

###### Proof.

We first notice that it suffices to show the comparison at the site
$X_{\mathrm{aff}}$ of affinoid open subsets of $X$.

To see this, we recall the equivalence of $\infty$-categories
$\mathscr{D}(X,R)\cong\mathrm{Sh}^{\mathrm{hyp}}(X,R)$ (cf. Paragraph 5),
where the right side is the full sub-$\infty$-category of contravariant
functors from $U\in X_{\mathrm{rig}}$ to the derived category
$\mathscr{D}(R)$. As $X_{\mathrm{aff}}$ is a basis of the rigid site
$X_{\mathrm{rig}}$, we may use the equivalence
$\mathrm{Sh}^{\mathrm{hyp}}(X,R)=\mathrm{Sh}^{\mathrm{hyp}}(X_{\mathrm{aff}},R)$
and regard objects in $\mathscr{D}(X,R)$ as contravariant functors on affinoid
open subsets of $X$. So a map or an isomorphism of sheaves of complexes can be
constructed via their evaluations at $X_{\mathrm{aff}}$. Moreover, by Lemma
5.5.4, it suffices to show the isomorphism over the site of smooth immersions
${\mathrm{SE}}_{X}$ of $X$, by applying the constant functor $\pi^{-1}$ to
both objects in the sequence $(\ast)$.

Now for each smooth immersion $i:U=\mathrm{Spa}(B)\rightarrow
Z=\mathrm{Spa}(P)$ for affinoid open subset $U\subset X$, as in the proof of
Theorem 5.5.1 we have the following natural map of cosimplicial objects in
$\widehat{\mathscr{DF}}({\mathrm{B^{+}_{dR,e}}})$

$\textstyle{{\widehat{\mathrm{dR}}^{\mathrm{an}}}_{B/{\mathrm{B^{+}_{dR,e}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\widehat{\mathrm{dR}}^{\mathrm{an}}}_{B/P^{\widehat{\otimes}\bullet+1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{O}_{D(\bullet)}}$$\textstyle{R\Gamma(U/\Sigma_{e\inf},\mathcal{O}_{X/\Sigma_{e}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

Here $D(n)$ is the envelope of the surjection

$P^{\widehat{\otimes}(n+1)=}P^{\widehat{\otimes}_{{\mathrm{B^{+}_{dR,e}}}}(n+1)}\rightarrow
B.$

Moreover, the induced maps of their underlying complexes above are all
isomorphisms, by the proof of the affinoid comparison in Theorem 5.5.1.

At last, note that the above maps are functorial with respect to smooth
immersions $i:U\rightarrow Z$, so we can improve the above map into the level
of sheaves over ${\mathrm{SE}}_{X}$

$\textstyle{R\pi_{*}{\widehat{\mathrm{dR}}^{\mathrm{an}}}_{X/\Sigma_{e}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widehat{\mathrm{dR}}_{{\mathrm{SE}}_{X}}^{\mathrm{an}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{O}_{{\mathrm{SE}}_{X}}}$$\textstyle{R\pi_{*}Ru_{X/\Sigma_{e}*},\mathcal{O}_{X/\Sigma_{e}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

Here where $\widehat{\mathrm{dR}}_{{\mathrm{SE}}_{X}}^{\mathrm{an}}$ is the
(cosimplicial) sheaf sending $i:U\rightarrow Z$ to the filtered complex
${\widehat{\mathrm{dR}}^{\mathrm{an}}}_{B/P^{\widehat{\otimes}\bullet+1}}$,
and $\mathcal{O}_{{\mathrm{SE}}_{X}}$ is the (cosimplicial) sheaf sending
$i:U\rightarrow Z$ to the structure sheaves $\mathcal{O}_{D(\bullet)}$ of
envelopes $D(\bullet)$. In this way, the isomorphism of the underlying
complexes over the site ${\mathrm{SE}}_{X}$ of smooth immersions

$R\pi_{*}\underline{\widehat{\mathrm{dR}}}_{X/{\mathrm{B^{+}_{dR,e}}}}^{\mathrm{an}}\longrightarrow
R\pi_{*}Ru_{X/\Sigma_{e}*},\mathcal{O}_{X/\Sigma_{e}}$

follows from the above computation at the sections $i:U\rightarrow Z$, and
thus we get the result by Lemma 5.5.4. So we are done.

∎

######  Remark 5.5.6.

The above comparison map, though functorial with respect to the rigid space
$X$, is constructed in an indirect way. It is natural to ask if we can
directly produce a natural morphism from the analytic derived de Rham complex
to infinitesimal cohomology sheaf. Here we want to mention that for algebraic
schemes over $\mathbb{C}$, this could be achieved by the universal property of
the derived de Rham complex in the $\infty$-category of filtered
$\mathbb{E}_{\infty}$-algebras.

##  6\. Éh descent

In this section, we prove the éh hyperdescent of infinitesimal cohomology of
crystals in vector bundles over $X/K$ and $X/{\mathrm{B^{+}_{dR,e}}}_{\inf}$
for a rigid space $X$. Our goal is to show the comparison between the éh de
Rham cohomology and infinitesimal cohomology of a crystal.

In order to extend a crystal to any space that maps to $X$ but is not
necessarily an open immersion, we will work mostly with coherent crystals over
the big site. Here we note that this is only for the technical convenience, as
crystals and their cohomology over $X/\Sigma_{e\inf}$ or
$X/\Sigma_{e{\mathrm{INF}}}$ are equivalent via pullback and restrictions
(Proposition 3.1.7, Proposition 4.1.2).

###  6.1. Descent for blowup coverings

We first deal with the descent for the blowup covering over the base
$\Sigma_{1}=\mathrm{Spa}(K)$, for an arbitrary complete non-archimedean
$p$-adic field $K$, not necessarily algebraically closed. The essential idea
follows from Hartshorne’s proof for algebraic de Rham cohomology [Har75, Chap
II, Section 4], where he provides a long exact sequence of the algebraic de
Rham cohomology for a blowup square.

We first give a Mayer-Vietories sequence for infinitesimal cohomology:

######  Proposition 6.1.1 (Mayer-Vietories sequence).

Let $X$ be a union of two closed analytic subspaces $X_{1}$ and $X_{2}$ over
$K$, and let $\mathcal{F}$ be a coherent crystal over $X/K_{{\mathrm{INF}}}$.
Then the map of rigid spaces $X_{1}\cap X_{2}\rightarrow X_{1}\cup
X_{2}\rightarrow X$ induces a natural distinguished triangle as below

$Ru_{X/K*}\mathcal{F}\longrightarrow
Ru_{X_{1}/K*}\mathcal{F}\operatorname*{\text{\raisebox{1.07639pt}{\scalebox{0.8}{$\bigoplus$}}}}Ru_{X_{2}/K*}\mathcal{F}\longrightarrow
Ru_{X_{1}\cap X_{2}/K*}\mathcal{F}.$

###### Proof.

As the functor $u_{X/K*}$ is the sheaf-version of the global section functor
$\Gamma(X/K_{{\mathrm{INF}}},-)$ (see Subsection 2.3), it suffices to show
that the maps in the statement above produce a natural distinguished triangle
after applying $R\Gamma(U,-)$, for every $U\subset X$ open affinoid. So we may
assume there exists a smooth affinoid rigid space $Z=\mathrm{Spa}(P)$ over
$\Sigma_{e}$, together with a closed immersion of $X=\mathrm{Spa}(P/I)$ into
$Z$, where $I$ is the defining ideal. Let $X_{i}$ be the closed analytic
subspace defined by the ideal $I_{i}$ in $P$. Then by assumption, $X$ is
defined by the ideal $I_{1}\cap I_{2}$, and the intersection $X_{3}:=X_{1}\cap
X_{2}$ is defined by $I_{3}:=I_{1}+I_{2}$. We denote by $D$ and $D_{i}$ to be
the envelope of $X$ and $X_{i}$ in $Z$ separately. Here we regard $D$ and
$D_{i}$ to be the ringed spaces, where the underlying topological spaces are
$X$ and $X_{i}$, and their (global sections of) structure sheaves are
$\mathcal{O}_{D}=\varprojlim_{n}P/I^{n}$ and $\mathcal{O}_{D_{i}}=\varprojlim
P/I_{i}^{n}$ separately (cf. Remark 2.2.2).

Now by Theorem 4.2.2, infinitesimal cohomology of the coherent crystal can be
functorially identified as the derived global sections of the following

$\mathcal{F}_{D}\otimes\Omega_{D}^{\bullet}\longrightarrow\mathcal{F}_{D_{1}}\otimes\Omega_{D_{1}}^{\bullet}\operatorname*{\text{\raisebox{1.07639pt}{\scalebox{0.8}{$\bigoplus$}}}}\mathcal{F}_{D_{2}}\otimes\Omega_{D_{2}}^{\bullet}\longrightarrow\mathcal{F}_{D_{3}}\otimes\Omega_{D_{3}}^{\bullet},$

where $\mathcal{F}_{D}\otimes\Omega_{D}^{\bullet}$ (resp.
$\mathcal{F}_{D_{i}}\otimes\Omega_{D_{i}}^{\bullet}$) is the restriction of
the de Rham complex of the coherent crystal $\mathcal{F}$ onto the envelope
$D_{X}(Z)$ along the closed immersion $X\rightarrow Z$ (resp. $D_{X_{i}}(Z)$
along the closed immersions $X_{i}\rightarrow Z$) separately. Moreover, by the
crystal condition, the coherent $\mathcal{O}_{D_{i}}$-module
$\mathcal{F}_{D_{i}}$ is equal to the tensor product
$\mathcal{F}_{D}\otimes_{\mathcal{O}_{D}}\mathcal{O}_{D_{i}}$, and each term
of the de Rham complex $\mathcal{F}_{D_{i}}\otimes\Omega_{D_{i}}^{\bullet}$ is
equal to the base change of terms of
$\mathcal{F}_{D}\otimes\Omega_{D}^{\bullet}$ along $D_{i}\rightarrow D$.444We
want to mention that the sheaf $\Omega_{D}^{j}$, defined as the inverse limit
$\varprojlim_{n}\Omega_{X_{n}/\Sigma_{e}}^{j}$ for $X_{n}$ being the $n$-th
infinitesimal neighborhood of $X$ into $Z$, is naturally isomorphic to the
tensor product $\Omega_{Z/\Sigma_{e}}^{j}\otimes\mathcal{O}_{D}$ by Lemma
3.3.3. This works similarly for $\Omega_{D_{i}}^{j}$, and in particular we get
$\Omega_{D_{i}}^{j}=\Omega_{D}^{j}\otimes_{\mathcal{O}_{D}}\mathcal{O}_{D_{i}}$.
In this way, by the compatibility of their Hodge filtrations, to show the
sequence above is distinguished, it suffices to show the following short exact
sequence of the rings

$0\longrightarrow\mathcal{O}_{D}\longrightarrow\mathcal{O}_{D_{1}}\operatorname*{\text{\raisebox{1.07639pt}{\scalebox{0.8}{$\bigoplus$}}}}\mathcal{O}_{D_{2}}\longrightarrow\mathcal{O}_{D_{3}}\longrightarrow
0.$

And since each structure sheaf of envelopes are given by the formal
completions of the ring $P$, we reduce the question to show that the following
sequence of inverse systems is exact

$0\longrightarrow\\{P/(I_{1}\cap
I_{2})^{n}\\}_{n}\longrightarrow\\{P/I_{1}^{n}\\}_{n}\operatorname*{\text{\raisebox{1.07639pt}{\scalebox{0.8}{$\bigoplus$}}}}\\{P/I_{2}^{n}\\}_{n}\longrightarrow\\{P/(I_{1}+I_{2})^{n}\\}_{n}\longrightarrow
0.$

Notice that for fixed $n$, we always have the following short exact sequence

$0\longrightarrow P/(I_{1}^{n}\cap I_{2}^{n})\longrightarrow
P/I_{1}^{n}\operatorname*{\text{\raisebox{1.07639pt}{\scalebox{0.8}{$\bigoplus$}}}}P/I_{2}^{n}\longrightarrow
P/(I_{1}^{n}+I_{2}^{n})\longrightarrow 0.$

The proof thus follows since the ring $P$ is noetherian, and the inverse
systems below are canonically isomorphic

$\\{P/(I_{1}^{n}+I_{2}^{n})\\}_{n}\longrightarrow\\{P/(I_{1}+I_{2})^{n}\\}_{n},~{}\\{P/(I_{1}\cap
I_{2})^{n}\\}_{n}\longrightarrow\\{P/(I_{1}^{n}\cap I_{2}^{n})\\}_{n}.$

∎

Here is our main theorem in this subsection.

######  Theorem 6.1.2.

Let $X$ be a rigid space over $K$, and let $Y$ be a smooth analytic closed
subset of $X$ over $K$, with the blowup map
$f:X^{\prime}:=\mathrm{Bl}_{X}(Y)\rightarrow X$ and the preimage
$Y^{\prime}:=Y\times_{X}X^{\prime}$ in $X^{\prime}$. Then for any coherent
crystal $\mathcal{F}$ over $X/K_{{\mathrm{INF}}}$, the blowup square for
$X^{\prime}\rightarrow X$ induces the following distinguished triangle

$Ru_{X/K*}\mathcal{F}\longrightarrow
Rf_{*}Ru_{X^{\prime}/K*}\mathcal{F}\operatorname*{\text{\raisebox{1.07639pt}{\scalebox{0.8}{$\bigoplus$}}}}Ru_{Y/K*}\mathcal{F}\longrightarrow
Rf_{*}Ru_{Y^{\prime}/K*}\mathcal{F}.$

In particular, by taking the derived global sections at $X$, we get a
distinguished triangle of infinitesimal cohomology

$R\Gamma(X/K_{{\mathrm{INF}}},\mathcal{F})\longrightarrow
R\Gamma(X^{\prime}/K_{{\mathrm{INF}}},\mathcal{F})\operatorname*{\text{\raisebox{1.07639pt}{\scalebox{0.8}{$\bigoplus$}}}}R\Gamma(Y/K_{{\mathrm{INF}}},\mathcal{F})\longrightarrow
R\Gamma(Y^{\prime}/K_{{\mathrm{INF}}},\mathcal{F}).$

Before we prove the result, first we recall the formal function theorem for a
proper map of rigid spaces.

######  Theorem 6.1.3 ([Kie67], Theorem 3.7).

Let $f:X^{\prime}\rightarrow X$ be a proper map of rigid space over $K$, and
let $\mathcal{I}$ be a sheaf of ideal over $X$, with $Y=\mathrm{Spa}(A/I)$
being the analytic closed subset of $X$ defined by $\mathcal{I}$. Assume
$\mathcal{G}$ is a coherent sheaf over $X^{\prime}$. Then the following
natural map is an isomorphism:

$(R^{j}f_{*}\mathcal{G})^{\wedge}\longrightarrow
R^{j}f_{*}(\widehat{\mathcal{G}}).$

Here $(-)^{\wedge}$ is the classical completion of a sheaf of
$\mathcal{O}_{X}$-modules (or $\mathcal{O}_{X^{\prime}}$-modules) with respect
to the ideal $\mathcal{I}$ (resp.
$f^{-1}\mathcal{I}\cdot\mathcal{O}_{X^{\prime}}$).

######  Remark 6.1.4.

Here we note that we may get a more derived version of the above theorem using
the derived completion, as in [Sta, Tag 0A0H]. For our uses, we do not jump
into this generality.

The rest of this subsection is devoted to prove Theorem 6.1.2.

Special case: $X$ is smooth. First we deal with the special case, assuming $X$
itself is smooth over $K$.

When $X$ is smooth, as the blowup center $Y$ is assumed to be smooth, the
blowup $X^{\prime}$ itself is also smooth over $K$. By Theorem 4.2.2, the
derived direct image of the coherent crystal over $X/K_{\inf}$ and
$X^{\prime}/K_{\inf}$ can be computed by their de Rham complexes
$\mathcal{F}_{X}\otimes\Omega_{X/K}^{\bullet}$ and
$\mathcal{F}_{X^{\prime}}\otimes\Omega_{X^{\prime}/K}^{\bullet}=f^{*}\mathcal{F}_{X}\otimes\Omega_{X^{\prime}/K}^{\bullet}$
separately. On the other hand, the derived direct image $Ru_{Y/K*}\mathcal{F}$
and $Ru_{Y^{\prime}/K*}\mathcal{F}$ are naturally isomorphic to the de Rham
complex over the envelopes $D_{Y}(X)$ and $D_{Y^{\prime}}(X^{\prime})$; namely
the complexes

$\mathcal{F}_{D}\otimes\Omega_{D_{Y}(X)}^{\bullet},~{}~{}~{}\mathcal{F}_{D^{\prime}}\otimes\Omega_{D_{Y^{\prime}}(X^{\prime})}^{\bullet},$

which are compatible with the Hodge filtrations of $\Omega_{X/K}^{\bullet}$
and $\Omega_{X^{\prime}/K}^{\bullet}$. So the sequence $(\ast)$ in Theorem
6.1.2 is naturally isomorphic to the following sequence of de Rham complexes

$\mathcal{F}_{X}\otimes\Omega_{X/K}^{\bullet}\longrightarrow
Rf_{*}(f^{*}\mathcal{F}_{X}\otimes\Omega_{X^{\prime}/K}^{\bullet})\operatorname*{\text{\raisebox{1.07639pt}{\scalebox{0.8}{$\bigoplus$}}}}\mathcal{F}_{D}\otimes\Omega_{D_{Y}(X)}^{\bullet}\longrightarrow
Rf_{*}(\mathcal{F}_{D^{\prime}}\otimes\Omega_{D_{Y^{\prime}}(X^{\prime})}^{\bullet}).$

In fact, we want to show the following more general statement:

######  Proposition 6.1.5.

Let $f:X^{\prime}\rightarrow X$ be a proper morphism of smooth rigid spaces
over $K$, and let $Y$ be a closed analytic subset of $X$, with
$Y^{\prime}=f^{-1}(Y)$. Let $\mathcal{G}^{\prime}$ and $\mathcal{G}$ be
coherent sheaves over $X^{\prime}$ and $X$ separately, such that
$f:X^{\prime}\rightarrow X$ induces an injective map of
$\mathcal{O}_{X}$-modules $\mathcal{G}\rightarrow f_{*}\mathcal{G}^{\prime}$.
Assume $f$ induces an isomorphism between open subsets $X^{\prime}\backslash
Y^{\prime}$ and $X\backslash Y$, and also induces an isomorphism of the
restriction of $\mathcal{G}\rightarrow f_{*}\mathcal{G}^{\prime}$ on
$X\backslash Y$. Then the following natural sequence is a distinguished
triangle

$\mathcal{G}\otimes\Omega_{X/K}^{i}\longrightarrow
Rf_{*}(\mathcal{G}^{\prime}\otimes\Omega_{X^{\prime}/K}^{i})\operatorname*{\text{\raisebox{1.07639pt}{\scalebox{0.8}{$\bigoplus$}}}}\mathcal{G}\otimes\Omega_{D_{Y}(X)/K}^{i}\longrightarrow
Rf_{*}(\mathcal{G}^{\prime}\otimes\Omega_{D_{Y^{\prime}}(X^{\prime})/K}^{i}),~{}i\in\mathbb{N}.$

###### Proof.

Let $\mathcal{M}$ be the coherent sheaf $\mathcal{G}\otimes\Omega_{X/K}^{i}$
over $X$, and let $\mathcal{M}^{\prime}$ be the coherent sheaf
$\mathcal{G}^{\prime}\otimes\Omega_{X^{\prime}/K}^{i}$ over $X^{\prime}$. By
the assumption of smoothness, both $\Omega_{X/K}^{i}$ and
$\Omega_{X^{\prime}/K}^{i}$ are locally free, and the natural map
$\mathcal{M}\rightarrow f_{*}\mathcal{M}^{\prime}$ is injective. Furthermore,
the restriction of the map $\mathcal{M}\rightarrow f_{*}\mathcal{M}^{\prime}$
on the open subset $X\backslash Y$ is an isomorphism.

Recall the $i$-th differential sheaf $\Omega_{D_{Y}(X)/K}^{i}$ of $D_{Y}(X)$,
as a sheaf over $X$, is defined as the inverse limit
$\varprojlim_{m}\Omega_{X_{m}/K}^{i}$, where $X_{m}$ is the $m$-th
infinitesimal neighborhood of $Y$ in $X$. As is shown in Lemma 3.3.3, the
sheaf $\Omega_{D_{Y}(X)/K}^{i}$ is naturally isomorphic to the formal
completion of the coherent sheaf $\Omega_{X/K}^{i}$ along $Y\rightarrow X$,
which is also equal to the tensor product
$\Omega_{X/K}^{i}\otimes\mathcal{O}_{D_{Y}(X)}$. Moreover, since $\mathcal{G}$
is coherent over $X$, the tensor product
$\mathcal{G}\otimes\Omega_{D_{Y}(X)}^{i}$ is isomorphic to the formal
completion $\widehat{\mathcal{M}}$ of
$\mathcal{M}=\mathcal{G}\otimes\Omega_{X/K}^{i}$ along $Y\rightarrow X$. The
same also holds for $X^{\prime},Y^{\prime}$ and $\mathcal{M}^{\prime}$.

We denote by $C_{1}$ and $C_{2}$ to be cones of the map
$\mathcal{M}\rightarrow Rf_{*}\mathcal{M}$ and
$\widehat{\mathcal{M}}\rightarrow Rf_{*}\widehat{\mathcal{M}}^{\prime}$
separately. Consider the following commutative diagram

$\textstyle{\mathcal{M}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Rf_{*}\mathcal{M}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widehat{\mathcal{M}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Rf_{*}\widehat{\mathcal{M}}^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C_{2}.}$

Here both rows are distinguished.

Now we make the following claim.

######  Claim 6.1.6.

The natural map $C_{1}\rightarrow C_{2}$ of cones above is an isomorphism.

###### Proof of the Claim.

First we notice that since the map $f:X^{\prime}\rightarrow X$ is isomorphic
on the open subsets $X^{\prime}\backslash Y^{\prime}\rightarrow X\backslash
Y$, while both $X^{\prime}$ and $X$ are smooth, the sheaves of differentials
$\Omega_{X/K}^{i}$ and $\Omega_{X^{\prime}/K}^{i}$ are vector bundles, and the
induced map $\Omega_{X/K}^{i}\rightarrow f_{*}\Omega_{X^{\prime}/k}^{i}$ is
injective. On the other hand, the map $\mathcal{G}\rightarrow
f_{*}\mathcal{G}^{\prime}$ is assumed to be an injective map of coherent
sheaves in the Proposition. Combine the above two conditions, we see the map
$\mathcal{M}\rightarrow f_{*}\mathcal{M}^{\prime}$ is injective, and the cone
lives in cohomological degree $[0,+\infty)$.

By the Formal Function Theorem 6.1.3, the cohomology sheaf
$R^{j}f_{*}\widehat{\mathcal{M}}^{\prime}$ is naturally isomorphic to the
formal completion $(R^{j}f_{*}\mathcal{M}^{\prime})^{\wedge}$ of
$R^{j}f_{*}\mathcal{M}^{\prime}$ along $Y\rightarrow X$. Moreover, by the
exactness of the formal completion on coherent sheaves, the natural map
$\widehat{\mathcal{M}}\rightarrow(f_{*}\mathcal{M}^{\prime})^{\wedge}$ is
injective, and we have a short exact sequence

$0\longrightarrow\widehat{\mathcal{M}}\longrightarrow(f_{*}\mathcal{M}^{\prime})^{\wedge}\longrightarrow\mathcal{H}^{0}(C_{2})\longrightarrow
0.$

This implies that $C_{2}$ lives also in cohomological degree no smaller than
zero. Furthermore, by the exactness of the above sequence, the cohomology
sheaf $\mathcal{H}^{0}(C_{2})$ is isomorphic to the formal completion of
$\mathcal{H}^{0}(C_{1})$ at $Y$. But since $\mathcal{H}^{0}(C_{1})$ is
coherent and is already supported at $Y$, we have

$\mathcal{H}^{0}(C_{1})=\mathcal{H}^{0}(C_{1})^{\wedge}\cong\mathcal{H}^{0}(C_{2}).$

This finishes the degree zero part.

For the higher cohomology, we consider the following diagram of cohomologies

$\textstyle{R^{j}f_{*}\mathcal{M}^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{H}^{j}(C_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(R^{j}f_{*}\mathcal{M}^{\prime})^{\wedge}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{H}^{j}(C_{2}).}$

As $\mathcal{M}$ and $\widehat{\mathcal{M}}$ are living in cohomological
degree zero, the horizontal maps above are isomorphisms, and it suffices to
show for each $i>0$, the left vertical map above is an isomorphism. But notice
that since $f$ induces an isomorphism between $\mathcal{M}$ and
$\mathcal{M}^{\prime}$ over $X\backslash Y$, the higher cohomology sheaf
$R^{j}f_{*}\mathcal{M}^{\prime}$ is coherent and is supported over $Y$. In
particular, the formal completion of $R^{j}f_{*}\mathcal{M}^{\prime}$ along
$Y\rightarrow X$ is equal to itself; namely the natural map below is an
isomorphism

$R^{j}f_{*}\mathcal{M}^{\prime}\longrightarrow(R^{j}f_{*}\mathcal{M}^{\prime})^{\wedge}.$

This leads to the isomorphism

$\mathcal{H}^{j}(C_{1})\cong\mathcal{H}^{j}(C_{2}),~{}\forall j\geq 1,$

and we finish the isomorphism between $C_{1}$ and $C_{2}$.

∎

We change the notation back to Proposition 6.1.5. Then we get two rows of
distinguished triangles

$\textstyle{\mathcal{G}\otimes\Omega^{i}_{X/K}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Rf_{*}(\mathcal{G}^{\prime}\otimes\Omega^{i}_{X^{\prime}/K})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{G}\otimes\Omega^{i}_{D_{Y}(X)/K}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Rf_{*}(\mathcal{G}^{\prime}\otimes\Omega^{i}_{D_{Y^{\prime}}(X^{\prime})/K})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C_{2},}$

the third vertical map is an isomorphism.

Finally, we consider the following two maps,

$\displaystyle\phi$ $\displaystyle:\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
21.81915pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathcal{G}\otimes\Omega_{X/K}^{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
18.32848pt\raise 6.5pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.75pt\hbox{$\scriptstyle{(+,+)~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 45.81915pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
45.81915pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{Rf_{*}(\mathcal{G}^{\prime}\otimes\Omega^{i}_{X^{\prime}/K})\operatorname*{\text{\raisebox{1.07639pt}{\scalebox{0.8}{$\bigoplus$}}}}(\mathcal{G}\otimes\Omega_{D_{Y}(X)/K}^{i}}$}}}}}}}\ignorespaces}}}}\ignorespaces);$
$\displaystyle\psi$ $\displaystyle:\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
63.67902pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{Rf_{*}(\mathcal{G}^{\prime}\otimes\Omega^{i}_{X^{\prime}/K})\operatorname*{\text{\raisebox{1.07639pt}{\scalebox{0.8}{$\bigoplus$}}}}(\mathcal{G}\otimes\Omega_{D_{Y}(X)/K}^{i})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
45.88922pt\raise 6.5pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.75pt\hbox{$\scriptstyle{~{}~{}~{}~{}~{}~{}~{}(-,+)}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 87.67902pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
87.67902pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{Rf_{*}(\mathcal{G}^{\prime}\otimes\Omega_{D_{Y^{\prime}}(X^{\prime})/K}^{i})}$}}}}}}}\ignorespaces}}}}\ignorespaces,$

where $+$ and $-$ are indicating the signs of the map. As the composition of
the above two maps is equal to zero, the map $\psi$ factors though a morphism
$\operatorname{Cone}(\phi)\rightarrow
Rf_{*}(\mathcal{G}^{\prime}\otimes\Omega_{D_{Y^{\prime}}(X^{\prime})/K}^{i})$
([Sta, Tag 08RI]). In this way, by chasing diagrams and the Claim above, the
map $\operatorname{Cone}(\phi)\rightarrow
Rf_{*}(\mathcal{G}^{\prime}\otimes\Omega_{D_{Y^{\prime}}(X^{\prime})/K}^{i})$
is an isomorphism, and we get the distinguished triangle we want.

∎

General case. We then deal with the general case of Theorem 6.1.2.

###### Proof of Theorem 6.1.2..

As the theorem is a local statement, by passing to an open covering if
necessary it suffices to assume $X$ is affinoid and admits a closed immersion
into a smooth affinoid rigid space $Z$. Moreover, as any coherent crystal over
$X/K_{{\mathrm{INF}}}$ is a crystal in vector bundles (Corollary 3.2.5), by
further taking open subsets we may assume
$\mathcal{F}_{X}\cong\mathcal{O}_{X}^{\oplus m}$ is trivial over $X$.

As $Y$ is smooth over $K$, the blowup $Z^{\prime}=\mathrm{Bl}_{Z}(Y)$ of the
smooth rigid space $Z$ at the center $Y$ is also smooth. Moreover, as
$X\rightarrow Z$ is a closed immersion while $X^{\prime}$ is the blowup of $X$
at $Y$, the natural map $X^{\prime}\rightarrow Z^{\prime}$ is also a closed
immersion, which is equal to the preimage of $X$ along the blowup map
$f:Z^{\prime}\rightarrow Z$. So we get the following commutative diagrams of
rigid spaces over $K$ with both of the square being cartesian

$\textstyle{Y^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Z^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Z.}$

As the restriction $\mathcal{F}_{D_{X}(Z)}$ is a vector bundle over
$\mathcal{O}_{D_{X}(Z)}$ whose pullback along $X\rightarrow D_{X}(Z)$ is
trivial, by Nakayama’s lemma, we may let $\mathcal{G}$ be a trivial vector
bundle over $Z$ such that the tensor product
$\mathcal{G}\otimes_{\mathcal{O}_{Z}}\mathcal{O}_{D_{X}(Z)}$ is equal to
$\mathcal{F}_{D_{X}(Z)}$. Let $\mathcal{G}^{\prime}$ be the pullback
$f^{*}\mathcal{G}$, as a trivial bundle over $Z^{\prime}$. Then by the crystal
condition of $\mathcal{F}$ we have

$\mathcal{G}^{\prime}\otimes_{\mathcal{O}_{Z^{\prime}}}\mathcal{O}_{D_{X^{\prime}}(Z^{\prime})}=\mathcal{F}_{D_{X^{\prime}}(Z^{\prime})}.$

Here we note that by our choices and the diagram above, the map
$\mathcal{G}\rightarrow f_{*}\mathcal{G}^{\prime}$ is injective and is an
isomorphism when restricted to open subsets $Z/Y$ and $Z/X$. Similar to the
proof of Proposition 6.1.5, we let $\mathcal{M}$ and $\mathcal{M}^{\prime}$ be
the tensor products $\mathcal{G}\otimes\Omega_{Z/K}^{i}$ and
$\mathcal{G}^{\prime}\otimes\Omega_{Z^{\prime}/K}^{i}$ over $Z$ and
$Z^{\prime}$ separately.

Now by the proof of Proposition 6.1.5, we have the following natural
commutative diagrams, with each row being distinguished

$\textstyle{\mathcal{M}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Rf_{*}\mathcal{M}^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widehat{\mathcal{M}_{/X}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Rf_{*}\widehat{\mathcal{M}^{\prime}_{/X^{\prime}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widehat{\mathcal{M}_{/Y}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Rf_{*}\widehat{\mathcal{M}^{\prime}_{/Y^{\prime}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C_{3}.}$

Here the sheaf $\widehat{\mathcal{M}_{/X}}$ and similar for the others is
denoted to be the formal completion of $\mathcal{M}$ along $X\rightarrow Z$.
Thanks to Claim 6.1.6, the map $C_{1}\rightarrow C_{2}$ and the
$C_{1}\rightarrow C_{3}$ are isomorphisms. In particular, the map
$C_{2}\rightarrow C_{3}$ is an isomorphism. In this way, as in the last part
of the proof for Proposition 6.1.5, the second and the third rows above
produces the following distinguished triangle

$\textstyle{\widehat{\mathcal{M}_{/X}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(+,+)~{}~{}~{}~{}~{}~{}~{}~{}~{}}$$\textstyle{Rf_{*}\widehat{\mathcal{M}^{\prime}_{/X^{\prime}}}\bigoplus\widehat{\mathcal{M}_{/Y}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{~{}~{}~{}~{}~{}(-,+)}$$\textstyle{Rf_{*}\widehat{\mathcal{M}^{\prime}_{/Y^{\prime}}}.}$

Hence by Lemma 3.3.3, we may replace those formal completions by their
corresponding sheaves over envelopes, and obtain the distinguished triangle
below

$\textstyle{\mathcal{F}_{D_{X}(Z)}\otimes\Omega_{D_{X}(Z)}^{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Rf_{*}(\mathcal{F}_{D_{X^{\prime}}(Z^{\prime})}\otimes\Omega_{D_{X^{\prime}}(Z^{\prime})}^{i})\operatorname*{\text{\raisebox{1.07639pt}{\scalebox{0.8}{$\bigoplus$}}}}\mathcal{F}_{D_{Y}(Z)}\otimes\Omega_{D_{Y}(Z)}^{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Rf_{*}(\mathcal{F}_{D_{Y^{\prime}}(Z^{\prime})}\otimes\Omega_{D_{Y^{\prime}}(Z^{\prime})}^{i}),}$

which implies the theorem by taking different $i$ and Theorem 4.2.2.

∎

######  Remark 6.1.7.

With the help of the blowup triangle in Theorem 6.1.2, we could show the
following: for a universal homeomorphism of rigid spaces
$f:X^{\prime}\rightarrow X$ over $K$ and a coherent crystal $\mathcal{F}$ over
$X/K_{{\mathrm{INF}}}$, there exists a natural isomorphism of cohomology
sheaves as below

$Ru_{X/K*}\mathcal{F}\longrightarrow Rf_{*}Ru_{X^{\prime}/K*}\mathcal{F}.$

###  6.2. Éh-hyperdescent in general

Now we are ready to prove the éh descent for a crystal over the infinitesimal
site. We first deal with the case for infinitesimal cohomology over an
arbitrary $p$-adic field $K$, where the strategy is to use the blowup square
for the éh-topology in [Guo19] and the descents in the first subsection. After
that, we generalize to the case over ${\mathrm{B^{+}_{dR,e}}}$.

Recall that the éh site $X_{\mathrm{\acute{e}h}}$ is defined over the category
${\mathrm{Rig}}_{K}|_{X}$ of all $K$-rigid spaces over $X$ and is equipped
with the éh-topology (cf. [Guo19, Section 2]). For an object
$X^{\prime}\rightarrow X$ in ${\mathrm{Rig}}_{K}|_{X}$, we denote by
$\pi_{X^{\prime}}:X_{\mathrm{\acute{e}h}}\rightarrow
X^{\prime}_{\mathrm{rig}}$ to be the map from the éh site of $X$ to the small
rigid site of $X^{\prime}$.

We first associate an infinitesimal crystal together with its de Rham complex
an analogous construction over the éh topology.

######  Construction 6.2.1 (éh de Rham complex).

Let $\mathcal{F}$ be an infinitesimal sheaf of $\mathcal{O}_{X/K}$-modules
over the big infinitesimal site $X/K_{{\mathrm{INF}}}$. We then associate a
sheaf $\mathcal{F}_{\mathrm{rig}}:=i_{X/K}^{-1}\mathcal{F}$ of
$\mathcal{O}_{X}=i_{X/K}^{-1}\mathcal{O}_{X/K}$-modules over the big rigid
site ${\mathrm{Rig}}_{K}|_{X}$, where
$i_{X/K}:\mathrm{Sh}({\mathrm{Rig}}_{K}|_{X})\rightarrow\mathrm{Sh}(X/K_{{\mathrm{INF}}})$
is the morphism of topoi as in Subsection 2.3. Here the section of
$\mathcal{F}_{\mathrm{rig}}$ at an object $f:X^{\prime}\rightarrow X$ in
${\mathrm{Rig}}_{K}|_{X}$ is the $\mathcal{O}_{X^{\prime}}(X^{\prime})$-module

$\mathcal{F}(X^{\prime},X^{\prime}),$

where $(X^{\prime},X^{\prime})\in X/\Sigma_{e{\mathrm{INF}}}$ is the trivial
thickening of $X^{\prime}$. We could then sheafify it with the éh-topology,
and thus get an éh-sheaf $\mathcal{F}_{\mathrm{\acute{e}h}}$ over the éh site
$X_{\mathrm{\acute{e}h}}$.

Now we specify $\mathcal{F}$ to be a coherent crystal over big the
infinitesimal site. As in the discussion of Paragraph 3.1, we could associate
$\mathcal{F}$ its de Rham complex over the big infinitesimal site

$\mathcal{F}\otimes_{\mathcal{O}_{X/K}}\Omega_{X/K_{{\mathrm{INF}}}}^{\bullet}.$

This allows us to get a complex of sheaves over ${\mathrm{Rig}}_{K}|_{X}$ and
$X_{\mathrm{\acute{e}h}}$ separately

$\displaystyle\mathcal{F}_{\mathrm{rig}}\longrightarrow\mathcal{F}_{\mathrm{rig}}\otimes_{\mathcal{O}_{\mathrm{rig}}}\Omega_{\mathrm{rig}}^{1}\longrightarrow\mathcal{F}_{\mathrm{rig}}\otimes_{\mathcal{O}_{\mathrm{rig}}}\Omega_{\mathrm{rig}}^{2}\longrightarrow\cdots;$
$\displaystyle\mathcal{F}_{\mathrm{\acute{e}h}}\longrightarrow\mathcal{F}_{\mathrm{\acute{e}h}}\otimes_{\mathcal{O}_{\mathrm{\acute{e}h}}}\Omega_{\mathrm{\acute{e}h}}^{1}\longrightarrow\mathcal{F}_{\mathrm{\acute{e}h}}\otimes_{\mathcal{O}_{\mathrm{\acute{e}h}}}\Omega_{\mathrm{\acute{e}h}}^{2}\longrightarrow\cdots.$

Here the sheaf $\Omega_{\mathrm{rig}}^{i}$ is the usual continuous Kähler
differential sheaf over the rigid site, and $\Omega_{\mathrm{\acute{e}h}}^{i}$
is the éh continuous Kähler differential, which is equal to the éh
sheafification of the usual continuous differential. The complex
$\mathcal{F}_{\mathrm{\acute{e}h}}\otimes\Omega_{\mathrm{\acute{e}h}}^{\bullet}$
is called _éh de Rham complex associated with the crystal $\mathcal{F}$_.

The Construction 6.2.1 produces two maps of objects in the derived category of
the big rigid topos:

$\textstyle{Ru_{X/K*}(\mathcal{F}\otimes_{\mathcal{O}_{X/K}}\Omega_{X/K_{{\mathrm{INF}}}}^{\bullet})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{F}_{\mathrm{rig}}\otimes_{\mathcal{O}_{\mathrm{rig}}}\Omega_{\mathrm{rig}}^{\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{R\pi_{*}(\mathcal{F}_{\mathrm{\acute{e}h}}\otimes_{\mathcal{O}_{\mathrm{\acute{e}h}}}\Omega_{\mathrm{\acute{e}h}}^{\bullet}).}$

Formally the first map is given by the sheafified version of the natural
transformation

${\mathrm{Rig}}_{K}|_{X}\ni
X^{\prime}\longmapsto\left(R\Gamma(X^{\prime}/\Sigma_{e{\mathrm{INF}}},-)\rightarrow
R\Gamma((X^{\prime},X^{\prime}),-)\right)$

at the complex of infinitesimal sheaves
$\mathcal{F}\otimes_{\mathcal{O}_{X/K}}\Omega_{X/K_{{\mathrm{INF}}}}^{\bullet})$.
The second map above comes from the counit morphism for the adjoint pair
$(\pi^{-1},\pi_{*})$, where $\pi^{-1}$ is the éh-sheafification functor.
Moreover, the above map can be improved into the filtered derived category,
where the left side is equipped with the infinitesimal filtration, and the
rest two complexes are equipped with their Hodge filtrations.

We also note that the natural map from the de Rham complex
$\mathcal{F}\otimes_{\mathcal{O}_{X/K}}\Omega_{X/K_{{\mathrm{INF}}}}^{\bullet}$
to the crystal $\mathcal{F}$ itself induces a natural isomorphism as below

$\textstyle{Ru_{X/K*}(\mathcal{F}\otimes_{\mathcal{O}_{X/K}}\Omega_{X/K_{{\mathrm{INF}}}}^{\bullet})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{~{}~{}~{}~{}~{}~{}~{}~{}\sim}$$\textstyle{Ru_{X/K*}\mathcal{F},}$

which is proved in Proposition 4.2.1.

Now we can state the descent result.

######  Theorem 6.2.2.

Let $X$ be a rigid space over $K$, and let $\mathcal{F}$ be a coherent crystal
over the big infinitesimal site $X/K_{{\mathrm{INF}}}$. Then the natural map
of $K$-linear complexes below is an isomorphism.

$Ru_{X/K*}(\mathcal{F}\otimes_{\mathcal{O}_{X/K}}\Omega_{X/K_{{\mathrm{INF}}}}^{\bullet})\longrightarrow
R\pi_{X*}(\mathcal{F}_{\mathrm{\acute{e}h}}\otimes_{\mathcal{O}_{\mathrm{\acute{e}h}}}\Omega_{\mathrm{\acute{e}h}}^{\bullet}).$

In particular, infinitesimal cohomology of the coherent crystal $\mathcal{F}$
satisfies the éh-hyperdescent.

###### Proof.

  * Step 1

In the Step 1, we show that by restricting to a smooth rigid space
$X^{\prime}$ over $K$ that admits a map to $X$, the morphism in the statement
is an isomorphism. Namely the natural morphism below is an isomorphism

$Ru_{X^{\prime}/K*}(\mathcal{F}\otimes_{\mathcal{O}_{X/K}}\Omega_{X/K_{{\mathrm{INF}}}}^{\bullet})\longrightarrow
R\pi_{X^{\prime}*}(\mathcal{F}_{\mathrm{\acute{e}h}}\otimes_{\mathcal{O}_{\mathrm{\acute{e}h}}}\Omega_{\mathrm{\acute{e}h}}^{\bullet}).$

We first apply $R\Gamma(X^{\prime},-)$ for the smooth rigid space
$X^{\prime}$. On the one hand, we apply Theorem 4.2.2 to the trivial closed
immersion $X^{\prime}\rightarrow X^{\prime}$ to get

$\displaystyle R\Gamma(X^{\prime}/K_{\mathrm{INF}},\mathcal{F})$
$\displaystyle\cong
R\Gamma(X^{\prime}/K_{{\mathrm{INF}}},\mathcal{F}\otimes_{\mathcal{O}_{X/K}}\Omega_{X/K_{{\mathrm{INF}}}}^{\bullet})$
$\displaystyle\cong
R\Gamma(X^{\prime},\mathcal{F}_{D}\otimes\Omega_{D}^{\bullet})$
$\displaystyle=R\Gamma(X^{\prime}_{\mathrm{rig}},\mathcal{F}_{X^{\prime}}\otimes\Omega_{X^{\prime}/K}^{\bullet}),$

where the envelope $D$ for the trivial closed immersion $X^{\prime}\rightarrow
X^{\prime}$ is just $X^{\prime}$ itself.

On the other hand, the éh-cohomology
$R\Gamma(X^{\prime}_{\mathrm{\acute{e}h}},\mathcal{F}_{\mathrm{\acute{e}h}}\otimes\Omega_{\mathrm{\acute{e}h}}^{\bullet})$
is in fact isomorphic to the cohomology of the de Rham complex of
$\mathcal{F}_{X^{\prime}}$ given by the restriction of
$\mathcal{F}_{\mathrm{rig}}\otimes\Omega_{\mathrm{rig}}^{\bullet}$ at
$X^{\prime}$. To see this, we notice that as the natural map of complexes
$\mathcal{F}_{\mathrm{rig}}\otimes\Omega_{\mathrm{rig}}^{\bullet}\rightarrow
R\pi_{X^{\prime}*}(\mathcal{F}_{\mathrm{\acute{e}h}}\otimes\Omega_{\mathrm{\acute{e}h}}^{\bullet})$
is filtered with respect to the Hodge filtrations, it suffices to show for
each $i\in\mathbb{N}$ the following map is an isomorphism

$R\Gamma(X^{\prime}_{\mathrm{rig}},\mathcal{F}_{X^{\prime}}\otimes\Omega_{X^{\prime}/K}^{i})\longrightarrow
R\Gamma(X^{\prime}_{\mathrm{\acute{e}h}},\mathcal{F}_{\mathrm{\acute{e}h}}\otimes\Omega_{\mathrm{\acute{e}h}}^{i}).$

Here the restriction $\mathcal{F}_{\mathrm{\acute{e}h}}|_{X^{\prime}}$ at the
éh site of $X^{\prime}$ can be given by the éh-sheafification of the rigid
sheaf $\mathcal{F}_{\mathrm{rig}}|_{X^{\prime}}$ over the big site
${\mathrm{Rig}}_{K}|_{X^{\prime}}$. Moreover, as $\mathcal{F}$ is a crystal in
vector bundles (Corollary 3.2.5) and the statement is local on $X^{\prime}$
(namely both of the above two complexes satisfies the hyperdescent for rigid
topology), by passing to an open rigid subspace of $X^{\prime}$ if necessary
we may assume the restriction $\mathcal{F}_{\mathrm{rig}}|_{X^{\prime}}$ of
$\mathcal{F}$ at $X^{\prime}$ is isomorphic to the direct sum
$\mathcal{O}_{X^{\prime}}^{m}$ of structure sheaves. Thus we reduce to show
that the natural map of cohomology of differentials below for a smooth
$K$-rigid space $X^{\prime}$ is an isomorphism

$R\Gamma(X^{\prime}_{\mathrm{rig}},\Omega_{X^{\prime}/K}^{i})\longrightarrow
R\Gamma(X^{\prime}_{\mathrm{\acute{e}h}},\Omega_{\mathrm{\acute{e}h}}^{i}),$

which is proved in [Guo19, Theorem 4.0.2].

In this way, as the map in the statement is given by the composition

$R\Gamma(X^{\prime}/K_{{\mathrm{INF}}},\mathcal{F}\otimes_{\mathcal{O}_{X/K}}\Omega_{X/K_{{\mathrm{INF}}}}^{\bullet})\longrightarrow
R\Gamma(X^{\prime}_{\mathrm{rig}},\mathcal{F}_{X^{\prime}}\otimes\Omega_{X^{\prime}/K}^{\bullet})\longrightarrow
R\Gamma(X^{\prime}_{\mathrm{\acute{e}h}},\mathcal{F}_{\mathrm{\acute{e}h}}\otimes_{\mathcal{O}_{\mathrm{\acute{e}h}}}\Omega_{\mathrm{\acute{e}h}}^{\bullet}),$

we see both maps above are isomorphisms when $X^{\prime}$ is smooth over $K$.

At last, notice that the cofiber $C$ of the map
$Ru_{X^{\prime}/K*}(\mathcal{F}\otimes_{\mathcal{O}_{X/K}}\Omega_{X/K_{{\mathrm{INF}}}}^{\bullet})\longrightarrow
R\pi_{X^{\prime}*}(\mathcal{F}_{\mathrm{\acute{e}h}}\otimes_{\mathcal{O}_{\mathrm{\acute{e}h}}}\Omega_{\mathrm{\acute{e}h}}^{\bullet})$
is a bounded below complex of sheaves over the small rigid site
$X^{\prime}_{\mathrm{rig}}$. If $C$ is not acyclic, then there would exist an
open subspace $U$ of $X^{\prime}$ such that the cohomology
$R\Gamma(U_{\mathrm{rig}},C)$ does not vanishes, which contradicts to the
computation above. So we get the isomorphism for smooth $K$-rigid space
$X^{\prime}$ that admits a map to $X$:

$Ru_{X^{\prime}/K*}(\mathcal{F}\otimes_{\mathcal{O}_{X/K}}\Omega_{X/K_{{\mathrm{INF}}}}^{\bullet})\longrightarrow
R\pi_{X^{\prime}*}(\mathcal{F}_{\mathrm{\acute{e}h}}\otimes_{\mathcal{O}_{\mathrm{\acute{e}h}}}\Omega_{\mathrm{\acute{e}h}}^{\bullet}).$

  * Step 2

Now we prove the isomorphism of the derived push-forwards as in the statement.

Let $i:X_{\mathrm{red}}\rightarrow X$ be the closed immersion by the reduced
sub rigid space of $X$. We first notice that the natural map below is an
isomorphism

$Ru_{X/K*}\mathcal{F}\longrightarrow
i_{*}Ru_{X_{\mathrm{red}}/K*}(\mathcal{F}),$

This is because locally both of them are computed using the de Rham complex
$\mathcal{F}_{D}\otimes\Omega_{D}^{\bullet}$, where $D$ is the envelope of $X$
in a smooth rigid spaces (Theorem 4.2.2). The same isomorphism holds for the
derived direct image of the infinitesimal de Rham complex
$\mathcal{F}\otimes\Omega_{X_{\mathrm{red}}/K_{{\mathrm{INF}}}}^{\bullet}$ by
Proposition 4.2.1. On the other hand, We notice that as the closed immersion
$i:X_{\mathrm{red}}\rightarrow X$ is an éh-covering ([Guo19, Section 2.4]),
which forms a constant éh-hypercovering as the product
$X_{\mathrm{red}}\times_{X}X_{\mathrm{red}}$ is equal to $X_{\mathrm{red}}$
itself, we get

$R\pi_{X*}(\mathcal{F}_{\mathrm{\acute{e}h}}\otimes\Omega_{\mathrm{\acute{e}h}}^{\bullet})\cong
i_{*}R\pi_{X_{\mathrm{red}}*}(\mathcal{F}_{\mathrm{\acute{e}h}}\otimes\Omega_{\mathrm{\acute{e}h}}^{\bullet}).$

Thus the above two isomorphisms allow us to assume $X$ is generically reduced,
and by passing to an open subset if necessary we may assume $X$ is quasi-
compact.

Now we can do the induction on the dimension of $X$. When $\dim(X)$ is of
dimension zero, as $X$ is quasi-compact and generically reduced, it is then
equal to a disjoint union of finite points and in particular is smooth over
$K$, where the statement follows from the Step 1.

In general, by the Temkin’s resolution of singularities for rigid spaces
([Tem12]), we can find a finite compositions $X_{n}\rightarrow\cdots
X_{1}\rightarrow X_{0}=X$, where each $X_{i}\rightarrow X_{i-1}$ is a blowup
at a smooth nowhere dense closed subspace $Y_{i}\subset X_{i-1}$, such that in
the last step $X_{n}$ is smooth over $K$. We denote by $Y_{i}^{\prime}$ to be
the preimage $Y_{i}\times_{X_{i-1}}X_{i}$ in $X_{i}$, which is of dimension
strictly smaller than $\dim(X_{i})=\dim(X)$, and we let $f_{i}$ be the blowup
map $X_{i}\rightarrow X_{i-1}$. Then for each $1\leq i\leq n$, we get a
natural distinguished triangle by Theorem 6.1.2

$Ru_{X_{i-1}/K*}(\mathcal{F}\otimes\Omega_{{\mathrm{INF}}}^{\bullet})\longrightarrow
Rf_{i*}Ru_{X_{i}/K*}(\mathcal{F}\otimes\Omega_{{\mathrm{INF}}}^{\bullet})\operatorname*{\text{\raisebox{1.07639pt}{\scalebox{0.8}{$\bigoplus$}}}}Ru_{Y_{i}/K*}(\mathcal{F}\otimes\Omega_{{\mathrm{INF}}}^{\bullet})\longrightarrow
Rf_{i*}Ru_{Y^{\prime}_{i}/K*}(\mathcal{F}\otimes\Omega_{{\mathrm{INF}}}^{\bullet}).$

Again here we use Proposition 4.2.1 to replace the $\mathcal{F}$ by its de
Rham complex. On the other hand, by the sheafified version of the blowup
square in the éh-topology ([Guo19, Proposition 5.1.4]), we have a natural
distinguished triangle

$R\pi_{X_{i-1}*}(\mathcal{F}_{\mathrm{\acute{e}h}}\otimes\Omega_{\mathrm{\acute{e}h}}^{\bullet})\longrightarrow
R\pi_{X_{i}*}(\mathcal{F}_{\mathrm{\acute{e}h}}\otimes\Omega_{\mathrm{\acute{e}h}}^{\bullet})\operatorname*{\text{\raisebox{1.07639pt}{\scalebox{0.8}{$\bigoplus$}}}}R\pi_{Y_{i}*}(\mathcal{F}_{\mathrm{\acute{e}h}}\otimes\Omega_{\mathrm{\acute{e}h}}^{\bullet})\longrightarrow
R\pi_{Y_{i}^{\prime}*}(\mathcal{F}_{\mathrm{\acute{e}h}}\otimes\Omega_{\mathrm{\acute{e}h}}^{\bullet}).$

The functoriality of the map in the statement allows us to produce a map from
the triangle $(\ast)$ to the triangle $(\ast\ast)$. Moreover, by the dimension
assumption and the induction assumption, we know the statement in the Theorem
is true for $X_{n}$ and all of $Y_{i}$ and $Y_{i}^{\prime}$. In this way,
since $X_{0}=X$, by a finite step of descending inductions via comparing the
above two triangles $(\ast)$ and $(\ast\ast)$, the natural map below is then
an isomorphism

$Ru_{X/K*}(\mathcal{F}\otimes\Omega_{X/K_{\mathrm{INF}}}^{\bullet})\longrightarrow
R\pi_{X*}(\mathcal{F}_{\mathrm{\acute{e}h}}\otimes\Omega_{\mathrm{\acute{e}h}}^{\bullet}).$

So we are done.

∎

######  Remark 6.2.3.

By the construction of the map in Theorem 6.2.2, we see infinitesimal
cohomology of a coherent crystal $\mathcal{F}$ over $X/K_{{\mathrm{INF}}}$ is
a direct summand of the cohomology
$R\Gamma(X_{\mathrm{rig}},\mathcal{F}_{X}\otimes\Omega_{X/K}^{\bullet})$ of
the usual de Rham complex over $X_{\mathrm{rig}}$.

######  Remark 6.2.4.

The isomorphism in the Theorem above cannot always be improved into a filtered
isomorphism. The discrepancy already appears in the schematic theory (see
[Bha12, Example 5.6]).

Now we are able to generalize the éh-hyperdescent to coherent crystals over
$X/\Sigma_{e{\mathrm{INF}}}$ for general $e$, not just $K$-linear crystals. We
assume $K$ is complete and algebraically closed in the next Theorem, so
${\mathrm{B^{+}_{dR,e}}}$ is well-defined for $K$.

######  Theorem 6.2.5.

Let $X$ be a rigid space over $X$, and let $\mathcal{F}$ be a crystal in
vector bundles over the big infinitesimal site $X/\Sigma_{e{\mathrm{INF}}}$.
Then infinitesimal cohomology of $\mathcal{F}$ over
$X/\Sigma_{e{\mathrm{INF}}}$ satisfies the éh-hyperdescent. Namely for an éh-
hypercovering $X_{\bullet}^{\prime}\rightarrow X^{\prime}$ of $K$-rigid spaces
over $X$, the following natural map is an isomorphism

$R\Gamma(X^{\prime}/\Sigma_{e{\mathrm{INF}}},\mathcal{F})\longrightarrow
R\lim_{\Delta^{\mathrm{op}}}\left(R\Gamma(X^{\prime}_{\bullet}/\Sigma_{e{\mathrm{INF}}},\mathcal{F})\right).$

###### Proof.

We prove the result by induction on $e$. For $e=1$, it is Theorem 6.2.2. In
general, we take the derived tensor product of short exact sequence
$0\rightarrow
K\rightarrow{\mathrm{B^{+}_{dR,e}}}\rightarrow\mathrm{B}_{\mathrm{dR},e-1}^{+}\rightarrow
0$ with the complex of sheaves $Ru_{X/\Sigma_{e}*}\mathcal{F}$. By the big
site version of the base change formula in Proposition 4.2.3 (cf. Corollary
2.2.8), we get a natural distinguished triangle

$Ru_{X/K*}\mathcal{F}_{1}\longrightarrow
Ru_{X/\Sigma_{e}*}\mathcal{F}\longrightarrow
Ru_{X/\Sigma_{e-1}*}\mathcal{F}_{e-1},$

where $\mathcal{F}_{1}$ and $\mathcal{F}_{e-1}$ are pullbacks of $\mathcal{F}$
along maps of sites $X/K_{{\mathrm{INF}}}\rightarrow
X/\Sigma_{e{\mathrm{INF}}}$ and $X/\Sigma_{e-1,{\mathrm{INF}}}\rightarrow
X/\Sigma_{e{\mathrm{INF}}}$ separately. In this way, applying the natural
transformation $R\Gamma(X^{\prime},-)\rightarrow
R\lim_{\Delta^{\mathrm{op}}}R\Gamma(X^{\prime}_{\bullet},-)$ to the above
triangle, we get the result by induction.

∎

###  6.3. Finiteness

With the use of the éh-hyperdescent, we show in this subsection the finiteness
and cohomological boundedness of infinitesimal cohomology, assuming the
properness of the rigid space

######  Theorem 6.3.1.

Let $X$ be a proper rigid space over $K$, and let $\mathcal{F}$ be a crystal
in vector bundles over the big infinitesimal site
$X/\Sigma_{e{\mathrm{INF}}}$. Then infinitesimal cohomology
$R\Gamma(X/\Sigma_{e{\mathrm{INF}}},\mathcal{F})$ is a bounded complex
supported in the cohomological degrees $[0,2n]$, where each cohomology is a
finite ${\mathrm{B^{+}_{dR,e}}}$-module.

###### Proof.

  * Smooth case

We first notice that when $X$ is smooth, infinitesimal cohomology
$R\Gamma(X/K_{{\mathrm{INF}}},\mathcal{F})$ is computed by the cohomology of
the de Rham complex
$\mathcal{F}\otimes_{\mathcal{O}_{X}}\Omega_{X/K}^{\bullet}$ via Theorem
4.2.2; namely we have

$R\Gamma(X/K_{{\mathrm{INF}}},\mathcal{F})\cong
R\Gamma(X,\mathcal{F}\otimes\Omega_{X/K}^{\bullet}).$

Moreover, each term $\mathcal{F}\otimes\Omega_{X/K}^{i}$ of the de Rham
complex is a coherent sheaf over $X$. Notice that the cohomology of a coherent
sheaf over a quasi-compact rigid space vanishes when the degree is above the
dimension ([dJvdP, Proposition 2.5.8]). Thus by the Hodge–de Rham spectral
sequence for $\mathcal{F}\otimes\Omega_{X/K}^{\bullet}$, we get the result for
$R\Gamma(X/K_{{\mathrm{INF}}},\mathcal{F})$ with smooth proper $X$.

In general, we use the base change formula in Proposition 4.2.3. By taking the
derived tensor product of $Ru_{X/\Sigma_{e}*}\mathcal{F}$ with the short exact
sequence $0\rightarrow
K\rightarrow{\mathrm{B^{+}_{dR,e}}}\rightarrow\mathrm{B}_{\mathrm{dR},e-1}\rightarrow
0$, we get a distinguished triangle

$R\Gamma(X/K_{{\mathrm{INF}}},\mathcal{F})\longrightarrow
R\Gamma(X/\Sigma_{e{\mathrm{INF}}},\mathcal{F})\longrightarrow
R\Gamma(X/\Sigma_{e-1{\mathrm{INF}}},\mathcal{F}).$

In this way, the claim for smooth proper $X$ follows from the induction on
$e$.

  * General case

In general, we prove by induction on the dimension of $X$. When $X$ is of
dimension zero, it is equal to a nilpotent extension of several points
$\mathrm{Spa}(K)$. So the result follows from the éh-hyperdescent along closed
immersions by reduced subspaces in Theorem 6.2.5 (in other words, we apply to
the Čech nerve at closed immersion of the reduced subspace).

Now assume $X$ is reduced of dimension $n$, and the claim is true for any
rigid space of smaller dimension. By the resolution of singularities of rigid
space in [Tem18], there exists a finite sequence of maps
$X_{m}\rightarrow\cdots X_{1}\rightarrow X_{0}=X$, where $X_{m}$ is smooth,
and each map $X_{i}\rightarrow X_{i-1}$ is a blowup at a closed analytic
subspace $Y_{i-1}$ of $X_{i-1}$, such that each $Y_{i-1}$ is nowhere dense in
$X_{i-1}$. We denote $E_{i}$ to be the exceptional locus
$Y_{i-1}\times_{X_{i-1}}X_{i}$ of the $i$-th blowup. We could then apply the
éh-hyperdescent in Theorem 6.2.5 to the Čech nerve associated with the blowup
covering $X_{i}\coprod Y_{i-1}\rightarrow X_{i-1}$. The limit
$R\lim_{\Delta^{\mathrm{op}}}$ of infinitesimal cohomology for the
hypercovering is isomorphic to the fiber of the blowup square

$R\Gamma(X_{i}/K_{{\mathrm{INF}}},\mathcal{F})\operatorname*{\text{\raisebox{1.07639pt}{\scalebox{0.8}{$\bigoplus$}}}}R\Gamma(Y_{i-1}/K_{{\mathrm{INF}}},\mathcal{F})\longrightarrow
R\Gamma(Y_{i}/K_{{\mathrm{INF}}},\mathcal{F}),$

and thus we get a long exact sequence

$\cdots\longrightarrow\mathrm{H}^{j}(X_{i-1}/K_{{\mathrm{INF}}},\mathcal{F})\longrightarrow\mathrm{H}^{j}(X_{i}/K_{{\mathrm{INF}}},\mathcal{F})\operatorname*{\text{\raisebox{1.07639pt}{\scalebox{0.8}{$\bigoplus$}}}}\mathrm{H}^{j}(Y_{i-1}/K_{{\mathrm{INF}}},\mathcal{F})\longrightarrow\mathrm{H}^{j}(Y_{i}/K_{{\mathrm{INF}}},\mathcal{F})\longrightarrow\cdots.$

In this way, with the help of the induction assumption for all $Y_{i}$, a
further descending induction from $X_{m}$ to $X_{0}=X$ finishes the proof.

∎

###  6.4. Algebraic and analytic infinitesimal cohomology

At the end of the section, we prove the comparison between the algebraic
infinitesimal cohomology and the analytic infinitesimal cohomology, for a
proper algebraic variety.

Recall that for an algebraic variety555For our purpose, a _variety_ is defined
to be a locally of finite type scheme over a field in the article.
$\mathscr{X}$ over a $p$-adic field $K$, we can define its _(algebraic)
infinitesimal site $\mathscr{X}/K_{\inf}$_, whose objects are schematic
infinitesimal thickenings $(\mathscr{U},\mathscr{T})$, where $\mathscr{U}$ is
an Zariski open subset of $\mathscr{X}$. The infinitesimal site
$\mathscr{X}/K_{\inf}$ is equipped with a structure sheaf
$\mathcal{O}_{\mathscr{X}/K}$, and its cohomology is called the _algebraic
infinitesimal cohomology_. Moreover, the infinitesimal structure sheaf admits
a surjection $\mathcal{O}_{\mathscr{X}/K}\rightarrow\mathcal{O}_{\mathscr{X}}$
to the Zariski structure sheaf, whose kernel $\mathcal{J}_{\mathscr{X}/K}$
defines a natural filtration on
$R\Gamma(\mathscr{X}/K_{\inf},\mathcal{O}_{\mathscr{X}/K})$. Similar to the
analytic theory, we call this filtration the _(algebraic) infinitesimal
filtration_.

Let $X=\mathscr{X}^{\mathrm{an}}$ be the rigid space over $K$ defined as the
analytification of a variety $\mathscr{X}$. As the analytification functor
${\mathrm{Sch}}_{K}\rightarrow{\mathrm{Rig}}_{K}$ preserves open and closed
immersions, it induces a natural map of ringed sites

$(X/K_{\inf},\mathcal{O}_{X/K})\longrightarrow(\mathscr{X}/K_{\inf},\mathcal{O}_{\mathscr{X}/K}).$

Moreover, as the surjection
$\mathcal{O}_{\mathscr{X}/K}\rightarrow\mathcal{O}_{\mathscr{X}}$ is
compatible with $\mathcal{O}_{X/K}\rightarrow\mathcal{O}_{X}$, the natural map
of infinitesimal structure sheaves above is then a filtered map. As a
consequence, by passing to their cohomology, we get a natural filtered
morphism in derived category

$R\Gamma(\mathscr{X}/K_{\inf},\mathcal{O}_{\mathscr{X}/K})\longrightarrow
R\Gamma(X/K_{\inf},\mathcal{O}_{X/K}).$

Our main result in this subsection is the following.

######  Theorem 6.4.1.

Let $\mathscr{X}$ be a proper algebraic variety over $K$, and let $X$ be its
analytifiation. Then the analytification functor induces a filtered
isomorphism of infinitesimal cohomology

$R\Gamma(\mathscr{X}/K_{\inf},\mathcal{O}_{\mathscr{X}/K})\longrightarrow
R\Gamma(X/K_{\inf},\mathcal{O}_{X/K}).$

Before the proof, we first recall that there is a natural map of ringed sites
$(X_{\mathrm{rig}},\mathcal{O}_{X})\rightarrow(\mathscr{X}_{{\mathrm{Zar}}},\mathcal{O}_{\mathscr{X}})$.
Here the rigid structure sheaf is flat over the Zariski structure sheaf, and
the pullback along the map induces a fully faithful functor from coherent
$\mathcal{O}_{\mathscr{X}}$-modules to coherent $\mathcal{O}_{X}$-modules.

Moreover, the above map of sites is compatible with the infinitesimal topos.
Recall that there exists a natural map of topoi

$\displaystyle u_{\mathscr{X}/K}:\mathrm{Sh}(\mathscr{X}/K_{\inf})$
$\displaystyle\longrightarrow\mathrm{Sh}(\mathscr{X}_{{\mathrm{Zar}}});$
$\displaystyle\mathcal{F}$
$\displaystyle\longmapsto(\mathscr{U}\mapsto\Gamma(\mathscr{U}/K_{\inf},\mathcal{F}|_{\mathscr{U}})).$

By construction, this functor is compatible with its rigid version
$u_{X/K}:\mathrm{Sh}(X/K_{\inf})\rightarrow\mathrm{Sh}(X_{\mathrm{rig}})$ (cf.
Subsection 2.3). Namely, the following diagram is commutative

$\textstyle{\mathrm{Sh}(X/K_{\inf})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{u_{X/K}}$$\textstyle{\mathrm{Sh}(\mathscr{X}/K_{\inf})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{u_{\mathscr{X}/K}}$$\textstyle{\mathrm{Sh}(X_{\mathrm{rig}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{Sh}(\mathscr{X}_{\mathrm{Zar}}).}$

We then claim the following result.

######  Proposition 6.4.2.

Let $\mathscr{X}$ be an algebraic variety over $K$, and let $X$ be its
analytifcation. Then the complex of coherent $\mathcal{O}_{X}$-modules
$Ru_{X/K*}(\mathcal{J}_{X/K}^{n}/\mathcal{J}_{X/K}^{n+1})$ is naturally
isomorphic to the analytification of the complex of coherent
$\mathcal{O}_{\mathscr{X}}$-modules
$Ru_{\mathscr{X}/K*}(\mathcal{J}_{\mathscr{X}/K}^{n}/\mathcal{J}_{\mathscr{X}/K}^{n+1})$.

###### Proof.

We denote the complex of coherent $\mathcal{O}_{\mathscr{X}}$-modules
$Ru_{\mathscr{X}/K*}(\mathcal{J}_{\mathscr{X}/K}^{n}/\mathcal{J}_{\mathscr{X}/K}^{n+1})$
by $C$, and $Ru_{X/K*}(\mathcal{J}_{X/K}^{n}/\mathcal{J}_{X/K}^{n+1})$ by
$C^{\prime}$. Then it suffices to show that the natural map below induced from
the pullback from $\mathscr{X}_{\mathrm{Zar}}$ to $X_{\mathrm{rig}}$ is an
isomorphism of complexes of $\mathcal{O}_{X}$-modules

$C\otimes_{\mathcal{O}_{\mathscr{X}}}\mathcal{O}_{X}\longrightarrow
C^{\prime}.$

As the result is a local statement for $\mathscr{X}$, let us assume
$\mathscr{X}=\operatorname{Spec}(A)$ is a finite type affine scheme over $K$
and $\mathscr{X}\rightarrow\mathscr{Y}=\operatorname{Spec}(A^{\prime})$ be a
closed immersion into an affine space over $K$. Moreover, notice that the
isomorphism could be checked locally on $X$, so we may take an open affinoid
disc of certain radius $\mathrm{Spa}(B^{\prime})$ in
$\mathscr{Y}^{\mathrm{an}}$, with the open subset
$X\cap\mathrm{Spa}(B^{\prime})=\mathrm{Spa}(B)$ in $X$. From our choices, we
get a cartesian diagram as below, where horizontal maps are surjective and
vertical maps are flat

$\textstyle{B^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B}$$\textstyle{A^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

So it suffices to show that

$R\Gamma(\operatorname{Spec}(A)/K_{\inf},\mathcal{J}_{\mathscr{X}/K}^{n}/\mathcal{J}_{\mathscr{X}/K}^{n+1})\otimes_{A}B\cong
R\Gamma(\mathrm{Spa}(B)/K_{\inf},\mathcal{J}_{X/K}^{n}/\mathcal{J}_{X/K}^{n+1}).$

We then recall from [BdJ, Section 2] that the algebraic infinitesimal
cohomology can be computed by the Čech-Alexander complex as below

$D\longrightarrow D(1)\longrightarrow D(2)\longrightarrow\cdots,$

where $D(m)$ is the formal completion of
$A^{\prime}(m):={A^{\prime}}^{\underset{K}{\otimes}m+1}$ along the surjection
$A^{\prime}(m)\rightarrow A$. We take the $n$-th graded piece for the
algebraic infinitesimal filtration, then the cohomology group
$R\Gamma(\operatorname{Spec}(A)/K_{\inf},\mathcal{J}_{\mathscr{X}/K}^{n}/\mathcal{J}_{\mathscr{X}/K}^{n+1})$
is isomorphic to the following map of $A$-linear cosimplicial complexes

$J_{D}^{n}/J_{D}^{n+1}\longrightarrow
J_{D(1)}^{n}/J_{D(n)}^{n+1}\longrightarrow
J_{D(2)}^{n}/J_{D(2)}^{n+1}\longrightarrow\cdots,$

where $J_{D(m)}$ is the kernel of the surjection $A^{\prime}(m)\rightarrow A$.
On the other hand, by the Čech-Alexander complex for rigid spaces in
Proposition 4.1.3, we have

$R\Gamma(\mathrm{Spa}(B)/K_{\inf},\mathcal{J}_{X/K}^{n}/\mathcal{J}_{X/K}^{n+1})\cong\left(J_{\mathcal{D}}^{n}/J_{\mathcal{D}}^{n+1}\longrightarrow
J_{\mathcal{D}(1)}^{n}/J_{\mathcal{D}(1)}^{n+1}\longrightarrow
J_{\mathcal{D}(2)}^{n}/J_{\mathcal{D}(2)}^{n+1}\longrightarrow\cdots\right),$

where $\mathcal{D}(m)$ is the formal completion for the surjection
$B^{\prime}(m):={B^{\prime}}^{\underset{K}{\widehat{\otimes}}m+1}\rightarrow
B$, and $J_{\mathcal{D}(m)}$ is the kernel of the map
$B^{\prime}(m)\rightarrow B$. Thus we are left to show the quasi-isomorphism
for the canonical map of $B$-linear cosimplicial complexes below

$\left(J_{D}^{n}/J_{D}^{n+1}\longrightarrow
J_{D(1)}^{n}/J_{D(n)}^{n+1}\longrightarrow
J_{D(2)}^{n}/J_{D(2)}^{n+1}\longrightarrow\cdots\right)\otimes_{A}B\longrightarrow\left(J_{\mathcal{D}}^{n}/J_{\mathcal{D}}^{n+1}\longrightarrow
J_{\mathcal{D}(1)}^{n}/J_{\mathcal{D}(1)}^{n+1}\longrightarrow
J_{\mathcal{D}(2)}^{n}/J_{\mathcal{D}(2)}^{n+1}\longrightarrow\cdots\right).$

At last notice that by our choices, the rigid space $\mathrm{Spa}(B^{\prime})$
is an open disc of some radius in the affine space
$\operatorname{Spec}(A^{\prime})^{\mathrm{an}}$. In particular, the following
map of rings is a cartesian diagram such that vertical maps are flat

$\textstyle{B^{\prime}(m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B^{\prime}}$$\textstyle{A^{\prime}(m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

In this way, combining this with the cartesian diagram in the first paragraph,
we see the kernel $J_{\mathcal{D}(m)}$ of the surjection
$B^{\prime}(m)\rightarrow B$ is equal to the base change of $J_{D(m)}$ along
the flat map $A^{\prime}(m)\rightarrow B^{\prime}(m)$. Hence we get the
natural equalities

$\displaystyle
J_{D(m)}B^{\prime}(m)=J_{D(m)}\otimes_{A^{\prime}(m)}B^{\prime}(m)=J_{\mathcal{D}(m)};$
$\displaystyle(J_{D(m)}^{n}/J_{D(m)}^{n+1})\otimes_{A}B=(J_{D(m)}^{n}/J_{D(m)}^{n+1})\otimes_{A^{\prime}(m)}B^{\prime}(m)=J_{\mathcal{D}(m)}^{n}/J_{\mathcal{D}(m)}^{n+1}.$

So we are done.

∎

At last, we finish the proof of Theorem 6.4.1.

###### Proof of Theorem 6.4.1.

To show the natural map in the statement is a filtered isomorphism, it
suffices to show the isomorphisms for their underlying complexes and each
graded pieces separately, as both of them are filtered complete.

For the underlying complexes, this follows from the éh descent. To see this,
we first notice that when $\mathscr{X}$ is smooth and proper over $K$, then
the algebraic and analytic infinitesimal cohomology are isomorphic to the
algebraic and analytic de Rham cohomology separately ([Gr68], Theorem 4.2.2),
which are isomorphic to each other by applying the GAGA theorem to their
Hodge-filtrations (cf. [Con06, Appendix A.1]). In general, we may assume
$\mathscr{X}_{\bullet}\rightarrow\mathscr{X}$ is a simplicial smooth varieties
by resolving singularities. Then its analytification $X_{\bullet}\rightarrow
X$ is an éh-hypercovering by smooth rigid spaces, and we get the isomorphism

$\displaystyle R\Gamma(\mathscr{X}/K_{\inf},\mathcal{O}_{\mathscr{X}/K})$
$\displaystyle\cong
R\lim_{[n]\in\Delta^{\mathrm{op}}}R\Gamma(\mathscr{X}_{n}/K_{\inf},\mathcal{O}_{\mathscr{X}_{n}/K})$
$\displaystyle\cong
R\lim_{[n]\in\Delta^{\mathrm{op}}}R\Gamma(X_{n}/K_{\inf},\mathcal{O}_{X_{n}/K})$
$\displaystyle\cong R\Gamma(X/K_{\inf},\mathcal{O}_{X/K}),$

where the first equality is the h-hyperdescent of algebraic de Rham cohomology
for blowups in [Har75], and the last is the éh-hyperdescent for analytic
infinitesimal cohomology in 6.2.2.

For the graded pieces, by Proposition 6.4.2 we have

$R\Gamma(X/K_{\inf},\mathcal{J}_{X/K}^{n}/\mathcal{J}_{X/K}^{n+1})=R\Gamma(X_{\mathrm{rig}},Ru_{X/K*}(\mathcal{J}_{X/K}^{n}/\mathcal{J}_{X/K}^{n+1}))\cong
R\Gamma(X_{\mathrm{rig}},\left(Ru_{\mathscr{X}/K*}(\mathcal{J}_{\mathscr{X}/K}^{n}/\mathcal{J}_{\mathscr{X}/K}^{n+1})\right)^{\mathrm{an}}).$

We denote $C$ to be the bounded below complex of coherent
$\mathcal{O}_{X}$-modules
$Ru_{\mathscr{X}/K*}(\mathcal{J}_{\mathscr{X}/K}^{n}/\mathcal{J}_{\mathscr{X}/K}^{n+1})$.
As $R\Gamma(X_{\mathrm{Zar}},\tau^{>n}C)$ lives in cohomogical degree larger
than $n$, we have the natural equalities

$R\Gamma(X_{\mathrm{Zar}},C)={\mathrm{colim}}_{n}R\Gamma(X_{\mathrm{Zar}},\tau^{\leq
n}C).$

Similarly we have

$R\Gamma(X_{\mathrm{rig}},C^{\mathrm{an}})={\mathrm{colim}}_{n}R\Gamma(X_{\mathrm{rig}},\tau^{\leq
n}(C^{\mathrm{an}})).$

On the other hand, as the rigid structure sheaf $\mathcal{O}_{X}$ is flat over
$\mathcal{O}_{\mathscr{X}}$, the analytification functor
$(-)^{\mathrm{an}}=-\otimes_{\mathcal{O}_{\mathscr{X}}}\mathcal{O}_{X}$ on
coherent complexes is an exact functor. So for each $n\in\mathbb{N}$, there
exists a natural equality

$\tau^{\leq n}C^{\mathrm{an}}=(\tau^{\leq n}C)^{\mathrm{an}}.$

Notice that for each bounded complex $\tau^{\leq n}C$ of coherent sheaves, by
rigid GAGA theorem ([Con06, Appendix A.1]) we have

$R\Gamma(X_{\mathrm{Zar}},\tau^{\leq n}C)\cong
R\Gamma(X_{\mathrm{rig}},(\tau^{\leq n}C)^{\mathrm{an}}).$

In this way, combining all of the isomorphisms above, we get

$\displaystyle R\Gamma(X_{\mathrm{Zar}},C)$
$\displaystyle\cong{\mathrm{colim}}_{n}R\Gamma(X_{\mathrm{Zar}},\tau^{\leq
n}C)$
$\displaystyle\cong{\mathrm{colim}}_{n}R\Gamma(X_{\mathrm{rig}},(\tau^{\leq
n}C)^{\mathrm{an}})$
$\displaystyle={\mathrm{colim}}_{n}R\Gamma(X_{\mathrm{rig}},\tau^{\leq
n}(C^{\mathrm{an}}))$ $\displaystyle\cong
R\Gamma(X_{\mathrm{rig}},C^{\mathrm{an}}).$

At last, substituting back the definition of $C$ and Proposition 6.4.2, we
then obtain the formula for graded piece of infinitesimal filtrations:

$R\Gamma(\mathscr{X}/K_{\inf},\mathcal{J}_{\mathscr{X}/K}^{n}/\mathcal{J}_{\mathscr{X}/K}^{n+1})\cong
R\Gamma(X/K_{\inf},\mathcal{J}_{X/K}^{n}/\mathcal{J}_{X/K}^{n+1}).$

∎

As an application, we get the comparison with singular cohomology when $K$ is
abstractly isomorphic to the field of complex numbers, proving Theorem 1.2.1
(v).

######  Corollary 6.4.3.

Assume there exists an abstract isomorphism of fields
$K\rightarrow\mathbb{C}$. Then for any proper algebraic variety
$\mathscr{X}/K$ with its analytification $X$, there exists a filtered
isomorphism of cohomology

$\mathrm{H}^{i}(X/K_{\inf},\mathcal{O}_{X/K})\cong\mathrm{H}^{i}_{\textrm{Sing}}(\mathscr{X}(\mathbb{C}),\mathbb{C}),$

where singular cohomology of $\mathscr{X}(\mathbb{C})$ is filtered by the
algebraic infinitesimal filtration.

###### Proof.

This follows from Theorem 6.4.1 and the classical result of Hartshorne in
[Har75]. ∎

Using the same idea of the proof for Proposition 6.4.2 and Theorem 6.4.1, we
can prove the base extension formula for infinitesimal cohomology.

######  Corollary 6.4.4.

Let $K_{0}$ be a complete $p$-adic extension of $\mathbb{Q}_{p}$, and let $K$
be a complete extension of $K_{0}$. Assume $X$ is a proper rigid space over
$K_{0}$, and let $\mathcal{F}$ be a coherent crystal over $X/K_{0,\inf}$. Then
the following natural map of filtered complexes is an isomorphism

$R\Gamma(X/K_{0,\inf},\mathcal{F})\otimes_{K_{0}}K\longrightarrow
R\Gamma(X_{K}/K_{\inf},\mathcal{F}_{K}).$

##  7\. Cohomology over ${\mathrm{B_{dR}^{+}}}$

In this section, we extend previous results to the infinitesimal site over
${\mathrm{B_{dR}^{+}}}$, for a rigid space $X$ over $\Sigma_{r}$ for some
fixed $r\in\mathbb{N}$. Our goal is to show Theorem 1.2.7 from the
introduction.

###  7.1. Infinitesimal sites and topoi over ${\mathrm{B_{dR}^{+}}}$

We fix a complete algebraic closed $p$-adic field $K$. Let $X$ be a rigid
space over ${\mathrm{B_{dR}^{+}}}/\xi^{r}$ for some fixed $r\in\mathbb{N}$. To
build an infinitesimal cohomology theory with the coefficient being
${\mathrm{B_{dR}^{+}}}=\varprojlim_{e\in\mathbb{N}}{\mathrm{B^{+}_{dR,e}}}$,
we construct an infinitesimal site $X/\Sigma_{\inf}$ as a union of all
$X/\Sigma_{e\inf}$ for $e\in\mathbb{N}_{\geq r}$, and consider its relation to
each infinitesimal site $X/\Sigma_{e\inf}$.

The site $X/\Sigma_{\inf}$.

We first give the definition of the infinitesimal site over
$\Sigma=\varinjlim_{e\in\mathbb{N}}\Sigma_{e}$.

######  Definition 7.1.1.

Let $X$ be a rigid space over
$\Sigma_{r}=\mathrm{Spa}({\mathrm{B_{dR}^{+}}}/\xi^{r})$, for some fixed
$r\in\mathbb{N}$. The _infinitesimal site $X/\Sigma_{\inf}$ over
${\mathrm{B_{dR}^{+}}}$_ is defined as follows:

  * •

The underlying category of $X/\Sigma_{\inf}$ is the category of pairs $(U,T)$,
for $(U,T)$ being an thickening in $X/\Sigma_{e\inf}$ for some $e\geq r$.

A morphism between $(U_{1},T_{1})$ and $(U_{2},T_{2})$ is a morphism of
objects in $X/\Sigma_{e\inf}$, for $e$ large enough such that both pairs are
objects in $X/\Sigma_{e\inf}$.

  * •

A collection of morphism $(U_{i},T_{i})\rightarrow(U,T)$ in $X/\Sigma_{\inf}$
is a covering if $\\{T_{i}\rightarrow T\\}$ is an open covering for the rigid
space $T$.

As a category, $X/\Sigma_{\inf}$ is the union of $X/\Sigma_{e\inf}$ for all
$e\geq r$. It is clear that the topology is locally rigid over each object in
$X/\Sigma_{\inf}$. Thus the description of a sheaf over $X/\Sigma_{\inf}$ is
similar to that of a sheaf over $X/\Sigma_{e\inf}$ as in Section 2.

######  Remark 7.1.2.

Similarly to the Discussion in Section 2, we could define the big version
infinitesimal site $X/\Sigma_{{\mathrm{INF}}}$, where the objects are
infinitesimal thickenings $(U,T)$ for $U$ being a rigid space over $X$ and
$U\rightarrow T$ a nil-extension over ${\mathrm{B_{dR}^{+}}}$. The relation
between the big infinitesimal sites $X/\Sigma_{{\mathrm{INF}}}$ and the small
one $X/\Sigma_{\inf}$, including the constructions in the rest of the
subsection, are exactly identical to the case over ${\mathrm{B^{+}_{dR,e}}}$
in Paragraph 2.1, and we will not duplicate again here.

Functoriality of $\mathrm{Sh}(X/\Sigma_{\inf})$.

The infinitesimal topos $\mathrm{Sh}(X/\Sigma_{\inf})$ is functorial with
respect to the rigid space $X$. Namely, for a map of
${\mathrm{B_{dR}^{+}}}$-rigid space $f:X\rightarrow Y$ where $\xi$ is
nilpotent, we have a natural map of topoi

$f_{\inf}:\mathrm{Sh}(X/\Sigma_{\inf})\longrightarrow\mathrm{Sh}(Y/\Sigma_{\inf}).$

The corresponding adjoint pair of functors are given by the following:

  * •

For a sheaf $\mathcal{G}\in\mathrm{Sh}(Y/\Sigma_{\inf})$, the inverse image
$f_{\inf}^{-1}\mathcal{G}$ is given by the restriction of
$\mu_{Y}^{-1}\mathcal{G}$ to the category $X/\Sigma_{\inf}$ along the map $f$,
and is equal to the sheaf associated with the presheaf

$X/\Sigma_{\inf}\ni(U,T)\longmapsto\varinjlim_{\begin{subarray}{c}(U,T)\rightarrow(V,S)\\\
(V,S)\in Y/\Sigma_{\inf},\\\ U\rightarrow
V~{}compatible~{}with~{}f\end{subarray}}\mathcal{G}(V,S).$

  * •

The direct image functor $f_{\inf*}$ sends a sheaf
$\mathcal{F}\in\mathrm{Sh}(X/\Sigma_{\inf})$ to the sheaf

$f_{\inf*}\mathcal{F}(V,S)=\varprojlim_{\begin{subarray}{c}(U,T)\rightarrow(V,S)\\\
(U,T)\in X/\Sigma_{{\mathrm{INF}}}\\\ U\rightarrow
V~{}compatible~{}with~{}f\end{subarray}}\mathcal{F}(U,T).$

We want to remind the reader that the construction of those two functors are
identical with the construction of the functoriality morphism
$\mathrm{Sh}(X/\Sigma_{e\inf})\rightarrow\mathrm{Sh}(Y/\Sigma_{e^{\prime}\inf})$
for the map of rigid spaces

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Sigma_{e}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Sigma_{e^{\prime}},}$

as in Subsection 2.4.

Relation with $X/\Sigma_{e\inf}$. Topologically, the infinitesimal site
$X/\Sigma_{\inf}$ is the limit of $X/\Sigma_{e\inf}$ for $e\geq r$. To make
this precise, we consider the following morphism of sites:

$u_{e}:X/\Sigma_{\inf}\longrightarrow X/\Sigma_{e\inf},$

whose corresponding functor is the canonical inclusion functor that sends
$(U,T)\in X/\Sigma_{e\inf}$ to the object $(U,T)\in X/\Sigma_{\inf}$. Note
that by construction, this cocontinuous functor is a fully faithful embedding.

This morphism induces an adjoint pair of functors $(u_{e}^{-1},u_{e*})$ given
as follows:

  * •

The functor $u_{e*}$ is the restriction functor, in a way that for a sheaf
$\mathcal{F}\in\mathrm{Sh}(X/\Sigma_{\inf})$ we have

$(u_{e*}\mathcal{F})_{T}=\mathcal{F}_{T}.$

  * •

For a sheaf $\mathcal{G}\in\mathrm{Sh}(X/\Sigma_{e\inf})$, the sheaf
$u_{e}^{-1}\mathcal{G}$ is the sheaf associated with the presheaf

$\displaystyle(V,S)$
$\displaystyle\mapsto\varinjlim_{\begin{subarray}{c}(V,S)\rightarrow(U,T)\\\
(U,T)\in X/\Sigma_{e\inf}\end{subarray}}\mathcal{G}(U,T)$
$\displaystyle=\begin{cases}\varnothing,~{}S\notin{\mathrm{Rig}}_{\Sigma_{e}};\\\
\mathcal{G}(V,S),~{}S\in{\mathrm{Rig}}_{\Sigma_{e}}.\end{cases}$

So by the definition of the site $X/\Sigma_{\inf}$, the restriction of
$u_{e}^{-1}\mathcal{G}$ at $(V,S)$ is

$(u_{e}^{-1}\mathcal{G})_{S}=\begin{cases}\varnothing,~{}S\notin{\mathrm{Rig}}_{\Sigma_{e}};\\\
\mathcal{G}_{S},~{}S\in{\mathrm{Rig}}_{\Sigma_{e}}.\end{cases}$

Here we notice that when $\mathcal{G}=h_{(U,T)}$ is the representable sheaf
for some object $(U,T)\in X/\Sigma_{e\inf}$, the inverse image
$u_{e}^{-1}h_{(U,T)}$ is nothing but the representable sheaf $h_{(U,T)}$ in
$\mathrm{Sh}(X/\Sigma_{\inf})$.

The morphism of site $u_{e}:X/\Sigma_{\inf}\rightarrow X/\Sigma_{e\inf}$
induces a map of topoi

$u_{e}:\mathrm{Sh}(X/\Sigma_{\inf})\longrightarrow\mathrm{Sh}(X/\Sigma_{e\inf}).$

It admits a section
$i_{e}:\mathrm{Sh}(X/\Sigma_{e\inf})\longrightarrow\mathrm{Sh}(X/\Sigma_{\inf})$,
where the corresponding adjoint pair of functors is given as follows:

  * •

For a sheaf $\mathcal{G}\in\mathrm{Sh}(X/\Sigma_{inf})$, the inverse image
$i_{e}^{-1}\mathcal{G}$ is the sheaf associated with the presheaf

$X/\Sigma_{e\inf}\ni(U,T)\longmapsto\varinjlim_{\begin{subarray}{c}(U,T)\rightarrow(U,S)\\\
(U,S)\in X/\Sigma_{\inf}\end{subarray}}\mathcal{G}(U,S)=\mathcal{G}(U,T).$

Namely, $i_{e}^{-1}=u_{e*}$ is the restriction functor.

  * •

The direct image functor $i_{e*}$ sends a sheaf
$\mathcal{F}\in\mathrm{Sh}(X/\Sigma_{e\inf})$ to the sheaf

$i_{e*}\mathcal{F}(V,S)=\varprojlim_{\begin{subarray}{c}(V,T)\rightarrow(V,S)\\\
(V,T)\in
X/\Sigma_{e\inf}\end{subarray}}\mathcal{F}(V,T)=\mathcal{F}(V,S\times_{\Sigma}\Sigma_{e}).$

It is clear that the composition $u_{e}\circ i_{e}$ is equal to the identity.
We also note that the above functors are functorial with respect to $e$.

######  Remark 7.1.3.

Here we notice that the map $i_{e}$ is in fact induced from a natural map of
sites

$\displaystyle i_{e}:X/\Sigma_{e\inf}$ $\displaystyle\longrightarrow
X/\Sigma_{\inf};$ $\displaystyle(U,T\times_{\Sigma}\Sigma_{e})$
$\displaystyle\reflectbox{$\longmapsto$}(U,T).$

This is analogous to the nilpotent bases situation, as in the Remark 2.4.3

######  Remark 7.1.4.

We also want to reminder the reader that the construction of map $i_{e}$ could
be regarded as the functoriality morphism of infinitesimal topoi associated
with the following diagram

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{id}}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Sigma_{e}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Sigma.}$

######  Remark 7.1.5.

The construction of $u_{e}$ and $i_{e}$ is compatible with the functoriality
morphism of infinitesimal topoi
$f_{\inf}:\mathrm{Sh}(X/\Sigma_{e\inf})\rightarrow\mathrm{Sh}(Y/\Sigma_{e^{\prime}\inf})$
for a map of rigid spaces $f:X/\Sigma_{e}\rightarrow Y/\Sigma_{e^{\prime}}$.
Namely we have the following commutative diagrams among infinitesimal topoi

$\textstyle{\mathrm{Sh}(X/\Sigma_{\inf})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{u_{e}}$$\scriptstyle{f_{\inf}}$$\textstyle{\mathrm{Sh}(X/\Sigma_{e\inf})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{\inf}}$$\textstyle{\mathrm{Sh}(Y/\Sigma_{\inf})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{u_{e^{\prime}}}$$\textstyle{\mathrm{Sh}(Y/\Sigma_{e^{\prime}\inf});}$
$\textstyle{\mathrm{Sh}(X/\Sigma_{e\inf})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{e}}$$\scriptstyle{f_{\inf}}$$\textstyle{\mathrm{Sh}(X/\Sigma_{\inf})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{\inf}}$$\textstyle{\mathrm{Sh}(Y/\Sigma_{e^{\prime}\inf})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{e^{\prime}}}$$\textstyle{\mathrm{Sh}(Y/\Sigma_{\inf}).}$

Relation to the rigid topos $\mathrm{Sh}(X_{\mathrm{rig}})$. Analogous to
Subsection 2.3, there exists a natural map of topoi to the rigid site
$X_{\mathrm{rig}}$ as below

$u_{X/\Sigma}:\mathrm{Sh}(X/\Sigma_{\inf})\longrightarrow\mathrm{Sh}(X_{\mathrm{rig}}).$

The corresponding preimage and the direct image functors are given as below

  * •

$\displaystyle u_{X/\Sigma*}:\mathrm{Sh}(X/\Sigma_{\inf})$
$\displaystyle\longrightarrow\mathrm{Sh}(X_{\mathrm{rig}});$
$\displaystyle\mathcal{F}$
$\displaystyle\longmapsto(U\mapsto\Gamma(U/\Sigma_{\inf},\mathcal{F})).$

  * •

$\displaystyle u_{X/\Sigma}^{-1}:\mathrm{Sh}(X_{\mathrm{rig}})$
$\displaystyle\longrightarrow\mathrm{Sh}(X/\Sigma_{\inf});$
$\displaystyle\mathcal{E}$
$\displaystyle\longmapsto((U,T)\mapsto\mathcal{E}(U)).$

Namely the push-forward functor $u_{X/\Sigma*}$ is the sheafified version of
the infinitesimal global section functor.

######  Remark 7.1.6.

The functor $u_{X/\Sigma}$ is functorial with respect to the rigid space $X$.
Precisely, given a map of rigid spaces $f:X\rightarrow Y$ over
${\mathrm{B_{dR}^{+}}}$ where $\xi$ is nilpotent, we have the following
commutative diagram

$\textstyle{\mathrm{Sh}(X/\Sigma_{\inf})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{u_{X/\Sigma}}$$\scriptstyle{f_{\inf}}$$\textstyle{\mathrm{Sh}(X_{\mathrm{rig}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{\mathrm{Sh}(Y/\Sigma_{\inf})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{u_{Y/\Sigma}}$$\textstyle{\mathrm{Sh}(Y_{\mathrm{rig}}).}$

######  Remark 7.1.7.

The functor $u_{X/\Sigma}$ is also compatible with
$u_{e}:\mathrm{Sh}(X/\Sigma_{\inf})\rightarrow\mathrm{Sh}(X/\Sigma_{e\inf})$
and
$i_{e}:\mathrm{Sh}(X/\Sigma_{e\inf})\rightarrow\mathrm{Sh}(X/\Sigma_{\inf})$.
Namely, the following diagrams commute:

$\textstyle{\mathrm{Sh}(X/\Sigma_{\inf})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{u_{e}}$$\scriptstyle{u_{X/\Sigma}}$$\textstyle{\mathrm{Sh}(X_{\mathrm{rig}})}$$\textstyle{\mathrm{Sh}(X/\Sigma_{e\inf})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{u_{X/\Sigma_{e}}}$;
$\textstyle{\mathrm{Sh}(X/\Sigma_{\inf})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{u_{X/\Sigma}}$$\textstyle{\mathrm{Sh}(X_{\mathrm{rig}})}$$\textstyle{\mathrm{Sh}(X/\Sigma_{e\inf})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{e}}$$\scriptstyle{u_{X/\Sigma_{e}}}$.

Here
$u_{X/\Sigma_{e}}:\mathrm{Sh}(X/\Sigma_{e\inf})\rightarrow\mathrm{Sh}(X_{\mathrm{rig}})$
is the analogous functor of $u_{X/\Sigma}$ onto the rigid site defined in
Subsection 2.3.

###  7.2. Cohomology of crystals over $X/\Sigma_{\inf}$

In this section, we consider the cohomology of a crystal $\mathcal{F}$ over
the infinitesimal site $X/\Sigma_{\inf}$. Our strategy is to interpret the
cohomology of $\mathcal{F}$ as the derived inverse limit of the cohomology of
the pullback $i_{e}^{*}\mathcal{F}$, where $i_{e}^{*}\mathcal{F}$ is a crystal
over the site $X/\Sigma_{e\inf}$.

To start with, we first describe a crystal over the infinitesimal site
$X/\Sigma_{\inf}$.

######  Definition 7.2.1.

Let $X$ be a rigid space over $\Sigma_{\inf}$ where $\xi$ is nilpotent.

  1. (i)

The _infinitesimal structure sheaf_ over $X/\Sigma_{\inf}$, denoted as
$\mathcal{O}_{X/\Sigma}$, is a sheaf over $X/\Sigma_{\inf}$ sending a
thickening $(U,T)\in X/\Sigma_{\inf}$ onto the global section of
$\mathcal{O}_{T}$ at $T$ as below

$\mathcal{O}_{X/\Sigma}:(U,T)\longmapsto\mathcal{O}_{T}(T).$

  2. (ii)

A _coherent crystal_ over $X/\Sigma_{\inf}$ is a
$\mathcal{O}_{X/\Sigma}$-coherent sheaf $\mathcal{F}$ over $X/\Sigma_{\inf}$
satisfies the crystal condition as in Definition 3.1.2. It is called _a
crystal in vector bundle_ if the restriction $\mathcal{F}_{T}$ at each
infinitesimal thickening $(U,T)\in X/\Sigma_{\inf}$ is a vector bundle over
$\mathcal{O}_{T}$.

Here we mention that similar to Proposition 3.1.7, it can be shown the
categories crystals over big and small sites are equivalent.

We notice that the morphism of sites $i_{e}:X/\Sigma_{e\inf}\rightarrow
X/\Sigma_{\inf}$ in the last subsection is naturally a morphism of ringed
sites for their structure sheaves. Moreover, since the preimage functor
$i_{e}^{-1}$ is equal to the restriction functor onto the subcategory
$X/\Sigma_{e\inf}$, we get

$i_{e}^{-1}\mathcal{O}_{X/\Sigma}=\mathcal{O}_{X/\Sigma_{e}}.$

So we can define the _pullback functor_
$i_{e}^{*}\mathcal{F}:=i_{e}^{-1}\mathcal{F}\otimes_{i_{e}^{-1}\mathcal{O}_{X/\Sigma}}\mathcal{O}_{X/\Sigma_{e}}$,
which is the same as the restriction functor $i_{e}^{-1}\mathcal{F}$ itself;
namely for an infinitesimal thickening $(U,T)\in X/\Sigma_{e\inf}$, we have

$(i_{e}^{*}\mathcal{F})_{T}=(i_{e}^{-1}\mathcal{F})_{T}=\mathcal{F}_{T}.$

Here we want to remark that the pullback functor $i_{e}^{*}=i_{e}^{-1}$ is
compatible with the pullback functor $f_{\inf}^{*}$ of the morphism
$f_{\inf}:\mathrm{Sh}(X/\Sigma_{e\inf})\rightarrow\mathrm{Sh}(Y/\Sigma_{e^{\prime}\inf})$
for a map of rigid spaces $f:X/\Sigma_{e}\rightarrow Y/\Sigma_{e^{\prime}}$.

The main tool of the subsection is the following lemma, relating a coherent
crystal over $X/\Sigma_{\inf}$ with those over $X/\Sigma_{e\inf}$ of
$\xi$-nilpotent coefficients.

######  Lemma 7.2.2.

Let $\mathcal{F}$ be a coherent crystal over the infinitesimal site
$X/\Sigma_{\inf}$ (resp. $X/\Sigma_{{\mathrm{INF}}}$), and let $X$ be defined
over $\Sigma_{r}$ for some $r\in\mathbb{N}$. Then we have the following.

  1. (i)

The pullback $i_{e}^{*}\mathcal{F}$ for each $e\in\mathbb{N}_{\geq r}$ is a
crystal over $X/\Sigma_{e\inf}$. When $\mathcal{F}$ is a crystal in vector
bundles, so is $\mathcal{F}$ over $X/\Sigma_{e\inf}$.

  2. (ii)

The counit map for the adjoint pairs $(i_{e}^{*},i_{e*})$ induces the
following isomorphism

$\mathcal{F}/\xi^{e}\longrightarrow Ri_{e*}i_{e}^{*}\mathcal{F}.$

In parituclar, we have the natural equivalences as below

$\mathcal{F}\longrightarrow R\varprojlim_{e\geq
r}\mathcal{F}/\xi^{e}\longrightarrow R\varprojlim_{e\geq
r}Ri_{e*}i_{e}^{*}\mathcal{F}.$

Here the transition maps in the last limit are given by the map of
infinitesimal sites $X/\Sigma_{e\inf}\rightarrow X/\Sigma_{e+1\inf}$ (resp.
$X/\Sigma_{e{\mathrm{INF}}}\rightarrow X/\Sigma_{e+1{\mathrm{INF}}}$) for the
closed immersions of bases.

###### Proof.

  1. (i)

The proof of the (i) follows from the definition of the crystal condition.

  2. (ii)

We recall from the last subsection that the push-forward functor
$i_{e*}\mathcal{G}$ is given by

$(i_{e*}\mathcal{G})(U,T)=\mathcal{G}(U,T\times_{\Sigma}\Sigma_{e}),$

for a sheaf $\mathcal{G}\in\mathrm{Sh}(X/\Sigma_{e\inf})$. We denote the fiber
product $T\times_{\Sigma}\Sigma_{e}$ by $T_{e}$, which is an infinitesimal
thickening of $U$ that is defined over $\Sigma_{e}$. Apply the above to the
pullback $\mathcal{G}=i_{e}^{*}\mathcal{F}$ of the crystal $\mathcal{F}$, and
notice that $i_{e}^{*}$ is the restriction functor, we get

$\displaystyle(Ri_{e*}i_{e}^{*}\mathcal{F})(U,T)$
$\displaystyle=R\Gamma((U,T_{e}),\mathcal{F})$
$\displaystyle=R\Gamma(T_{e},\mathcal{F}_{T_{e}})$
$\displaystyle=R\Gamma(T_{e},\mathcal{F}_{T}/\xi^{e})$
$\displaystyle=R\Gamma(T,\mathcal{F}_{T}/\xi^{e}),$

where the last equality follows from the observation that $T_{e}\rightarrow T$
has the same underlying topological spaces. Hence the cone of
$\mathcal{F}/\xi^{e}\rightarrow Ri_{e*}i_{e}^{*}\mathcal{F}$, which is bounded
below and has no cohomology, vanishes in the derived category.

At last, notice that for a coherent sheaf $\mathcal{F}$ of
$\mathcal{O}_{X/\Sigma}$-modules over $X/\Sigma_{\inf}$, we always have

$\mathcal{F}\cong R\varprojlim_{e\geq
r}\mathcal{F}/\xi^{e}\cong\varprojlim_{e\geq r}\mathcal{F}/\xi^{e}.$

So the last claim in $(ii)$ follows.

∎

Now we are able to give the main result about the cohomology of crystals over
the infinitesimal site $X/\Sigma_{\inf}$.

######  Theorem 7.2.3.

Let $X$ be a rigid space over some $\Sigma_{r}$, and let $\mathcal{F}$ be a
coherent crystal over $X/\Sigma_{\inf}$.

  1. (i)

There exists a natural isomorphism of complexes of sheaves of
${\mathrm{B_{dR}^{+}}}$-modules as below:

$Ru_{X/\Sigma*}\mathcal{F}\longrightarrow R\varprojlim_{e\geq
r}Ru_{X/\Sigma_{e}*}(i_{e}^{*}\mathcal{F}).$

In particular, by applying the derived global section functor, we get

$R\Gamma(X/\Sigma_{\inf},\mathcal{F})\cong R\varprojlim_{e\geq
r}R\Gamma(X/\Sigma_{e\inf},i_{e}^{*}\mathcal{F}).$

  2. (ii)

Let $\\{Y_{e}\\}_{e\geq r}$ be a direct system of rigid spaces over
$\Sigma_{e}$, such that each $Y_{e}$ is smooth over $\Sigma_{e}$ with
$Y_{e+1}\times_{\Sigma_{e+1}}\Sigma_{e}\cong Y_{e}$. Assume $X$ admits a
closed immersions into $Y_{r}$. Then we have natural isomorphisms of complexes
of sheaves of ${\mathrm{B_{dR}^{+}}}$-modules as below

$Ru_{X/\Sigma*}\mathcal{F}\longrightarrow\mathcal{F}_{D}\otimes\Omega_{D}^{\bullet}\cong
R\varprojlim_{e\geq
r}(\mathcal{F}_{D_{X}(Y_{e})}\otimes\Omega_{D_{X}(Y_{e})}^{\bullet}),$

where $D=\varinjlim_{e\geq r}D_{X}(Y_{e})$ is the colimit of envelopes, and
$\mathcal{F}_{D}\otimes\Omega_{D}^{\bullet}$ is the de Rham complex of
$\mathcal{F}$ over $D$.

  3. (iii)

Suppose $\mathcal{F}$ is a crystal in vector bundles over $X/\Sigma_{\inf}$.
We equip the rings ${\mathrm{B_{dR}^{+}}}$ and ${\mathrm{B^{+}_{dR,e}}}$ with
their $\xi$-adic filtrations. Then for each $e\geq r$, the natural maps below
are filtered isomorphisms

$\displaystyle(Ru_{X/\Sigma*}\mathcal{F})\otimes_{{\mathrm{B_{dR}^{+}}}}^{L}{\mathrm{B^{+}_{dR,e}}}$
$\displaystyle\longrightarrow Ru_{X/\Sigma_{e}*}(i_{e}^{*}\mathcal{F});$
$\displaystyle Ru_{X/\Sigma*}\mathcal{F}$ $\displaystyle\longrightarrow
R\varprojlim_{e}\left((Ru_{X/\Sigma*}\mathcal{F})\otimes_{{\mathrm{B_{dR}^{+}}}}^{L}{\mathrm{B^{+}_{dR,e}}}\right).$

In particular, when $X$ is quasi-compact quasi-separated, by applying the
derived global section functor we obtain the following canonical filtered
equivalences

$\displaystyle
R\Gamma(X/\Sigma_{\inf},\mathcal{F})\otimes_{{\mathrm{B_{dR}^{+}}}}^{L}{\mathrm{B^{+}_{dR,e}}}$
$\displaystyle\cong R\Gamma(X/\Sigma_{e\inf},i_{e}^{*}\mathcal{F});$
$\displaystyle R\Gamma(X/\Sigma_{\inf},\mathcal{F})$ $\displaystyle\cong
R\varprojlim_{e}\left(R\Gamma(X/\Sigma_{\inf},\mathcal{F})\otimes_{{\mathrm{B_{dR}^{+}}}}{\mathrm{B^{+}_{dR,e}}}\right).$

Before we prove, we want to remark that the result for crystals over the big
site $X/\Sigma_{{\mathrm{INF}}}$ are true and the proof is identical to the
small site case.

###### Proof.

  1. (i)

This follows from applying $Ru_{X/\Sigma*}$ to the equivalences
$\mathcal{F}\longrightarrow R\varprojlim_{e\geq r}Ri_{e*}i_{e}^{*}\mathcal{F}$
in Lemma 7.2.2. Here we use the identity of maps of topoi in the last
subsection

$u_{X/\Sigma}\circ i_{e}=u_{X/\Sigma_{e}}.$

  2. (ii)

For each $e\geq r$, by Theorem 4.2.2 there exists a natural isomorphism of
complexes of sheaves of ${\mathrm{B^{+}_{dR,e}}}$-modules

$Ru_{X/\Sigma_{e}*}i_{e}^{*}\mathcal{F}\longrightarrow\mathcal{F}_{D_{X}(Y_{e})}\otimes\Omega_{D_{X}(Y_{e})}^{\bullet}.$

So the map of ringed sites $X/\Sigma_{e\inf}\rightarrow X/\Sigma_{e+1\inf}$
induced from the closed immersion of the bases
$\Sigma_{e}\rightarrow\Sigma_{e+1}$ together with (i) produces the inverse
limits

$\displaystyle Ru_{X/\Sigma*}\mathcal{F}$ $\displaystyle\cong
R\varprojlim_{e\geq
r}(\mathcal{F}_{D_{X}(Y_{e})}\otimes\Omega_{D_{X}(Y_{e})}^{\bullet})$
$\displaystyle\cong\mathcal{F}_{D}\otimes\Omega_{D}^{\bullet},$

where we use the compatibility of the de Rham complexes
$\mathcal{F}_{D_{X}(Y_{e})}\otimes\Omega_{D_{X}(Y_{e})}^{\bullet}$ for
different $e$, by our choices of the direct system of smooth rigid spaces
$\\{Y_{e}\\}_{e}$.

  3. (iii)

We first notice that the second half of the statement follows from its sheaf
version, by the following equality

$R\Gamma(U,(Ru_{X/\Sigma*}\mathcal{F})\otimes_{{\mathrm{B_{dR}^{+}}}}^{L}{\mathrm{B^{+}_{dR,e}}})\cong
R\Gamma(U/\Sigma_{\inf},\mathcal{F})\otimes_{{\mathrm{B_{dR}^{+}}}}{\mathrm{B^{+}_{dR,e}}}.$

Here the equivalence follows from applying $R\Gamma(U,-)$ to the distinguished
triangle resolving ${\mathrm{B^{+}_{dR,e}}}$ over ${\mathrm{B_{dR}^{+}}}$ as
below

$\textstyle{Ru_{X/\Sigma*}\mathcal{F}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cdot\xi^{e}}$$\textstyle{Ru_{X/\Sigma*}\mathcal{F}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(Ru_{X/\Sigma*}\mathcal{F})\otimes_{{\mathrm{B_{dR}^{+}}}}^{L}{\mathrm{B^{+}_{dR,e}}}.}$

Moreover the above equality shows that the sheaf version isomorphism is local
with respect to $X$, so it suffices to assume that $X$ admits a closed
immersion into a direct system of smooth rigid spaces $\\{Y_{e}\\}_{e}$ over
$\Sigma_{e}$, where the results follow from the explicit calculation of the
completed de Rham complexes as in item (ii) and Theorem 4.2.2. So we are done.

∎

######  Remark 7.2.4.

Recall that for a smooth affinoid rigid space $X=\mathrm{Spa}(R)$ over $K$,
crystalline cohomology of $X$ over ${\mathrm{B_{dR}^{+}}}$, introduced in
[BMS, Section 13], is defined as the inverse limit

$\varprojlim_{e\in\mathbb{N}}\Omega_{D_{X}(Y_{e})}^{\bullet},$

where $X\rightarrow Y_{e}=\mathrm{Spa}({\mathrm{B^{+}_{dR,e}}}\langle
T_{i}^{\pm 1}\rangle)$ is a closed immersion. So Theorem 7.2.3 implies that
infinitesimal cohomology $R\Gamma(X/\Sigma_{\inf},\mathcal{O}_{X/\Sigma})$
coincides with crystalline cohomology of $X$ over ${\mathrm{B_{dR}^{+}}}$ in
the sense of [BMS].

With the help of Theorem 7.2.3, we can compare infinitesimal cohomology of $X$
over ${\mathrm{B_{dR}^{+}}}$ with the derived de Rham complex.

######  Definition 7.2.5.

Let $X$ be a rigid space over $\Sigma_{r}$. Then the _analytic derived de Rham
complex of $X$ over ${\mathrm{B_{dR}^{+}}}$_, denoted as
$\widehat{\mathrm{dR}}^{\mathrm{an}}_{X/\Sigma}$, is defined to be the derived
inverse limit of the filtered complexes

$\widehat{\mathrm{dR}}^{\mathrm{an}}_{X/\Sigma}:=R\varprojlim_{e\geq
r}\widehat{\mathrm{dR}}^{\mathrm{an}}_{X/\Sigma_{e}}.$

Apply Theorem 7.2.3 (i) to the infinitesimal structure sheaf
$\mathcal{O}_{X/\Sigma}$ and the comparison in Theorem 5.5.5, we get the
following.

######  Corollary 7.2.6.

Let $X$ be a rigid space over $\Sigma_{r}$. There exists a natural filtered
map between the analytic derived de Rham complex and infinitesimal cohomology
sheaves, inducing an isomorphism on the underlying complexes

$\widehat{\mathrm{dR}}^{\mathrm{an}}_{X/\Sigma}\longrightarrow
Ru_{X/\Sigma*}\mathcal{O}_{X/\Sigma}.$

In particular, applying the derived global section on the underlying
complexes, we get the following comparison of cohomology

$R\Gamma(X,\widehat{\mathrm{dR}}^{\mathrm{an}}_{X/\Sigma})\cong
R\Gamma(X/\Sigma_{\inf},\mathcal{O}_{X/\Sigma}).$

The next result concerns the éh descent for cohomology of crystals over the
big infinitesimal site $X/\Sigma_{{\mathrm{INF}}}$, where $X$ is a rigid space
over $K$.

######  Proposition 7.2.7.

Let $X$ be a rigid space over $K$, and let $\mathcal{F}$ be a crystal in
vector bundles over the big infiniteismal site $X/\Sigma_{{\mathrm{INF}}}$.
Then the cohomology sheaf $Ru_{X/\Sigma*}\mathcal{F}$ satisfies the éh-
hyperdescent. Namely for an éh-hypercovering $X_{\bullet}^{\prime}\rightarrow
X^{\prime}$ of $K$-rigid spaces over $X$, the following natural map is an
isomorphism

$R\Gamma(X^{\prime}/\Sigma_{{\mathrm{INF}}},\mathcal{F})\longrightarrow
R\lim_{[n]\in\Delta^{\mathrm{op}}}\left(R\Gamma(X^{\prime}_{n}/\Sigma_{{\mathrm{INF}}},\mathcal{F})\right).$

###### Proof.

By Lemma 7.2.2, (i), the pullback $i_{e}^{*}\mathcal{F}$ over
$X/\Sigma_{e{\mathrm{INF}}}$ is a crystal in vector bundles. Thanks to Theorem
6.2.5, we know the natural map $X_{\bullet}^{\prime}\rightarrow X^{\prime}$
induces a natural isomorphism as below

$R\Gamma(X^{\prime}/\Sigma_{{\mathrm{INF}}},i_{e}^{*}\mathcal{F})\longrightarrow
R\lim_{[n]\in\Delta^{\mathrm{op}}}\left(R\Gamma(X^{\prime}_{n}/\Sigma_{{\mathrm{INF}}},i_{e}^{*}\mathcal{F})\right).$

Thus the result we want follows from taking the derived limit over all $e$, by
Theorem 7.2.3 (i). ∎

We want to mention that thanks to the Corollary 2.2.8, it is safe to replace
the cohomology of $\mathcal{F}$ over the big infinitesimal site by the
cohomology $R\Gamma(X^{\prime}/\Sigma_{\inf},\iota^{-1}\mathcal{F})$ of the
restriction $\iota^{-1}\mathcal{F}$ over the small infinitesimal site
$X/\Sigma_{\inf}$. In particular, by applying the above result to the
infinitesimal structure sheaf $\mathcal{O}_{X/\Sigma}$, we see infinitesimal
cohomology over ${\mathrm{B_{dR}^{+}}}$ satisfies the éh-hyperdescent.

######  Corollary 7.2.8.

Let $X$ be a rigid space over $K$. Then infinitesimal cohomology
$R\Gamma(X/\Sigma_{\inf},\mathcal{O}_{X/\Sigma})$ satisfies the éh-
hyperdescent.

Another quick upshot is the finiteness of infinitesimal cohomology for a
proper rigid space $X$.

######  Proposition 7.2.9.

Let $X$ be a proper rigid space of dimension $n$ over $K$, and let
$\mathcal{F}$ be a coherent crystal. The infinitesimal cohomology
$R\Gamma(X/\Sigma_{{\mathrm{INF}}},\mathcal{F})$ is then a perfect
${\mathrm{B_{dR}^{+}}}$-complex in cohomological degrees $[0,2n]$.

###### Proof.

Thanks to Theorem 7.2.3 (i), we can write
$R\Gamma(X/\Sigma_{{\mathrm{INF}}},\mathcal{F})$ as the derived limit of
$R\Gamma(X\Sigma_{e{\mathrm{INF}}},i_{e}^{*}\mathcal{F})$. Here each
$R\Gamma(X/\Sigma_{e{\mathrm{INF}}},i_{e}^{*}\mathcal{F})$ is a bounded
complex in cohomological degree $[0,2n]$ such that each cohomology group is
finite over ${\mathrm{B^{+}_{dR,e}}}$ (Proposition 6.3.1). So the result then
follows from the short exact sequence

$0\longrightarrow
R^{1}\varprojlim_{e}\mathrm{H}^{i-1}(X/\Sigma_{e{\mathrm{INF}}},i_{e}^{*}\mathcal{F})\longrightarrow\mathrm{H}^{i}(X/\Sigma_{{\mathrm{INF}}},\mathcal{F})\longrightarrow\varprojlim_{e}\mathrm{H}^{i}(X/\Sigma_{e{\mathrm{INF}}},i_{e}^{*}\mathcal{F})\longrightarrow
0.$

Here we note that the inverse system
$\\{\mathrm{H}^{2n}(X/\Sigma_{e{\mathrm{INF}}},i_{e}^{*}\mathcal{F})\\}_{e}$
satisfies the Mittag-Leffler condition, by the finiteness of each
$\mathrm{H}^{2n}(X/\Sigma_{e{\mathrm{INF}}},i_{e}^{*}\mathcal{F})$ over
${\mathrm{B^{+}_{dR,e}}}$. ∎

###  7.3. Comparison with pro-étale cohomology

In this subsection, we compare infinitesimal cohomology of $X/\Sigma_{\inf}$
with the pro-étale cohomology of the de Rham period sheaf
${\mathbb{B}_{\mathrm{dR}}}$. As an application, we show the degeneracy of the
Hodge–de Rham spectral sequence, together with a torsion freeness of
infinitesimal cohomology
$\mathrm{H}^{i}(X/\Sigma,\mathcal{O}_{X/\Sigma_{\inf}})$ over
${\mathrm{B_{dR}^{+}}}$. Throughout the section, we will assume the basics of
the pro-étale topology defined in [Sch13].

Comparison theorem. Let $X$ be a rigid space over $K$, and let
$X_{\mathrm{pro\acute{e}t}}$ be the pro-étale site of $X$. The pro-étale site
admits a basis, which consists of affinoid adic spaces
$U=\mathrm{Spa}(B,B^{+})$ that are pro-étale over $X$ and are _affinoid
perfectoid_ (namely, the Huber pair $(B,B^{+})$ is a perfectoid algebra over
$K$). Over the pro-étale site, we can associate the complete structure sheaf
$\widehat{\mathcal{O}}_{X}$, whose section at an affinoid perfectoid space
$U=\mathrm{Spa}(B,B^{+})$ is the $K$-algebra $B$. Denote
$\nu:X_{\mathrm{pro\acute{e}t}}\rightarrow X_{{\mathrm{rig}}}$ to be the
canonical morphism from the pro-étale site to the rigid site of $X$.

We recall from [Sch13] that the _de Rham period sheaf
${\mathbb{B}^{+}_{\mathrm{dR}}}$_, defined as a sheaf of
${\mathrm{B_{dR}^{+}}}$-algebras over $X_{\mathrm{pro\acute{e}t}}$, sending an
affinoid perfectoid space $U=\mathrm{Spa}(B,B^{+})$ onto the ring

${\mathbb{B}^{+}_{\mathrm{dR}}}(B,B^{+}):=\varprojlim_{m}\left(W(\varprojlim_{x\mapsto
x^{p}}B^{+}/p)[\frac{1}{p}]/\xi^{m}\right).$

The sheaf ${\mathbb{B}^{+}_{\mathrm{dR}}}$ admits a canonical surjection
$\theta:{\mathbb{B}^{+}_{\mathrm{dR}}}\rightarrow\widehat{\mathcal{O}}_{X}$
that are compatible with the surjection map
$\theta:{\mathrm{B_{dR}^{+}}}\rightarrow K$ for the period ring
${\mathrm{B_{dR}^{+}}}$. It can be shown that $\xi$ is a nonzero-divisor in
${\mathbb{B}^{+}_{\mathrm{dR}}}$, and the ideal
$\ker(\theta)\subset{\mathbb{B}^{+}_{\mathrm{dR}}}$ is generated by
$\xi\in{\mathrm{B_{dR}^{+}}}$. So we could invert the element $\xi$ to get a
sheaf of
${\mathrm{B}_{\mathrm{dR}}}={\mathrm{B_{dR}^{+}}}[\frac{1}{\xi}]$-algebras
over $X_{\mathrm{pro\acute{e}t}}$, which we denote by
${\mathbb{B}_{\mathrm{dR}}}$. The sheaf of rings ${\mathbb{B}_{\mathrm{dR}}}$
then admits a natural descending filtration defined by
${\mathrm{Fil}}^{i}{\mathbb{B}_{\mathrm{dR}}}:=\xi^{i}{\mathbb{B}^{+}_{\mathrm{dR}}}$,
where each graded piece ${\mathrm{gr}}^{i}{\mathbb{B}_{\mathrm{dR}}}$, which
is equal to $\widehat{\mathcal{O}}_{X}\cdot\xi^{i}$, is canonically isomorphic
to the pro-étale structure sheaf.

We first recall the comparison between infinitesimal cohomology and the pro-
étale cohomology of ${\mathbb{B}_{\mathrm{dR}}}$ for smooth rigid spaces.

######  Theorem 7.3.1 ([BMS], Theorem 13.1).

Let $X$ be a smooth rigid space over $K$. Then there exists a natural map of
complexes of sheaves of ${\mathrm{B_{dR}^{+}}}$-modules over $X$

$Ru_{X/\Sigma*}\mathcal{O}_{X/\Sigma}\longrightarrow
R\nu_{*}{\mathbb{B}^{+}_{\mathrm{dR}}}.$

It is an isomorphism after inverting $\xi$.

###### Proof.

This is essentially proved in the [BMS], Theorem 13.1, and we explain here the
relation of their result with our statement.

Let $X$ be a smooth rigid space over $K$ of dimension $d$. Assume
$U=\mathrm{Spa}(R)$ is a _very small_ affinoid open subset in $X$; namely it
admits an étale morphism onto a torus $\mathbb{T}^{d}_{K}$, where the map can
be extended to a closed immersion into a larger torus
$\mathbb{T}^{n}=\mathrm{Spa}(K\langle T_{i}^{\pm 1}\rangle)$. For any such
closed immersion, we could associate the torus $\mathbb{T}^{n}$ an affinoid
perfectoid space $\mathbb{T}^{n,\infty}=\mathrm{Spa}(K\langle
T_{i}^{\pm\frac{1}{p^{\infty}}}\rangle)$. The canonical map
$\mathbb{T}^{n,\infty}\rightarrow\mathbb{T}^{n}$ is pro-étale, and its
pullback along $U\rightarrow\mathbb{T}^{n}$ produces a pro-étale morphism from
an affinoid perfectoid space $\mathrm{Spa}(R_{\infty},R^{+}_{\infty})$ over
$U=\mathrm{Spa}(R)$.

We denote by $D$ to be the envelope of $U$ inside of the direct system
$\\{\mathbb{T}^{n}_{{\mathrm{B^{+}_{dR,e}}}}\\}_{e}$ of tori over
$\\{{\mathrm{B^{+}_{dR,e}}}\\}_{e}$. Then for any such choice of morphisms
$(U\rightarrow\mathbb{T}^{d}\rightarrow\mathbb{T}^{n})$, we could construct
two ${\mathrm{B_{dR}^{+}}}$-linear complexes

  * •

The de Rham complex $\Omega_{D}^{\bullet}$ of $U$ in
$\\{\mathbb{T}^{n}_{{\mathrm{B^{+}_{dR,e}}}}\\}_{e}$, that computes
infinitesimal cohomology $R\Gamma(U/\Sigma_{\inf},\mathcal{O}_{X/\Sigma})$ by
Theorem 7.2.3.

  * •

The Koszul complex
$K_{{\mathbb{B}^{+}_{\mathrm{dR}}}(R_{\infty})}=K_{{\mathbb{B}^{+}_{\mathrm{dR}}}(R_{\infty})}(\gamma_{u_{i}}-1)$,
that computes the pro-étale cohomology
$R\Gamma(U_{\mathrm{pro\acute{e}t}},{\mathbb{B}^{+}_{\mathrm{dR}}})$.

As in the proof of the [BMS, Theorem 13.1], for any choice of
$(U\rightarrow\mathbb{T}^{d}\rightarrow\mathbb{T}^{n})$, there exists a
natural map of actual complexes

$\Omega_{D}^{\bullet}\longrightarrow
K_{{\mathbb{B}^{+}_{\mathrm{dR}}}(R_{\infty})},$

which is functorial with respect to the choices of triples, such that it
becomes an isomorphism after inverting $\xi$. Notice that the set of triples
for a fixed $U$ is filtered, and the transition map of both complexes are
isomorphisms. In this way, the induced isomorphism

$R\Gamma(U/\Sigma_{\inf},\mathcal{O}_{X/\Sigma})[\frac{1}{\xi}]\rightarrow
R\Gamma(U_{\mathrm{pro\acute{e}t}},{\mathbb{B}_{\mathrm{dR}}})$

is independent of the triples
$(U\rightarrow\mathbb{T}^{d}\rightarrow\mathbb{T}^{n})$. Since the collection
of very small open subsets of $X$ form a basis in the rigid topology, we could
then get a natural isomorphism as below

$Ru_{X/\Sigma*}\mathcal{O}_{X/\Sigma}[\frac{1}{\xi}]\longrightarrow
R\nu_{*}{\mathbb{B}_{\mathrm{dR}}}.$

∎

Using the éh-hyperdescent, we could improve the above result into the non-
smooth situations.

######  Theorem 7.3.2.

Let $X$ be a rigid space over $K$. Then there exists a natural map of
complexes of sheaves of ${\mathrm{B_{dR}^{+}}}$-modules over $X$ as below

$Ru_{X/\Sigma*}\mathcal{O}_{X/\Sigma}\longrightarrow
R\nu_{*}{\mathbb{B}^{+}_{\mathrm{dR}}}.$

It is an isomorphism after inverting $\xi$.

###### Proof.

In the proof, we use $\nu_{X}:X_{\mathrm{pro\acute{e}t}}\rightarrow
X_{\mathrm{rig}}$ to denote the natural map of sites associated with the rigid
space $X$.

We first notice that the pro-étale cohomology sheaf
$R\nu_{X*}{\mathbb{B}^{+}_{\mathrm{dR}}}$ and
$R\nu_{X*}{\mathbb{B}_{\mathrm{dR}}}$ satisfy the éh-hyperdescent. To see
this, we recall from [Guo19, Section 4] that the derived push-forward
$R\nu_{X*}\widehat{\mathcal{O}}_{X}$ is naturally isomorphic to $R\pi_{X*}C$,
where $C=R\alpha_{*}\widehat{\mathcal{O}}_{v}\in D^{\geq
0}(X_{\mathrm{\acute{e}h}})$ is the derived push-forward of the completed
$v$-structure sheaf (see [Guo19, Section 3.2]), and
$\pi_{X}:X_{\mathrm{\acute{e}h}}\rightarrow X_{\mathrm{rig}}$ is the natural
map of sites. As an upshot, since $C$ is a bounded below complex of éh-
sheaves, its direct image $R\pi_{X*}C$ in the rigid site naturally satisfies
the éh-hyperdescent; namely for an éh-hypercovering
$\rho:X_{\bullet}\rightarrow X$ over $K$, the induced map below is an
isomorphism

$R\pi_{X*}C\longrightarrow R\rho_{*}R\pi_{X_{\bullet}*}C.$

We could then replace the above by the derived push-forward of the pro-étale
structure sheaf to get a natural isomorphism

$R\nu_{X*}\widehat{\mathcal{O}}_{X}\longrightarrow
R\rho_{*}R\nu_{X_{\bullet}*}\widehat{\mathcal{O}}_{X_{\bullet}}.$

On the other hand, notice the de Rham period sheaf
${\mathbb{B}^{+}_{\mathrm{dR}}}$ is completed under the $\xi$-adic topology
such that the $i$-th graded piece is equal to the complete structure sheaf
$\widehat{\mathcal{O}}_{X}\cdot\xi^{i}$ up to a twist. In this way, by the
hyperdescent for graded pieces and the induction on $e$, we get

$\displaystyle R\nu_{X*}{\mathbb{B}^{+}_{\mathrm{dR}}}$
$\displaystyle=R\varprojlim_{e\in\mathbb{N}}R\nu_{X*}{\mathbb{B}^{+}_{\mathrm{dR}}}/\xi^{e}$
$\displaystyle\cong
R\varprojlim_{e\in\mathbb{N}}R\rho_{*}R\nu_{X_{\bullet}*}{\mathbb{B}^{+}_{\mathrm{dR}}}/\xi^{e}$
$\displaystyle\cong
R\rho_{*}R\varprojlim_{e\in\mathbb{N}}R\nu_{X_{\bullet}*}{\mathbb{B}^{+}_{\mathrm{dR}}}/\xi^{e}$
$\displaystyle=R\rho_{*}R\nu_{X_{\bullet}*}{\mathbb{B}^{+}_{\mathrm{dR}}}.$

Namely the pro-étale cohomology of ${\mathbb{B}^{+}_{\mathrm{dR}}}$ hence
${\mathbb{B}_{\mathrm{dR}}}={\mathbb{B}^{+}_{\mathrm{dR}}}[\frac{1}{\xi}]$
satisfies the éh-hyperdescent.

At last, notice that the collection of maps $f:X^{\prime}\rightarrow X$ for
smooth rigid spaces $X^{\prime}$ form a basis of the éh-site
$X_{\mathrm{\acute{e}h}}$. In this way, the natural comparison map
$Ru_{X^{\prime}/\Sigma*}\mathcal{O}_{X^{\prime}/\Sigma}\rightarrow
R\nu_{X^{\prime}*}{\mathbb{B}^{+}_{\mathrm{dR}}}$ for smooth $X^{\prime}$
extends to a map for $X$ via the éh-hyperdescent (for infinitesimal cohomology
sheaf, this is Theorem 7.2.7), and by inverting $\xi$ we get the isomorphism

$Ru_{X/\Sigma*}\mathcal{O}_{X/\Sigma}[\frac{1}{\xi}]\longrightarrow
R\nu_{*}{\mathbb{B}_{\mathrm{dR}}}.$

∎

######  Remark 7.3.3.

The morphism between infinitesimal cohomology and the pro-étale cohomology is
constructed in an indirect way. In fact, by enlarging the infinitesimal site
$X/\Sigma_{\inf}$ to a bigger site that allows all (adic spectra of) complete
Huber rings as in [Yao19, Construction 5.11], the de Rham period ring
${\mathbb{B}^{+}_{\mathrm{dR}}}(R_{\infty})$ for a perfectoid algebra
$R_{\infty}$ can be then regarded as a pro-thickening in this enlarged
category. In this way, the arrow from the associated ind-object to the final
object in the enlarged infinitesimal topos will induce a map on their
cohomology, and it can be checked via computations in smooth case and the éh-
hyperdescent that this coincides with our morphism.

Consider a special case when $X$ comes from a small subfield below. Precisely,
let $K_{0}$ be a discretely valued subfield of $K$ such that the residue field
of $K_{0}$ is perfect. Assume $Y$ is a proper rigid space over $K_{0}$, and
$X=Y\times_{K_{0}}K$ is the base field extension of $X_{0}$. We recall from
[Guo19, Theorem 8.2.2] that there exists a
$\operatorname{Gal}(K/K_{0})$-equivariant filtered comparison between the pro-
étale cohomology
$R\Gamma(X_{\mathrm{pro\acute{e}t}},{\mathbb{B}_{\mathrm{dR}}})$ and the
tensor product

$R\Gamma(Y_{{\mathrm{\acute{e}h}}},\Omega_{{\mathrm{\acute{e}h}},/K_{0}}^{\bullet})\otimes_{K_{0}}\mathrm{B}_{\mathrm{dR}}.$

Here $\Omega_{{\mathrm{\acute{e}h}},/K_{0}}^{i}$ is the éh-differential for
rigid spaces over $K_{0}$, and the filtration is defined by the product
filtration, where the éh de Rham cohomology
$R\Gamma(Y_{{\mathrm{\acute{e}h}}},\Omega_{{\mathrm{\acute{e}h}},/K_{0}}^{\bullet})$
is equipped with a natural descending filtration by
${\mathrm{Fil}}^{i}=R\Gamma(Y_{{\mathrm{\acute{e}h}}},\Omega_{{\mathrm{\acute{e}h}},/K_{0}}^{\geq
i})$. Moreover, by taking the zero-th graded pieces, we get

$R\Gamma(X_{\mathrm{pro\acute{e}t}},\widehat{\mathcal{O}}_{X})\cong\operatorname*{\text{\raisebox{1.07639pt}{\scalebox{0.8}{$\bigoplus$}}}}_{i}R\Gamma(Y_{{\mathrm{\acute{e}h}}},\Omega_{{\mathrm{\acute{e}h}},/K_{0}}^{i})\otimes_{K_{0}}K(-i).$

From this, we get the following:

######  Corollary 7.3.4.

Let $Y$ be a proper rigid space over the discretely valued subfield $K_{0}$ of
$K$ as above, and let $X$ be its base extension to $K$. Then we have a
canonical isomorphism

$R\Gamma(X/\Sigma_{\inf},\mathcal{O}_{X/\Sigma})[\frac{1}{\xi}]\cong
R\Gamma(Y_{{\mathrm{\acute{e}h}}},\Omega_{{\mathrm{\acute{e}h}},/K_{0}}^{\bullet})\otimes_{K_{0}}\mathrm{B}_{\mathrm{dR}}.$

In particular, infinitesimal cohomology of $Y\times_{K_{0}}K$ over
$\mathrm{B}_{\mathrm{dR}}$ admits a $\operatorname{Gal}(K/K_{0})$-equivariant
filtration such that the zero-th graded factor is equal to

$\operatorname*{\text{\raisebox{1.07639pt}{\scalebox{0.8}{$\bigoplus$}}}}_{i}R\Gamma(Y_{{\mathrm{\acute{e}h}}},\Omega_{{\mathrm{\acute{e}h}},/K_{0}}^{i})\otimes_{K_{0}}K(-i).$

Torsion freeness and Hodge–de Rham degeneracy. For the rest of the subsection,
we prove infinitesimal cohomology
$\mathrm{H}^{n}(X/\Sigma_{\inf},\mathcal{O}_{X/\Sigma})$ is torsion free over
${\mathrm{B_{dR}^{+}}}$ (namely Theorem 1.2.7 (vi)), and show the degeneracy
for the éh Hodge–de Rham spectral sequence.

######  Theorem 7.3.5.

Let $X$ be a proper rigid space over $K$. Then we have the following

  1. (i)

infinitesimal cohomology
$\mathrm{H}^{n}(X/\Sigma_{\inf},\mathcal{O}_{X/\Sigma})$ is torsion free over
${\mathrm{B_{dR}^{+}}}$, for each $n\in\mathbb{N}$.

  2. (ii)

the éh Hodge–de Rham spectral sequence over $K$ below degenerates at its first
page:

$E_{1}^{i,j}=\mathrm{H}^{j}(X_{\mathrm{\acute{e}h}},\Omega_{{\mathrm{\acute{e}h}},X/K}^{i})\Longrightarrow\mathrm{H}^{i+j}(X_{\mathrm{\acute{e}h}},\Omega_{{\mathrm{\acute{e}h}},X/K}^{\bullet}).$

######  Remark 7.3.6.

Note that the part (ii) generalizes the degeneracy result in [Guo19,
Proposition 8.0.8], where the latter needs the assumption for $X$ to be
defined over a discretely valued subfield.

###### Proof.

We first notice that pro-étale cohomology
$\mathrm{H}^{n}(X_{\mathrm{pro\acute{e}t}},{\mathbb{B}^{+}_{\mathrm{dR}}})$ is
finite free over ${\mathrm{B_{dR}^{+}}}$. Recall the Primitive Comparison
Theorem over ${\mathrm{B_{dR}^{+}}}$ as below ([Sch12])

$\mathrm{H}^{n}(X_{\mathrm{\acute{e}t}},\mathbb{Q}_{p})\otimes_{\mathbb{Q}_{p}}{\mathrm{B_{dR}^{+}}}\cong\mathrm{H}^{n}(X_{\mathrm{pro\acute{e}t}},{\mathbb{B}^{+}_{\mathrm{dR}}}).$

As étale cohomology $\mathrm{H}^{n}(X_{\mathrm{\acute{e}h}},\mathbb{Q}_{p})$
is a finite dimensional vector space over $\mathbb{Q}_{p}$, this in particular
implies the finite freeness of
$\mathrm{H}^{n}(X_{\mathrm{pro\acute{e}t}},{\mathbb{B}^{+}_{\mathrm{dR}}})$
over ${\mathrm{B_{dR}^{+}}}$. In particular, by Theorem 7.3.2 and the
finiteness in Proposition 7.2.9, we get the following relations

$\displaystyle\dim_{\mathbb{Q}_{p}}\mathrm{H}^{n}(X_{\mathrm{\acute{e}t}},\mathbb{Q}_{p})$
$\displaystyle=\operatorname{rank}_{\mathrm{B_{dR}^{+}}}\mathrm{H}^{n}(X_{\mathrm{pro\acute{e}t}},{\mathbb{B}^{+}_{\mathrm{dR}}})$
$\displaystyle=\dim_{\mathrm{B}_{\mathrm{dR}}}\mathrm{H}^{n}(X_{\mathrm{pro\acute{e}t}},{\mathbb{B}_{\mathrm{dR}}})$
$\displaystyle=\dim_{\mathrm{B}_{\mathrm{dR}}}\mathrm{H}^{n}(X/\Sigma_{\inf},\mathcal{O}_{X/\Sigma})[\frac{1}{\xi}]$
$\displaystyle=\operatorname{rank}_{\mathrm{B_{dR}^{+}}}\mathrm{H}^{n}(X/\Sigma_{\inf},\mathcal{O}_{X/\Sigma})/torsion$
$\displaystyle\leq\dim_{K}\mathrm{H}^{n}(X/\Sigma_{\inf},\mathcal{O}_{X/\Sigma})/\xi.$

On the other hand, by the base change formula in Theorem 7.2.3, (iii), we get
the following short exact sequence of $K$-vector spaces

$0\longrightarrow\mathrm{H}^{n+1}(X/\Sigma_{\inf},\mathcal{O}_{X/\Sigma})[\xi]\longrightarrow\mathrm{H}^{n}(X/K_{\inf},\mathcal{O}_{X/K})\longrightarrow\mathrm{H}^{n}(X/\Sigma_{\inf},\mathcal{O}_{X/\Sigma})/\xi.$

This implies the inequalities

$\dim_{K}\mathrm{H}^{n}(X/\Sigma_{\inf},\mathcal{O}_{X/\Sigma})/\xi\leq\dim_{K}\mathrm{H}^{n}(X/K_{\inf},\mathcal{O}_{X/K}).$

Now using the comparison between infinitesimal cohomology and éh de Rham
cohomology for the trivial crystal $\mathcal{F}=\mathcal{O}_{X/K}$ in Theorem
6.2.2, we have

$\dim_{K}\mathrm{H}^{n}(X/K_{\inf},\mathcal{O}_{X/K})=\dim_{K}\mathrm{H}^{n}(X_{\mathrm{\acute{e}h}},\Omega_{\mathrm{\acute{e}h}}^{\bullet}).$

Note that the natural Hodge filtration on the éh de Rham complex
$\Omega_{\mathrm{\acute{e}h}}^{\bullet}$ induces the $E_{1}$ spectral sequence

$E_{1}^{i,j}=\mathrm{H}^{j}(X_{\mathrm{\acute{e}h}},\Omega_{\mathrm{\acute{e}h}}^{i})\Longrightarrow\mathrm{H}^{i+j}(X_{\mathrm{\acute{e}h}},\Omega_{\mathrm{\acute{e}h}}^{\bullet}).$

As a consequence, we get

$\dim_{K}\mathrm{H}^{n}(X/K_{\inf},\mathcal{O}_{X/K})\leq\sum_{i+j=n}\dim_{K}\mathrm{H}^{j}(X_{\mathrm{\acute{e}h}},\Omega_{\mathrm{\acute{e}h}}^{i}).$

However, by Hodge–Tate decomposition in [Guo19, Theorem 1.1.3], we have

$\dim_{\mathbb{Q}_{p}}\mathrm{H}^{n}(X_{\mathrm{\acute{e}t}},\mathbb{Q}_{p})=\sum_{i+j=n}\dim_{K}\mathrm{H}^{j}(X_{\mathrm{\acute{e}h}},\Omega_{\mathrm{\acute{e}h}}^{i}).$

Hence combining all of the relations of dimensions above, we see all of the
inequalities should be equalities. In this way, the $E_{1}$ spectral sequence
degenerates at the first page, and for any $n\in\mathbb{N}$ we have

$\displaystyle\mathrm{H}^{n+1}(X/\Sigma_{\inf},\mathcal{O}_{X/\Sigma})[\xi]=0;$
$\displaystyle\mathrm{H}^{n}(X/\Sigma_{\inf},\mathcal{O}_{X/\Sigma})/\xi=0.$

∎

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