# Prismatic cohomology of rigid analytic spaces over de Rham period ring

###### Abstract.

Inspired by Bhatt-Scholze [BS19], in this article, we introduce prismatic
cohomology for rigid analytic spaces with l.c.i singularities, with
coefficients over Fontaine’s de Rham period ring ${\mathrm{B_{dR}^{+}}}$.

Haoyang Guo

###### Contents

  1. 1 Introduction
    1. 1.1 Background and main theorems
    2. 1.2 Simpson’s functor
    3. 1.3 Simplicial resolution
    4. 1.4 Other comments
    5. 1.5 Acknowledgements
  2. 2 Prismatic site over ${\mathrm{B_{dR}^{+}}}$
    1. 2.1 Algebraic preliminary
    2. 2.2 Prismatic site
  3. 3 Prismatic crystal and log connection
  4. 4 From infinitesimal to prismatic
    1. 4.1 A variant of Simpson’s functor
    2. 4.2 Prisms associated to infinitesimal thickenings
    3. 4.3 Local calculation for l.c.i singularities
  5. 5 Hodge-Tate filtration
    1. 5.1 Smooth case
    2. 5.2 Simplicial resolution for affinoid algebras
    3. 5.3 Left Kan extension and Hodge-Tate filtration
  6. 6 Infinitesimal comparison
  7. 7 Pro-étale comparison
    1. 7.1 Künneth formula and étale localization
    2. 7.2 Pro-étale comparison
  8. 8 Galois invariant
    1. 8.1 Condensed mathematics and condensed group cohomology
    2. 8.2 Galois invariant of cohomology

##  1\. Introduction

###  1.1. Background and main theorems

Let $k$ be a complete discretely valued $p$-adic field that has a perfect
residue field, and let $K$ be its completed algebraic closure. In the seminal
work of Bhatt and Scholze [BS19], they introduce the notion of prisms and
prismatic cohomology for a $p$-adic smooth formal scheme $X$ over
$\mathcal{O}_{k}$. A prism can be defined as a diagram of algebras

$(A\longrightarrow A/I\longleftarrow R),$

where $\operatorname{Spf}(R)$ is an open subset of $X$, and $A$ is equipped
with an additional structure of a lift of Frobenius operator that satisfies
various conditions ([BS19, Section 2, 3]). The resulted cohomology theory,
namely prismatic cohomology, turns out to specialize to many other important
$p$-adic cohomology in the $p$-adic Hodge theory, and thus provides a powerful
tool to study the relation of different cohomology associated to the given
formal scheme $X$.

Let $X$ now be a rigid space over $K$, and let $\xi$ be a fixed generator of
the kernel ideal for the natural surjection
$\theta:{\mathrm{B_{dR}^{+}}}\rightarrow K$, where ${\mathrm{B_{dR}^{+}}}$ is
Fontaine’s de Rham period ring ([Fon94]). In this article, we study a rational
analogue of prismatic cohomology for rigid spaces over $K$, with coefficients
in ${\mathrm{B_{dR}^{+}}}$. We start by defining the notion of a prism in our
setting.

######  Definition 1.1.1.

A ${\mathrm{B_{dR}^{+}}}$-prism over $X$ is defined as a diagram of
${\mathrm{B_{dR}^{+}}}$-algebras as below

$(B\longrightarrow B/\xi\longleftarrow R),$

where $\mathrm{Spa}(R)$ is an affinoid open subset of $X$, and $B$ is a
$\xi$-adic complete, $\xi$-torsionfree algebra over ${\mathrm{B_{dR}^{+}}}$
satisfies the finite type condition as in Definition 2.2.1 (i).

One essential difference between our notion of prisms and that of Bhatt-
Scholze in [BS19] is that there is _no_ Frobenius structure in our definition.
Nonetheless, the resulted cohomology theory over the opposite category of
prisms still enjoys many good properties when $X$ has mild singularities.

Now we state our main results.

######  Theorem 1.1.2.

There is a ${\mathrm{B_{dR}^{+}}}$-linear cohomology theory on the category of
$K$-rigid spaces

$X\longmapsto
R\Gamma_{{\mathbbl{\Delta}}}(X/{\mathrm{B_{dR}^{+}}}):=R\Gamma(X/{\mathrm{B_{dR}^{+}}}_{{\mathbbl{\Delta}}},\mathcal{O}_{{\mathbbl{\Delta}}}),$

such that for an affinoid rigid space $X$ that has l.c.i singularities, it
satisfies the following property (as $\mathbb{E}_{\infty}$-algebras over
${\mathrm{B_{dR}^{+}}}$):

  1. (i)

(Pro-étale comparison) There is a natural map

$R\Gamma_{{\mathbbl{\Delta}}}(X/{\mathrm{B_{dR}^{+}}})\longrightarrow
R\Gamma_{\mathrm{pro\acute{e}t}}(X,{\mathbb{B}^{+}_{\mathrm{dR}}}),$

where the right hand side is pro-étale cohomology of the deRham period sheaf
as in [Sch13]. 111First introduced by Brinon in [Bri08]. The map is an
isomorphism when $X$ is smooth.

  2. (ii)

(infinitesimal comparison) There is a natural map

$R\Gamma_{\inf}(X/{\mathrm{B_{dR}^{+}}})\longrightarrow
R\Gamma_{{\mathbbl{\Delta}}}(X/{\mathrm{B_{dR}^{+}}}),$

which factors through the subcomplex
$L\eta_{\xi}R\Gamma_{{\mathbbl{\Delta}}}(X/{\mathrm{B_{dR}^{+}}})$. Here the
left hand side is infinitesimal cohomology of $X/{\mathrm{B_{dR}^{+}}}$ as in
[Guo21, Section 7] (see also Section 4.2).

Moreover, if $X$ is smooth, the above induces an isomorphism
$R\Gamma_{\inf}(X/{\mathrm{B_{dR}^{+}}})\rightarrow
L\eta_{\xi}R\Gamma_{{\mathbbl{\Delta}}}(X/{\mathrm{B_{dR}^{+}}})$.

  3. (iii)

(Hodge-Tate filtration) There is a natural $\mathbb{N}^{\mathrm{op}}$-indexed
increasing exhaustive filtration over the reduction
$R\Gamma_{{\mathbbl{\Delta}}}(X/{\mathrm{B_{dR}^{+}}})\otimes^{L}_{\mathrm{B_{dR}^{+}}}K$,
whose $i$-th graded piece is
$R\Gamma(X,L\wedge^{i}\mathbb{L}^{\mathrm{an}}_{X/K}(-i)[-i])$.

  4. (iv)

(Galois invariant) Assume $X=Y_{K}$ for some affinoid rigid space $Y$ over
$k$. Then there is a natural condensed algebra
$\underline{R\Gamma}_{{\mathbbl{\Delta}}}(X/{\mathrm{B_{dR}^{+}}})$ underlying
$R\Gamma_{{\mathbbl{\Delta}}}(X/{\mathrm{B_{dR}^{+}}})$, satisfying

$R\Gamma(G_{k},\underline{R\Gamma}_{{\mathbbl{\Delta}}}(X/{\mathrm{B_{dR}^{+}}}))\cong
R\Gamma(G_{k},{\mathrm{B_{dR}^{+}}})\otimes^{L}_{k}R\Gamma_{\inf}(Y/k),$

where $R\Gamma(G_{k},-)$ is defined as the condensed group cohomology as in
[Bos21, Appendix 2].

We give several comments on the result above.

######  Remark 1.1.3 (Cohomology of ${\mathbb{B}^{+}_{\mathrm{dR}}}$).

Let $X$ be a smooth rigid space over $K$ of dimension $n$. In this case,
infinitesimal cohomology $R\Gamma_{\inf}(X/{\mathrm{B_{dR}^{+}}})$ was first
introduced in [BMS, Section 13] (denoted there as
$R\Gamma_{\mathrm{crys}}(X/{\mathrm{B_{dR}^{+}}})$ and is called _crystalline
cohomology_). It is locally computed by the formal completion of the de Rham
complex of a smooth ambient space with respect to a closed immersion ([BMS,
Section 13.1], [Guo21, Theorem 4.1.1]). Shown in [BMS, Theorem 13.1],
infinitesimal cohomology admits a map to pro-étale cohomology
$R\Gamma_{\mathrm{pro\acute{e}t}}(X,{\mathbb{B}^{+}_{\mathrm{dR}}})$, which is
an isomorphism after tensoring with $\mathrm{B_{dR}}$. This in particular
allows us to compute pro-étale cohomology of
${\mathbb{B}_{\mathrm{dR}}}={\mathbb{B}^{+}_{\mathrm{dR}}}[\frac{1}{\xi}]$
using the de Rham complex. It is then a natural question if one can also
compute pro-étale cohomology of ${\mathbb{B}^{+}_{\mathrm{dR}}}$ using a de
Rham style cohomology and within the category of noetherian algebras.

Our Theorem 1.1.2.(i) then gives an answer to the question. In fact, if
$X=\mathrm{Spa}(R)$ admits a compatible system of smooth
${\mathrm{B_{dR}^{+}}}/\xi^{e}$-algebras $\widetilde{R}_{e}$ (which always
exists locally), and let $\widetilde{R}$ be their inverse limit. Then
$R\Gamma_{{\mathbbl{\Delta}}}(R/{\mathrm{B_{dR}^{+}}})$ is isomorphic to the
following $\xi$-divided de Rham complex

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
5.77779pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\widetilde{R}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
5.29778pt\raise 5.43056pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.43056pt\hbox{$\scriptstyle{d~{}~{}~{}~{}~{}~{}~{}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 29.77779pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
29.77779pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\frac{\widetilde{\Omega}^{1}}{\xi}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
50.49776pt\raise 5.43056pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.43056pt\hbox{$\scriptstyle{d}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 67.33336pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
67.33336pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
104.83336pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
104.83336pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\frac{\widetilde{\Omega}^{n}}{\xi^{n}}}$}}}}}}}\ignorespaces}}}}\ignorespaces.$

Here
$\widetilde{\Omega}^{1}:=\varprojlim_{e}\Omega_{(\widetilde{R}/\xi^{e})/{\mathrm{B^{+}_{dR,e}}}}^{1}$
is the inverse limit of analytic differentials of affinoid algebras
$\widetilde{R}_{e}$ over ${\mathrm{B^{+}_{dR,e}}}$, and the map $d$ is defined
as the composition

$\widetilde{R}\longrightarrow\widetilde{\Omega}^{1}\longrightarrow\frac{\widetilde{\Omega}^{1}}{\xi},$

where the first map is the continuous differential over
${\mathrm{B_{dR}^{+}}}$, and the second map is the natural inclusion.

######  Remark 1.1.4 (Prismatic ${\mathbb{B}^{+}_{\mathrm{dR}}}$).

In the recent work [BS21, Construction 6.4] Bhatt-Scholze introduce the notion
of _de Rham period sheaf_ ${{\mathbbl{\Delta}}}_{\bullet}[1/p]^{\wedge}_{I}$
over the category of quasi-regular semiperfectoid algebras over a smooth
$p$-complete $\mathcal{O}_{K}$-algebra $R$ (with quasi-syntomic topology).
Using ${\mathrm{A_{inf}}}$-comparison in [BS19, Theorem 17.2] together with
pro-étale comparison in Theorem 1.1.2 (i), one can show that two prismatic
cohomology over ${\mathrm{B_{dR}^{+}}}$ are compatible, namely cohomology of
${{\mathbbl{\Delta}}}_{\bullet}[1/p]^{\wedge}_{I}$ of a smooth formal scheme
of $\mathscr{X}$ is isomorphic to
$R\Gamma_{{\mathbbl{\Delta}}}(\mathscr{X}_{\eta}/{\mathrm{B_{dR}^{+}}})$
(after replacing the ideal $I=(\phi(\xi))$ by $(\xi)$). Thus our prismatic
cohomology coincides with that of [BS21] for rigid spaces that have good
reduction.

######  Remark 1.1.5 (Globalization).

Theorem 1.1.2 naturally extends to proper rigid spaces that has l.c.i
singularities (with
$L\eta_{\xi}R\Gamma_{{\mathbbl{\Delta}}}(X/{\mathrm{B_{dR}^{+}}})$ in Theorem
1.1.2 (ii) replaced by a sheaf version).

When $X$ is proper smooth, by Hodge-Tate filtration and the finiteness of
coherent cohomology, each
$\mathrm{H}^{i}_{{\mathbbl{\Delta}}}(X/{\mathrm{B_{dR}^{+}}})$ is finite
generated over ${\mathrm{B_{dR}^{+}}}$ and lives in cohomological degree
$0\leq i\leq 2\dim(X)$. In fact, using the primitive comparison theorem
([Sch12]) one can show that each
$\mathrm{H}^{i}_{{\mathbbl{\Delta}}}(X/{\mathrm{B_{dR}^{+}}})$ is finite free
over ${\mathrm{B_{dR}^{+}}}$.

However, if $X$ is not smooth, the Hodge-Tate filtration in Theorem 1.1.2
(iii) has infinite graded pieces. In the special case when $X=Y_{K}$ for a
proper rigid space $Y$ over a discretely valued subfield $k$, the Hodge-Tate
filtration splits into an infinite direct sum
$\operatorname*{\text{\raisebox{1.07639pt}{\scalebox{0.8}{$\bigoplus$}}}}_{i\in\mathbb{N}}R\Gamma(X,L\wedge^{i}\mathbb{L}^{\mathrm{an}}_{X/K}(-i)[-i])$,
which is in general infinite dimensional for proper rigid spaces that have
l.c.i singularities. Thus one loses the finiteness if we go beyond the smooth
proper assumption.

On the other hand, combine Theorem 1.1.2 (i) and (ii), one obtain the
following composition

$R\Gamma_{\inf}(X/{\mathrm{B_{dR}^{+}}})\longrightarrow
R\Gamma_{{\mathbbl{\Delta}}}(X/{\mathrm{B_{dR}^{+}}})\longrightarrow
R\Gamma_{\mathrm{pro\acute{e}t}}(X,{\mathbb{B}^{+}_{\mathrm{dR}}}).$

It can be shown that the composition is an isomorphism after inverting $\xi$
([Guo21, Theoren 7.3.2]), and thus
$R\Gamma_{\inf}(X/{\mathrm{B_{dR}^{+}}})[\frac{1}{\xi}]$ is a direct summand
of $R\Gamma_{{\mathbbl{\Delta}}}(X/{\mathrm{B_{dR}^{+}}})[\frac{1}{\xi}]$, and
contains all Galois invariants when $X=Y_{K}$ is defined over $k$ (notice the
latter is finite when $Y$ is proper). One can then ask if the complement of
$R\Gamma_{\inf}(X/{\mathrm{B_{dR}^{+}}})$ in
$R\Gamma_{{\mathbbl{\Delta}}}(X/{\mathrm{B_{dR}^{+}}})$ has bounded torsion,
which would imply the following:

######  Question 1.1.6.

Are the maps in Theorem 1.1.2 (i) and (ii) inducing isomorphisms after
inverting $\xi$?

In the case when $X=Y_{K}$ is defined over the discretely valued subfield $k$,
it can be shown that the boundedness of torsion in
$R\Gamma_{{\mathbbl{\Delta}}}(X/{\mathrm{B_{dR}^{+}}})$ follows from the
following question on analytic derived de Rham cohomology of $Y/k$.

######  Question 1.1.7.

Let $Y$ be a rigid space over $k$ that has l.c.i singularities. Is there a
positive integer $N$, such that the following map of cohomology is always zero
for any $n\in\mathbb{N}$ and $i\in\mathbb{Z}$

$\mathrm{H}^{i}(Y,{\mathrm{Fil}}^{n+N}{\widehat{\mathrm{dR}}^{\mathrm{an}}}_{Y/k})\longrightarrow\mathrm{H}^{i}(Y,{\mathrm{Fil}}^{n}{\widehat{\mathrm{dR}}^{\mathrm{an}}}_{Y/k}).$

Here ${\mathrm{Fil}}^{\bullet}$ is the natural filtration on the analytic
derived de Rham complex.

######  Remark 1.1.8.

The question has a more classical variant in the complex algebraic geometry.
Namely for a complex algebraic variety $Y$ that has l.c.i singularities, can
we find a positive integer $N$ so that the following map is always zero:

$\mathrm{H}^{i}(Y,{\mathrm{Fil}}^{n+N}\widehat{\mathrm{dR}}_{Y/\mathbb{C}})\longrightarrow\mathrm{H}^{i}(Y,{\mathrm{Fil}}^{n}\widehat{\mathrm{dR}}_{Y/\mathbb{C}}),$

where ${\mathrm{Fil}}^{\bullet}$ is the natural filtration on the Hodge-
completed derived de Rham complex $\widehat{\mathrm{dR}}_{Y/\mathbb{C}}$.

If $Y$ is of dimension zero, one can show the above by hand, using explicit
calculation for a closed immersion into $\mathbb{A}^{1}$. On the other hand,
it is shown in [Bha12, Corollary 5.4] that the map of cohomology
${\mathrm{Fil}}^{n}\widehat{\mathrm{dR}}_{Y/\mathbb{C}}\rightarrow\widehat{\mathrm{dR}}_{Y/\mathbb{C}}$
is the zero map for $n>>0$.

######  Remark 1.1.9.

(Condensed mathematics) Due to the fact that there is no derived category of
topological groups, we instead use the notion of condensed mathematics of
Clausen-Scholze to study Galois invariant of prismatic cohomology in Theorem
1.1.2 (iv). As byproducts, we also compute Galois invariant of infinitesmal
cohomology, and give a condensed base extension formula of infinitesimal
cohomology, as in Theorem 8.2.3 and Theorem 8.2.3 (see also the filtered
versions as in Corollary 8.2.5 and Remark 8.2.6).

We now briefly discuss the proof in next two subsections.

###  1.2. Simpson’s functor

Recall from [Simp] that Simpson introduced an equivalence between the category
of $\mathbb{Z}^{\mathrm{op}}$-filtered vector spaces over a field and the
category of modules over $\mathbb{A}^{1}/\mathbb{G}_{m}$, using a Rees algebra
construction. The functor is defined as follows

$(V,{\mathrm{Fil}}^{\bullet})\longmapsto\underset{i\in\mathbb{Z}}{{\mathrm{colim}}}\frac{{\mathrm{Fil}}^{i}V}{t^{i}},$

where $t$ is the coordinate of $\mathbb{A}^{1}$, and admits a natural action
by $\mathbb{G}_{m}$. The functor can be extended to an equivalence in the
derived level ([Mou19]), and its pullback along
$\\{t=0\\}\rightarrow\mathbb{A}^{1}$ is the direct sum of graded pieces for
$(V,{\mathrm{Fil}}^{\bullet})$.

Analogous to the above construction, we introduce a variant of Simpson’s
functor, sending a $\mathbb{N}^{\mathrm{op}}$-filtered
${\mathrm{B_{dR}^{+}}}$-complex onto the ${\mathrm{B_{dR}^{+}}}$-complex as
below

$\Psi:(M,{\mathrm{Fil}}^{\bullet})\longmapsto(\underset{i\in\mathbb{N}}{{\mathrm{colim}}}\frac{{\mathrm{Fil}}^{i}M}{\xi^{i}})^{\wedge},$

where $(-)^{\wedge}$ is the derived $\xi$-completion. One of our main
observations is that prismatic cohomology over ${\mathrm{B_{dR}^{+}}}$ can be
computed by applying Simpson’s functor at infinitesimal cohomology of
$X/{\mathrm{B_{dR}^{+}}}$ with a lifted infinitesimal filtration.

######  Theorem 1.2.1.

Let $X$ be an affinoid rigid space over $K$ that has l.c.i singularities. Then
we have the following formula

$\Psi(R\Gamma_{\inf}(X/{\mathrm{B_{dR}^{+}}}),\widetilde{\mathrm{Fil}}^{\bullet})\cong
R\Gamma_{{\mathbbl{\Delta}}}(X/{\mathrm{B_{dR}^{+}}}).$

Here $\widetilde{\mathrm{Fil}}^{\bullet}$ is a lift of the infinitesimal
filtration of $X/K$ along ${\mathrm{B_{dR}^{+}}}\rightarrow K$, and is not
canonical in general (see discussion around Definition 4.3.9). We use the
following two examples to illustrate what the formula looks like.

######  Example 1.2.2.

Assume $X=\mathrm{Spa}(R)$ is smooth affinoid and admits a compatible system
of lifts $\mathrm{Spa}(\widetilde{R}_{e})$ as in Remark 1.1.3. Then
$R\Gamma_{\inf}(X/{\mathrm{B_{dR}^{+}}})$ is isomorphic to
$\widetilde{\Omega}^{\bullet}$, and the lifted infinitesimal filtration is
defined as the Hodge filtration

$\widetilde{\mathrm{Fil}}^{i}\widetilde{\Omega}^{\bullet}=(\widetilde{\Omega}^{i}\longrightarrow\cdots\longrightarrow\widetilde{\Omega}^{n})[-i].$

In particular, applying Simpson’s functor at this filtered object, we get

$\Psi(R\Gamma_{\inf}(X/{\mathrm{B_{dR}^{+}}}),\widetilde{\mathrm{Fil}}^{\bullet})=\widetilde{\Omega}^{\bullet}\cup\frac{\widetilde{\Omega}^{\geq
1}}{\xi}\cup\cdots\cup\frac{\widetilde{\Omega}^{d}}{\xi^{d}},$

which is exactly the $\xi$-divided de Rham complex as in Remark 1.1.3. Here
the union is taken inside of $\widetilde{\Omega}^{\bullet}[\frac{1}{\xi}]$.

######  Example 1.2.3.

Another special case is when $X=Y_{K}$ for an affinoid rigid space $Y$ over
the discretely valued subfield $k$. Under the assumption, we can take the
completed base extension of the infinitesimal filtration of $Y/k$ along a
fixed embedding $k\rightarrow{\mathrm{B_{dR}^{+}}}$

$\widetilde{\mathrm{Fil}}^{\bullet}:={\mathrm{Fil}}^{\bullet}_{\inf}(Y/k)\widehat{\otimes}_{k}{\mathrm{B_{dR}^{+}}},$

which in particular admits a $G_{k}$-action. We warn the reader that this is
_not_ the same as the infinitesimal filtration for
$R\Gamma_{\inf}(Y_{K}/{\mathrm{B_{dR}^{+}}})$.

###  1.3. Simplicial resolution

Another ingredient of the proof is the simplicial resolution and the left Kan
extension for affinoid algebras. Not as in the algebraic setting, where one
can use polynomial rings of infinite variables to resolve any algebra; in the
rigid geometry a typical algebraic object is topologically generated by finite
variables. It is then natural to ask if we can resolve an affinoid algebra by
convergent power series rings of finite variables, thus within the rigid
geometry. We give a positive answer to this question, and show in Section 5.2
that an affinoid algebra admits a simplicial resolution of Tate algebras with
finite variables. Moreover, we use this to show that the analytic cotangent
complex $\mathbb{L}^{\mathrm{an}}$ and the analytic derived de Rham complex
${\widehat{\mathrm{dR}}^{\mathrm{an}}}$ for an affinoid algebra can be
computed using the aforementioned resolution (Proposition 5.3.1).

With the help of the simplicial resolution, we can apply the left Kan
extension at prismatic cohomology of smooth affinoid algebras. This allows us
to define a notion of _derived prismatic cohomology_
$L{{\mathbbl{\Delta}}}_{R/{\mathrm{B_{dR}^{+}}}}$ for an arbitrary affinoid
algebra $R$ over $K$ (see Definition 5.3.4), analogous to [BS19, Section 7].
As expected, derived prismatic cohomology and usual prismatic cohomology
coincide for l.c.i singularities.

######  Theorem 1.3.1.

Let $X$ be a rigid space over $K$ that has l.c.i singularities. There is a
natural isomorphism of ${\mathrm{B_{dR}^{+}}}$-$\mathbb{E}_{\infty}$-algebras

$L{{\mathbbl{\Delta}}}_{R/{\mathrm{B_{dR}^{+}}}}\longrightarrow
R\Gamma_{{\mathbbl{\Delta}}}(X/{\mathrm{B_{dR}^{+}}}).$

This in particular allows us to define the Hodge-Tate filtration on reduced
prismatic cohomology
$R\Gamma_{{\mathbbl{\Delta}}}(X/{\mathrm{B_{dR}^{+}}})\otimes^{L}_{\mathrm{B_{dR}^{+}}}K$
for rigid spaces with l.c.i singularities, which is Theorem 1.1.2 (iii).

######  Remark 1.3.2.

For our application, we consider a slightly more general situation. Namely we
show the existence of the finite type resolutions for a pair $(B,I)$,
consisting of a topologically finite type algebra $B$ over
${\mathrm{A_{inf}}}/\xi^{e}$ (resp. ${\mathrm{B_{dR}^{+}}}/\xi^{e}$, or
${\mathrm{B_{dR}^{+}}}$) together with an ideal $I$. This roughly means that
any closed immersion of topologically finite type affine formal schemes (resp.
affinoid rigid spaces) can be simplicially resolved by regular closed
immersions. In the special case when $I$ is the zero ideal, we get the
aforementioned simplicial resolution for a topological algebra itself.

######  Remark 1.3.3.

As both derived prismatic cohomology and the usual prismatic cohomology are
well-defined for general rigid spaces, it is a natural question if Theorem
1.3.1 holds without the assumption on $X$, or if the cone of the comparing map
always has bounded $\xi$-torsion. We hope to investigate the question in the
future.

###  1.4. Other comments

######  Remark 1.4.1.

We want to mention that one can define an analogous notion of prismatic
cohomology for complex algebraic varieties, replacing ${\mathrm{B_{dR}^{+}}}$
by $\mathbb{C}[[t]]$. In the smooth case, this was studied independently in an
unpublished article by Nils Waßmuth ([Waß19]), where he proved the de Rham
comparison and the Hodge-Tate comparison for affine smooth varieties over
$\mathbb{C}$. In general, similar to what we do in this article, one can use
Simpson’s equivalence to study prismatic cohomology for complex varieties that
has l.c.i singularities, using derived de Rham cohomology ([Ill71], [Bha12])
and Hartshorne’s de Rham cohomology over $\mathbb{C}$ ([Har75]).

######  Remark 1.4.2.

(Crystal) As in [BS21], our prismatic site
$X/{\mathrm{B_{dR}^{+}}}_{{\mathbbl{\Delta}}}$ admits a natural notion of
_crystals_ , which we study in Section 3. Analogous to the crystal-connection
translation in the crystalline theory, a prismatic crystal corresponds to a
quasi-nilpotent integrable connection that have log poles at $\\{\xi=0\\}$
(Definition 3.0.3, Proposition 3.0.12). In fact, important resources of
prismatic crystals are those coming from the infinitesimal site
$X/{\mathrm{B_{dR}^{+}}}_{\inf}$, whose associated log-connections are
integral. Moreover, our Theorem 1.2.1 can be extended to prismatic crystals
that come from the infinitesimal site.

######  Remark 1.4.3.

(Non-noetherian enlargement) As mentioned in the beginning, our prismatic site
(Definition 1.1.1) consists only of those ${\mathrm{B_{dR}^{+}}}$-algebras
that are noetherian. However, one can still ask if we could allow other
interesting but non-noetherian rings. In fact, to compare prismatic cohomology
with pro-étale cohomology as in Theorem 1.1.2 (i), we consider an enlarged
version of the prismatic site (with the same indiscrete topology) in Section
7.2, adding algebras coming from perfectoid rings. There we prove that
cohomology of the enlarged site coincides with the smaller one in Definition
1.1.1. Similarly, adding ${\mathbb{B}^{+}_{\mathrm{dR}}}(R)$ for quasi-regular
semiperfectoid rings $R$ over $\mathcal{O}_{K}$ would not change the
cohomology.

#### Leitfaden of the paper

The article is structured as follows. We start with Section 2 about the basics
on the prismatic site over ${\mathrm{B_{dR}^{+}}}$. Then we study the notion
of prismatic crystals in Section 3, where we relate it to log connections, and
compute their cohomology using Čech-Alexander complex. In Section 4, we give a
local calculation of cohomology for an affinoid rigid space that admit a
regular immersion. Here we first introduce the ${\mathrm{B_{dR}^{+}}}$-variant
of Simpson’s functor in Section 4.1. Then in Section 4.2 we construct a
natural functor from infinitesimal crystals to prismatic crystals. In Section
4.3 we compute prismatic cohomology using regular immersions and Simpson’s
functor, and in particular prove Theorem 1.2.1 (in Theorem 4.3.12). Next in
Section 5, we prove the Hodge-Tate filtration theorem (i.e. Theorem 1.1.2
(iii)) in Theorem 5.1.1 and Corollary 5.3.6. We start by proving the version
for smooth rigid spaces in Theorem 5.1.1. Then we prove the existence of
simplicial resolution for affinoid algebras paired with ideals in Section 5.2.
With the help of the simplicial resolution, we are able to define the notion
of derived prismatic cohomology and its Hodge-Tate filtration (Definition
5.3.4). Finally we show derived prismatic cohomology and usual prismatic
cohomology coincide for l.c.i singularities (Theorem 5.3.5), thus finishes the
proof of Theorem 1.2.1 (iii). In Section 6, we prove the infinitesimal
comparison theorem (i.e. Theorem 1.1.2 (ii)) in Theorem 6.0.1. In Section 7,
we prove the pro-étale comparison theorem (i.e. Theorem 1.1.2 (i)) in Theorem
7.2.1. As a byproduct we also give a Künneth formula for reduced prismatic
cohomology in Theorem 7.1.2. Finally we prove the formula on Galois invariant
(i.e. Theorem 1.1.2 (iv)) in Section 8. We start with a brief discussion in
Section 8.1 recalling basic results we need from condensed mathematics. Then
we calculate Galois invariant of infinitesimal cohomology and prismatic
cohomology in Theorem 8.2.3, Corollary 8.2.5, and Theorem 8.2.9.

#### Conventions and notations

We fix a complete perfectoid field extension $K/\mathbb{Q}_{p}$ such that $K$
contains all $p^{\infty}$ roots of unity. We also fix a generator
$\xi=\frac{[\epsilon]-1}{\varphi^{-1}[\epsilon]-1}$ for the canonical
surjection $\theta:{\mathrm{B_{dR}^{+}}}\rightarrow K$, following [Fon94]. For
a non-negative integer $e$, we denote ${\mathrm{A_{inf,e}}}$ to be the
quotient ring ${\mathrm{A_{inf}}}/\xi^{e}$, and ${\mathrm{B^{+}_{dR,e}}}$ to
be ${\mathrm{B_{dR}^{+}}}/\xi^{e}$ with the $p$-adic rational topology. We
assume the basics of quasi-compact quasi-separated topologically of finite
type adic spaces over a $p$-adic field or ${\mathrm{B^{+}_{dR,e}}}$ (which we
call $K$-rigid space or ${\mathrm{B^{+}_{dR,e}}}$-rigid spaces separately),
and refer the reader to [Hu96] for a detailed foundational study. We also use
the notation ${\mathrm{B_{dR}^{+}}}\langle
T_{i}\rangle={\mathrm{B_{dR}^{+}}}\langle T_{1},\ldots,T_{l}\rangle$
throughout the article to denote the inverse limit of affinoid rings
${\mathrm{B^{+}_{dR,e}}}\langle T_{1},\ldots,T_{l}\rangle$ with respect $e$.
For a ${\mathrm{B_{dR}^{+}}}$-module $M$, we use $M_{\mathrm{tf}}$ to denote
its $\xi$-torsionfree quotient $M/(\xi^{\infty}\text{-torsion})$.

We use mildly the language of infinity category and
$\mathbb{E}_{\infty}$-algebras throughout the article, and refer the reader to
[Lu09] and [Lu17] for the foundations.

###  1.5. Acknowledgements

The project was initiated during a conversation with Arthur-César Le Bras. I
thank Le Bras heartily for encouraging me to work on this topic at the very
beginning. It is obvious that the article is heavily influenced by the
foundational work of Bhatt-Scholze [BS19]. I am grateful for various helpful
discussions with Bhargav Bhatt on construction of prismatic sites. Many thanks
go to Sebastian Bartling and Shizhang Li for communications in the early stage
of the work, and to Peter Scholze for explaining condensed group cohomology. I
also thank Dmitry Kubrak and Emanuel Reinecke for asking questions in an
informal seminar, which motivates me to consider the left Kan extension for
rigid spaces; and thank Guido Bosco for useful comments on the draft. At last,
during the writing process I am funded by National Science Foundation FRG
grant no. DMS-1952399, and by Max Planck Institute for Mathematics.

##  2\. Prismatic site over ${\mathrm{B_{dR}^{+}}}$

In this section, we define the prismatic site over ${\mathrm{B_{dR}^{+}}}$ for
an affinoid rigid space over $K$.

###  2.1. Algebraic preliminary

We start with an algebraic preliminary on power-bounded functions of a finite
type scheme over a rigid space. The result here will be used later to show the
weak finality of the prismatic envelope (Proposition 2.2.11).

######  Proposition 2.1.1.

Let $R$ be a topologically finite type algebra over $K$, and let $S$ be a
finite type algebra over $R$. Assume there exists a $K$-linear homomorphism
$\phi:K\langle T\rangle\rightarrow S$. Then the image $\phi(T)$ is finite over
$R$.

When $R$ is the $p$-adic field $K$, this is proved in [HL20, Lemma 5.5].

###### Proof.

  * Step 1

We first reduce to the case when $S$ is normal. Let $S_{\mathrm{red}}$ be the
reduced quotient of the noetherian ring $S$, and let $S_{\mathrm{nor}}$ be the
normalization of $S_{\mathrm{red}}$. We then have the diagram

$\textstyle{K\langle{T}\rangle\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\textstyle{S\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S_{\mathrm{red}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S_{\mathrm{nor}}}$$\textstyle{R\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$.

Denote $t$, $t_{1}$, and $t_{2}$ to be the image of $T$ in $S$,
$S_{\mathrm{red}}$ and $S_{\mathrm{nor}}$ separately. Assume there exists a
monic polynomial $g(X)\in R[X]$ over $R$ such that $g(t_{2})=0$. As the ring
$S_{\mathrm{red}}$ is noetherian, the normalization
$S_{\mathrm{red}}\rightarrow S_{\mathrm{nor}}$ is injective. In particular,
$g(t_{2})=0$ implies $g(t_{1})$ is zero, which then implies that $g(t)$ is
nilpotent in $S$. Thus by taking $h(X):=g(X)^{N}\in R[X]$ for some large $N$,
we can find a monic polynomial $h(X)$ over $R$ with $h(t)=0$. So it suffices
to show that by passing to the normalization of $S$, the image of $T\in
K\langle T\rangle$ is finite over $R$.

  * Step 2

Denote by $X$ the scheme $\operatorname{Spec}(S)$, and $X^{\mathrm{an}}$ the
relative analytification of $S$ over the affinoid ring $R$, as a rigid space
over $\mathrm{Spa}(R)$ (c.f [Con06]). Here we note that $X^{\mathrm{an}}$ is
also a normal rigid space, as the normality is checked around each $K$-valued
point of the scheme $X$ ([Berk93, Theorem 2.2.1]). For each affinoid open
subset $U=\mathrm{Spa}(A)$ of $X^{\mathrm{an}}$, the composition of rings
$K\langle T\rangle\rightarrow S\rightarrow A$ is automatically continuous by
[BGR, 6.1.3/1]. In particular, this induces a map of rigid spaces
$X^{\mathrm{an}}\rightarrow\mathbb{B}^{1}$, where
$\mathbb{B}^{1}=\mathrm{Spa}(K\langle T\rangle)$ is the unit disc over $K$.
Here we note that the ring $S$ is contained in the global section
$\mathcal{O}_{X^{\mathrm{an}}}(X^{\mathrm{an}})$ of the structure sheaf.
Moreover, to give a map from the rigid space $Y=X^{\mathrm{an}}$ to
$\mathbb{B}^{1}$ is equivalent to give a power-bounded global analytic
function over $Y$, i.e. an element in $\mathcal{O}_{Y}^{+}(Y)$. 222To see
this, it suffices to use the criterion of a power-bounded function over a
rigid space: an element $f\in\mathcal{O}_{Y}(Y)$ is power-bounded if and only
if $|f|_{y}\leq 1$ for any rigid point $y\in Y$ ([BGR, 6.2.3/1]). So the image
$t=\phi(T)$ is a power-bounded elements in
$\mathcal{O}_{X^{\mathrm{an}}}(X^{\mathrm{an}})$.

  * Step 3

As the affine scheme $X=\operatorname{Spec}(S)$ is finite type over the
noetherian scheme $\operatorname{Spec}(R)$, by Nagata’s compactification
theorem we could find a schematic open immersion $X\rightarrow\overline{X}$
with dense image, where $\overline{X}$ is a proper
$\operatorname{Spec}(R)$-scheme. In particular, by taking the relative
analytification $\overline{X}^{\mathrm{an}}$ of $\overline{X}$, the induced
morphism $X^{\mathrm{an}}\rightarrow\overline{X}^{\mathrm{an}}$ is an open
immersion of rigid spaces over $\mathrm{Spa}(R)$ whose complement is Zariski
closed of codimension $\geq 1$ in $\overline{X}$. Moreover, notice that since
$X^{\mathrm{an}}$ is normal, it is also an open dense subspace in the
normalization of $\overline{X}^{\mathrm{an}}$, so we can thus assume
$\overline{X}^{\mathrm{an}}$ is normal ([Con99, Theorem 2.1.2]). We apply the
first Riemann extension theorem ([Bar76, $\S$3, Theorem]) for the open
immersion $X^{\mathrm{an}}\rightarrow\overline{X}^{\mathrm{an}}$ into the
proper normal rigid space, to get the isomorphism

$\mathcal{O}^{+}_{X^{\mathrm{an}}}(X^{\mathrm{an}})=\mathcal{O}^{+}_{\overline{X}^{\mathrm{an}}}(\overline{X}^{\mathrm{an}}).$

But since $\overline{X}^{\mathrm{an}}$ is proper over $\mathrm{Spa}(R)$, the
global section of the structure sheaf
$\mathcal{O}_{\overline{X}^{\mathrm{an}}}(\overline{X}^{\mathrm{an}})$ is a
finite $R$-algebra. In this way, the element
$t=\phi(T)\in\mathcal{O}^{+}_{X^{\mathrm{an}}}(X^{\mathrm{an}})=\mathcal{O}^{+}_{\overline{X}^{\mathrm{an}}}(\overline{X}^{\mathrm{an}})\subset\mathcal{O}_{\overline{X}^{\mathrm{an}}}(\overline{X}^{\mathrm{an}})$
is also finite over $R$, and we are done.

∎

The proof above also shows the following result.

######  Corollary 2.1.2.

Let $Y$ be the relative analytification of a finite type scheme over an
affinoid rigid space $\mathrm{Spa}(R)$. Then any power-bounded function of $Y$
is finite over $R$.

The analogous result of Proposition 2.1.1 holds for ${\mathrm{B^{+}_{dR,m}}}$
as well.

######  Corollary 2.1.3.

Let $R$ be a topologically finite type algebra over ${\mathrm{B^{+}_{dR,m}}}$,
and let $S$ be a finite type algebra over $R$. Assume there exists a
$K$-linear homomorphism $\phi:{\mathrm{B^{+}_{dR,m}}}\langle
T\rangle\rightarrow S$. Then the image $\phi(T)$ is finite over $R$.

###### Proof.

By Proposition 2.1.1, the image $\phi(T)$ mod $\xi$ is finite over $R/\xi$. In
particular, there exists a polynomial $g(X)\in{\mathrm{B^{+}_{dR,m}}}[X]$,
such that $g(\phi(T))=0$ mod $\xi$. Thus by taking a power of $g(X)$ we get
the result. ∎

###  2.2. Prismatic site

In this subsection, we introduce the basics on the prismatic site over
${\mathrm{B_{dR}^{+}}}$.

We start by defining the prismatic site.

######  Definition 2.2.1.

Let $X=\mathrm{Spa}(R)$ be an affinoid rigid space over $K$.

  1. (i)

An ${\mathrm{B_{dR}^{+}}}$-_prism_ (or in short a _prism_) $(B,\xi B)$ over
$X$ is defined by the following diagram of maps

$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B/\xi}$$\textstyle{R\ignorespaces\ignorespaces\ignorespaces\ignorespaces,}$$\scriptstyle{f}$

where $B$ is a noetherian ${\mathrm{B_{dR}^{+}}}$-algebra satisfying the
conditions below:

     * •

it is $\xi$-torsion free and $\xi$-adic complete;

     * •

each quotient $B/\xi^{m}$ is a finite type algebra over some
${\mathrm{B^{+}_{dR,m}}}\langle T_{i}\rangle$.

The map $B\rightarrow B/\xi$ is the natural surjection, and $f:R\rightarrow
B/\xi$ is a map of $K$-algebras.

  2. (ii)

A map of prisms $(B,\xi B)\rightarrow(B^{\prime},\xi B^{\prime})$ over $R$ is
defined as a commutative diagram between the data above.

  3. (iii)

The _prismatic site of $X=\mathrm{Spa}(R)$ over ${\mathrm{B_{dR}^{+}}}$_,
denoted as $X/{\mathrm{B_{dR}^{+}}}_{{\mathbbl{\Delta}}}$, is defined on the
opposite category of all ${\mathrm{B_{dR}^{+}}}$-prisms over $X$ and is
equipped with the indiscrete topology.

######  Remark 2.2.2.

By definition, given a prism $(B,\xi B)$ over $R$, the only covering of the
prism $(B,\xi B)$ is the identity map from itself.

######  Remark 2.2.3.

Except for everything is in the rational level (namely over $\mathbb{Q}$), the
most obvious simplification from [BS19] is that a prism does not come with a
delta structure defined integrally. Namely there is no Frobenius lift
structure.

######  Remark 2.2.4.

To save us from complicated notations, we will abbreviate notation of a prism
$(B,\xi B)$ simply into $B$ from time to time, when the meaning is clear.

Next we define the notion of the prismatic envelope.

######  Definition 2.2.5.

Let $B\rightarrow B^{\prime}$ be a surjection of complete noetherian
${\mathrm{B_{dR}^{+}}}$-algebras, and let $I$ be the kernel ideal. The
_prismatic envelope_ of $B^{\prime}$ in $B$, is defined as the
${\mathrm{B_{dR}^{+}}}$-algebra

$B[\frac{I}{\xi}]^{\wedge}_{\mathrm{tf}}:=\left(B[x_{1},\ldots,x_{m}]/(\xi
x_{i}-f_{i},1\leq i\leq m)\right)^{\wedge}/(\xi^{\infty}\text{-}torsion).$

where $(f_{i})$ is a finite set of generators for the ideal $I$, and
$(-)^{\wedge}$ is the classical $\xi$-adic completion.

To make sense of the notation that is independent of the choice of generators,
we have the following easy property.

######  Lemma 2.2.6.

Let $I$ be an ideal of a noetherian ${\mathrm{B_{dR}^{+}}}$-algebra $B$, and
let $(f_{i})\subset(f_{i},g)$ be two finite sets of generators of $I$. Then
the natural map of rings below is an isomorphism

$\left(B[x_{i}]/(\xi
x_{i}-f_{i})\right)^{\wedge}/(\xi^{\infty}\text{-}torsion)\longrightarrow\left(B[x_{i},y]/(\xi
x_{i}-f_{i};\xi y-g)\right)^{\wedge}/(\xi^{\infty}\text{-}torsion).$

In particular, the ring
$B[\frac{I}{\xi}]^{\wedge}_{\mathrm{tf}}/(\xi^{\infty}\text{-}torsion)$ in
Definition 2.2.5 is independent of the choice of generators of $I$.

###### Proof.

As a finite type algebra over $B$ is noetherian, its formal completion along
$\xi=0$ identifies the $\xi^{\infty}$-elements before and after the completion
([Sta, Tag 05E9]). In particular, we can switch the order of the
$\xi$-completion and the $\xi$-torsionfree quotient, and it suffices to show
the isomorphism of the map between incomplete algebras

$\left(B[x_{i}]/(\xi
x_{i}-f_{i})\right)/(\xi^{\infty}\text{-}torsion)\longrightarrow\left(B[x_{i},y]/(\xi
x_{i}-f_{i};\xi y-g)\right)/(\xi^{\infty}\text{-}torsion).$

By assumption, since $(f_{i})$ generates the ideal, we can write $g=\sum
a_{j}f_{j}$. So we get the formula in the second ring above

$\xi\cdot(y-\sum a_{j}x_{j})=0.$

Thus by the $\xi$-torsion freeness of the target ring, the map above is
surjective. Moreover, we can define a section of the above map by mapping $y$
onto $\sum a_{j}x_{j}$, which is possible by the $\xi$-torsion freeness of the
quotient. Hence we get the isomorphism. ∎

######  Remark 2.2.7.

In the above setting, we note that as a $B$-algebra, the envelope is
automatically complete $I$-adically, by the $\xi$-adic completeness and the
$\xi$-divisibility of the image of $I$ in the envelope. In particular, the map
$B\rightarrow B[\frac{I}{\xi}]^{\wedge}_{\mathrm{tf}}$ factors through the
$I$-adic completion of $B$. As a consequence, mod any power of $\xi$, there is
a natural isomorphism of $B$-algebras

$B[\frac{I}{\xi}]^{\wedge}_{\mathrm{tf}}\cong
D[\frac{I}{\xi}]^{\wedge}_{\mathrm{tf}},$

where $D$ is the formal completion of $B$ along the ideal $I$.

######  Remark 2.2.8.

Let $B$ and $I$ be as in Definition 2.2.5, and let $R=B/(I,\xi)$. As the image
of $(I,\xi)$ under the composition

$B\rightarrow B[\frac{I}{\xi}]^{\wedge}_{\mathrm{tf}}\rightarrow
B[\frac{I}{\xi}]^{\wedge}_{\mathrm{tf}}/\xi$

is zero, it naturally induces a continuous map $f:R=B/(I,\xi)\rightarrow
B[\frac{I}{\xi}]^{\wedge}_{\mathrm{tf}}/\xi$. This in particular produces a
prism $B[\frac{I}{\xi}]^{\wedge}_{\mathrm{tf}}$ over $R$. Furthermore, for a
prism $(B^{\prime},\xi B^{\prime})$ in
$X/{\mathrm{B_{dR}^{+}}}_{{\mathbbl{\Delta}}}$, to give a continuous
${\mathrm{B_{dR}^{+}}}$-map from the complete ${\mathrm{B_{dR}^{+}}}$-algebra
$B$ to $B^{\prime}$ such that the composition with $B^{\prime}\rightarrow
B^{\prime}/\xi$ factors through the surjection $B\rightarrow R$, it is
equivalent to give a map of prisms from
$(B[\frac{I}{\xi}]^{\wedge}_{\mathrm{tf}}),B[\frac{I}{\xi}]^{\wedge}_{\mathrm{tf}}/\xi)$
to $(B^{\prime},\xi B^{\prime})$, by the $\xi$-torsion free assumption on
$B^{\prime}$.

######  Definition 2.2.9.

We define the _prism $D_{{\mathbbl{\Delta}}}$ associated to the envelope_ for
the surjection $B\rightarrow I$ to be the one associated to the ring
$B[\frac{I}{\xi}]^{\wedge}_{\mathrm{tf}}$.

To simplify the notation, we also use the same symbol $D_{{\mathbbl{\Delta}}}$
to denote the ring $B[\frac{I}{\xi}]^{\wedge}_{\mathrm{tf}}$ itself.

######  Example 2.2.10.

Let $B$ be the limit ring ${\mathrm{B_{dR}^{+}}}\langle
T,S\rangle=\varprojlim_{n}{\mathrm{B_{dR}^{+}}}/\xi^{n}\langle T,S\rangle$,
and assume $R$ is the quotient ring $K\langle T\rangle$ of $B$. Then the
envelope of the surjection $B\rightarrow{\mathrm{B_{dR}^{+}}}\langle T\rangle$
is given by the following formula

${\mathrm{B_{dR}^{+}}}\langle
T\rangle[\frac{S}{\xi}]^{\wedge}=\varprojlim_{n}\left({\mathrm{B^{+}_{dR,n}}}\langle
T\rangle[\frac{S}{\xi}]\right).$

Thanks to Proposition 2.1.1, the envelope of a ${\mathrm{B_{dR}^{+}}}$-lift of
$R$ into a formally smooth topological algebra over ${\mathrm{B_{dR}^{+}}}$
covers the final object in the prismatic topos.

######  Proposition 2.2.11.

Let $P$ be the ring ${\mathrm{B_{dR}^{+}}}\langle T_{i}\rangle$ and $I$ be an
ideal of $P$, and $f_{0}:P\rightarrow R:=P/(I,\xi)$ be the quotient. Then the
prism $D_{{\mathbbl{\Delta}}}$ associated to the envelope for $P\rightarrow R$
covers the final object of the topos
$\mathrm{Sh}(X/{\mathrm{B_{dR}^{+}}}_{{\mathbbl{\Delta}}})$.

###### Proof.

Let $(B,\xi B)$ be a prism over $R$. We need to show that it admits a map from
the prism associated to the envelope
$P[\frac{I}{\xi}]^{\wedge}_{\mathrm{tf}}$. We first notice the following
commutative diagram

$\textstyle{P={\mathrm{B_{dR}^{+}}}\langle
T_{i}\rangle\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{0}}$$\textstyle{B/\xi}$$\textstyle{R~{}~{}.\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

Then we show the diagram lifts to a homomorphism $P\rightarrow B$ over
${\mathrm{B_{dR}^{+}}}$. By the assumption of a prism in Definition 2.2.1
(ii), the ring $B/\xi^{m}$ admits a surjection from
${\mathrm{B^{+}_{dR,m}}}\langle a_{j_{m}}\rangle[b_{l_{m}}]$ for finite sets
of $j_{m}$ and $l_{m}$, compatible with $m\in\mathbb{N}$. Moreover, by
enlarging the set of $j_{m}$ and $l_{m}$ inductively for each $m$, we may
assume a compatible system of surjections as below

$\textstyle{\mathrm{B_{dR,m+1}^{+}}\langle
a_{j_{m+1}}\rangle[b_{l_{m+1}}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B/\xi^{m+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathrm{B^{+}_{dR,m}}}\langle
a_{j_{m}}\rangle[b_{l_{m}}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B/\xi^{m}.}$

In particular, when $m=1$, the quotient ring $B/\xi$ is a finite type algebra
over an affinoid ring $K\langle a_{j_{1}}\rangle$, and it admits a map from
$P/\xi=K\langle T_{i}\rangle$ by the diagram above. Note that by Proposition
2.1.1 the image $t_{i}$ of each $T_{i}$ in $B/\xi$ is integral over $K\langle
a_{j_{1}}\rangle$. Moreover, by Corollary 2.1.3 any lift $\widetilde{t}_{i}$
of $t_{i}$ in $B/\xi^{m}$ is integral over ${\mathrm{B^{+}_{dR,m}}}\langle
a_{j_{m}}\rangle$. So by choosing compatible lifts $\widetilde{t}_{i}$ of
$t_{i}$ in $B/\xi^{m}$ for each $m\in\mathbb{N}$, we could extend the map
${\mathrm{B^{+}_{dR,m}}}\langle a_{j_{m}}\rangle\rightarrow B/\xi$ to
${\mathrm{B^{+}_{dR,m}}}\langle a_{j_{m}};c_{i}\rangle\rightarrow B/\xi^{m}$,
where $c_{i}$ maps onto lifts $\widetilde{t}_{i}$ of $t_{i}\in B/\xi$. As a
consequence, we can enlarge variables $\\{a_{j_{m}}\\}$ into
$\\{a_{j_{m}},c_{i}\\}$ to get a surjection ${\mathrm{B^{+}_{dR,m}}}\langle
a_{j_{m},c_{i}}\rangle[b_{l_{m}}]\rightarrow B/\xi^{m}\rightarrow B/\xi$. In
particular, we may assume the map $P={\mathrm{B_{dR}^{+}}}\langle
T_{i}\rangle\rightarrow B/\xi$ above factors through the surjections as below:

$\textstyle{P\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{T_{i}\mapsto
c_{i}}$$\textstyle{{\mathrm{B^{+}_{dR,m}}}\langle
a_{j_{m}},c_{i}\rangle[b_{l_{m}}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B/\xi^{m}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B/\xi}$$\textstyle{R~{}~{},\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

which is compatible with $m\in\mathbb{N}$. So by taking an inverse limit with
respect to $m$, we get the commutative diagram

$\textstyle{P\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B/\xi}$$\textstyle{R~{}~{}.\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

At last, since the image of the ideal $I\subset P$ in $B$ is killed by $\xi$
and the ring $B$ is $\xi$-torsion free, the homomorphism $P\rightarrow B$
induces a natural commutative diagram as below

$\textstyle{P[\frac{I}{\xi}]^{\wedge}_{\mathrm{tf}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P[\frac{I}{\xi}]^{\wedge}_{\mathrm{tf}}/\xi=R[\frac{I}{\xi}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{~{}~{}R~{}~{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B/\xi}$$\textstyle{~{}~{}R~{}~{},\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

where $P[\frac{I}{\xi}]^{\wedge}_{\mathrm{tf}}$ is the $\xi$-adic completion
of the ${\mathrm{B_{dR}^{+}}}$-algebra $P[\frac{I}{\xi}]$. Hence the ring
$P[\frac{I}{\xi}]^{\wedge}_{\mathrm{tf}}$ together with the $R$-algebra
structure of its quotient forms a prism $D_{{\mathbbl{\Delta}}}$, and we get a
map of prisms $(D_{{\mathbbl{\Delta}}},\xi
D_{{\mathbbl{\Delta}}})\rightarrow(B,\xi B)$.

∎

##  3\. Prismatic crystal and log connection

There are two natural _structure sheaves_ over
$X/{\mathrm{B_{dR}^{+}}}_{{\mathbbl{\Delta}}}$, defined as follows:

$\displaystyle\mathcal{O}_{{\mathbbl{\Delta}}}$ $\displaystyle:(B,\xi
B)\longmapsto B;$ $\displaystyle\overline{\mathcal{O}}_{{\mathbbl{\Delta}}}$
$\displaystyle:(B,\xi B)\longmapsto B/\xi.$

This allows us to define the notion of the crystal over the prismatic site,
analogous to the infinitesimal story.

######  Definition 3.0.1.

A (coherent) _prismatic crystal_ over $X$ is defined as a sheaf $\mathcal{F}$
of $\mathcal{O}_{{\mathbbl{\Delta}}}$-modules over
$X/{\mathrm{B_{dR}^{+}}}_{{\mathbbl{\Delta}}}$, satisfying

  1. (a)

each $\mathcal{F}(B,\xi B)$ is a finite module over $B$, for every prism
$(B,\xi B)\in X/{\mathrm{B_{dR}^{+}}}_{{\mathbbl{\Delta}}}$;

  2. (b)

for every map of prisms $(B,\xi B)\rightarrow(B^{\prime},\xi B^{\prime})$, the
natural base change map below is an isomorphism of $B^{\prime}$-modules

$\mathcal{F}(B,\xi
B)\otimes_{B}B^{\prime}\longrightarrow\mathcal{F}(B^{\prime},\xi B^{\prime}).$

Notice that given any prismatic crystal $\mathcal{F}$, there is a way to
associate a _reduction_ $\overline{\mathcal{F}}$ to it, by assigning

$(B,\xi B)\longmapsto\mathcal{F}(B,\xi B)\otimes_{B}B/\xi.$

This in particular gives a prismatic crystal over the reduced structure sheaf
$\overline{\mathcal{O}}_{{\mathbbl{\Delta}}}=\mathcal{O}_{{\mathbbl{\Delta}}}\otimes_{\mathrm{B_{dR}^{+}}}K$.

######  Definition 3.0.2.

A prismatic crystal $\mathcal{F}$ is called _flat_ if each $B$-module
$\mathcal{F}(B,\xi B)$ is locally free over $B$. It is called _reduced_ if
each $B$-module $\mathcal{F}(B,\xi B)$ is locally free over $B/\xi$.

Analogous to the infinitesimal theory as in [Guo21, Theorem 3.3.1], a crystal
can be described as an _integrable log connection_ over the envelope.

To start, let $B=\varprojlim_{m}B_{m}$ be an inverse limit of flat noetherian
${\mathrm{B^{+}_{dR,m}}}$-algebras $B_{m}$, such that the transition maps
induce isomorphisms
$B_{m}\otimes_{{\mathrm{B^{+}_{dR,m}}}}\mathrm{B_{dR,m-1}^{+}}\rightarrow
B_{m-1}$, and $B_{1}$ is isomorphic to a formal completion of some Tate
algebra $K\langle T_{1},\ldots,T_{l}\rangle$. Let $I$ be an ideal of $B$.
Denote $D_{{\mathbbl{\Delta}}}$ to be $\xi$-torsion free quotient of the
classical $\xi$-completion of the noetherian ring $B[\frac{I}{\xi}]$.

Fix a map $P:={\mathrm{B_{dR}^{+}}}\langle T_{i}\rangle\rightarrow B$ lifting
$K\langle T_{i}\rangle\rightarrow B_{1}$ as above. We first notice that there
is a natural diagram of ${\mathrm{B_{dR}^{+}}}$-linear continuous
differentials

$\textstyle{P\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d~{}~{}~{}~{}~{}~{}~{}~{}~{}}$$\textstyle{\Omega_{P/{\mathrm{B_{dR}^{+}}}}^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{D_{{\mathbbl{\Delta}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}}$$\textstyle{D_{{\mathbbl{\Delta}}}\otimes_{P}\frac{\Omega_{P/{\mathrm{B_{dR}^{+}}}}^{1}}{\xi}.}$

Here $\Omega_{P/{\mathrm{B_{dR}^{+}}}}^{1}$ is the inverse limit of the
$p$-adic continuous Kähler differential of $P_{m}/{\mathrm{B^{+}_{dR,m}}}$,
and horizontal maps are continuous with respect to $\xi$-adic topology, such
that their mod $\xi$-reduction are continuous differentials with respect to
$K\langle T_{i}\rangle$ and $B/\xi$. On the other hand, the vertical maps are
natural maps induced from the inclusions

$\Omega_{P/{\mathrm{B_{dR}^{+}}}}^{1}\subset\frac{\Omega_{P/{\mathrm{B_{dR}^{+}}}}^{1}}{\xi}\subset\Omega_{P/{\mathrm{B_{dR}^{+}}}}^{1}\otimes_{\mathrm{B_{dR}^{+}}}\mathrm{B_{dR}}.$

In particular, for an element $\frac{f_{1}\cdots f_{i}}{\xi^{i}}$ in
$D_{{\mathbbl{\Delta}}}=P[\frac{I}{\xi}]^{\wedge}_{\mathrm{tf}}$ with
$f_{1},\ldots,f_{i}\in I$, we have

$d(\frac{f_{1}\cdots
f_{i}}{\xi^{i}})=\frac{\sum_{j=1}^{i}f_{1}\cdots\hat{f_{j}}\cdots f_{i}\cdot
df_{j}}{\xi^{i}}=\sum_{j=1}^{i}\frac{f_{1}\cdots\hat{f_{j}}\cdots
f_{i}}{\xi^{i-1}}\cdot\frac{df_{j}}{\xi}\in
D_{{\mathbbl{\Delta}}}\otimes_{P}\frac{\Omega_{P/{\mathrm{B_{dR}^{+}}}}^{1}}{\xi}.$

Now we can consider a generalization to the coefficient theory.

######  Definition 3.0.3.

For a finite generated module $M$ over $D_{{\mathbbl{\Delta}}}$, a _log
connection_ of $M$ is defined as an ${\mathrm{B_{dR}^{+}}}$-linear morphism

$\nabla:M\longrightarrow
M\otimes_{D_{{\mathbbl{\Delta}}}}\frac{\Omega^{1}_{D_{{\mathbbl{\Delta}}}}}{\xi}:=M\otimes_{P}\frac{\Omega_{P/{\mathrm{B_{dR}^{+}}}}^{1}}{\xi},$

such that for $x\in M$ and $f\in D_{{\mathbbl{\Delta}}}$ we have

$\nabla(fx)=f\nabla(x)+x\otimes df,$

where $df$ is the continuous relative differential of $f\in
D_{{\mathbbl{\Delta}}}$ over ${\mathrm{B_{dR}^{+}}}$ constructed above.

The connection is called _integrable_ if the induced composition below is zero

$\nabla^{1}\circ\nabla:M\longrightarrow
M\otimes_{D_{{\mathbbl{\Delta}}}}\frac{\Omega^{2}_{D_{{\mathbbl{\Delta}}}}}{\xi^{2}}:=M\otimes_{P}\frac{\Omega_{P/{\mathrm{B_{dR}^{+}}}}^{2}}{\xi^{2}}.$

Here the map
$\nabla^{1}:M\otimes_{D_{{\mathbbl{\Delta}}}}\frac{\Omega^{1}_{D_{{\mathbbl{\Delta}}}}}{\xi}\longrightarrow
M\otimes_{D_{{\mathbbl{\Delta}}}}\frac{\Omega^{2}_{D_{{\mathbbl{\Delta}}}}}{\xi^{2}}$
is given by the formula

$x\otimes\frac{\omega}{\xi}\longmapsto\nabla(x)\wedge\frac{\omega}{\xi}+x\otimes\frac{d\omega}{\xi}.$

######  Remark 3.0.4.

When $B$ is the $J$-adic completion of $P={\mathrm{B_{dR}^{+}}}\langle
T_{i}\rangle$ for some ideal $J$ in $P$, and $I$ is the zero ideal, the above
definition extends the notion of connections in the infinitesimal theory
defined in [Guo21, Definition 3.1.8], allowing a log pole at $\\{\xi=0\\}$.

######  Remark 3.0.5.

In the special case when $\nabla:M\rightarrow
M\otimes_{D_{{\mathbbl{\Delta}}}}\frac{\Omega_{P/{\mathrm{B_{dR}^{+}}}}^{1}}{\xi}$
factors through
$M\otimes_{D_{{\mathbbl{\Delta}}}}\Omega_{P/{\mathrm{B_{dR}^{+}}}}^{1}$, we
call the connection _integral_. We will see soon that log connections coming
from the infinitesimal site are all integral.

Different from the infinitesimal theory, not all of the integrable log
connections come from crystal. This is more similar to the schematic theory in
mixed characteristic, where _quasi-nilpotence_ condition is needed in order to
make the transition power series associated to a connection to be convergent
(c.f. [BO78]). We now define the ${\mathrm{B_{dR}^{+}}}$-analogue of the
quasi-nilpotence as below.

######  Definition 3.0.6.

Let $B,~{}I,~{}P={\mathrm{B_{dR}^{+}}}\langle T_{1},\ldots,T_{l}\rangle$ be as
above, and $(M,\nabla)$ be a log connection over
$D_{{\mathbbl{\Delta}}}=B[\frac{I}{\xi}]^{\wedge}_{\mathrm{tf}}$. We say
$(M,\nabla)$ is _quasi-nilpotent_ if for any element $x\in M$, the following
power series are convergent in the module
$M\otimes_{D_{{\mathbbl{\Delta}}}}B[\frac{I}{\xi},\widetilde{\delta}_{1},\ldots,\widetilde{\delta}_{l}]^{\wedge}$

$\sum_{E=(e_{i})}\prod_{i=1}^{l}\widetilde{\nabla}_{i}^{e_{i}}(x)\otimes\frac{\widetilde{\delta}_{i}^{e_{i}}}{e_{i}!}.$

Here $\widetilde{\nabla}_{i}:M\rightarrow M$ is defined as the composition

$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\nabla~{}~{}~{}~{}~{}~{}~{}~{}}$$\textstyle{M\otimes_{P}\frac{\Omega_{P/{\mathrm{B_{dR}^{+}}}}^{1}}{\xi}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{~{}~{}~{}~{}~{}~{}~{}x\otimes\frac{dT_{i}}{\xi}\mapsto
x}$$\scriptstyle{~{}~{}~{}~{}~{}~{}~{}~{}x\otimes\frac{dT_{j}}{\xi}\mapsto
0}$$\textstyle{M,}$

where $j$ above is not equals to $i$.

######  Remark 3.0.7.

As an exercise of Taylor expansion in calculus, it can be shown that in order
to show the quasi-nilpotence, it is enough to check the convergence for a set
of generators in Definition 3.0.6.

######  Remark 3.0.8.

The map $\widetilde{\nabla}_{i}$ should be thought as the multiplication of
$\xi$ with the classically defined map $\nabla_{i}$, where the latter is the
composition of $\nabla$ and the paring with $\frac{\partial}{\partial T_{i}}$.
More explicitly the pairing map sends $\Omega_{P/{\mathrm{B_{dR}^{+}}}}^{1}$
to $P$ by the formula

$dT_{i}\longmapsto 1;~{}dT_{j}\longmapsto 0,~{}\forall j\neq i.$

This could be made precise either by inverting $\xi$ or when $M$ has no
$\xi$-torsion.

######  Example 3.0.9.

As a simple example of a quasi-nilpotent log connection, let $R=K\langle
T\rangle,$ $P={\mathrm{B_{dR}^{+}}}\langle T\rangle$ and $I=(0)$, so that
$D_{{\mathbbl{\Delta}}}=P$. Let $(M,\nabla)=({\mathrm{B_{dR}^{+}}}\langle
T\rangle,d)$ be the natural continuous differential operator on the structure
sheaf with $df(T)=f^{\prime}(T)dT$. Then for an element
$f(T)\in{\mathrm{B_{dR}^{+}}}\langle T\rangle$, the power series in Definition
3.0.6 is

$\sum_{e}f^{(e)}(T)\cdot\xi^{e}\otimes\frac{\widetilde{\delta}^{e}}{e!}=f(T+\xi\widetilde{\delta}),$

which is convergent in the ring
$P[{\widetilde{\delta}}]^{\wedge}={\mathrm{B_{dR}^{+}}}\langle
T\rangle[\widetilde{\delta}]^{\wedge}$, by checking mod $\xi^{n}$ for every
$n\in\mathbb{N}$. Thus the canonical continuous differential on $P$ is quasi-
nilpotent.

In fact, as we will see, this is the (log) connection corresponds to the
prismatic structure sheaf $\mathcal{O}_{{\mathbbl{\Delta}}}$ over
$R/{\mathrm{B_{dR}^{+}}}_{{\mathbbl{\Delta}}}$.

######  Example 3.0.10.

If a log connection is integral, then it is quasi-nilpotent. To show this, it
suffices to notice that by assumption in Remark 3.0.5 when $(M,\nabla)$ is
integral, each $\widetilde{\nabla}_{i}$(x) is divisible by $\xi$.

######  Example 3.0.11.

To give an example that is not quasi-nilpotent, still consider the special
setting as above when $R=K\langle T\rangle$, $P={\mathrm{B_{dR}^{+}}}\langle
T\rangle$ and $I$ is the zero ideal, in which case the envelope
$D_{{\mathbbl{\Delta}}}$ is equal to the ring $P$. Let $M$ be
$D_{{\mathbbl{\Delta}}}$ itself, and let $\nabla$ be the twisted differential
map

$\nabla=\frac{d}{\xi}:{\mathrm{B_{dR}^{+}}}\langle
T\rangle\longrightarrow{\mathrm{B_{dR}^{+}}}\langle
T\rangle\otimes\frac{dT}{\xi};~{}f(T)\longmapsto
f^{\prime}(T)\otimes\frac{dT}{\xi}.$

Then this is an integrable log connection of $M$ over $D_{{\mathbbl{\Delta}}}$
but is not nilpotent. To see the latter, it suffices to notice that given
$f(T)\in{\mathrm{B_{dR}^{+}}}\langle T\rangle$, the formal power series in
Definition 3.0.6 is equal to

$\sum_{e}f^{(e)}(T)\otimes\frac{\widetilde{\delta}^{e}}{e!}=f(T+\widetilde{\delta}),$

which after mod $\xi$ is not convergent in $K\langle
T\rangle[\widetilde{\delta}]$ whenever $f$ mod $\xi$ is an infinite series.

######  Proposition 3.0.12.

Let $P={\mathrm{B_{dR}^{+}}}\langle T_{i}\rangle$, let $I$ be an ideal of $P$,
let $R$ be the topologically finite type algebra $P/(I,\xi)$ and let
$D_{{\mathbbl{\Delta}}}=P[\frac{I}{\xi}]^{\wedge}_{\mathrm{tf}}$ be the
associated envelope. Then the following two categories are equivalent:

$\displaystyle\\{coherent~{}prismatic~{}crystals~{}over~{}R\\}$
$\displaystyle\longrightarrow\\{(M,\nabla)~{}|~{}M\in{\mathrm{Coh}}(D_{{\mathbbl{\Delta}}}),~{}\nabla~{}integrable~{}quasi\text{-}nilpotent~{}log~{}connection\\}$
$\displaystyle\mathcal{F}$
$\displaystyle\longmapsto(\mathcal{F}(D_{{\mathbbl{\Delta}}}),\nabla_{D_{{\mathbbl{\Delta}}}}).$

###### Proof.

The proof is similar to that of [Guo21, Theorem 3.3.1]. We first notice that
by Proposition 2.2.11 the prism $D_{{\mathbbl{\Delta}}}$ associated to the
prismatic envelope for the surjection $P\rightarrow P/I$ covers the final
object of the topos. Let $\mathcal{F}$ be a coherent prismatic crystal over
$R$. By Taylor expansion, the formula in Definition 3.0.6 is exactly the
transition isomorphism of the pullbacks along the two projections

$\phi:\mathrm{pr}_{1}^{*}\mathcal{F}(D_{{\mathbbl{\Delta}}})\rightarrow\mathrm{pr}_{0}^{*}\mathcal{F}(D_{{\mathbbl{\Delta}}}),$

where $\mathrm{pr}_{i}$ are two canonical maps of rings
$D(1)_{{\mathbbl{\Delta}}}=P[\frac{I,\Delta(1)}{\xi}]^{\wedge}_{\mathrm{tf}}\leftleftarrows
P[\frac{I}{\xi}]$ induced by $P\langle\Delta(1)\rangle\leftleftarrows P$. Here
$\Delta(1)$ is the kernel ideal for the diagonal surjection
$P^{\widehat{\otimes}_{{\mathrm{B_{dR}^{+}}}}2}\rightarrow P$. To define the
log connection on $D_{{\mathbbl{\Delta}}}$, we note that the following map has
the image in
$\mathcal{F}(D_{{\mathbbl{\Delta}}})\otimes_{P}\frac{\Delta(1)/\Delta(1)^{2}}{\xi}$
(which is isomorphic to
$\cong\mathcal{F}(D_{{\mathbbl{\Delta}}})\otimes_{P}\frac{\Omega_{P/{\mathrm{B_{dR}^{+}}}}^{1}}{\xi}$)

$\operatorname{id}\otimes
1-\phi\circ(1\otimes\operatorname{id}):\mathcal{F}(D_{{\mathbbl{\Delta}}})\longrightarrow\mathrm{pr}_{0}^{*}\mathcal{F}(D_{{\mathbbl{\Delta}}})/(\frac{\Delta(1)^{2}}{\xi^{2}}).$

This allows us to define the log connection as

$\nabla:\mathcal{F}(D_{{\mathbbl{\Delta}}})\longrightarrow\mathcal{F}(D_{{\mathbbl{\Delta}}})\otimes_{P}\frac{\Omega_{P/{\mathrm{B_{dR}^{+}}}}^{1}}{\xi}.$

Then as a consequence of the crystal condition, both
$\mathrm{pr}_{i}^{*}\mathcal{F}(D_{{\mathbbl{\Delta}}})$ are isomorphic to
$\mathcal{F}(D(1)_{{\mathbbl{\Delta}}})$, and the existence of the isomorphism
implies the convergence of the power series. Moreover, integrability for the
log connection coming from the crystal follows from the crystal conditions
again, and can be proved as in [Sta][07J6].

Conversely, given an integrable, quasi-nilpotent log connection $(M,\nabla)$
over an envelope, Lemma 2.2.11 allows us to define a finite module over any
prism $(B,\xi B)$ via the pullback along a map
$D_{{\mathbbl{\Delta}}}\rightarrow B$. Then the rest is to show the
independence of the choice of maps, which follows from the quasi-nilpotence
and the integrability, and can be checked as in the proof of [Guo21, Theorem
3.3.1] (especially [Guo21, Claim 3.3.4]). Here we note that the quasi-
nilpotence assumption is used to show the formal power series associated to
the transition morphism converges, thus exists. ∎

We also compute the Čech-Alexander complex under the prismatic setting. As a
preparation, we introduce the following notations. As before, let
$P={\mathrm{B_{dR}^{+}}}\langle T_{1},\ldots,T_{m}\rangle\rightarrow
P/I\rightarrow R=P/(I,\xi)$ be surjections, and let $D_{{\mathbbl{\Delta}}}$
be the associated envelope. Denote $P(l)$ to be the inverse limit

$P(l):=\varprojlim_{n}\left({\mathrm{B^{+}_{dR,n}}}\langle
T_{i}\rangle^{\widehat{\otimes}_{{\mathrm{B^{+}_{dR,n}}}}(l+1)}\right)\cong\varprojlim_{n}{\mathrm{B^{+}_{dR,n}}}\langle
T_{i},\delta_{j,i}\rangle,$

where $\delta_{j,i}$ for $1\leq j\leq l,~{}1\leq i\leq m$ correspond to
natural generators of the kernel ideal $\Delta(l)$ for the diagonal surjection
$P(l)\rightarrow P$, and the complete tensor product
$\widehat{\otimes}_{{\mathrm{B^{+}_{dR,n}}}}$ above is defined as the
integrally $p$-complete tensor product.

######  Proposition 3.0.13.

Let $P,R$ and $P(l)$ be as above. The $(l+1)$-th self fiber product of the
prism $D_{{\mathbbl{\Delta}}}$ over the final object in the prismatic topos is
representable by the prism associated to the prismatic envelope
$D(l)_{{\mathbbl{\Delta}}}$ for $P(l)\rightarrow P(l)/(\Delta(l),I)=P/I$.

###### Proof.

Let $(B,\xi B)$ be a prism over $R$. We first notice that by Remark 2.2.8, to
give a map of prisms from $(D_{{\mathbbl{\Delta}}},\xi
D_{{\mathbbl{\Delta}}})$ to $(B,\xi B)$, it is equivalent to give a continuous
${\mathrm{B_{dR}^{+}}}$-linear morphism $g:P={\mathrm{B_{dR}^{+}}}\langle
T_{i}\rangle\rightarrow B$, satisfying the following commutative diagram

$\textstyle{P\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{R\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B/\xi.}$

This implies that a map from $(l+1)$-th self product of the prism
$(D_{{\mathbbl{\Delta}}},\xi D_{{\mathbbl{\Delta}}})$ to $(B,\xi B)$ is
equivalent to $(l+1)$ continuous ${\mathrm{B_{dR}^{+}}}$-linear morphisms
$g_{0},\ldots,g_{l}:P\rightarrow B$, such that the induced compositions
$P\rightarrow B\rightarrow B/\xi$ all factor through the morphism
$f:R\rightarrow B/\xi$.

We then consider the integral level. To give $(l+1)$ continuous maps from $P$
to $B$, it suffices to give a sequence of ${\mathrm{A_{inf,n}}}$-linear (for
${\mathrm{A_{inf,n}}}={\mathrm{A_{inf}}}/\xi^{n}$) continuous maps
$g_{0,n},\ldots,g_{l,n}:{\mathrm{A_{inf,n}}}\langle T_{i}\rangle\rightarrow
B/\xi^{n}$, compatible with $n\in\mathbb{N}$ via natural surjections, and with
the commutative diagrams $(\ast)$ above (after inverting $p$). When $n=1$, the
maps $g_{j,n}$ all factors through the same map $\mathcal{O}_{K}\langle
T_{i}\rangle\rightarrow R_{0}\rightarrow B/\xi$, where $R_{0}$ is a ring of
definition of $R$ and $R_{0}\rightarrow B/\xi$ is induced from $f:R\rightarrow
B/\xi$. This is equivalent to give a map $\mathcal{O}_{K}\langle
T_{i}\rangle^{\otimes_{\mathcal{O}_{K}}(l+1)}\rightarrow B/\xi$ that factors
through the diagonal and the above surjection $\mathcal{O}_{K}\langle
T_{i}\rangle\rightarrow R_{0}\rightarrow B/\xi$. Notice that the tensor
product is a priori only the ordinary tensor product of
$\mathcal{O}_{K}$-rings, but since $R_{0}$ is $p$-complete, the above induces
uniquely a continuous map $\mathcal{O}_{K}\langle
T_{i}\rangle^{\widehat{\otimes}_{\mathcal{O}_{K}}(l+1)}\rightarrow B/\xi$ for
the $p$-adic completed tensor product. For general $n\in\mathbb{N}$, the
choice of $\\{g_{0,n},\ldots,g_{l,n}\\}$ amounts to give a map from
${\mathrm{A_{inf,n}}}\langle
T_{i}\rangle^{\otimes_{{\mathrm{A_{inf,n}}}}(l+1)}\rightarrow B/\xi^{n}$,
whose composition with $B/\xi^{n}\rightarrow B/\xi$ factors through the
diagonal surjection of ${\mathrm{A_{inf,n}}}\langle
T_{i}\rangle^{\otimes_{{\mathrm{A_{inf,n}}}}(l+1)}$ and
${\mathrm{A_{inf,n}}}\langle T_{i}\rangle\rightarrow R_{0}\rightarrow B/\xi$.
Using Corollary 2.1.3, the images of incomplete tensor product in $B/\xi^{n}$
are within an affinoid algebra over ${\mathrm{B^{+}_{dR,n}}}$, so the map
uniquely factors through a map ${\mathrm{A_{inf,n}}}\langle
T_{i}\rangle^{\widehat{\otimes}_{{\mathrm{A_{inf,n}}}}(l+1)}\rightarrow
B/\xi^{n}$, where the tensor product of the source is $p$-completed. In this
way, by inverting $p$ and taking the inverse limit with respect to $n$, we see
to give $(l+1)$ continuous ${\mathrm{B_{dR}^{+}}}$-linear map from
$P\rightarrow B$ satisfying the diagram $(\ast)$, it is the same as giving the
following commutative diagram

$\textstyle{P(l)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P={\mathrm{B_{dR}^{+}}}\langle
T_{i}\rangle\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{R\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B/\xi}$.

In this way, by the construction of the prismatic envelope in Definition 2.2.5
and Remark 2.2.8, we see this is equivalent to give a map of prisms from
$(D(l)_{{\mathbbl{\Delta}}},\xi D(l)_{{\mathbbl{\Delta}}})$ to $(B,\xi B)$,
where $D(l)_{{\mathbbl{\Delta}}}$ is the prismatic envelope.

∎

######  Corollary 3.0.14.

Let $P={\mathrm{B_{dR}^{+}}}\langle T_{i}\rangle\rightarrow P/I\rightarrow
R=P/(I,\xi)$ be surjections with $R$ being topologically finite type over $K$.
Let $\mathcal{F}$ be a sheaf over
$X/{\mathrm{B_{dR}^{+}}}_{{\mathbbl{\Delta}}}$, and let
$D(l)_{{\mathbbl{\Delta}}}$ be the prismatic envelope as in Lemma 3.0.13. Then
the following natural map from the homotopy limit over the simplicial diagram
$\Delta^{\mathrm{op}}$ is a quasi-isomorphism

$R\varprojlim_{[l]\in\Delta^{\mathrm{op}}}\mathcal{F}(D(l)_{{\mathbbl{\Delta}}},\xi
D(l)_{{\mathbbl{\Delta}}})\longrightarrow
R\Gamma(X/{\mathrm{B_{dR}^{+}}}_{{\mathbbl{\Delta}}},\mathcal{F}).$

###### Proof.

This follows from the general category theory and Proposition 3.0.13, as the
the simplicial diagram of prisms $[l]\mapsto(D(l)_{{\mathbbl{\Delta}}},\xi
D(l)_{{\mathbbl{\Delta}}})$ in the prismatic site
$X/{\mathrm{B_{dR}^{+}}}_{{\mathbbl{\Delta}}}$ forms the Čech nerve of the
covering map $(D_{{\mathbbl{\Delta}}},\xi D_{{\mathbbl{\Delta}}})\rightarrow
1_{X/{\mathrm{B_{dR}^{+}}}_{{\mathbbl{\Delta}}}}$ (Lemma 2.2.11), where
$1_{X/{\mathrm{B_{dR}^{+}}}_{{\mathbbl{\Delta}}}}$ is the final object in the
prismatic topos $\mathrm{Sh}(X/{\mathrm{B_{dR}^{+}}}_{{\mathbbl{\Delta}}})$. ∎

As an application, we obtain a tensor product formula relating the cohomology
of a crystal and its reduction.

######  Corollary 3.0.15.

Let $\mathcal{F}$ be a flat prismatic crystal over $X$, and let
$\overline{\mathcal{F}}=\mathcal{F}\otimes_{\mathrm{B_{dR}^{+}}}K$ be its
reduction. Then the natural map below is an isomorphism

$R\Gamma(X/{\mathrm{B_{dR}^{+}}}_{{\mathbbl{\Delta}}},\mathcal{F})\otimes^{L}_{\mathrm{B_{dR}^{+}}}K\longrightarrow
R\Gamma(X/{\mathrm{B_{dR}^{+}}}_{{\mathbbl{\Delta}}},\overline{\mathcal{F}}).$

###### Proof.

First notice the map of cohomology above is induced by the reduction map of
crystals $\mathcal{F}\rightarrow\overline{\mathcal{F}}$. So it suffices to
check that the map of cohomology is an isomorphism, assuming $R$ admits a
surjection as in Corollary 3.0.14. Then we notice that by the local freeness
of each $\mathcal{F}(D(l)_{{\mathbbl{\Delta}}},\xi D(l)_{{\mathbbl{\Delta}}})$
over $D(l)$ and the $\xi$-torsionfreeness of $D(l)$ over
${\mathrm{B_{dR}^{+}}}$, the derived tensor product of the cosimplicial
diagram $[l]\mapsto\mathcal{F}(D(l)_{{\mathbbl{\Delta}}},\xi
D(l)_{{\mathbbl{\Delta}}})$ with $K$ is equal to

$[l]\longmapsto\overline{\mathcal{F}}(D(l)_{{\mathbbl{\Delta}}},\xi
D(l)_{{\mathbbl{\Delta}}})=\mathcal{F}(D(l)_{{\mathbbl{\Delta}}},\xi
D(l)_{{\mathbbl{\Delta}}})/\xi.$

Thus the rest follows from Corollary 3.0.14. ∎

##  4\. From infinitesimal to prismatic

In this section, we show how to compute cohomology of a prismatic crystal
locally by using infinitesimal cohomology, for rigid spaces that has l.c.i.
singularity.

###  4.1. A variant of Simpson’s functor

We start this subsection with introducing a variant of Simpson’s functor in
[Simp, Lemma 19], over the ($\infty$) category of filtered objects (rings,
modules, or complexes) over a ${\mathrm{B_{dR}^{+}}}$-algebra. This will be
used to connect the infinitesimal theory to the prismatic theory.

######  Construction 4.1.1.

Let $(M,{\mathrm{Fil}}^{\bullet}M)$ be a $\mathbb{N}^{\mathrm{op}}$-indexed
descending filtered complex of ${\mathrm{B_{dR}^{+}}}$-modules. Namely
${\mathrm{Fil}}^{\bullet}M$ is an object in
${\mathscr{DF}}({\mathrm{Mod}}_{{\mathrm{B_{dR}^{+}}}})=\textrm{Fun}(\mathbb{N}^{\mathrm{op}},{\mathrm{Mod}}_{{\mathrm{B_{dR}^{+}}}})$
as in the beginning of [Guo21, Section 5]. Then we define the functor
$\Psi(M,{\mathrm{Fil}}^{\bullet}M)$ (or $\Psi(M)$ in short if there is no
confusion) as an object in $\mathscr{D}({\mathrm{B_{dR}^{+}}})$ by the
following formula

$\Psi(M,{\mathrm{Fil}}^{\bullet}M):=\left({\mathrm{colim}}_{i\in\mathbb{N}}\frac{Fil^{i}M}{\xi^{i}}\right)^{\wedge}.$

Here $(-)^{\wedge}$ above is the derived $\xi$-adic completion, and precisely
the colimit is taken over the following diagram of
${\mathrm{B_{dR}^{+}}}$-complexes

$\textstyle{{\mathrm{Fil}}^{0}M}$$\textstyle{{\mathrm{Fil}}^{1}M\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cdot\xi}$$\textstyle{{\mathrm{Fil}}^{1}M}$$\textstyle{{\mathrm{Fil}}^{2}M\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cdot\xi}$$\textstyle{{\mathrm{Fil}}^{2}M}$$\textstyle{{\mathrm{Fil}}^{3}M\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cdot\xi}$$\textstyle{\vdots,}$

where all horizontal arrows are defining morphisms of the filtration.

######  Remark 4.1.2.

The idea of the diagram in Construction 4.1.1 is that we want to add the
denominator $\xi^{i}$ to the $i$-th filtration. To see this, we change the
notation and denote $\frac{{\mathrm{Fil}}^{i}M}{\xi^{i}}$ to be the
${\mathrm{B_{dR}^{+}}}$-complex ${\mathrm{Fil}}^{i}M$ but formally adjoining
the denominator $\frac{1}{\xi^{i}}$. Then the diagram in 4.1.1 can be
rewritten as the following

$\textstyle{{\mathrm{Fil}}^{0}M}$$\textstyle{{\mathrm{Fil}}^{1}M\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\frac{{\mathrm{Fil}}^{1}M}{\xi}}$$\textstyle{\frac{{\mathrm{Fil}}^{2}M}{\xi}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\frac{{\mathrm{Fil}}^{2}M}{\xi^{2}}}$$\textstyle{\frac{{\mathrm{Fil}}^{3}M}{\xi^{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\vdots,}$

where each vertical map identifies $\frac{Fil^{i}M}{\xi^{i-1}}$ with
$\xi\cdot\frac{Fil^{i}M}{\xi^{i}}$ via the multiplication by $\xi$. Note that
in the special case when each ${\mathrm{Fil}}^{i}M$ is a complex of
$\xi$-torsionfree ${\mathrm{B_{dR}^{+}}}$-modules, the vertical maps are all
inclusion maps.

######  Remark 4.1.3.

When $(M,{\mathrm{Fil}}^{\bullet}M)=(A,I^{\bullet})$ is a noetherian
$\xi$-torsionfree ${\mathrm{B_{dR}^{+}}}$-algebra $A$ together with a
filtration given by a power of its ideal $I$, the module $\Psi(A,I^{\bullet})$
is also equipped with a natural ${\mathrm{B_{dR}^{+}}}$-algebra structure, by

$\frac{I^{i}}{\xi^{i}}\otimes\frac{I^{j}}{\xi^{j}}\longrightarrow\frac{I^{i+j}}{\xi^{i+j}}.$

More generally, under the Day convolution on
${\mathscr{DF}}({\mathrm{Mod}}_{{\mathrm{B_{dR}^{+}}}})$ (see for example
[GP18]), the functor
$\Psi:{\mathscr{DF}}({\mathrm{Mod}}_{{\mathrm{B_{dR}^{+}}}})\rightarrow\mathscr{D}({\mathrm{B_{dR}^{+}}})$
is lax-symmetric monoidal, thus inducing a functor on $\mathbb{E}_{\infty}$:

$\Psi:\mathrm{CAlg}({\mathscr{DF}}({\mathrm{Mod}}_{{\mathrm{B_{dR}^{+}}}}))\longrightarrow\mathrm{CAlg}(\mathscr{D}({\mathrm{Mod}}_{{\mathrm{B_{dR}^{+}}}})).$

Intuitively, using the notation in the above remark, this is to say the
following multiplicative structure in $\Psi(M,{\mathrm{Fil}}^{\bullet}M)$

$\frac{{\mathrm{Fil}}^{i}M}{\xi^{i}}\otimes\frac{{\mathrm{Fil}}^{j}M}{\xi^{j}}\longrightarrow\frac{{\mathrm{Fil}}^{i+j}M}{\xi^{i+j}}.$

######  Remark 4.1.4.

Moreover, when $(M,{\mathrm{Fil}}^{\bullet}M)$ is a filtered complex of
$A$-modules for a $\xi$-adic complete ${\mathrm{B_{dR}^{+}}}$-algebra $A$, as
the arrows in the diagram of 4.1.1 above are all $A$-linear, the object
$\Psi(M,{\mathrm{Fil}}^{\bullet}M)$ is also a complex of $A$-modules.

In the following, for a $K$-module $M$, we denote
$M\otimes_{K}\xi^{i}/\xi^{i+1}$ as $M(i)$, called the _$i$ -th Tate twist_ of
$M$. This would not change the underlying $K$-module structure, but its Galois
structure is replaced by the $i$-th Tate twist when $M$ itself is equipped
with one.

######  Remark 4.1.5.

Given a filtered ${\mathrm{B_{dR}^{+}}}$-complex
$(M,{\mathrm{Fil}}^{\bullet}M)$ such that ${\mathrm{Fil}}^{\bullet}M$ is
uniformly bounded to the right, we note that the derived mod $\xi$ reduction
has the following formula

$\Psi(M,{\mathrm{Fil}}^{\bullet}M)\otimes^{L}_{{\mathrm{B_{dR}^{+}}}}K\cong\bigoplus_{n\in\mathbb{N}}{\mathrm{gr}}^{n}(M\otimes^{L}_{\mathrm{B_{dR}^{+}}}K)(-n),$

where $M\otimes^{L}_{\mathrm{B_{dR}^{+}}}K$ is the filtered complex of
$K$-modules endowed with filtration
${\mathrm{Fil}}^{i}M\otimes^{L}_{\mathrm{B_{dR}^{+}}}K$. This follows from
[Sta, Tag 0EEV], the commutativity of colimit and derived tensor product, and
the diagram in Construction 4.1.1.

###  4.2. Prisms associated to infinitesimal thickenings

We now construct a functor from the infinitesimal site to the prismatic site,
over the de Rham period ring ${\mathrm{B_{dR}^{+}}}$.

To start, we consider the following pro-version analogue of the infinitesimal
site considered in [Guo21].

######  Definition 4.2.1.

Let $X$ be a rigid space over $K$. The _pro-infinitesimal site_ of
$X/{\mathrm{B_{dR}^{+}}}$, denoted as
$X/{\mathrm{B_{dR}^{+}}}_{\mathrm{pinf}}$, is defined as the category of
$(U,T_{\ast}):=(U,T_{e})$ for $e\in\mathbb{N}$, such that

  * •

each $(U,T_{e})$ is an infinitesimal thickening of affinoid rigid spaces over
${\mathrm{B^{+}_{dR,e}}}$ (in the sense of [Guo21, Section 2.1], with $T_{e}$
flat over ${\mathrm{B^{+}_{dR,e}}}$ and $U$ an affinoid open subspace of $X$;

  * •

transition maps $T_{e}\rightarrow T_{e+1}$ are also infiniteisimal thickenings
that are compatible with maps from $U$, and there is a compatible system of
closed immersions $T_{e}\rightarrow\mathrm{Spa}({\mathrm{B^{+}_{dR,e}}}\langle
x_{1},\ldots,x_{l}\rangle)$ for some fixed $l\in\mathbb{N}$.

The category is equipped with indiscrete topology.

######  Remark 4.2.2.

Let $R$ be a topologically of finite type $K$-algebra, and let $P\rightarrow
R$ be a surjection for a smooth ${\mathrm{B_{dR}^{+}}}$-algebra $P$ (in the
sense that each $P/\xi^{e}$ is smooth over ${\mathrm{B^{+}_{dR,e}}}$ as an
adic space). Then the induced pro-infinitesimal thickening
$(U=\mathrm{Spa}(R),T_{e}=\mathrm{Spa}(P/I^{e}))$ is weakly final over
$\mathrm{Spa}(R)/{\mathrm{B_{dR}^{+}}}_{\mathrm{inf}}$, where
$I=\ker(P\rightarrow R)$.

######  Remark 4.2.3.

Analogous to the classical infinitesimal theory, we can define the
infinitesimal structure sheaf $\mathcal{O}_{X/{\mathrm{B_{dR}^{+}}}}$ and the
notion of crystals over $X/{\mathrm{B_{dR}^{+}}}_{\mathrm{pinf}}$, as in
[Guo21]. Here for a pro-infinitesimal thickening
$(U,T_{e}=\mathrm{Spa}(B_{e}))$ we have

$\mathcal{O}_{X/{\mathrm{B_{dR}^{+}}}}(U,T_{e})=\varprojlim_{e}B_{e}.$

Moreover, by [Guo21, Theorem 7.2.3, Theorem 3.3.1], the category of crystals
over $X/{\mathrm{B_{dR}^{+}}}_{\mathrm{inf}}$ in the sense of [Guo21] is
naturally equivalent to that over $X/{\mathrm{B_{dR}^{+}}}_{\mathrm{pinf}}$,
inducing natural isomorphisms between their cohomology.

Next we define a natural functor from the infinitesimal site to the prismatic
site. Let $R$ be a fixed topologically finite type algebra over $K$, and let
$X=\mathrm{Spa}(R)$.

######  Definition 4.2.4.

Let $(U=\mathrm{Spa}(A),T_{e}=\mathrm{Spa}(B_{e}))\in
X/{\mathrm{B_{dR}^{+}}}_{\mathrm{pinf}}$ be a pro-infinitesimal thickening
over the rigid space $X=\mathrm{Spa}(R)$ over $K$, and let
$B=\varprojlim_{e}B_{e}$, with the ideal $I=\ker(B\rightarrow A)$. The
_associated prism_ of $(U,T_{e})$ is defined as

$\left(B^{\prime}\rightarrow B^{\prime}/\xi\leftarrow R\right),$

where $B^{\prime}=B[\frac{I}{\xi}]^{\wedge}_{\mathrm{tf}}$, the symbol
$(-)^{\wedge}$ is the classical $\xi$-adic completion, and the arrow on the
right is the natural composition $R\rightarrow A\rightarrow
B[\frac{I}{\xi}]^{\wedge}_{\mathrm{tf}}/\xi$.

######  Remark 4.2.5.

When $(U,T_{e})$ is a pro-infinitesimal thickening as in Remark 4.2.2, by the
construction above together with Proposition 2.2.11, the associated prism
covers the final object of the prismatic site.

By construction, this defines a map of ringed sites

$\pi:(X/{\mathrm{B_{dR}^{+}}}_{{\mathbbl{\Delta}}},\mathcal{O}_{{\mathbbl{\Delta}}})\longrightarrow(X/{\mathrm{B_{dR}^{+}}}_{\mathrm{pinf}},\mathcal{O}_{X/{\mathrm{B_{dR}^{+}}}}).$

Moreover, by taking the coherent pullback, we can produce a prismatic crystal
from an infinitesimal crystal.

######  Definition/Lemma 4.2.6.

There is a natural functor between categories of crystals

$\pi^{*}:\mathrm{Cry}_{X/{\mathrm{B_{dR}^{+}}}_{\mathrm{pinf}}}\rightarrow\mathrm{Cry}_{X/{\mathrm{B_{dR}^{+}}}_{{\mathbbl{\Delta}}}},$

such that for an infinitesimal thickening
$(U=\mathrm{Spa}(A),T_{e}=\mathrm{Spa}(B_{e}))$ with $(B^{\prime}\rightarrow
B^{\prime}/\xi\leftarrow R)$ being the associated prism as in Definition
4.2.4, we have

$\pi^{*}\mathcal{F}(B^{\prime})=\mathcal{F}(U,T_{e})\otimes_{B}B^{\prime}.$

###### Proof.

For each $(U=\mathrm{Spa}(A),T_{e}=\mathrm{Spa}(B_{e}))$, we construct a
module over $B^{\prime}=B[\frac{I}{\xi}]^{\wedge}_{\mathrm{tf}}$ by taking the
tensor product as in the formula above. Then by the crystal condition of
$\mathcal{F}\in\mathrm{Cry}_{X/{\mathrm{B_{dR}^{+}}}_{\mathrm{inf}}}$, this
satisfies the crystal condition with respect to prisms coming from
infinitesimal thickenings. Then it suffices to notice that by Remark 4.2.5 the
category of associated prisms covers the final objects of the prismatic topos.
∎

###  4.3. Local calculation for l.c.i singularities

In this subsection, we give some concrete local formulae computing prismatic
cohomology of crystals using Simpson’s functor, for affinoid rigid space that
has l.c.i singularities (in the sense of [GL20, Appendix]). Precisely, we
assume the following throughout the subsection.

######  Assumption 4.3.1.

We let $R$ be a fixed topologically finite type $K$-algebra, and let $P$ be
the ring ${\mathrm{B_{dR}^{+}}}\langle T_{i}\rangle$. Assume $I$ is a Koszul-
regular ideal of $P$ (in the sense of [Sta, Tag 07CU]), such that $P/I$ is
flat over ${\mathrm{B_{dR}^{+}}}$, and $P/(I,\xi)$ is isomorphic to $R$.

Slightly abuse the language, we call any such surjection $P\rightarrow P/I$ as
above a _regular closed immersion_.

######  Example 4.3.2.

Assume $R$ is a topologically finite type algebra over $K$ that has l.c.i
singularities. By definition in [GL20, Proposition 5.3], the kernel ideal
$\overline{I}$ of any surjection from $K\langle T_{i}\rangle$ onto $R$ is a
Koszul-regular ideal in $K\langle T_{i}\rangle$. In particular, by [Sta, Tag
0669], as the union of $\xi$ with a lift of any (Zariski) local Koszul-regular
sequence of $\overline{I}$ is a Koszul-regular sequence, the ideal
$\ker(P\rightarrow R)$ is also a Koszul-regular ideal of $P$. So by taking any
lift of Koszul-regular sequence of $\overline{I}$, we see Zariski (thus
analytic) locally the affinoid algebra $R$ always admits a setup as in 4.3.1.

######  Remark 4.3.3.

It is worth mentioning the category of (regular) closed immersions are sifted.
This is because given any two surjections $P_{i}\rightarrow
P_{i}/I_{i}\rightarrow R$ for $i=1,2$ and $P_{i}$ being of the form
${\mathrm{B_{dR}^{+}}}\langle T_{j}\rangle$, we can form the product

$P:=\varprojlim_{e}\left((P_{1}\otimes_{\mathrm{B_{dR}^{+}}}{\mathrm{B^{+}_{dR,e}}})\otimes_{\mathrm{B^{+}_{dR,e}}}(P_{2}\otimes_{\mathrm{B_{dR}^{+}}}{\mathrm{B^{+}_{dR,e}}})\right),$

which naturally admits a surjection to $R$ together with maps from $P_{i}$.
Moreover, by taking the ideal $I\subset P$ generated by the image of $I_{1}$
and $I_{2}$, the quotient ring $P/I$ is a flat ${\mathrm{B_{dR}^{+}}}$-lift of
$R$ along ${\mathrm{B_{dR}^{+}}}\rightarrow K$. At last, when both surjections
$P_{i}\rightarrow R$ are locally complete intersection, we can use the
analytic cotangent complex to check that $P\rightarrow R$ is also regular
([GL20, Appendix]).

We start by computing the prismatic envelope using Simpson’s functor.

######  Lemma 4.3.4.

Let $P$, $I$, and $R$ be as in 4.3.1. Then the prismatic envelope
$P[\frac{I}{\xi}]^{\wedge}_{\mathrm{tf}}$ for $P\rightarrow P/I$ is naturally
isomorphic to $\Psi(P,I^{\bullet})$. In particular, $\Psi(P,I^{\bullet})$
lives in cohomological degree zero and is a $\xi$-adically complete,
$\xi$-torsionfree noetherian algebra over $P$.

###### Proof.

We first assume that $I$ admits a finite set of generators $(f_{i})$ such that
$(\xi,f_{i})$ is a Koszul-regular sequence, which is always true Zariski
locally around $\operatorname{Spec}(R)$ in $\operatorname{Spec}(P)$ (c.f.
Example 4.3.2). Then there exists a natural map from the polynomial algebra
$P[x_{i}]$ to $\Psi(P,I^{\bullet})$, sending $x_{i}$ onto
$\frac{f_{i}}{\xi}\in\frac{I}{\xi}$. This induces a natural homomorphism from
the noetherian ring $P[\frac{f_{i}}{\xi}]=P[x_{i}]/(\xi\cdot x_{i}-f_{i})$ to
$\Psi(P,I^{\bullet})$, and factors through the derived $\xi$-completion of
$P[\frac{f_{i}}{\xi}]$. Note by [Sta, Tag 0A06] the classical $\xi$-completion
and the derived $\xi$-completion of a noetherian
${\mathrm{B_{dR}^{+}}}$-algebra are the same. Thus the above leads to a
natural map from the actual noetherian algebra $P[\frac{f_{i}}{\xi}]^{\wedge}$
to $\Psi(P,I^{\bullet})$, where the former is classically $\xi$-completed.
Moreover, as $(\xi,f_{i})$ forms a regular sequence of the ring $P$, the map
of rings ${\mathrm{B_{dR}^{+}}}[y_{i}]\rightarrow P$, $y_{i}\mapsto f_{i}$ is
flat after localizing at the ideal $(\xi)$ (c.f. [Har66, Proposition 1]). As a
consequence, the ring $P[\frac{f_{i}}{\xi}]$ has no $\xi$-torsion, and we get
a map from $P[\frac{I}{\xi}]^{\wedge}_{\mathrm{tf}}$ to $\Psi(P,I^{\bullet})$.

To show the isomorphism, by the derived Nakayama lemma ([Sta, Tag 0G1U]) for
${\mathrm{B_{dR}^{+}}}$, it suffices to check the above is an isomorphism
after a derived tensor product with $K$. On the one hand, by Remark 4.1.5 we
have

$\Psi(P,I^{\bullet})\otimes^{L}_{\mathrm{B_{dR}^{+}}}K\cong\bigoplus_{n\in\mathbb{N}}{\mathrm{gr}}^{n}(P\otimes^{L}_{\mathrm{B_{dR}^{+}}}K).$

By the assumption of $P$, each $I^{n}/I^{n+1}$ is Tor-independent of $K$ over
${\mathrm{B_{dR}^{+}}}$. So the above tensor product is equal to the discrete
module as below

$\bigoplus_{n\in\mathbb{N}}(\overline{I})^{n}/(\overline{I})^{n+1},$

where $\overline{I}$ is the tensor product $I\otimes_{\mathrm{B_{dR}^{+}}}K$.
On the other hand, the assumption of $P$ and $I=(f_{i})$ implies the ring
$P[x_{1},\ldots,x_{m}]/(\xi x_{i}-f_{i},1\leq i\leq m)$ is flat over
${\mathrm{B_{dR}^{+}}}$. In particular, we have

$\displaystyle P[\frac{f_{i}}{\xi}]\otimes^{L}_{\mathrm{B_{dR}^{+}}}K$
$\displaystyle\cong P[\frac{f_{i}}{\xi}]\otimes_{\mathrm{B_{dR}^{+}}}K$
$\displaystyle\cong P/(\xi,f_{i})[x_{i}],$

which by assumption is exactly the direct sum
$\bigoplus_{n\in\mathbb{N}}(\overline{I})^{n}/(\overline{I})^{n+1}$. In this
way, as the above two reductions are abstractly isomorphic to each other, to
finish the proof, it suffices to notice that the map we constructed identifies
the generators of each direct summand.

In general, one can still form the map from
$P[\frac{I}{\xi}]^{\wedge}_{\mathrm{tf}}$ to $\Psi(P,I^{\bullet})$, replacing
$(f_{i})$ above by any finite set of generators of $I$. As the ring $P$ is
noetherian, we can switch the order of the torsionfree quotient and the
$\xi$-completion to get

$P[\frac{I}{\xi}]^{\wedge}_{\mathrm{tf}}=\left(P[\frac{f_{i}}{\xi}]/(\xi^{\infty}\text{-torsion})\right)^{\wedge}.$

So to show the isomorphism, it suffices to take the derived reduction mod
$\xi$ and consider the following

$\left(P[\frac{f_{i}}{\xi}]/(\xi^{\infty}\text{-torsion})\right)\otimes^{L}_{\mathrm{B_{dR}^{+}}}K\rightarrow\Psi(P,I^{\bullet})\otimes^{L}_{\mathrm{B_{dR}^{+}}}K.$

But notice that the map commutes with any Zariski localization of $P$, and for
each $\mathfrak{p}\in\operatorname{Spec}(R)$ there is a Zariski open
neighborhood of $\mathfrak{p}$ in $\operatorname{Spec}(P)$ such that $I$
admits a set of Koszul-regular generators and the above is an isomorphism.
Thus a Čech complex argument for an open covering finishes the proof.

∎

From now on towards the end of the subsection, we use $D_{\inf}$ to denote the
infinitesimal envelope $\varprojlim P/I^{m}$ for the surjection $P\rightarrow
R$ (c.f. [Guo21, Section 2.2]), and use
$D_{{\mathbbl{\Delta}}}=P[\frac{I}{\xi}]^{\wedge}_{\mathrm{tf}}$ for the
prismatic envelope, which by Lemma 4.3.4 is isomorphic to
$\Psi(P,I^{\bullet})$.

Next we use Simpson’s functor to relate a connection over $D_{\inf}$ to a log
connection over $D_{{\mathbbl{\Delta}}}$.

######  Theorem 4.3.5.

Let $P,I,R$ and $D_{\inf},D_{{\mathbbl{\Delta}}}$ be as above, let
$\mathcal{F}$ be a flat infinitesimal crystal over
$R/{\mathrm{B_{dR}^{+}}}_{\mathrm{pinf}}$, and let $(M,\nabla)$ the be the
integrable connection of $\mathcal{F}$ at $D_{\inf}$ (as in [Guo21, Section
3]). Then the log connection of $\pi^{*}\mathcal{F}$ at
$D_{{\mathbbl{\Delta}}}$ is integral, integrable, and is naturally isomorphic
to the unique extension of $\nabla$ on
$M\otimes_{D_{\inf}}D_{{\mathbbl{\Delta}}}$ by the Leibniz rule.

By 4.2.6, we can denote the log connection as $(\pi^{*}M,\nabla)$, with
$\pi^{*}M\cong M\otimes_{D_{\inf}}D_{{\mathbbl{\Delta}}}$. Note that by
Example 3.0.10, the log connection $(\pi^{*}M,\nabla)$ is in particular quasi-
nilpotent.

###### Proof.

By proof of Proposition 3.0.12, to describe the log connection structure on
$\pi^{*}M=M\otimes_{D_{\inf}}D_{{\mathbbl{\Delta}}}$ (see 4.2.6), it suffices
to consider the difference map

$\phi_{{\mathbbl{\Delta}}}:\mathrm{pr}_{1}^{*}(\pi^{*}\mathcal{F})(D_{{\mathbbl{\Delta}}})\longrightarrow\mathrm{pr}_{0}^{*}(\pi^{*}\mathcal{F})(D_{{\mathbbl{\Delta}}}),$

where $\phi_{{\mathbbl{\Delta}}}$ is the isomorphism of
$D(1)_{{\mathbbl{\Delta}}}$-modules induced from the crystal condition. Here
by 4.2.6, the isomorphism $\phi_{{\mathbbl{\Delta}}}$ is naturally isomorphic
to $\phi_{\inf}\otimes_{D(1)_{\inf}}D(1)_{{\mathbbl{\Delta}}}$, with

$\phi_{\inf}=\left(\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
23.75629pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathrm{pr}_{1}^{*}\mathcal{F}(D_{\inf})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
30.03406pt\raise 4.28406pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.28406pt\hbox{$\scriptstyle{\sim}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 47.75629pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
47.75629pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\mathrm{pr}_{0}^{*}\mathcal{F}(D_{\inf})}$}}}}}}}\ignorespaces}}}}\ignorespaces\right).$

On the other hand, notice that the isomorphism $\phi_{\inf}$ is filtered under
the $(I,\Delta(1))$-adic filtration, and in particular by applying
$\Psi(-,(I,\Delta(1))^{\bullet})$ and Lemma 4.3.4 we have

$\phi_{{\mathbbl{\Delta}}}=\Psi(\phi_{\inf},(I,\Delta(1))^{\bullet}).$

Thus by taking the difference between $\operatorname{id}\otimes 1$ and
$\phi_{{\mathbbl{\Delta}}}$ and mod the ideal $(\frac{\Delta(1)}{\xi})^{2}$,
we get a natural log connection on
$\pi^{*}M=M\otimes_{D_{\inf}}D_{{\mathbbl{\Delta}}}$.

Moreover, by the property of colimits, there is a natural commutative diagram
sending the connection $(M,\nabla)$ to the log connection on $\pi^{*}M$,
compatible with the natural structure map $D_{\inf}\rightarrow
D_{{\mathbbl{\Delta}}}$:

$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\nabla}$$\textstyle{M\otimes_{P}\Omega_{P/{\mathrm{B_{dR}^{+}}}}^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M\otimes_{D_{\inf}}D_{{\mathbbl{\Delta}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(M\otimes_{D_{\inf}}D_{{\mathbbl{\Delta}}})\otimes_{P}\frac{\Omega_{P/{\mathrm{B_{dR}^{+}}}}^{1}}{\xi},}$

where the vertical maps are natural maps compatible with the inclusion
$\Omega_{P/{\mathrm{B_{dR}^{+}}}}^{1}\rightarrow\frac{\Omega_{P/{\mathrm{B_{dR}^{+}}}}^{1}}{\xi}$
over the ring $P$. From the diagram, we see the log connection on $\pi^{*}M$
coincides with $\nabla$ when restricted to $M$, and is uniquely extended by
$\nabla$ using the Leibniz rule of the log connection.

At last, the integrability and the quasi-nilpotence either follow from the
general result in Proposition 3.0.12, or the above commutative diagram and
Leibniz rule, as $\widetilde{\nabla}_{i}(x)$ is $\xi$-divisible for any $i$
and $x\in M$, and thus by Definition 3.0.6 the connection $\Psi(\nabla)$ is
integral and thus quasi-nilpotent.

∎

The following lemma is useful when relating the log connection $\pi^{*}M$ with
the filtration.

######  Lemma 4.3.6.

Assume the same as in Theorem 4.3.5. Then $\pi^{*}M$ is naturally isomorphic
to $\Psi(M,I^{\bullet}M)$.

###### Proof.

There is a natural map below

$M\otimes_{D_{\inf}}D_{{\mathbbl{\Delta}}}=M\otimes_{D_{\inf}}^{L}\Psi(D_{\inf},I^{\bullet})\longrightarrow\Psi(M,I^{\bullet}M).$

By the commutativity between the derived tensor product functor and the
colimit, the above is an isomorphism before the derived $\xi$-adic completion
for $\Psi(-,-)$ as in 4.1.1. So thanks to the derived $\xi$-completeness of
$D_{{\mathbbl{\Delta}}}=\Psi(D_{\inf},I^{\bullet})$ and the finiteness of $M$
over $D_{\inf}$, we see the left hand side above is automatically
$\xi$-complete, and hence isomorphic to the right. ∎

In the l.c.i case, we also get the faithfulness of the functor from
infinitesimal crystals to prismatic crystals.

######  Proposition 4.3.7.

Let $X$ be an affinoid rigid space over $K$ that is a local complete
intersection. The functor $\pi^{*}$ in 4.2.6 from the category of flat
infinitesimal crystals over $X/{\mathrm{B_{dR}^{+}}}_{\mathrm{pinf}}$ to the
category of flat prismatic crystals over
$X/{\mathrm{B_{dR}^{+}}}_{{\mathbbl{\Delta}}}$ is faithful.

###### Proof.

By the tensor product formula of $\pi^{*}M$ and the compatiblility between
connections and log connections, it suffices to show that under Assumption
4.3.1 for the ring $P={\mathrm{B_{dR}^{+}}}\langle T_{i}\rangle$, the ideal
$I$ and the ring $R=P/(I,\xi)$, the natural map $D_{\inf}\rightarrow
D_{{\mathbbl{\Delta}}}=P[\frac{I}{\xi}]^{\wedge}_{\mathrm{tf}}$ is an
inclusion.

By definition and Lemma 4.3.4, we have $D_{\inf}$ is the $I$-adic completion
of $P$, and $D_{{\mathbbl{\Delta}}}$ is the $\xi$-adic completion of
$P[\frac{I}{\xi}]_{\mathrm{tf}}$. So thanks to the left exactness of the
inverse limit, it suffices to show the injectivity for each $n\in\mathbb{N}$
below

$P/(I^{n},\xi^{n})\longrightarrow P[\frac{I}{\xi}]_{\mathrm{tf}}/\xi^{n},$

where we use the equality $\varprojlim P/(I^{n},\xi^{n})=\varprojlim P/I^{n}$
by the $\xi$-adic completeness of $P$. This can be checked Zariski locally,
where we can assume $I$ admits a finite set of regular generators
$(f_{1},\ldots,f_{l})$. In this case, the natural map below is flat

$\textstyle{{\mathrm{B_{dR}^{+}}}[y_{j}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{y_{j}\mapsto
f_{j}~{}~{}~{}~{}}$$\textstyle{P={\mathrm{B_{dR}^{+}}}\langle T_{i}\rangle}$

Thus the injectivity of $(\ast)$ for each $n$ follows from that of

${\mathrm{B^{+}_{dR,n}}}[y_{j}]\rightarrow{\mathrm{B^{+}_{dR,n}}}[x_{j}]/(\xi\cdot
x_{j}-y_{j}),$

by base change along the flat map ${\mathrm{B_{dR}^{+}}}[y_{j}]\rightarrow P$.

∎

######  Remark 4.3.8.

From Theorem 4.3.5, the log connection associated to the image of $\pi^{*}$ is
integral and integrable. One might also be able to use the explicit formula of
the connection to describe the essential image of the functor $\pi^{*}$.

We then give a formula computing prismatic cohomology of a crystal that comes
from the infinitesimal site, using Simpson’s functor and the de Rham complex.

Assume there is a system of affinoid rigid spaces $\\{X_{e}\\}$, with each
$X_{e}$ flat over ${\mathrm{B^{+}_{dR,e}}}$, lifting a given $K$-rigid space
$X$. Let $\mathcal{F}$ be a crystal over
$X/{\mathrm{B_{dR}^{+}}}_{\mathrm{pinf}}$, and let $\mathcal{F}_{e}$ be its
restriction to $X_{e}$, as a crystal over the infinitesimal site
$X_{e}/{\mathrm{B^{+}_{dR,e}}}_{\inf}$ (c.f. [Guo21, Section 2.4]). Following
[Guo21, Section 2], the $I_{X_{e}/{\mathrm{B^{+}_{dR,e}}}}$-adic filtration on
an infinitesimal crystal $\mathcal{F}_{e}$ induces a natural filtration of
$R\Gamma(X_{e}/{\mathrm{B^{+}_{dR,e}}}_{\inf},\mathcal{F}_{e})$, compatible
with the closed immersion $X_{e}/{\mathrm{B^{+}_{dR,e}}}_{\inf}\rightarrow
X_{e+1}/\mathrm{B^{+}_{dR,e+1}}_{\inf}$. In particular, by the limit formula
$R\Gamma(X/{\mathrm{B_{dR}^{+}}}_{\mathrm{pinf}},\mathcal{F})=R\varprojlim_{e}R\Gamma(X_{e}/{\mathrm{B^{+}_{dR,e}}}_{\inf},\mathcal{F}_{e})$,
the above defines a filtration on infinitesimal cohomology of
$X/{\mathrm{B_{dR}^{+}}}$ as below

######  Definition 4.3.9.

Let $X$ be a rigid space over $K$, and let $\\{X_{e}\\}$ be a compatible
system of ${\mathrm{B^{+}_{dR,e}}}$-flat lifts of $X$. For a crystal
$\mathcal{F}$ over $X/{\mathrm{B_{dR}^{+}}}_{\mathrm{pinf}}$, the $i$-th
_lifted (infinitesimal) filtration_ on its cohomology is

$\widetilde{\mathrm{Fil}}^{i}R\Gamma(X/{\mathrm{B_{dR}^{+}}}_{\mathrm{pinf}},\mathcal{F}):=\varprojlim_{e}R\Gamma(X_{e}/{\mathrm{B^{+}_{dR,e}}}_{\inf},\mathcal{I}_{X_{e}/{\mathrm{B^{+}_{dR,e}}}}^{i}\mathcal{F}_{e}).$

Note that similar to [Guo21, Theorem 7.2.3 (iii)], the derived tensor product
of the lifted filtration with ${\mathrm{B^{+}_{dR,e}}}$ over
${\mathrm{B_{dR}^{+}}}$ is isomorphic to the filtration of
$R\Gamma(X_{e}/{\mathrm{B^{+}_{dR,e}}}_{\inf},\mathcal{F}_{e})$.

######  Remark 4.3.10.

When $P/I$ is smooth over ${\mathrm{B_{dR}^{+}}}$ (in the sense that each
reduction mod $\xi^{e}$ is a smooth affinoid algebra over
${\mathrm{B^{+}_{dR,e}}}$), the $i$-th lifted filtration coincides with the
$i$-th Hodge filtration of the de Rham complex of $\mathcal{F}$ over $P/I$. In
particular, this is the ${\mathrm{B_{dR}^{+}}}$-linear lift of $i$-th the
Hodge filtration of the de Rham complex of $\mathcal{F}(R)$ over $K$, hence
the name.

######  Remark 4.3.11.

When $X$ is a rigid space defined over a discretely valued subfield $k$, it
admits a canonical (depending on a choice embedding
$k\rightarrow{\mathrm{B_{dR}^{+}}}$) $G_{k}$-equivariant lifts $\\{X_{e}\\}$,
where each $X_{e}$ is the complete base extension of the $k$-rigid space with
${\mathrm{B^{+}_{dR,e}}}$. In particular the lifted filtration admits a
natural action by Galois group.

######  Theorem 4.3.12.

Let $P,I,R$ be as in 4.3.1 with $X=\mathrm{Spa}(R)$, and let $\mathcal{F}$ be
a flat infinitesimal crystal over $X/{\mathrm{B_{dR}^{+}}}_{\mathrm{pinf}}$.
There is a natural isomorphism, functorial in $\mathcal{F}$ and the surjection
$P\rightarrow P/I$, computing cohomology of the prismatic crystal
$\pi^{*}\mathcal{F}$

$\Psi(R\Gamma(X/{\mathrm{B_{dR}^{+}}}_{\mathrm{pinf}},\mathcal{F}),\widetilde{\mathrm{Fil}}^{\bullet})\cong
R\Gamma(X/{\mathrm{B_{dR}^{+}}}_{{\mathbbl{\Delta}}},\pi^{*}\mathcal{F}).$

###### Proof.

The idea is to compare Čech-Alexander complexes of two sides. Denote
$D(n)_{\inf}$ to be the formal completion of the complete tensor product
$P(n)=P^{\widehat{\otimes}_{{\mathrm{B_{dR}^{+}}}}n+1}:={\mathrm{B_{dR}^{+}}}\langle
T_{i},\delta_{i,j};1\leq i\leq l,1\leq j\leq n\rangle$ along the surjection
onto $P/I$, and let $\Delta(n)$ be the ideal of $D(n)_{\inf}$ defined by
$(\delta_{i,j})$. Let
$D(n)_{{\mathbbl{\Delta}}}=P(n)[\frac{I,\Delta(n)}{\xi}]^{\wedge}_{\mathrm{tf}}\cong\Psi(P(n),(I,\Delta(n))^{\bullet})$
be the prismatic envelope of $P/I$ inside of
$P^{\widehat{\otimes}_{{\mathrm{B_{dR}^{+}}}}n+1}$ (Lemma 4.3.4). Note that by
Remark 2.2.7, the ring $D(n)_{{\mathbbl{\Delta}}}$ is also naturally
isomorphic to $D(n)_{\inf}[\frac{I,\Delta(n)}{\xi}]^{\wedge}_{\mathrm{tf}}$.

We then turn to the Čech-Alexander complexes of $\mathcal{F}$ and
$\pi^{*}\mathcal{F}$. By [Guo21, Proposition 2.2.7] and its
${\mathrm{B_{dR}^{+}}}$-linear version [Guo21, Theorem 7.2.3], we have a
natural filtered isomorphism

$R\Gamma(X/{\mathrm{B_{dR}^{+}}}_{\mathrm{pinf}},\mathcal{F})\cong
R\varprojlim_{[n]\in\Delta^{\mathrm{op}}}\mathcal{F}(D(n)_{\inf}),$

where the left side is endowed with the lifted filtration, and each term of
the right side is equipped with the $(I,\Delta(n))$-adic filtration (as
$(I,\Delta(n))$ is the kernel ideal of the pro-infiniteisimal thickening
$D(n)_{\inf}\rightarrow P/I$). On the other hand, by Corollary 3.0.14 we have
the natural isomorphism

$R\Gamma(X/{\mathrm{B_{dR}^{+}}}_{{\mathbbl{\Delta}}},\pi^{*}\mathcal{F})\cong
R\varprojlim_{[n]\in\Delta^{\mathrm{op}}}\mathcal{F}(D(n)_{{\mathbbl{\Delta}}}),$

Moreover, by the construction of $\pi^{*}\mathcal{F}$ in Theorem 4.3.5 and
Lemma 4.3.6 the right hand side above is equal to

$R\varprojlim_{[n]\in\Delta^{\mathrm{op}}}\Psi(\mathcal{F}(D(n)_{\inf}),(I,\Delta(n))^{\bullet}).$

In this way, by applying $\Psi$ at the sequence (1) using the lifted
filtration, and notice that homotopy limit along $\Delta^{\mathrm{op}}$
commuts with $\Psi$, we get

$\displaystyle
R\Gamma(X/{\mathrm{B_{dR}^{+}}}_{{\mathbbl{\Delta}}},\pi^{*}\mathcal{F})$
$\displaystyle\cong
R\varprojlim_{[n]\in\Delta^{\mathrm{op}}}\Psi(\mathcal{F}(D(n)_{{\mathbbl{\Delta}}}),(I,\Delta(n))^{\bullet})$
$\displaystyle\cong\Psi(R\varprojlim_{[n]\in\Delta^{\mathrm{op}}}\mathcal{F}(D(n)_{\inf}),\widetilde{\mathrm{Fil}}^{\bullet})$
$\displaystyle\cong\Psi(R\Gamma(X/{\mathrm{B_{dR}^{+}}}_{\mathrm{pinf}},\mathcal{F}),\widetilde{\mathrm{Fil}}^{\bullet}).$

∎

As a corollary, we have the following explicit formula computing prismatic
cohomology.

######  Corollary 4.3.13.

Let $P$, $I$, $R$ be as in 4.3.1. Let $\mathcal{F}$ be a flat infinitesimal
crystal over $X/{\mathrm{B_{dR}^{+}}}_{\mathrm{pinf}}$, and let $M$ be its
section at the pro infinitesimal thickening $D_{\inf}=\varprojlim_{n}P/I^{n}$.
Then there is an isomorphism, functorial in $\mathcal{F}$, as below

$R\Gamma(X/{\mathrm{B_{dR}^{+}}}_{{\mathbbl{\Delta}}},\pi^{*}\mathcal{F})\cong\left(\widetilde{M}\rightarrow\widetilde{M}\underset{P}{\operatorname*{\text{\raisebox{1.07639pt}{\scalebox{0.8}{$\bigotimes$}}}}}\frac{\Omega_{P/{\mathrm{B_{dR}^{+}}}}^{1}}{\xi}\rightarrow\cdots\rightarrow\widetilde{M}\underset{P}{\operatorname*{\text{\raisebox{1.07639pt}{\scalebox{0.8}{$\bigotimes$}}}}}\frac{\Omega_{P/{\mathrm{B_{dR}^{+}}}}^{d}}{\xi^{d}}\right),$

where
$\widetilde{M}=\pi^{*}\mathcal{F}(P[\frac{I}{\xi}]^{\wedge}_{\mathrm{tf}})\cong
M\otimes_{D_{\inf}}P[\frac{I}{\xi}]^{\wedge}_{\mathrm{tf}}$ is the section of
$\pi^{*}\mathcal{F}$ at the prismatic envelope
$P[\frac{I}{\xi}]^{\wedge}_{\mathrm{tf}}$, and the right hand side is the de
Rham complex associated to the log connection of $\widetilde{M}$

###### Proof.

Using Theorem 4.3.12, cohomology of $\pi^{*}\mathcal{F}$ is isomorphic to
$\Psi(R\Gamma(X/{\mathrm{B_{dR}^{+}}}_{\mathrm{pinf}},\mathcal{F}),\widetilde{\mathrm{Fil}}^{\bullet})$.
Then by taking an inverse limit of [Guo21, Theorem 4.1.1] with respect to
$\mathrm{Spa}(P/(I,\xi^{e}))$ over ${\mathrm{B^{+}_{dR,e}}}$, we have

$\Psi(R\Gamma(X/{\mathrm{B_{dR}^{+}}}_{\mathrm{pinf}},\mathcal{F}),\widetilde{\mathrm{Fil}}^{\bullet})\cong\Psi(M\underset{P}{\otimes}\Omega_{P/{\mathrm{B_{dR}^{+}}}}^{\bullet},\widetilde{\mathrm{Fil}}^{\bullet}),$

with $\widetilde{\mathrm{Fil}}^{\bullet}$ is the lifted infinitesimal
filtration on the de Rham complex. Thus by writing out the right hand side
above explicitly, we get the conclusion. ∎

In the special case when $R$ is smooth over $K$, we can obtain a simpler
formula of cohomology of $\pi^{*}\mathcal{F}$ using the de Rham complex.

######  Corollary 4.3.14.

Assume $X=\mathrm{Spa}(R)$ is smooth of dimension $d$ over $K$, and
$\widetilde{R}=\varprojlim_{e}\widetilde{R}/\xi^{e}$ is a smooth lift over
${\mathrm{B_{dR}^{+}}}$. Let $\mathcal{F}$ be a flat infinitesimal crystal
over $X/{\mathrm{B_{dR}^{+}}}_{\mathrm{pinf}}$, and let $M$ be its section at
the pro infinitesimal thickening $\widetilde{R}\rightarrow R$. Then we have

$R\Gamma(X/{\mathrm{B_{dR}^{+}}}_{{\mathbbl{\Delta}}},\pi^{*}\mathcal{F})\cong\left(M\rightarrow
M\underset{\widetilde{R}}{\otimes}\frac{\Omega_{\widetilde{R}/{\mathrm{B_{dR}^{+}}}}^{1}}{\xi}\rightarrow\cdots\rightarrow
M\underset{\widetilde{R}}{\otimes}\frac{\Omega_{\widetilde{R}/{\mathrm{B_{dR}^{+}}}}^{d}}{\xi^{d}}\right).$

###### Proof.

As a smooth lift $\widetilde{R}/\xi^{e}$ itself is an infinitesimal envelope
of $R$ into a smooth rigid space over ${\mathrm{B^{+}_{dR,e}}}$, we can apply
Theorem 4.3.12 at the filtered isomorphism
$R\Gamma(X/{\mathrm{B^{+}_{dR,e}}}_{\inf},\mathcal{F})\cong(M/\xi^{e}\otimes_{\widetilde{R}/\xi^{e}}\Omega_{(\widetilde{R}/\xi^{e})/{\mathrm{B^{+}_{dR,e}}}}^{\bullet})$
in [Guo21, Theorem 4.1.1] to get the result. ∎

Another special case is when $\mathcal{F}$ is the infinitesimal structure
sheaf, in which case $\pi^{*}\mathcal{F}$ is the prismatic structure sheaf
$\mathcal{O}_{{\mathbbl{\Delta}}}$. In this case, we can compute prismatic
cohomology using Simpson’s functor and the analytic derived de Rham complex.

######  Corollary 4.3.15.

Let $P$, $I$, $R$ be as in 4.3.1. Then there is an isomorphism as below

$R\Gamma(X/{\mathrm{B_{dR}^{+}}}_{{\mathbbl{\Delta}}},\mathcal{O}_{{\mathbbl{\Delta}}})\cong\Psi({\widehat{\mathrm{dR}}^{\mathrm{an}}}_{(P/I)/{\mathrm{B_{dR}^{+}}}}).$

###### Proof.

Recall the analytic derived de Rham complex
${\widehat{\mathrm{dR}}^{\mathrm{an}}}_{(P/I)/{\mathrm{B_{dR}^{+}}}}$ over
${\mathrm{B_{dR}^{+}}}$ is defined as the filtered object

$R\varprojlim_{e}{\widehat{\mathrm{dR}}^{\mathrm{an}}}_{(P/(I,\xi^{e}))/{\mathrm{B^{+}_{dR,e}}}},$

where
${\widehat{\mathrm{dR}}^{\mathrm{an}}}_{(P/(I,\xi^{e}))/{\mathrm{B^{+}_{dR,e}}}}$
is constructed as in [Guo21, Section 5] for affinoid algebras over
${\mathrm{B^{+}_{dR,e}}}$. By [Guo21, Corollary 5.5.2], assuming 4.3.1, the
derived de Rham complex
${\widehat{\mathrm{dR}}^{\mathrm{an}}}_{(P/(I,\xi^{e}))/{\mathrm{B^{+}_{dR,e}}}}$
is filtered isomorphic to
$R\Gamma(\mathrm{Spa}(P/(I,\xi^{e}))/{\mathrm{B^{+}_{dR,e}}}_{\inf},\mathcal{O}_{e\inf})$.
Thus the formula follows from Theorem 4.3.12. ∎

######  Corollary 4.3.16.

Assume $X=\mathrm{Spa}(R)$ is smooth of dimension $d$ over $K$, and
$\widetilde{R}=\varprojlim_{e}\widetilde{R}/\xi^{e}$ is a smooth lift over
${\mathrm{B_{dR}^{+}}}$. Then we have

$R\Gamma(X/{\mathrm{B_{dR}^{+}}}_{{\mathbbl{\Delta}}},\mathcal{O}_{{\mathbbl{\Delta}}})\cong\left(\widetilde{R}\rightarrow\frac{\Omega_{\widetilde{R}/{\mathrm{B_{dR}^{+}}}}^{1}}{\xi}\rightarrow\cdots\rightarrow\frac{\Omega_{\widetilde{R}/{\mathrm{B_{dR}^{+}}}}^{d}}{\xi^{d}}\right).$

To finish this section, we show that a choice of regular immersion induces a
splitting of reduced prismatic cohomology.

######  Proposition 4.3.17.

Let $X$ be a rigid space over $K$, and let $\\{Y_{e}\\}_{e\in\mathbb{N}}$ be a
system of smooth rigid spaces over ${\mathrm{B^{+}_{dR,e}}}$ with
$Y_{e+1}\times_{\mathrm{B_{dR,e+1}^{+}}}{\mathrm{B^{+}_{dR,e}}}\cong Y_{e}$,
and $X\rightarrow Y_{0}$ is a regular closed immersion. Let $\mathcal{F}$ be a
flat infinitesimal crystal over $X/{\mathrm{B_{dR}^{+}}}_{\mathrm{pinf}}$.
Then the choice of $\\{Y_{e}\\}$ induces an isomorphism as below

$R\Gamma(X/{\mathrm{B_{dR}^{+}}}_{{\mathbbl{\Delta}}},\overline{\pi^{*}\mathcal{F}})\cong\bigoplus_{n}{\mathrm{gr}}^{n}R\Gamma(X/K_{\mathrm{inf}},\overline{\mathcal{F}}),$

where the latter is the direct sum of graded pieces for the infinitesimal
filtration over cohomology of the crystal
$\overline{\mathcal{F}}=\mathcal{F}/\xi$ over $X/K_{\mathrm{pinf}}$.

###### Proof.

By the global version of Theorem 4.3.12 and Corollary 3.0.15, the regular
immersion induces an isomorphism between the left hand side above and the
following

$\Psi(R\Gamma(X/{\mathrm{B_{dR}^{+}}}_{\mathrm{pinf}},\mathcal{F}),\widetilde{\mathrm{Fil}}^{\bullet})\otimes^{L}_{\mathrm{B_{dR}^{+}}}K,$

where $\widetilde{\mathrm{Fil}}^{\bullet}$ is the lifted infinitesimal
filtration. By Remark 4.1.5 it is further equal to

$\bigoplus_{n}{\mathrm{gr}}^{n}\left(R\Gamma(X/{\mathrm{B_{dR}^{+}}}_{\mathrm{pinf}},\mathcal{F})\otimes^{L}_{\mathrm{B_{dR}^{+}}}K\right)(-n).$

Here the filtration is the infinitesimal filtration. At last, by [Guo21,
Theorem 7.2.3], the base change
$R\Gamma(X/{\mathrm{B_{dR}^{+}}}_{\mathrm{pinf}},\mathcal{F})\otimes^{L}_{\mathrm{B_{dR}^{+}}}K$
is isomorphic to infinitesimal cohomology
$R\Gamma(X/K_{\mathrm{inf}},\overline{\mathcal{F}})$ with the canonical
infinitesimal filtration. ∎

Note that in the special case when
$\mathcal{F}=\mathcal{O}_{X/{\mathrm{B_{dR}^{+}}}}$ is the infinitesimal
structure sheaf, the $i$-th graded piece above is isomorphic to
$R\Gamma(X,L\wedge^{i}\mathbb{L}^{\mathrm{an}}_{R/K})(-i)[-i]$.

##  5\. Hodge-Tate filtration

Using the local computation from the last section and the simplicial
resolutions, we are able to prove the Hodge-Tate filtration theorem.

###  5.1. Smooth case

In the smooth case, we first construct a natural map from Hodge-Tate
cohomology to prismatic cohomology, using the Bockstein operator as in [BS19].
Let $R$ be a topological finite type algebra over $K$. Consider the short
exact sequence of ${\mathrm{B_{dR}^{+}}}$-modules

$\xi^{i+1}/\xi^{i+2}\longrightarrow\xi^{i}/\xi^{i+2}\longrightarrow\xi^{i}/\xi^{i+1}.$

We take the derived tensor product of the sequence with
$R\Gamma(R/{\mathrm{B_{dR}^{+}}}_{{\mathbbl{\Delta}}},\mathcal{O}_{{\mathbbl{\Delta}}})$
over ${\mathrm{B_{dR}^{+}}}$, then the connecting morphism induces the
following

$R\Gamma(R/{\mathrm{B_{dR}^{+}}}_{{\mathbbl{\Delta}}},\mathcal{O}_{{\mathbbl{\Delta}}})\otimes^{L}_{\mathrm{B_{dR}^{+}}}K\cdot\xi^{i}\longrightarrow
R\Gamma(R/{\mathrm{B_{dR}^{+}}}_{{\mathbbl{\Delta}}},\mathcal{O}_{{\mathbbl{\Delta}}})\otimes^{L}_{\mathrm{B_{dR}^{+}}}K\cdot\xi^{i+1}[1].$

By the reduction isomorphism in Corollary 3.0.15 we have

$R\Gamma(R/{\mathrm{B_{dR}^{+}}}_{{\mathbbl{\Delta}}},\mathcal{O}_{{\mathbbl{\Delta}}})\otimes^{L}_{\mathrm{B_{dR}^{+}}}K\cong
R\Gamma(R/{\mathrm{B_{dR}^{+}}}_{{\mathbbl{\Delta}}},\overline{\mathcal{O}}).$

Thus we can rewrite the connecting morphism as below

$\beta_{\xi}:R\Gamma(R/{\mathrm{B_{dR}^{+}}}_{{\mathbbl{\Delta}}},\overline{\mathcal{O}}_{{\mathbbl{\Delta}}})(i)\longrightarrow
R\Gamma(R/{\mathrm{B_{dR}^{+}}}_{{\mathbbl{\Delta}}},\overline{\mathcal{O}}_{{\mathbbl{\Delta}}})(i+1)[1].$

As in [BS19, Section 4], by evaluating at the $i$-th cohomology group, the
above induces a $K$-linear cdga over $R$

$(\mathrm{H}^{\bullet}(R/{\mathrm{B_{dR}^{+}}}_{{\mathbbl{\Delta}}},\overline{\mathcal{O}}_{{\mathbbl{\Delta}}})(\bullet),\beta_{\xi}).$

So by the universal property of the algebraic de Rham complex
$\Omega_{R/K}^{\bullet,alg}$ of $R/K$, we get a natural map of cdga

$(\Omega_{R/K}^{\bullet,alg},d)\longrightarrow(\mathrm{H}^{\bullet}(R/{\mathrm{B_{dR}^{+}}}_{{\mathbbl{\Delta}}},\overline{\mathcal{O}}_{{\mathbbl{\Delta}}})(\bullet),\beta_{\xi}).$

The next result proves that the map induces an isomorphism of of Hodge-Tate
cohomology with the analytic differentials when $R$ is smooth.

######  Theorem 5.1.1.

Assume $R$ is a smooth affinoid algebra over $K$. Then there is a natural
isomorphism

$\mathrm{H}^{i}(R/{\mathrm{B_{dR}^{+}}}_{{\mathbbl{\Delta}}},\overline{\mathcal{O}}_{{\mathbbl{\Delta}}})\cong\Omega_{R/K}^{i}(-i),$

where $\Omega_{R/K}^{i}$ is the sheaf of $i$-th analytic differentials.

Assume the above case when $R$ is smooth and affinoid, we call the Postnikov
filtration of
$R\Gamma(R/{\mathrm{B_{dR}^{+}}}_{{\mathbbl{\Delta}}},\overline{\mathcal{O}}_{{\mathbbl{\Delta}}})$
the _Hodge-Tate filtration_.

###### Proof.

As we have constructed a natural map from algebraic de Rham complex of $R/K$
to the cdga
$(\mathrm{H}^{\bullet}(R/{\mathrm{B_{dR}^{+}}}_{{\mathbbl{\Delta}}},\overline{\mathcal{O}}_{{\mathbbl{\Delta}}})(\bullet),\beta_{\xi})$,
it suffices to show that the map factors through the analytic de Rham complex
of $R/K$, and induces a termwise isomorphism with the target.

Let us assume $R$ admits a surjection from $P={\mathrm{B_{dR}^{+}}}\langle
T_{i}\rangle$, together with an ideal $I\in P$ satisfying 4.3.1. By Corollary
4.3.13, prismatic cohomology is isomorphic to $\xi$-divided de Rham cohomology

$C_{P}:=\left(D_{{\mathbbl{\Delta}}}\rightarrow
D_{{\mathbbl{\Delta}}}\underset{P}{\operatorname*{\text{\raisebox{1.07639pt}{\scalebox{0.8}{$\bigotimes$}}}}}\frac{\Omega_{P/{\mathrm{B_{dR}^{+}}}}^{1}}{\xi}\rightarrow\cdots\rightarrow
D_{{\mathbbl{\Delta}}}\underset{P}{\operatorname*{\text{\raisebox{1.07639pt}{\scalebox{0.8}{$\bigotimes$}}}}}\frac{\Omega_{P/{\mathrm{B_{dR}^{+}}}}^{d}}{\xi^{d}}\right),$

where each differential map is the continuous differential with respect to
$(\xi,p)$-adic topology. Denote its termwise reduction by $\overline{C}_{P}$,
which by Theorem 4.3.12 and Remark 4.1.5 is isomorphic to

$\operatorname*{\text{\raisebox{1.07639pt}{\scalebox{0.8}{$\bigoplus$}}}}_{i\in\mathbb{N}}L\wedge^{i}\mathbb{L}_{R/K}(-i)[-i].$

As $R$ is smooth over $K$, each $L\wedge^{i}\mathbb{L}_{R/K}$ is isomorphic to
$i$-th analytic differential $\Omega_{R/K}^{i}$. So by applying the Bockstein
operator $\beta_{\xi}$ and its associated complex, we have

$(\mathrm{H}^{\bullet}(\overline{C}_{P})(\bullet),\beta_{\xi})=(\Omega_{R/K}^{\bullet},d),$

where $d$ is the $p$-adic continuous differential. Thus the universal map from
algebraic de Rham complex $(\Omega_{R/K}^{\bullet,alg},d)$ to
$(\mathrm{H}^{\bullet}(\overline{C}_{P})(\bullet),\beta_{\xi})$ identifies its
analytic quotient with the target. At last, notice that since the category of
regular surjections $P\rightarrow R$ is sifted as in Remark 4.3.3 (c.f.
[Guo21, Lemma 5.5.4]), the isomorphism above is independent of the choice of
$P\rightarrow R$. So we are done.

∎

###  5.2. Simplicial resolution for affinoid algebras

To extend the Hodge-Tate filtration to non-smooth rigid spaces, we will need
the simplicial resolution of affinoid algebras by smooth ones. Here we mention
that different from the algebraic setting, we cannot use the standard
polynomial resolution where variables are infinite. To resolve topologically
finite type algebras over a $p$-adic field, we will need a version of
simplicial resolution such that each term satisfies the finite generatedness
condition, thus within the category of topologically finite type affinoid
algebras.

We start by recalling the following general construction resolving a given
class of a given simplicial algebra using finite type polynomials. As a
convention, for a simplicial abelian group $A=A_{\bullet}$ and a co-degeneracy
map $s^{j}:[n+1]\rightarrow[n]$, we use $s_{j}$ to denote the corresponding
map $A_{n}\rightarrow A_{n+1}$ for $A$. Similarly for (co-) face maps.

######  Construction 5.2.1.

Let $R$ be a commutative ring, and $A$ be a simplicial $R$-algebra. Let
$d\in\mathbb{N}$ be an integer, and $\omega\in A_{d-1}$ be a cycle of degree
$d-1$.

For each $n\in\mathbb{N}$, let $X_{n}$ be the finite set

$X_{n}:=\\{x_{t}|t:[n]\rightarrow[d]\text{ is surjective and monotone}\\}.$

Then by [[Iy07], 4.10], there is a simplicial $A$-algebra structure on
$B=A[\\{x\\}|\partial x=\omega]$. defined as the free simplicial $A$-algebra
such that $B_{n}=A_{n}[X_{n}]$.

The lemma below describes the properties we need for this resolution.

######  Lemma 5.2.2.

Let $R$ be a ring, and $A$ be a simplicial $R$-algebra. Then the simplicial
$A$-algebra $B=A[\\{x\\}|\partial x=\omega]$ satisfies the following:

  1. (i)

For for $n\leq d-1$, we have $A_{n}=B_{n}$.

  2. (ii)

For each $n\in\mathbb{N}$, $B_{n}$ is a polynomial of finite variables over
$A_{n}$.

  3. (iii)

The natural map $A\rightarrow B$ induces a short exact sequence

$0\longrightarrow
A_{d-1}\cdot\omega\longrightarrow\pi_{d-1}(A)\longrightarrow\pi_{d-1}(B)\longrightarrow
0.$

In particular the cycle $\omega\in\pi_{d-1}(A)$ is killed in $B$.

Now we apply the above construction to the $p$-adic setting, to show the
existence of topologically finite type resolutions for an affinoid algebra.

######  Theorem 5.2.3.

Let $(B,I)$ be a topologically finitely presented algebra over
${\mathrm{A_{inf,e}}}$ with a finitely generated ideal, such that $B/I$ is
$p$-torsion free. Then there exists a quasi-isomorphism from simplicial pairs
of ${\mathrm{A_{inf,e}}}$-algebras with ideals,

$({\mathrm{A_{inf,e}}}\langle
X_{\bullet},Y_{\bullet}\rangle,(Y_{\bullet}))\longrightarrow(B,I),$

where both $X_{i}$ and $Y_{i}$ are finite for each $i\in\mathbb{N}$.

###### Proof.

We proceed the construction by induction as below.

  * Step 1

By assumption, $B$ is topologically finitely presented and $I$ is finitely
generated. We pick two finite sets $X^{(0)}$ and $Y^{(0)}$ that corresponds to
the generators of $B$ (over ${\mathrm{A_{inf,e}}}$ as a topological algebra)
and $I$ (over $B$ as a module) separately, with surjections
$(f^{(0)},g^{(0)}):{\mathrm{A_{inf,e}}}\langle
X^{(0)},Y^{(0)}\rangle\rightarrow B$ and $(Y^{(0)})\rightarrow I$. We denote
$({\mathrm{A_{inf,e}}}\langle
X^{(0)}_{\bullet},Y^{(0)}_{\bullet}\rangle,(Y^{(0)}_{\bullet}))$ to be the
associated constant simplicial algebra with the ideal.

  * Step 2

We now construct the simplcial pair $({\mathrm{A_{inf,e}}}\langle
X^{(1)}_{\bullet},Y^{(1)}_{\bullet}\rangle,(Y^{(1)}_{\bullet}))$ together with
a map

$f^{(1)}:({\mathrm{A_{inf,e}}}\langle
X^{(0)}_{\bullet},Y^{(0)}_{\bullet}\rangle,(Y^{(0)}_{\bullet}))\longrightarrow({\mathrm{A_{inf,e}}}\langle
X^{(1)}_{\bullet},Y^{(1)}_{\bullet}\rangle,(Y^{(1)}_{\bullet})),$

such that the image of $(\ker(f^{(0)}),\ker(g^{(0)}))$ is killed.

Both $\ker(f^{(0)})$ and $\ker(g^{(0)})$ are finitely generated
${\mathrm{A_{inf,e}}}\langle
X^{(0)}_{\bullet},Y^{(0)}_{\bullet}\rangle$-modules (this follows from [Guo21,
Corollary 5.1.3 (iii)]). So we can pick finite generators $u_{i}$ of
$\ker(f^{(0)})$ and $v_{j}$ of $\ker(g^{(0)})$ separately. Using 5.2.1 we form
a free simplicial algebra

${\mathrm{A_{inf,e}}}\langle
X^{(0)}_{\bullet},Y^{(0)}_{\bullet}\rangle[\partial U_{i}=u_{i}][\partial
V_{j}=v_{j}]$

over ${\mathrm{A_{inf,e}}}\langle X^{(1)}_{\bullet},Y^{(1)}_{\bullet}\rangle$,
such that the image of the following two maps vanish (Lemma 5.2.2 (iii)):

$\displaystyle\ker(f^{(0)})$
$\displaystyle\longrightarrow\pi_{0}({\mathrm{A_{inf,e}}}\langle
X^{(1)}_{\bullet},Y^{(1)}_{\bullet}\rangle[\partial U_{i}=u_{i}][\partial
V_{j}=v_{j}]),$ $\displaystyle\ker(g^{(0)})$
$\displaystyle\longrightarrow\pi_{0}((Y^{(0)}_{\bullet},V_{\bullet})).$

We take the term-wise $p$-adic completion at the simplicial ring
${\mathrm{A_{inf,e}}}\langle
X^{(0)}_{\bullet},Y^{(0)}_{\bullet}\rangle[\partial U_{i}=u_{i}][\partial
V_{j}=v_{j}]$. Then by Lemma 5.2.2 (ii), we get a new simplicial pair over
${\mathrm{A_{inf,e}}}$:

$({\mathrm{A_{inf,e}}}\langle
X^{(1)}_{\bullet},Y^{(1)}_{\bullet}\rangle,(Y^{(1)})),$

where each $X^{(1)}_{n}$ (resp. $Y^{(1)}_{n}$) is the finite set given by the
union of $X^{(0)}_{n}$ (resp. $Y^{(0)}_{n}$) with the newly added variables
from $[\partial U_{i}=u_{i}]$ (resp. $[\partial V_{j}=v_{j}]$), as in 5.2.1.
Moreover, the image of the pair of ideals $(\ker(f^{(0)}),\ker(g^{(0)})$ in
the $0$-th fundamental group vanishes, by the composition below

$\textstyle{(\ker(f^{(0)}),\ker(g^{(0)}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{({\mathrm{A_{inf,e}}}\langle
X^{(0)}_{\bullet},Y^{(0)}_{\bullet}\rangle[\partial U_{i}=u_{i}][\partial
V_{j}=v_{j}],(Y^{(0)}\cup\widetilde{V}_{\bullet}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{({\mathrm{A_{inf,e}}}\langle
X^{(1)}_{\bullet},Y^{(1)}_{\bullet}\rangle,(Y^{(1)})).}$

(Here we temporarily use $\widetilde{V}_{\bullet}$ to denote the newly added
variables in this step). Note that by Lemma 5.2.2 (iii) the zero-th
fundamental group of $({\mathrm{A_{inf,e}}}\langle
X^{(1)}_{\bullet},Y^{(1)}_{\bullet}\rangle,(Y^{(1)}))$ is exactly isomorphic
to $(A_{e},I_{e})$.

  * Step 3

As fundamental groups $\pi_{i}$ of $(B,I)$ are trivial for $i\geq 1$, we use
the above agreement to kill the cycles of $({\mathrm{A_{inf,e}}}\langle
X^{(1)}_{\bullet},Y^{(1)}_{\bullet}\rangle,(Y^{(1)}))$ inductively, starting
from $\pi_{1}$. We first make the following observation about the finiteness:

######  Claim 5.2.4.

Let $J=J_{\bullet}$ be a simplicial finitely generated ideal of a simplicial
topologically finitely presented $\mathrm{A_{inf,e}}$-algebra $A_{\bullet}$.
Then for each $n\in\mathbb{N}$, the $n$-th fundamental group
$\pi_{n}(J_{\bullet})$ is a finitely generated $A_{n}$-module.

Granting the Claim, we can apply it onto the unit ideal and the ideal
$(Y^{(n)}_{\bullet})$ in the simplicial ring ${\mathrm{A_{inf,e}}}\langle
X^{(n)}_{\bullet},Y^{(n)}_{\bullet}\rangle$. Then exactly as in Step 2, by
using 5.2.1 and taking the term-wise $p$-adic completion, we can produce the
simplicial pair $({\mathrm{A_{inf,e}}}\langle
X^{(n+1)}_{\bullet},Y^{(n+1)}_{\bullet}\rangle,(Y^{(n+1)}_{\bullet}))$, so
that $\pi_{n}$ of the simplicial pair ${\mathrm{A_{inf,e}}}\langle
X^{(n)}_{\bullet},Y^{(n)}_{\bullet}\rangle$ is killed. By Lemma 5.2.2 (ii),
this will preserve the finite sets $X_{i}$ and $Y_{i}$ for $i\leq n$. So we
can take the colimit to get a simplicial finitely presented
${\mathrm{A_{inf,e}}}$-pair

$({\mathrm{A_{inf,e}}}\langle
X_{\bullet},Y_{\bullet}\rangle,(Y_{\bullet})):=\underset{n\in\mathbb{N}}{{\mathrm{colim}}}({\mathrm{A_{inf,e}}}\langle
X^{(n)}_{\bullet},Y^{(n)}_{\bullet}\rangle,(Y^{(n)}_{\bullet})).$

Note that by the construction, as $\pi_{*}({\mathrm{A_{inf,e}}}\langle
X_{\bullet},Y_{\bullet}\rangle,(Y_{\bullet}))=\pi_{0}({\mathrm{A_{inf,e}}}\langle
X_{\bullet},Y_{\bullet}\rangle,(Y_{\bullet}))=(A,I)$, the natural map from
$({\mathrm{A_{inf,e}}}\langle X_{\bullet},Y_{\bullet}\rangle,(Y_{\bullet}))$
to its $0$-th fundamental group is a surjective quasi-isomorphism. This
finishes the construction we need.

###### Proof of the Claim.

As in [Iy07, 3.4], to compute the $i$-th fundamental group, we can take the
normalization of $J_{\bullet}$ by

$N(J)_{n}=\cap_{i=1}^{n}\ker(d_{i}).$

Then $\pi_{n}(J)$ is a quotient of the kernel ideal for the $A_{n}$-linear map

$\ker(d_{0}|_{N(J)_{n}}:N(J)_{n}\rightarrow N(J)_{n-1}).$

We note that the group $N(J)_{n}$ can be written as the kernel of the
$A_{n}$-linear map

$\textstyle{J_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\oplus_{i=1}^{n}d_{i}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}}$$\textstyle{\bigoplus_{i=1}^{n}J_{n-1},}$

where the $A_{n}$-linear structure on each component of the right side is
defined through the ring homomorphism $d_{i}:A_{n}\rightarrow A_{n-1}$. But
since $J_{n}$ is finitely generated, while $J_{n-1}$ is $p$-torsion free, by
[Sta, Tag 0519] and [Guo21, Corollary 5.1.3], the kernel of the map above is
finitely generated over $A_{n}$. This implies that the $p$-torsion free
$A_{n}$-module $N(J)_{n}$ is finitely presented over $A_{n}$. In this way, the
kernel of the $A_{n}$-linear map

$d_{0}:N(J)_{n}\longrightarrow N(J)_{n-1},$

with the source being finitely presented and the target being $p$-torsion
free, is finitely generated. This finishes the proof showing
$\pi_{n}(J_{\bullet})$ is finitely generated over $A_{n}$, and we are done.

∎

∎

Moreover, the above allows us to establish a rational version as well.

######  Proposition 5.2.5.

Let $(B,I)$ be a topologically finitely presented algebra over
${\mathrm{B^{+}_{dR,e}}}$ with a finitely generated ideal. Then there exists a
quasi-isomorphism from a simplicial pair of ${\mathrm{B^{+}_{dR,e}}}$-algebras
with ideals

$({\mathrm{B^{+}_{dR,e}}}\langle
X_{\bullet},Y_{\bullet}\rangle,(Y_{\bullet}))\longrightarrow(B,I),$

where both $X_{i}$ and $Y_{i}$ are finite for each $i\in\mathbb{N}$. Moreover,
the same holds if we replace ${\mathrm{B^{+}_{dR,e}}}$ by
${\mathrm{B_{dR}^{+}}}$.

###### Proof.

The case for ${\mathrm{B^{+}_{dR,e}}}$ follows from Theorem 5.2.3 by choosing
an integral model of the pair over ${\mathrm{A_{inf,e}}}$, since
${\mathrm{B^{+}_{dR,e}}}={\mathrm{A_{inf,e}}}[\frac{1}{p}]$. To get the result
for ${\mathrm{B_{dR}^{+}}}$-pair $(B,I)$, we need to adjust the proof of
Theorem 5.2.3, changing its base from ${\mathrm{A_{inf,e}}}$ to
${\mathrm{B_{dR}^{+}}}$. The Step 1 in Theorem 5.2.3 is identical. For Step 2,
we need to find a map

$f^{(1)}:({\mathrm{B_{dR}^{+}}}\langle
X_{\bullet}^{(0)},Y_{\bullet}^{(0)}\rangle,(Y_{\bullet}^{(0)}))\longrightarrow({\mathrm{B_{dR}^{+}}}\langle
X^{(1)}_{\bullet},Y^{(1)}_{\bullet}\rangle,(Y^{(1)}_{\bullet})),$

such that the image of $(\ker(f^{(0)}),\ker(g^{(0)}))$ vanishes. Let us choose
a finite set of generators $\\{h_{l}\\}$ of the kernel ideals so that their
mod $\xi$ reduction are power bounded (i.e. inside of $\mathcal{O}_{K}\langle
X_{\bullet}^{(0)},Y_{\bullet}^{(0)}\rangle$). Then imitating Step 2 in Theorem
5.2.3, we may choose a compatible set of maps

$({\mathrm{B^{+}_{dR,e}}}\langle
X^{(0)}_{\bullet},Y^{(0)}_{\bullet}\rangle,(Y_{\bullet}^{(0)}))\longrightarrow({\mathrm{B^{+}_{dR,e}}}\langle
X^{(1)}_{\bullet},Y^{(1)}_{\bullet}\rangle,(Y_{\bullet}^{(1)})),$

such that the image of the mod $\xi^{e}$ reduction of $\\{h_{l}\\}$ vanishes.
Thus by taking the inverse limit with respect to $e$, we get Step 2.

At last, to proceed Step 3 of Theorem 5.2.3, we notice that the finiteness in
5.2.4 holds automatically for ${\mathrm{B_{dR}^{+}}}$-pairs
$({\mathrm{B_{dR}^{+}}}\langle
X^{(n)}_{\bullet},Y^{(n)}_{\bullet}\rangle,(Y^{(n)}_{\bullet}))$ over
${\mathrm{B_{dR}^{+}}}$, as each ${\mathrm{B_{dR}^{+}}}\langle
X^{(n)}_{i},Y^{(n)}_{i}\rangle$ is noetherian. In this way, similar to the
above modification on Step 2, we can mod powers of $\xi$ and kills the
reduction of a fixed finite set of generators for
$\pi_{n}({\mathrm{B_{dR}^{+}}}\langle
X^{(n)}_{\bullet},Y^{(n)}_{\bullet}\rangle,(Y^{(n)}_{\bullet}))$. A further
inverse limit of this compatible system of simplicial pairs finishes the
proof. ∎

In the special case when $I$ is the zero ideal, we get the resolution of the
algebra itself.

######  Corollary 5.2.6.

Assume $R$ is either a $p$-torsionfree topologically finitely presented
${\mathrm{A_{inf,e}}}/{\mathrm{B^{+}_{dR,e}}}/{\mathrm{B_{dR}^{+}}}$-algebra.
There is a simplicial
${\mathrm{A_{inf,e}}}/{\mathrm{B^{+}_{dR,e}}}/{\mathrm{B_{dR}^{+}}}$-algebra
resolution $A_{\bullet}\rightarrow R$, such that each $A_{n}$ is isomorphic to
a Tate algebra of finite variables over
${\mathrm{A_{inf,e}}}/{\mathrm{B^{+}_{dR,e}}}/{\mathrm{B_{dR}^{+}}}$.

###  5.3. Left Kan extension and Hodge-Tate filtration

Now we are able to extend the Hodge-Tate filtration to general l.c.i rigid
spaces, using simplicial resolution developed last subsection. The idea is
similar to the left Kan extension used in the algebraic setting, but we do it
over affinoid algebras.

We start by observing that derived de Rham complex can be computed using
simplicial resolution.

######  Proposition 5.3.1.

Let $R$ be a topologically finite type ${\mathrm{B^{+}_{dR,e}}}$-algebra. Then
for a given simplicial resolution ${\mathrm{B^{+}_{dR,e}}}\langle
X_{\bullet}\rangle\rightarrow R$ where each set of variables $X_{n}$ is
finite, we have

$\displaystyle\underset{\Delta}{{\mathrm{colim}}}~{}\mathbb{L}^{\mathrm{an}}_{{\mathrm{B^{+}_{dR,e}}}\langle
X_{\bullet}\rangle/{\mathrm{B^{+}_{dR,e}}}}\cong\mathbb{L}^{\mathrm{an}}_{R/{\mathrm{B^{+}_{dR,e}}}},$
$\displaystyle\underset{\Delta}{{\mathrm{colim}}}~{}{\widehat{\mathrm{dR}}^{\mathrm{an}}}_{{\mathrm{B^{+}_{dR,e}}}\langle
X_{\bullet}\rangle/{\mathrm{B^{+}_{dR,e}}}}/{\mathrm{Fil}}^{m}\cong{\widehat{\mathrm{dR}}^{\mathrm{an}}}_{R/{\mathrm{B^{+}_{dR,e}}}}/{\mathrm{Fil}}^{m},$
$\displaystyle\text{filtered completion of
}\left(\underset{\Delta}{{\mathrm{colim}}}~{}{\widehat{\mathrm{dR}}^{\mathrm{an}}}_{{\mathrm{B^{+}_{dR,e}}}\langle
X_{\bullet}\rangle/{\mathrm{B^{+}_{dR,e}}}}\right)\cong{\widehat{\mathrm{dR}}^{\mathrm{an}}}_{R/{\mathrm{B^{+}_{dR,e}}}}.$

###### Proof.

We temporarily use the notation $(-)^{\wedge}_{p}$ to denote the derived
$p$-completion. Let $R_{0}$ be a ring of definition of $R$, which is
topologically finite type over ${\mathrm{A_{inf,e}}}$. By induction, we can
find a simplicial topologically finite type ${\mathrm{A_{inf,e}}}$-algebras
$A_{\bullet}\rightarrow R_{0}$, such that each $A_{n}$ is a ring of definition
of ${\mathrm{B^{+}_{dR,e}}}\langle X_{n}\rangle$. The triple of rings
${\mathrm{A_{inf,e}}}\rightarrow A_{\bullet}\rightarrow R_{0}$ then induces
the natural triangle of algebraic cotangent complexes

$\mathbb{L}_{A_{\bullet}/{\mathrm{A_{inf,e}}}}\otimes_{A_{\bullet}}^{L}R_{0}\longrightarrow\mathbb{L}_{R_{0}/{\mathrm{A_{inf,e}}}}\longrightarrow\mathbb{L}_{R_{0}/A_{\bullet}},$

where the final term is equal to the homotopy colimit
$\underset{[n]\in\Delta}{{\mathrm{colim}}}~{}\mathbb{L}_{R_{0}/A_{n}}$. We
then apply the derived $p$-completion at the triangle and then invert at $p$.
By [Sta, Tag 091V], the derived completion commutes with colimits. In
particular we get the equalities

$\displaystyle(\mathbb{L}_{R_{0}/A_{\bullet}})^{\wedge}_{p}[\frac{1}{p}]$
$\displaystyle=(\underset{[n]\in\Delta}{{\mathrm{colim}}}~{}\mathbb{L}_{R_{0}/A_{n}})^{\wedge}_{p}[\frac{1}{p}]$
$\displaystyle\cong\underset{[n]\in\Delta}{{\mathrm{colim}}}~{}(\mathbb{L}_{R_{0}/A_{n}})^{\wedge}_{p}[\frac{1}{p}]$
$\displaystyle=\underset{[n]\in\Delta}{{\mathrm{colim}}}~{}\mathbb{L}^{\mathrm{an}}_{R/{\mathrm{B^{+}_{dR,e}}}\langle
X_{n}\rangle}.$

Note that by assumption the map ${\mathrm{B^{+}_{dR,e}}}\langle