$(\Omega_{(B,J,\delta)/(A,I,\gamma)}^{\ast},\mathrm{d})$. The Whitehead tower
$(\tau_{\geq
n}\operatorname{dR}_{(B,J,\delta)/(A,I,\gamma)})_{n\in(\mathbb{Z},\geq)}$
defines a nonpositive202020In the literature, the conjugate filtration is
increasing. We make it decreasing by negating the sign. exhaustive filtration,
thus the map
$\varphi_{(A,I)}^{\ast}(B/J)\rightarrow(\Omega_{(B,J,\delta)/(A,I,\gamma)}^{\ast},\mathrm{d})$
is a map of filtered $\mathbb{F}_{p}$-CDGAs (where
$\varphi_{(A,I)}^{\ast}(B/J)$ is trivially filtered), which gives rise to a
map of filtered $\mathbb{E}_{\infty}$-$\mathbb{F}_{p}$-algebras. Combined with
the map above, we get a functor
$\operatorname{dRCon}_{\mathbb{F}_{p}}^{0}\rightarrow\operatorname{Fun}(\Delta^{1},\operatorname{CAlg}(\operatorname{DF}^{\leq
0}(\mathbb{F}_{p})))$.

######  Definition 4.37.

The conjugate-filtered derived de Rham cohomology functor
$\operatorname{Fil}_{\operatorname{conj}}^{\ast}\operatorname{dR}_{\cdot/\cdot}:\operatorname{dRCon}_{\mathbb{F}_{p}}\rightarrow\operatorname{CAlg}(\operatorname{DF}^{\leq
0}(\mathbb{F}_{p}))$ along with the structure map
$\varphi_{(A\twoheadrightarrow
A^{\prime\prime})}^{\ast}(B^{\prime\prime})\rightarrow\operatorname{Fil}_{\operatorname{conj}}^{\ast}\operatorname{dR}_{(B\twoheadrightarrow
B^{\prime\prime},\delta)/(A\twoheadrightarrow A^{\prime\prime},\gamma)}$ is
defined to be the left derived functor (Proposition A.10) of the functor
$\operatorname{dRCon}_{\mathbb{F}_{p}}^{0}\rightarrow\operatorname{Fun}(\Delta^{1},\operatorname{CAlg}(\operatorname{DF}^{\leq
0}(\mathbb{F}_{p})))$ above.

It follows either from Propositions A.10 and 2.43 or the fact that
$\operatorname{AniPair}\simeq\mathcal{P}_{\Sigma}(\mathcal{D}^{0})\subseteq\mathcal{P}(\mathcal{D}^{0})$
is stable under filtered colimits (Proposition A.7) that

######  Lemma 4.38.

The conjugate filtration on the derived de Rham cohomology is exhaustive.

We now prove the corresponding results of Section 4.1 for the conjugate
filtration.

######  Lemma 4.39.

The conjugate-filtered derived de Rham cohomology functor
$\operatorname{dRCon}_{\mathbb{F}_{p}}\rightarrow\operatorname{CAlg}(\operatorname{DF}^{\leq
0}(\mathbb{F}_{p}))$ preserves small colimits (note that so does the functor
$\operatorname{dRCon}_{\mathbb{F}_{p}}\rightarrow\operatorname{CAlg}_{\mathbb{F}_{p}},((A\twoheadrightarrow
A^{\prime\prime},\gamma)\rightarrow(B\twoheadrightarrow
B^{\prime\prime},\delta))\mapsto\varphi_{(A\twoheadrightarrow
A^{\prime\prime})}^{\ast}(B^{\prime\prime})$).

###### Proof.

First, we note that, for any connective $\mathbb{E}_{\infty}$-ring $A$, the
Whitehead-tower functor
$D(A)\rightarrow\operatorname{DF}(A),M\mapsto(\tau_{\geq
n}M)_{n\in(\mathbb{Z},\geq)}$ is canonically lax symmetric monoidal (recall
that $\operatorname{DF}(A)$ is endowed with the Day convolution). We give an
informal description: given $M,N\in D(A)$, for all $m,n\in\mathbb{Z}$, the
canonical map $\tau_{\geq m}M\rightarrow M$ and $\tau_{\geq n}N\rightarrow N$
gives rise to a map $(\tau_{\geq m}M)\otimes_{A}^{\mathbb{L}}(\tau_{\geq
n}N)\rightarrow M\otimes_{A}^{\mathbb{L}}N$. Since $(\tau_{\geq
m}M)\otimes_{A}^{\mathbb{L}}(\tau_{\geq n}N)$ is $(m+n)$-connective, this
gives rise to a map $(\tau_{\geq m}M)\otimes_{A}^{\mathbb{L}}(\tau_{\geq
n}N)\rightarrow\tau_{\geq m+n}(M\otimes_{A}^{\mathbb{L}}N)$. Assembling these
maps, we get the lax symmetric monoidal structure. Next, when $A$ is given by
a field, in particular, $A=\mathbb{F}_{p}$, the structure above is in fact
symmetric monoidal, since $(\tau_{\geq m}M)\otimes_{A}^{\mathbb{L}}(\tau_{\geq
n}N)\rightarrow\tau_{\geq m+n}(M\otimes_{A}^{\mathbb{L}}N)$ is an equivalence
for all $m,n\in\mathbb{Z}$.

Now recall that in a symmetric monoidal $\infty$-category, finite coproducts
of commutative algebra objects are given by tensor products. It follows from
Lemma 4.12 that the conjugate-filtered derived de Rham cohomology functor
$\operatorname{dRCon}_{\mathbb{F}_{p}}\rightarrow\operatorname{CAlg}(\operatorname{DF}^{\leq
0}(\mathbb{F}_{p}))$ is the left derived functor of a finite-coproduct-
preserving functor, and then the result follows from Proposition A.10. ∎

Note that by the divided power Poincaré’s lemma [Sta21, Tag 07LC], the
conjugate filtration on the divided power polynomial algebra is trivial. The
proof of Proposition 4.16 leads to

######  Proposition 4.40.

The natural transformation
$\operatorname{Fil}_{\operatorname{conj}}^{\ast}\operatorname{dR}_{(B\twoheadrightarrow
B^{\prime\prime},\delta)/(A\twoheadrightarrow
A^{\prime\prime},\gamma)}\rightarrow\operatorname{Fil}_{\operatorname{conj}}^{\ast}\operatorname{dR}_{(\operatorname{id}_{B^{\prime\prime}},0)/(A\rightarrow
A^{\prime\prime},\gamma)}$ of functors
$\operatorname{dRCon}\rightrightarrows\operatorname{CAlg}(\operatorname{DF}^{\leq
0}(\mathbb{Z}))$ induced by the counit map associated to
$((A\twoheadrightarrow A^{\prime\prime},\gamma)\rightarrow(B\twoheadrightarrow
B^{\prime\prime},\delta))\in\operatorname{dRCon}$ is an equivalence. In other
words, the conjugate-filtered de Rham cohomology functor
$\operatorname{dRCon}\rightarrow\operatorname{CAlg}(\operatorname{DF}^{\leq
0}(\mathbb{Z}))$ is
$(\operatorname{dRCon}\rightarrow\operatorname{CrysCon})$-invariant
(Definition 2.57) (note that so is the functor
$\operatorname{dRCon}_{\mathbb{F}_{p}}\rightarrow\operatorname{CAlg}_{\mathbb{F}_{p}},((A\twoheadrightarrow
A^{\prime\prime},\gamma)\rightarrow(B\twoheadrightarrow
B^{\prime\prime},\delta))\mapsto\varphi_{(A\twoheadrightarrow
A^{\prime\prime})}^{\ast}(B^{\prime\prime})$).

######  Definition 4.41.

The conjugate-filtered derived crystalline cohomology functor
$\operatorname{Fil}_{\operatorname{conj}}^{\ast}\operatorname{CrysCoh}:\operatorname{CrysCon}\rightarrow\operatorname{CAlg}(\operatorname{DF}^{\leq
0}(\mathbb{F}_{p}))$ along with the structure map
$\varphi_{(A\twoheadrightarrow
A^{\prime\prime},\gamma)}^{\ast}(R)\rightarrow\operatorname{CrysCoh}_{R/(A\twoheadrightarrow
A^{\prime\prime},\gamma)}$ is defined to be the composite
$\operatorname{CrysCon}\rightarrow\operatorname{dRCon}\rightarrow\operatorname{Fun}(\Delta^{1},\operatorname{CAlg}(\operatorname{DF}^{\leq
0}(\mathbb{F}_{p})))$, where the later functor is the conjugate-filtered
derived de Rham cohomology functor combined with the structure map.

By Lemma 4.38, we have

######  Lemma 4.42.

The conjugate filtration on the derived crystalline cohomology is exhaustive.

Similar to Proposition 4.19, we have

######  Proposition 4.43.

The conjugate-filtered derived crystalline cohomology functor
$\operatorname{CrysCon}\rightarrow\operatorname{CAlg}(\operatorname{DF}^{\leq
0}(\mathbb{F}_{p}))$ preserves small colimits.

Now we analyze the associated graded pieces of the conjugate filtration. Let
$(A,I,\gamma)\rightarrow(B,J,\delta)$ be an element in
$\operatorname{dRCon}^{0}$. We recall the inverse212121A priori, the “inverse”
Cartier map $C^{-1}$ is not defined to be the inverse of a map, but just
defined to be a map. Cartier map
$C^{-1}:\varphi_{(A,I)}^{\ast}(\Omega_{(B/J)/(A/I)}^{\star})\rightarrow
H^{\star}(\Omega_{(B,J)/(A,I)}^{\ast},\mathrm{d})$ of graded
$\varphi_{(A,I)}^{\ast}(B/J)$-algebras (where $\star$ is the grading), then we
deduce that this is in fact an isomorphism. Our presentation is adapted from
the proof of [Kat70, Thm 7.2].

$\mathord{\star}=0$:

This is the composite map $\varphi_{(A,I)}^{\ast}(B/J)\rightarrow B\rightarrow
H^{0}(\Omega_{(B,J)/(A,I)}^{\ast},\mathrm{d})$, i.e., the
$\varphi_{(A,I)}^{\ast}(B/J)$-algebra structure on
$H^{0}(\Omega_{(B,J)/(A,I)},\mathrm{d})$.

$\mathord{\star}=1$:

Consider the map $B\rightarrow H^{1}(\Omega_{(B,J)/(A,I)}^{\ast},\mathrm{d})$
of sets given by $f\mapsto[f^{p-1}\mathrm{d}f]$. We first check that this map
is additive: in $\Omega_{\mathbb{Z}[u,v]/\mathbb{Z}}^{1}$, we have

$\displaystyle(u+v)^{p-1}\mathrm{d}(u+v)-u^{p-1}\mathrm{d}u-v^{p-1}\mathrm{d}v$
$\displaystyle=$
$\displaystyle\frac{1}{p}(\mathrm{d}((u+v)^{p})-\mathrm{d}(u^{p})-\mathrm{d}(v^{p}))$
$\displaystyle=$
$\displaystyle\frac{1}{p}\mathrm{d}\left(\sum_{j=1}^{p-1}\binom{p}{j}u^{j}v^{p-1-j}\right)$
$\displaystyle=$
$\displaystyle\mathrm{d}\left(\sum_{j=1}^{p-1}\frac{1}{p}\binom{p}{j}u^{j}v^{p-1-j}\right)$

We deduce the additivity by the map $\mathbb{Z}[u,v]\rightarrow B,u\mapsto
f,v\mapsto g$.

Now we note that the map $f\mapsto[f^{p-1}\mathrm{d}f]$ satisfies Leibniz rule
(recall that $H^{1}(\Omega_{(B,J)/(A,I)}^{\ast},\mathrm{d})$ is a
$\varphi_{(A,I)}^{\ast}(B/J)$-module, therefore a $B/J$-module). Indeed,
$[(fg)^{p-1}\mathrm{d}(fg)]=f^{p}[g^{p-1}\mathrm{d}g]+g^{p}[f^{p-1}\mathrm{d}f]$.

Thus we get a derivation $B/J\rightarrow
H^{1}(\Omega_{(B,J)/(A,I)}^{\ast},\mathrm{d})$, which gives rise to a
$B/J$-linear map $\Omega_{(B/J)/(A/I)}^{1}\rightarrow
H^{1}(\Omega_{(B,J)/(A,I)}^{\ast},\mathrm{d})$ and after linearization, we get
$\varphi_{(A,I)}^{\ast}\Omega_{(B/J)/(A,I)}^{1}\rightarrow
H^{1}(\Omega_{(B,J)/(A,I)}^{\ast},\mathrm{d})$.

$\mathord{\star}\char 62\relax 1$:

Taking the exterior power of the map for $\mathord{\star}=1$.

Now we show the Cartier isomorphism:

######  Lemma 4.44.

Let $(A,I,\gamma)\rightarrow(B,J,\delta)$ be an element in
$\operatorname{CrysCon}_{\mathbb{F}_{p}}^{0}$. Then the inverse Cartier map
$C^{-1}:\varphi_{(A,I)}^{\ast}(\Omega_{(B/J)/(A/I)}^{\star})\rightarrow
H^{\star}(\Omega_{(B,J)/(A,I)}^{\ast},\mathrm{d})$ is an isomorphism of graded
$\varphi_{(A,I)}^{\ast}(B/J)$-algebras.

###### Proof.

Recall that $(B,J,\delta)$ is of the form
$(\Gamma_{A[X]}(Y)\twoheadrightarrow(A/I)[X],\delta)$. It is then direct to
check that the inverse Cartier map $C^{-1}$ factors as
$\varphi_{(A,I)}^{\ast}(\Omega_{(B/J)/(A/I)}^{\star})\rightarrow
H^{\star}(\Omega_{(A[X],IA[X])/(A,I)}^{\ast},\mathrm{d})\rightarrow
H^{\star}(\Omega_{(B,J)/(A,I)}^{\ast},\mathrm{d})$, where the first map is the
inverse Cartier map associated to $(A\twoheadrightarrow
A/I,\gamma)\rightarrow(A[X],IA[X],\gamma)$, and the second map is an
isomorphism by the divided power Poincaré’s lemma [Sta21, Tag 07LC].

Thus we can assume that $(B,J,\delta)=(A[X],IA[X],\gamma)$. In this case, the
inverse Cartier map is base-changed from that for
$(A,0,0)\rightarrow(A[X],0,0)$ along $(A,0,0)\rightarrow(A,I,\gamma)$, thus we
can assume that $I=0$, which is [Kat70, Thm 7.2]. ∎

It then follows from Proposition A.10 that

######  Proposition 4.45.

There exists a natural isomorphism222222To avoid the ambiguity of symbols, we
suppress the asterisk on $\operatorname{Fil}^{\ast}$ to avoid confusion with
the pullback symbol $\varphi^{\ast}$.

$C^{-1}:\varphi_{(A\twoheadrightarrow
A^{\prime\prime})}^{\ast}\left(\bigwedge_{B^{\prime\prime}}^{\star}\mathbb{L}_{B^{\prime\prime}/A^{\prime\prime}}\right)\left[-\mathord{\star}\right]\rightarrow\operatorname{gr}_{\operatorname{conj}}^{-\mathord{\star}}\operatorname{dR}_{(B\twoheadrightarrow
B^{\prime\prime})/(A\twoheadrightarrow A^{\prime\prime})}$

in $\operatorname{CAlg}(\operatorname{Gr}^{\leq
0}(D(\varphi_{A\twoheadrightarrow
A^{\prime\prime}}^{\ast}(B^{\prime\prime}))))$, called the derived Cartier
isomorphism (cf. [Bha12a, Prop 3.5]), which is functorial232323Here we use the
same convention as in Remark 3.61. in $((A\twoheadrightarrow
A^{\prime\prime},\gamma)\rightarrow(B\twoheadrightarrow
B^{\prime\prime},\delta))\in\operatorname{dRCon}_{\mathbb{F}_{p}}$.

Note that both functors are
$(\operatorname{dRCon}_{\mathbb{F}_{p}}\rightarrow\operatorname{CrysCon}_{\mathbb{F}_{p}})$-invariant
(Definition 2.57), it follows from Proposition 2.56 that

######  Proposition 4.46.

There exists a natural isomorphism

$C^{-1}:\varphi_{(A\twoheadrightarrow
A^{\prime\prime})}^{\ast}\left(\bigwedge_{R}^{\star}\mathbb{L}_{R/A^{\prime\prime}}\right)\left[-\mathord{\star}\right]\rightarrow\operatorname{gr}_{\operatorname{conj}}^{-\mathord{\star}}\operatorname{CrysCoh}_{R/(A\twoheadrightarrow
A^{\prime\prime})}$

in $\operatorname{CAlg}(\operatorname{Gr}^{\leq
0}(D(\varphi_{A\twoheadrightarrow A^{\prime\prime}}^{\ast}(R))))$, called the
derived Cartier isomorphism, which is functorial in $((A\twoheadrightarrow
A^{\prime\prime},\gamma),A^{\prime\prime}\rightarrow
R)\in\operatorname{CrysCon}_{\mathbb{F}_{p}}$.

###  4.3. Relative animated PD-envelope

As in the classical case [Sta21, Tag 07H9], there is a relative version of
animated PD-envelope, which is needed to study the derived crystalline
cohomology, which is defined by the adjunction of undercategories:

######  Lemma 4.47 (dual to [Lur09, Prop 5.2.5.1]).

Let
$\mathcal{C}\underset{G}{\overset{F}{\mathrel{\begin{subarray}{c}\longrightarrow\\\\[-2.58334pt]
\longleftarrow\end{subarray}}}}\mathcal{D}$ be an adjoint pair of
$\infty$-categories. Assume that the $\infty$-category $\mathcal{D}$ admits
pushouts and let $D\in\mathcal{D}$ be an object. Then

  1. (1)

The induced functor
$g:\mathcal{D}_{D\mathord{/}}\rightarrow\mathcal{C}_{GD\mathord{/}}$ admits a
left adjoint $f$.

  2. (2)

The functor $f$ is equivalent to the composition

$\mathcal{C}_{GD\mathord{/}}\xrightarrow{f^{\prime}}\mathcal{D}_{FGD\mathord{/}}\xrightarrow{f^{\prime\prime}}\mathcal{D}_{D\mathord{/}}$

where $f^{\prime}$ is induced by $F$ and $f^{\prime\prime}$ is induced by the
pushout along the counit map $FGD\rightarrow D$.

We note that this construction is functorial in $D\in\mathcal{D}$.

######  Notation 4.48.

We denote the comma category
$\operatorname{AniPDPair}\times_{\operatorname{AniPair}}\operatorname{Fun}(\Delta^{1},\operatorname{AniPair})$
by $\operatorname{PDEnvCon}$, an object of which is denoted by
$(A\twoheadrightarrow A^{\prime\prime},\gamma)\rightarrow(B\twoheadrightarrow
B^{\prime\prime})$, instead of the cumbersome notation $((A\twoheadrightarrow
A^{\prime\prime},\gamma),(A\twoheadrightarrow
A^{\prime\prime})\rightarrow(B\twoheadrightarrow B^{\prime\prime}))$.

######  Definition 4.49.

Let $(A\twoheadrightarrow A^{\prime\prime},\gamma)\in\operatorname{AniPDPair}$
be an animated PD-pair. The (relative) animated PD-envelope of an animated
pair in $\operatorname{AniPair}_{(A\twoheadrightarrow
A^{\prime\prime})\mathord{/}}$ is the image under the functor
$\operatorname{AniPair}_{(A\twoheadrightarrow
A^{\prime\prime})\mathord{/}}\rightarrow\operatorname{AniPDPair}_{(A\twoheadrightarrow
A^{\prime\prime},\gamma)\mathord{/}}$ induced by the animated PD-envelope
functor $\operatorname{AniPair}\rightarrow\operatorname{AniPDPair}$ by Lemma
4.47.

Concretely, let $B\twoheadrightarrow B^{\prime\prime}$ be an object in
$\operatorname{AniPair}_{(A\twoheadrightarrow A^{\prime\prime})\mathord{/}}$
and let $(C\twoheadrightarrow A^{\prime\prime},\gamma_{C})$ and
$(D\twoheadrightarrow B^{\prime\prime},\gamma_{D})$ denote the animated PD-
envelopes of $A\twoheadrightarrow A^{\prime\prime}$ and $B\twoheadrightarrow
B^{\prime\prime}$ respectively (we have tacitly used Lemma 3.37). Then the
relative animated PD-envelope of $B\twoheadrightarrow B^{\prime\prime}$ is
given by $(A\twoheadrightarrow
A^{\prime\prime},\gamma)\amalg_{(C\twoheadrightarrow
A^{\prime\prime},\gamma_{C})}(D\twoheadrightarrow
B^{\prime\prime},\gamma_{D})$ where the map $(C\twoheadrightarrow
A^{\prime\prime},\gamma_{C})\rightarrow(A\twoheadrightarrow
A^{\prime\prime},\gamma)$ is the counit map associated to
$(A\twoheadrightarrow A^{\prime\prime},\gamma)\in\operatorname{AniPDPair}$ and
the map $(C\twoheadrightarrow
A^{\prime\prime},\gamma_{C})\rightarrow(D\twoheadrightarrow
B^{\prime\prime},\gamma_{D})$ is the image of $(A\twoheadrightarrow
A^{\prime\prime})\rightarrow(B\twoheadrightarrow B^{\prime\prime})$ under the
animated PD-envelope functor.

This defines the (relative) animated PD-envelope functor
$\operatorname{RelPDEnv}:\operatorname{PDEnvCon}\rightarrow\operatorname{Fun}(\Delta^{1},\operatorname{AniPDPair})$.

######  Example 4.50.

Let $(A\twoheadrightarrow A^{\prime\prime},\gamma)\in\operatorname{AniPDPair}$
be an animated PD-pair. Then the animated PD-envelope of $A\twoheadrightarrow
A^{\prime\prime}$ relative to $(A\twoheadrightarrow A^{\prime\prime},\gamma)$
is given by $(A\twoheadrightarrow A^{\prime\prime},\gamma)$. This follows from
the fact that $A\twoheadrightarrow A^{\prime\prime}$ relative to
$(A\twoheadrightarrow A^{\prime\prime},\gamma)$ is the base change of
$\operatorname{id}_{\mathbb{Z}}:\mathbb{Z}\rightarrow\mathbb{Z}$ relative to
$(\operatorname{id}_{\mathbb{Z}}:\mathbb{Z}\rightarrow\mathbb{Z},0)$ along the
map
$(\operatorname{id}_{\mathbb{Z}}:\mathbb{Z}\rightarrow\mathbb{Z},0)\rightarrow(A\twoheadrightarrow
A^{\prime\prime},\gamma)$ of animated PD-pairs. Compare with Lemma 4.56.

######  Example 4.51.

Let $(A\twoheadrightarrow A^{\prime\prime},\gamma)\in\operatorname{AniPDPair}$
be an animated PD-pair. Then the animated PD-envelope of
$\operatorname{id}_{A^{\prime\prime}}:A^{\prime\prime}\twoheadrightarrow
A^{\prime\prime}$ relative to $(A\twoheadrightarrow A^{\prime\prime},\gamma)$
is given by $(\operatorname{id}_{A^{\prime\prime}}:A^{\prime\prime}\rightarrow
A^{\prime\prime},0)$. This follows from checking the universal property of the
unit map at
$\operatorname{id}_{A^{\prime\prime}}:A^{\prime\prime}\twoheadrightarrow
A^{\prime\prime}$ of the adjunction
$\operatorname{AniPair}_{(A\twoheadrightarrow
A^{\prime\prime})\mathord{/}}\rightleftarrows\operatorname{AniPDPair}_{(A\twoheadrightarrow
A^{\prime\prime},\gamma)\mathord{/}}$.

It follows immediately from Lemma 3.41 that

######  Lemma 4.52.

Let $(A\twoheadrightarrow A^{\prime\prime},\gamma)\in\operatorname{AniPDPair}$
be an animated PD-pair, $(B\twoheadrightarrow
B^{\prime\prime})\in\operatorname{AniPair}_{(A\twoheadrightarrow
A^{\prime\prime})\mathord{/}}$ an animated pair under $A\twoheadrightarrow
A^{\prime\prime}$. Let $(C\twoheadrightarrow B^{\prime\prime},\delta)$ denote
its relative animated PD-envelope. Then the unit map $(B\twoheadrightarrow
B^{\prime\prime})\rightarrow(C\twoheadrightarrow B^{\prime\prime})$ becomes an
equivalence after rationalization.

Recall that given a PD-pair $(A,I,\gamma)$ and a map $(A,I)\rightarrow(B,J)$
of pairs with $A\rightarrow B$ being flat, the PD-structure $\gamma$ extends
to $B$, i.e, there exists a unique PD-structure $\overline{\gamma}$ on
$(B,IB)$ such that the map $(A,I)\rightarrow(B,J)$ of pairs gives rise to a
map $(A,I,\gamma)\rightarrow(B,IB,\overline{\gamma})$ of PD-pairs. Then the
PD-envelope of $(B,J)$ with respect to $(A,I,\gamma)$ is the same as that with
respect to the PD-pair $(B,IB,\overline{\gamma})$, which corresponds to the
crystalline cohomology of $B/J$ with respect to $(B,IB,\overline{\gamma})$. We
now show an animated analogue (without flatness).

Let $\operatorname{CrysCon}_{\operatorname{surj}}$ denote the full subcategory
$\operatorname{AniPDPair}\times_{\operatorname{Ani}(\operatorname{Ring})}\operatorname{Fun}(\Delta^{1},\operatorname{Ani}(\operatorname{Ring}))_{\geq
0}\subseteq\operatorname{CrysCon}$ spanned by objects $((A\twoheadrightarrow
A^{\prime\prime},\gamma),A^{\prime\prime}\rightarrow R)$ such that
$A^{\prime\prime}\rightarrow R$ is also surjective. There is a canonical
functor
$\operatorname{PDEnvCon}\rightarrow\operatorname{CrysCon}_{\operatorname{surj}}$
given as follows: for every object $((A\twoheadrightarrow
A^{\prime\prime},\gamma)\rightarrow(B\twoheadrightarrow
B^{\prime\prime}))\in\operatorname{PDEnvCon}$, we get the commutative diagram

$\begin{array}[]{ccc}A&\relbar\joinrel\twoheadrightarrow&A^{\prime\prime}\\\
{\mbox{\rotatebox[origin={c}]{-90.0}{$\longrightarrow$}}}&&{\mbox{\rotatebox[origin={c}]{-90.0}{$\longrightarrow$}}}\\\
B&\relbar\joinrel\twoheadrightarrow&B^{\prime\prime}\end{array}$

in $\operatorname{Ani}(\operatorname{Ring})$, which gives rise to two
surjective maps $B\otimes_{A}^{\mathbb{L}}A^{\prime\prime}\twoheadrightarrow
B^{\prime\prime}$ and $B\twoheadrightarrow
B\otimes_{A}^{\mathbb{L}}A^{\prime\prime}$. Furthermore, the later admits a
PD-structure: it is the underlying animated pair of the pushout
$(\operatorname{id}_{B}:B\rightarrow
B,0)\amalg_{(\operatorname{id}_{A}:A\rightarrow A,0)}(A\twoheadrightarrow
A^{\prime\prime},\gamma)$ in $\operatorname{AniPDPair}$. We denote by
$(B\twoheadrightarrow B\otimes_{A}^{\mathbb{L}}A^{\prime\prime},\delta)$ this
pushout. Then we get an object $((B\twoheadrightarrow
B\otimes_{A}^{\mathbb{L}}A^{\prime\prime},\delta),B\otimes_{A}^{\mathbb{L}}A^{\prime\prime}\twoheadrightarrow
B^{\prime\prime})$ in $\operatorname{PDEnvCon}$.

One verifies that

######  Lemma 4.53.

The functor
$\operatorname{PDEnvCon}\rightarrow\operatorname{CrysCon}_{\operatorname{surj}}$
constructed above admits a fully faithful right adjoint
$\operatorname{CrysCon}_{\operatorname{surj}}\rightarrow\operatorname{PDEnvCon}$
given by $((A\twoheadrightarrow
A^{\prime\prime},\gamma),A^{\prime\prime}\twoheadrightarrow
R)\mapsto((A\twoheadrightarrow
A^{\prime\prime},\gamma)\rightarrow(A\twoheadrightarrow R))$.

Thus $\operatorname{CrysCon}_{\operatorname{surj}}$ could be seen as a
reflective subcategory (Definition 2.44) of $\operatorname{PDEnvCon}$. Now we
claim that

######  Lemma 4.54.

The relative animated PD-envelope functor
$\operatorname{PDEnvCon}\rightarrow\operatorname{Fun}(\Delta^{1},\operatorname{AniPDPair})$
is
$(\operatorname{PDEnvCon}\rightarrow\operatorname{CrysCon}_{\operatorname{surj}})$-invariant
(Definition 2.57).

###### Proof.

For every object $(A\twoheadrightarrow
A^{\prime\prime},\gamma)\rightarrow(B\twoheadrightarrow B^{\prime\prime})$ in
$\operatorname{PDEnvCon}$, we have a map $(A\twoheadrightarrow
A^{\prime\prime},\gamma)\rightarrow(B\twoheadrightarrow
A^{\prime\prime}\otimes_{A}^{\mathbb{L}}B,\delta)$ of animated PD-pairs. By
the concrete description of the relative animated PD-envelope functor, it
suffices to show that this map along with the counit maps forms a pushout
diagram of animated PD-pairs. As discussed above, $(B\twoheadrightarrow
A^{\prime\prime}\otimes_{A}^{\mathbb{L}}B,\delta)$ is the pushout
$(\operatorname{id}_{B}:B\rightarrow
B,0)\amalg_{(\operatorname{id}_{A}:A\rightarrow A,0)}(A\twoheadrightarrow
A^{\prime\prime},\gamma)$. The counit maps for $(\operatorname{id}_{A},0)$ and
$(\operatorname{id}_{B},0)$ are identities (Lemma 3.39). The result then
follows from Proposition 3.34, which implies that counit maps are compatible
with small colimits. ∎

Consequently, in order to study the relative animated PD-envelope functor, it
suffices to study the composite
$\operatorname{CrysCon}_{\operatorname{surj}}\rightarrow\operatorname{PDEnvCon}\rightarrow\operatorname{Fun}(\Delta^{1},\operatorname{AniPDPair})$.
By abuse of terminology, we will simply denote this functor as
$\operatorname{RelPDEnv}$ as well and call the image (or after evaluation at
$1\in\Delta^{1}$) the animated PD-envelope of an object $((A\twoheadrightarrow
A^{\prime\prime},\gamma),A^{\prime\prime}\twoheadrightarrow
R)\in\operatorname{CrysCon}_{\operatorname{surj}}$. We remark that the functor
$\operatorname{CrysCon}_{\operatorname{surj}}\rightarrow\operatorname{PDEnvCon}$
preserves small colimits by Proposition 3.34, therefore so does the composite
functor.

We note that $\operatorname{CrysCon}_{\operatorname{surj}}$ is projectively
generated: let
$\operatorname{CrysCon}_{\operatorname{surj}}^{0}\subseteq\operatorname{CrysCon}_{\operatorname{surj}}$
be the full subcategory spanned by objects
$((\Gamma_{\mathbb{Z}[Y,Z]}(X)\twoheadrightarrow\mathbb{Z}[Y,Z],\gamma),\mathbb{Z}[Y,Z]\twoheadrightarrow\mathbb{Z}[Z])$
for all finite sets $X,Y,Z$.

######  Lemma 4.55.

The full subcategory
$\operatorname{CrysCon}_{\operatorname{surj}}^{0}\subseteq\operatorname{CrysCon}_{\operatorname{surj}}$
constitutes a set of compact projective generators for
$\operatorname{CrysCon}_{\operatorname{surj}}$.

###### Proof.

We only sketch the proof, which is similar to that of Theorem 3.23. The key
observation is that the composite of forgetful functors
$\operatorname{CrysCon}_{\operatorname{surj}}^{0}\rightarrow\operatorname{Fun}(\Delta^{2},\operatorname{Ani}(\operatorname{Ring}))_{\operatorname{surj}}\rightarrow\operatorname{Fun}(\Delta^{2},D(\mathbb{Z})_{\geq
0})_{\operatorname{surj}},((A\twoheadrightarrow
A^{\prime\prime},\gamma),A^{\prime\prime}\twoheadrightarrow
R)\mapsto(A\twoheadrightarrow A^{\prime\prime}\twoheadrightarrow R)$, which
preserves filtered colimits and geometric realizations by Proposition 3.34,
admits a left adjoint, where
$\operatorname{Fun}(\Delta^{2},\mathcal{C})_{\operatorname{surj}}\subseteq\operatorname{Fun}(\Delta^{2},\mathcal{C})$
is the full subcategory spanned by $(X\rightarrow Y\rightarrow
Z)\in\operatorname{Fun}(\Delta^{2},\mathcal{C})$ such that $X\rightarrow Y$
and $Y\rightarrow Z$ are surjective, for $\mathcal{C}=D(\mathbb{Z})_{\geq 0}$
and $\mathcal{C}=\operatorname{Ani}(\operatorname{Ring})$.

The $\infty$-category $\operatorname{Fun}(\Delta^{2},D(\mathbb{Z})_{\geq
0})_{\operatorname{surj}}$ admits a set
$\\{\mathbb{Z}X\oplus\mathbb{Z}Y\oplus\mathbb{Z}Z\twoheadrightarrow\mathbb{Z}Y\oplus\mathbb{Z}Z\twoheadrightarrow\mathbb{Z}Z|X,Y,Z\in\operatorname{Fin}\\}$
of compact projective generators which spans the full subcategory
$\operatorname{Fun}(\Delta^{2},D(\mathbb{Z})_{\geq
0})_{\operatorname{surj}}^{0}$, which follows from the fact that the left
adjoint to the left derived functor
$\mathcal{P}_{\Sigma}(\operatorname{Fun}(\Delta^{2},D(\mathbb{Z})_{\geq
0})_{\operatorname{surj}}^{0})\rightarrow\operatorname{Fun}(\Delta^{2},D(\mathbb{Z})_{\geq
0})_{\operatorname{surj}}^{0}$ is conservative (cf. the proof of [Lur17, Prop
25.2.1.2]).

The result then follows from Proposition A.15. ∎

By Proposition A.10, the functor
$\operatorname{CrysCon}_{\operatorname{surj}}\rightarrow\operatorname{Fun}(\Delta^{1},\operatorname{AniPDPair})$
is the left derived functor of the restricted functor
$\operatorname{CrysCon}_{\operatorname{surj}}^{0}\rightarrow\operatorname{Fun}(\Delta^{1},\operatorname{AniPDPair})$,
which is concretely given as follows:

######  Lemma 4.56.

The relative animated PD-envelope of an object
$((\Gamma_{\mathbb{Z}[Y,Z]}(X)\twoheadrightarrow\mathbb{Z}[Y,Z],\gamma),\mathbb{Z}[Y,Z]\twoheadrightarrow\mathbb{Z}[Z])\in\operatorname{CrysCon}_{\operatorname{surj}}^{0}$
is functorially given by
$(\Gamma_{\mathbb{Z}[Z]}(X,Y)\twoheadrightarrow\mathbb{Z}[Z],\tilde{\gamma})\in\operatorname{AniPDPair}_{(\Gamma_{\mathbb{Z}[Y,Z]}(X)\twoheadrightarrow\mathbb{Z}[Y,Z],\gamma)\mathord{/}}$,
i.e. coincides with the classical relative PD-envelope.

###### Proof.

First, by the adjointness, there exists a functorial comparison map from the
relative animated PD-envelope to
$(\Gamma_{\mathbb{Z}[Z]}(X,Y)\twoheadrightarrow\mathbb{Z}[Z],\tilde{\gamma})$.
It suffices to show that this is an equivalence.

In this case,
$((\Gamma_{\mathbb{Z}[Y,Z]}(X)\twoheadrightarrow\mathbb{Z}[Y,Z],\gamma),\Gamma_{\mathbb{Z}[Y,Z]}(X)\twoheadrightarrow\mathbb{Z}[Z])\in\operatorname{PDEnvCon}$
is the base change of
$((\operatorname{id}_{\mathbb{Z}[Y,Z]},0),\mathbb{Z}[Y,Z]\twoheadrightarrow\mathbb{Z}[Z])\in\operatorname{PDEnvCon}$
along
$(\operatorname{id}_{\mathbb{Z}[Y,Z]},0)\rightarrow(\Gamma_{\mathbb{Z}[Y,Z]}(X)\twoheadrightarrow\mathbb{Z}[Y,Z],\gamma)$.
The result then follows from the base-change property of the relative
adjunction, along with the simple fact that the (absolute) animated PD-
envelope of $\mathbb{Z}[Y,Z]\twoheadrightarrow\mathbb{Z}[Z]$ is
$(\Gamma_{\mathbb{Z}[Z]}(Y)\twoheadrightarrow\mathbb{Z}[Z],\gamma)$. ∎

As a generalization of Definition 3.59, we now introduce the conjugate
filtration on the relative animated PD-envelope in $\operatorname{char}p$. Let
$(A\twoheadrightarrow
A^{\prime\prime},\gamma)\in\operatorname{PDPair}_{\mathbb{F}_{p}}$ be a PD-
pair and $I\subseteq A^{\prime\prime}$ an ideal. We recall that there is a
canonical “Frobenius” map $\varphi_{A\twoheadrightarrow
A^{\prime\prime}}:A^{\prime\prime}\rightarrow A$ by Lemma 4.36. We suppose
that $\varphi_{A\twoheadrightarrow A^{\prime\prime}}$ is flat242424This is
satisfied when $(A\twoheadrightarrow
A^{\prime\prime},\gamma)\in\mathcal{E}^{0}$, which is the only case that we
need to develop the theory. For more examples, see Remark 4.62.. Let
$(B,J,\delta)$ denote the classical PD-envelope of $(A\twoheadrightarrow
A^{\prime\prime}/I)$ relative to $(A\twoheadrightarrow
A^{\prime\prime},\gamma)$. We note that $B/J\cong A^{\prime\prime}/I$. As in
the absolute case, due to the PD-structure $(B,J,\delta)$, there is a
canonical $\varphi_{A\twoheadrightarrow
A^{\prime\prime}}^{\ast}(A^{\prime\prime}/I)$-algebra structure on $B$, and we
consider the nonpositive filtration on $B$ given by
$\operatorname{Fil}^{-n}(B)$ for $n\in\mathbb{N}$ being the
$\varphi_{A\twoheadrightarrow
A^{\prime\prime}}^{\ast}(A^{\prime\prime}/I)$-submodule of $B$ generated by
$\left\\{\gamma_{i_{1}p}(f_{1})\cdots\gamma_{i_{m}p}(f_{m})|i_{1}+\cdots+i_{m}\leq
n\text{ and }f_{1},\ldots,f_{m}\in I\right\\}$. We have the following relative
version of Lemma 3.58, for which the proof of [Bha12a, Lem 3.42] adapts:

######  Lemma 4.57.

Let $(A\twoheadrightarrow A^{\prime\prime},\gamma)$ be a
PD-$\mathbb{F}_{p}$-pair such that $\varphi_{A\twoheadrightarrow
A^{\prime\prime}}$ is flat, and let $I\subseteq A^{\prime\prime}$ be an ideal
such that $I/I^{2}$ is a flat $A^{\prime\prime}/I$-module. The relative PD-
envelope $(B,J,\delta)$ and the filtration $\operatorname{Fil}^{\ast}B$ are
constructed above.

Then there is a comparison map $\varphi_{A\twoheadrightarrow
A^{\prime\prime}}^{\ast}(\Gamma_{A^{\prime\prime}/I}^{i}(I/I^{2}))\rightarrow\operatorname{gr}^{-i}B$
of $\varphi_{A\twoheadrightarrow
A^{\prime\prime}}^{\ast}(A^{\prime\prime}/I)$-modules induced by the maps
$(\gamma_{ip})_{i\in\mathbb{N}}$ (as in Lemma 3.58) which is functorial in
$((A\twoheadrightarrow
A^{\prime\prime},\gamma),A^{\prime\prime}\twoheadrightarrow
A^{\prime\prime}/I)$ in a subcategory of
$\operatorname{CrysCon}_{\mathbb{F}_{p},\operatorname{surj}}$. Furthermore, if
$I\subseteq A^{\prime\prime}$ is generated by a Koszul-regular
sequence252525We only need the simple case that $((A\twoheadrightarrow
A^{\prime\prime},\gamma),A^{\prime\prime}\twoheadrightarrow
A^{\prime\prime}/I)\in\operatorname{CrysCon}_{\mathbb{F}_{p},\operatorname{conj}}$,
which “simplifies” the proof in the sense that a “brute-force” computation
suffices., then the comparison map above is an isomorphism.

######  Definition 4.58.

The conjugate filtration functor (on the animated PD-envelope)
$\mathbb{L}\operatorname{ConjFil}:\operatorname{CrysCon}_{\mathbb{F}_{p},\operatorname{surj}}\rightarrow\operatorname{CAlg}(\operatorname{DF}^{\leq
0}(\mathbb{F}_{p}))$ together with the structure map of functors
$\operatorname{CrysCon}_{\mathbb{F}_{p},\operatorname{surj}}\rightrightarrows\operatorname{CAlg}(\operatorname{DF}^{\leq
0}(\mathbb{F}_{p}))$ from $((A\twoheadrightarrow
A^{\prime\prime},\gamma),A^{\prime\prime}\twoheadrightarrow
R)\mapsto\varphi_{A\twoheadrightarrow
A^{\prime\prime}}^{\ast}(R)=R\otimes_{A^{\prime\prime},\varphi_{A\twoheadrightarrow
A^{\prime\prime}}}^{\mathbb{L}}A$ to $\mathbb{L}\operatorname{ConjFil}$ is
defined to be the left derived functor (Proposition A.10) of
$\operatorname{CrysCon}_{\mathbb{F}_{p},\operatorname{surj}}^{0}\ni((A\twoheadrightarrow
A^{\prime\prime},\gamma),A^{\prime\prime}\twoheadrightarrow
A^{\prime\prime}/I)\mapsto(\varphi_{A\twoheadrightarrow
A^{\prime\prime}}^{\ast}(A^{\prime\prime}/I)\rightarrow\operatorname{Fil}^{\ast}B)\in\operatorname{Fun}(\Delta^{1},\operatorname{CAlg}(\operatorname{DF}^{\leq
0}(\mathbb{F}_{p})))$ constructed above.

As in the absolute case (including Remark 3.61), it follows from Lemma 4.57
that

######  Corollary 4.59.

For every $((A\twoheadrightarrow
A^{\prime\prime},\gamma),A^{\prime\prime}\twoheadrightarrow
R)\in\operatorname{CrysCon}_{\mathbb{F}_{p},\operatorname{surj}}$, there
exists an equivalence

$\varphi_{A\twoheadrightarrow
A^{\prime\prime}}^{\ast}(\Gamma_{R}^{i}(\operatorname{gr}^{1}(\mathbb{L}\operatorname{AdFil}(A^{\prime\prime}\twoheadrightarrow
R))))\rightarrow\operatorname{gr}^{-i}(\mathbb{L}\operatorname{ConjFil}((A\twoheadrightarrow
A^{\prime\prime},\gamma),A^{\prime\prime}\twoheadrightarrow R))$

in $D(\varphi_{A\twoheadrightarrow A^{\prime\prime}}^{\ast}(R))_{\geq 0}$ for
all $i\in\mathbb{N}$ which is functorial in $((A\twoheadrightarrow
A^{\prime\prime},\gamma),A^{\prime\prime}\twoheadrightarrow
R)\in\operatorname{CrysCon}_{\mathbb{F}_{p},\operatorname{surj}}$.

As in the absolute case, we have

######  Corollary 4.60.

For every $((A\twoheadrightarrow
A^{\prime\prime},\gamma),A^{\prime\prime}\twoheadrightarrow
R)\in\operatorname{CrysCon}_{\mathbb{F}_{p},\operatorname{surj}}$ such that
$A^{\prime\prime}\twoheadrightarrow R$ is a quasiregular animated pair, let
$(B\twoheadrightarrow R,\delta)$ denote the relative animated PD-envelope.
Then $B$ is a flat $\varphi_{A\twoheadrightarrow
A^{\prime\prime}}^{\ast}(R)$-module.

Similar to Proposition 3.72, we have

######  Proposition 4.61.

Let $(A\twoheadrightarrow A^{\prime\prime},\gamma)\in\operatorname{PDPair}$ be
a PD-pair and $I\subseteq A^{\prime\prime}$ an ideal generated by a Koszul-
regular sequence. Let $(B\twoheadrightarrow B^{\prime\prime},\delta)$ denote
the relative animated PD-envelope of $((A\twoheadrightarrow
A^{\prime\prime},\gamma),A^{\prime\prime}\rightarrow
A^{\prime\prime}/I)\in\operatorname{CrysCon}$. Then $(B\twoheadrightarrow
B^{\prime\prime},\delta)$ is a PD-pair, therefore coincides with the classical
relative PD-envelope.

######  Remark 4.62.

More precisely, in Corollary 3.66, the map $\varphi_{A\twoheadrightarrow
A^{\prime\prime}}^{\ast}(R)\rightarrow B$ is induced by the Frobenius map
$\varphi_{B\twoheadrightarrow R}:R\rightarrow B$ (which could be seen by left
deriving the special case that $((A\twoheadrightarrow
A^{\prime\prime},\gamma),A^{\prime\prime}\rightarrow
R)\in\operatorname{CrysCon}_{\mathbb{F}_{p},\operatorname{surj}}^{0}$). In
particular, if the Frobenius map $\varphi_{A\twoheadrightarrow
A^{\prime\prime}}:A^{\prime\prime}\rightarrow A$ is flat, then so is the
Frobenius map $\varphi_{B\twoheadrightarrow R}:R\rightarrow B$.

For example, when $R$ is a quasiregular semiperfect $\mathbb{F}_{p}$-algebra
[BMS19, Def 8.8], we set $(A\twoheadrightarrow
A^{\prime\prime},\gamma)=(\operatorname{id}_{R^{\flat}}:R^{\flat}\rightarrow
R^{\flat},0)$ and the map $A^{\prime\prime}\rightarrow R$ to be the canonical
map, by definition, $R^{\flat}$ is a perfect $\mathbb{F}_{p}$-algebra
therefore $\varphi_{R^{\flat}}$ is flat. Then the animated PD-envelope
$B\twoheadrightarrow R$ of $A^{\prime\prime}\rightarrow R$ satisfies the
condition that the Frobenius map $\varphi_{B\rightarrow R}:R\rightarrow B$ is
flat and hence $B$ is static. It follows that $(R\twoheadrightarrow B,\delta)$
is a PD-pair (Proposition 3.32).

Note that the associated graded pieces of derived crystalline cohomology and
relative animated PD-envelope of a “surjective” crystalline context
$((A\twoheadrightarrow
A^{\prime\prime},\gamma),A^{\prime\prime}\twoheadrightarrow
R)\in\operatorname{CrysCon}_{\mathbb{F}_{p},\operatorname{surj}}$, with
respect to conjugate filtrations, are equivalent by Corollaries 4.59, 4.46 and
3.54. In fact, we have

######  Lemma 4.63.

There is a canonical equivalence

$\operatorname{Fil}_{\operatorname{conj}}\operatorname{CrysCoh}_{R/(A\twoheadrightarrow
A^{\prime\prime},\gamma)}\rightarrow\mathbb{L}\operatorname{ConjFil}((A\twoheadrightarrow
A^{\prime\prime},\gamma),A^{\prime\prime}\twoheadrightarrow R)$

in $\operatorname{CAlg}(\operatorname{DF}^{\leq
0}(\varphi_{A\twoheadrightarrow A^{\prime\prime}}^{\ast}(R)))$ which is
functorial262626Here we apply the same convention as in Remark 3.61. in
$((A\twoheadrightarrow
A^{\prime\prime},\gamma),A^{\prime\prime}\twoheadrightarrow
R)\in\operatorname{CrysCon}_{\mathbb{F}_{p},\operatorname{surj}}$.

###### Proof.

We first point out how to produce the comparison map of underlying
$\mathbb{E}_{\infty}$-$\mathbb{F}_{p}$-algebras, i.e., ignoring the
$\varphi_{A\twoheadrightarrow A^{\prime\prime}}^{\ast}(R)$-algebra structures
and conjugate filtrations. This is logically not necessary but it benefits our
understanding. Given $((A\twoheadrightarrow
A^{\prime\prime},\gamma),A^{\prime\prime}\twoheadrightarrow R)$, let
$(B\twoheadrightarrow R,\delta)$ denote its relative animated PD-envelope. It
follows from Proposition 4.16 that the crystalline cohomology
$\operatorname{CrysCoh}_{R/(A\twoheadrightarrow A^{\prime\prime},\gamma)}$ is
naturally equivalent to the derived de Rham cohomology
$\operatorname{dR}_{(B\twoheadrightarrow R,\delta)/(A\twoheadrightarrow
A^{\prime\prime},\gamma)}$, and by definition, it is equipped with a map
$\operatorname{dR}_{(B\twoheadrightarrow R,\delta)/(A\twoheadrightarrow
A^{\prime\prime},\gamma)}\rightarrow B$ of
$\mathbb{E}_{\infty}$-$\mathbb{F}_{p}$-algebras, which gives rise to the
underlying comparison map that we want.

By Lemmas 4.55 and A.10, it suffices to construct the equivalence restricted
to the full subcategory
$\operatorname{CrysCon}_{\mathbb{F}_{p},\operatorname{surj}}^{0}\subseteq\operatorname{CrysCon}_{\mathbb{F}_{p},\operatorname{surj}}$,
i.e., to establish the equivalence for all
$((\Gamma_{\mathbb{F}_{p}[Y,Z]}(X)\twoheadrightarrow\mathbb{F}_{p}[Y,Z],\gamma_{0}),\mathbb{F}_{p}[Y,Z]\twoheadrightarrow\mathbb{F}_{p}[Z])\in\operatorname{CrysCon}_{\mathbb{F}_{p},\operatorname{surj}}^{0}$.
This is essentially [Bha12a, Lem 3.29 & Thm 3.27]. We will briefly sketch the
argument. The preceding paragraph has already established a comparison map of
underlying $\mathbb{E}_{\infty}$-$\mathbb{F}_{p}$-algebras. The key point is
that both sides are static: the relative animated PD-envelope is static by
definition, and the derived crystalline cohomology is static by Cartier
isomorphism (Proposition 4.46) and the fact that static modules are closed
under extension and filtered colimits, see Corollary 3.68 for a similar
argument. Then the result follows from explicit simplicial resolution. ∎

We now deduce the integral version of the comparison above. We recall that
$\mathbb{L}\operatorname{PDFil}:\operatorname{AniPDPair}\rightarrow\operatorname{CAlg}(\operatorname{DF}^{\geq
0}(\mathbb{Z}))$ is the PD-filtration functor (Definition 3.73), and
$\operatorname{Fil}_{H}$ is the Hodge-filtration.

######  Proposition 4.64.

There is a canonical equivalence

$\operatorname{Fil}_{H}\operatorname{CrysCoh}\rightarrow\mathbb{L}\operatorname{PDFil}\circ\operatorname{RelPDEnv}$

of functors
$\operatorname{CrysCon}_{\operatorname{surj}}\rightrightarrows\operatorname{CAlg}(\operatorname{DF}^{\geq
0}(\mathbb{Z}))$.

###### Proof.

The comparison map is established in the same way as in the proof of Lemma
4.63. It suffices to show that this is an equivalence.

We first show that this becomes an equivalence after passing to underlying
$\mathbb{E}_{\infty}$-$\mathbb{Z}$-algebras, i.e. ignoring the Hodge
filtration. By conservativity of the forgetful functor
$\operatorname{CAlg}_{\mathbb{Z}}\rightarrow D(\mathbb{Z})$, it suffices to
show the equivalence for underlying $\mathbb{Z}$-module spectra, which follows
from Lemmas 4.63, 4.52 and 4.11.

To establish the equivalence of filtered
$\mathbb{E}_{\infty}$-$\mathbb{Z}$-algebras, it remains to show that the
comparison map induces equivalences after passing to associated graded pieces,
and by Lemma 2.1, it suffices to prove the result restricted to the full
subcategory
$\operatorname{CrysCon}_{\operatorname{surj}}^{0}\subseteq\operatorname{CrysCon}_{\operatorname{surj}}$,
which is essentially due to [Ill72, Cor VIII.2.2.8], see [Bha12a, Rem 3.33]. ∎

###  4.4. Affine crystalline site

We now turn to the site-theoretic aspects of the derived crystalline
cohomology by showing that the derived crystalline cohomology is equivalent to
the cohomology of the affine crystalline site under a mild smoothness
condition. We warn the reader again that our theory is non-completed. Fix a
crystalline context $((A\twoheadrightarrow
A^{\prime\prime},\gamma_{A}),A^{\prime\prime}\rightarrow
R)\in\operatorname{CrysCon}$.

######  Definition 4.65.

The affine crystalline site $\operatorname{Cris}(R/(A\twoheadrightarrow
A^{\prime\prime},\gamma_{A}))$ is defined to be the opposite $\infty$-category
of animated PD-pairs $(B\twoheadrightarrow B^{\prime\prime},\gamma_{B})$ under
$(A\twoheadrightarrow A^{\prime\prime},\gamma_{A})$ along with an equivalence
$R\xrightarrow{\simeq}B^{\prime\prime}$ of $A$-algebras, depicted by the
diagram

which we will simply denoted by
$\left(R\xrightarrow{\simeq}B^{\prime\prime}\twoheadleftarrow
B\right)\in\operatorname{Cris}(R/(A\twoheadrightarrow
A^{\prime\prime},\gamma_{A}))$. More formally, it is the homotopy fiber of the
functor $\operatorname{AniPDPair}_{(A\twoheadrightarrow
A^{\prime\prime})\mathord{/}}\rightarrow\operatorname{Ani}(\operatorname{Ring})_{A^{\prime\prime}\mathord{/}},(B\twoheadrightarrow
B^{\prime\prime},\gamma_{B})\mapsto B^{\prime\prime}$ at the object
$R\in\operatorname{Ani}(\operatorname{Ring})_{A^{\prime\prime}\mathord{/}}$.
The endowed Grothendieck topology is indiscrete.

The structure presheaf $\operatorname{Cris}(R/(A\twoheadrightarrow
A^{\prime\prime},\gamma_{A}))^{\operatorname{op}}\rightarrow\operatorname{CAlg}_{A}$,
denoted by $\mathcal{O}_{\operatorname{CRIS}(R/(A\twoheadrightarrow
A^{\prime\prime},\gamma_{A}))}$ (or simply $\mathcal{O}$ when there is no
ambiguity), is induced by the evaluation
$\operatorname{AniPDPair}_{(A\twoheadrightarrow
A^{\prime\prime},\gamma_{A})\mathord{/}}\rightarrow\operatorname{Ani}(\operatorname{Ring})_{A\mathord{/}}\rightarrow\operatorname{CAlg}_{A\mathord{/}}$.
Concretely, it is given by
$\left(R\xrightarrow{\simeq}B^{\prime\prime}\twoheadleftarrow B\right)\mapsto
B$.

Although the affine crystalline site is not small, the cohomology of the
structure presheaf exists in $\operatorname{CAlg}_{A}$ by Čech-Alexander
calculation (which we will reproduce in Proposition 4.70). We will simply call
it the cohomology of the crystalline site and denote by
$R\Gamma(\operatorname{Cris}(R/(A\twoheadrightarrow
A^{\prime\prime},\gamma_{A})),\mathcal{O})$. Furthermore, the structure sheaf
admits the PD-filtration (Definition 3.73), which gives rise to a filtration
on the cohomology of the crystalline site, called the Hodge-filtration and
denoted by $\operatorname{Fil}_{H}$. We now have a comparison between the
derived crystalline cohomology and the cohomology of the crystalline site,
which becomes an equivalence after Hodge-completion:

######  Proposition 4.66.

There is a natural comparison map

$\operatorname{Fil}_{H}\operatorname{CrysCoh}_{R/(A\twoheadrightarrow
A^{\prime\prime},\gamma_{A})}\rightarrow\operatorname{Fil}_{H}R\Gamma(\operatorname{Cris}(R/(A\twoheadrightarrow
A^{\prime\prime},\gamma_{A})),\mathcal{O})$

in the $\infty$-category $\operatorname{CAlg}(\operatorname{DF}^{\geq 0}(A))$.
After passing to the associated graded pieces, i.e. composition with the
functor $\operatorname{CAlg}(\operatorname{DF}^{\geq
0}(A))\rightarrow\operatorname{CAlg}(\operatorname{Gr}^{\geq 0}(A))$, the
comparison map above becomes an equivalence. Moreover, when $\pi_{0}(R)$ is a
finitely generated $\pi_{0}(A^{\prime\prime})$-algebra, then the comparison
map is an equivalence.

We need some preparation about cosimplicial objects in $\infty$-categories.

######  Definition 4.67 ([Lur09, Def 6.1.2.2]).

Let $\mathcal{C}$ be an $\infty$-category. A cosimplicial object of
$\mathcal{C}$ is a functor
$X^{\bullet}:\boldsymbol{\Delta}\rightarrow\mathcal{C}$. The value of this
functor at $[\nu]\in\boldsymbol{\Delta}$ is denoted by $X^{\nu}$. A map of
cosimplicial objects $X^{\bullet}\rightarrow Y^{\bullet}$ is simply a map of
functors.

We note that there are two inclusions
$\\{0\\}\hookrightarrow[1]\hookleftarrow\\{1\\}$ viewed as two maps
$[0]\rightrightarrows[1]$, and a constant map $[1]\rightarrow[0]$, which
induce three functors $i_{0}$, $i_{1}$:
$\boldsymbol{\Delta}\simeq\boldsymbol{\Delta}_{/[0]}\rightrightarrows\boldsymbol{\Delta}_{/[1]}$
and
$\delta:\boldsymbol{\Delta}_{/[1]}\rightarrow\boldsymbol{\Delta}_{/[0]}\simeq\boldsymbol{\Delta}$.
For any $\infty$-category $\mathcal{C}$, let $i_{0}^{\ast}$ (resp.
$i_{1}^{\ast}$) denote the induced functor
$\operatorname{Fun}(\boldsymbol{\Delta}_{/[1]},\mathcal{C})\rightarrow\operatorname{Fun}(\boldsymbol{\Delta},\mathcal{C})$,
and let $\delta^{\ast}$ denote the induced functor
$\operatorname{Fun}(\boldsymbol{\Delta},\mathcal{C})\rightarrow\operatorname{Fun}(\boldsymbol{\Delta}_{/[1]},\mathcal{C})$.

######  Definition 4.68 ([Lur17, Def 7.2.1.6]).

Let $\mathcal{C}$ be an $\infty$-category and let $f$ and $g$ be two maps
$X^{\bullet}\rightrightarrows Y^{\bullet}$ of cosimplicial objects. A
simplicial homotopy from $f$ to $g$ is a map
$h:\delta^{\ast}(X^{\bullet})\rightarrow\delta^{\ast}(Y^{\bullet})$ of
functors $\boldsymbol{\Delta}_{/[1]}\rightrightarrows\mathcal{C}$ such that
the map $i_{0}^{\ast}(h):X^{\bullet}\rightarrow Y^{\bullet}$ (resp.
$i_{1}^{\ast}(h):X^{\bullet}\rightarrow Y^{\bullet}$), being a map of
cosimplicial objects, is equivalent to $f$ (resp. $g$). When
$X^{\bullet}=Y^{\bullet}$, we say that the simplicial homotopy
$h:\delta^{\ast}(X^{\bullet})\rightarrow\delta^{\ast}(X^{\bullet})$ is
constant if it is equivalent to
$\operatorname{id}_{\delta^{\ast}(X^{\bullet})}$.

######  Lemma 4.69.

Let $\mathcal{C}$ be an $\infty$-category and $X^{\bullet},Y^{\bullet}$ two
cosimplicial objects of which the totalization exist in $\mathcal{C}$. Let $f$
and $g$ be two maps $X^{\bullet}\rightrightarrows Y^{\bullet}$ of cosimplicial
objects such that there exists a simplicial homotopy from $f$ to $g$. Then the
maps $f,g$ induces equivalent272727Or called “homotopic”. We avoid the
terminology “homotopic” to avoid confusion with the simplicial homotopy. maps
$\lim_{\boldsymbol{\Delta}}X^{\bullet}\rightrightarrows\lim_{\boldsymbol{\Delta}}Y^{\bullet}$
of totalizations.

###### Denis Nardin.

For every cosimplicial object $X^{\bullet}$ in $\mathcal{C}$, there are two
observations:

  1. (1)

The canonical map
$\lim_{\boldsymbol{\Delta}_{/[1]}}\delta^{\ast}(X^{\bullet})\rightarrow\lim_{\boldsymbol{\Delta}}X^{\bullet}$
is an equivalence (this involves the existence of the limit as the source).
Indeed, it suffices to show that the map
$\delta:\boldsymbol{\Delta}_{/[1]}\rightarrow\boldsymbol{\Delta}$ is
coinitial. By Joyal’s version of Quillen’s Theorem A [Lur09, Thm 4.1.3.1], it
suffices to show that, for every $[n]\in\boldsymbol{\Delta}$, the category
$\boldsymbol{\Delta}_{/[1]}\times_{\boldsymbol{\Delta}}\boldsymbol{\Delta}_{/[n]}$
is weakly contractible. Its geometric realization is
$\Delta^{1}\times\Delta^{n}$, which is known to be weakly contractible.

  2. (2)

The two maps
$\lim_{\boldsymbol{\Delta}}X=\lim_{\boldsymbol{\Delta}}i_{\nu}^{\ast}\delta^{\ast}(X^{\bullet})\rightarrow\lim_{\boldsymbol{\Delta}_{/[1]}}\delta^{\ast}(X^{\bullet})$
for $\nu=0,1$ are equivalences, and these two maps are equivalent. Indeed,
both are inverses of the equivalence
$\lim_{\boldsymbol{\Delta}_{/[1]}}\delta^{\ast}(X^{\bullet})\rightarrow\lim_{\boldsymbol{\Delta}}X^{\bullet}$
above.

Note that the map $\lim_{\boldsymbol{\Delta}}f$ (resp.
$\lim_{\boldsymbol{\Delta}}g$) could be identified with the composite

$\lim_{\boldsymbol{\Delta}}X=\lim_{\boldsymbol{\Delta}}i_{\nu}^{\ast}\delta^{\ast}(X^{\bullet})\longrightarrow\lim_{\boldsymbol{\Delta}_{/[1]}}\delta^{\ast}(X^{\bullet})\xrightarrow{\lim_{\boldsymbol{\Delta}_{/[1]}}(h)}\lim_{\boldsymbol{\Delta}_{/[1]}}\delta^{\ast}(Y^{\bullet})\longrightarrow\lim_{\boldsymbol{\Delta}}Y^{\bullet}$

for $\nu=0$ (resp. $\nu=1$). The result then follows. ∎

Proof of Proposition 4.66 There is a map from the constant presheaf
$\underline{\operatorname{Fil}_{H}\operatorname{CrysCoh}_{R/(A\twoheadrightarrow
A^{\prime\prime},\gamma_{A})}}$ on the affine crystalline site to the
structure presheaf $\mathcal{O}$ given by the canonical map in Definition
4.25, which induces the comparison map in question.

Now we show that this map becomes an equivalence after passing to the
associated graded pieces. We first note that, when the map
$A^{\prime\prime}\rightarrow R$ is surjective, i.e. $((A\twoheadrightarrow
A^{\prime\prime},\gamma_{A}),A^{\prime\prime}\rightarrow
R)\in\operatorname{CrysCon}_{\operatorname{surj}}$, the result follows
directly from Proposition 4.64. Our strategy is to reduce the general case to
this special case via Čech-Alexander computation.

We pick a polynomial $A$-algebra $P$ (of possibly infinitely many variables)
along with a surjection $P\twoheadrightarrow R$ of $A$-algebras. Let
$P^{\bullet}\rightarrow R$ denote the Čech conerve of the object $P\rightarrow
R$ in the $\infty$-category
$\left(\operatorname{Ani}(\operatorname{Ring})_{A\mathord{/}}\right)_{/R}$.
Concretely, it is given by $P^{\nu}:=P^{\otimes_{A}^{\mathbb{L}}(\nu+1)}$, and
the map $P^{\nu}\rightarrow R$ is simply given by the composite map
$P^{\nu}\rightarrow P\rightarrow R$ which is surjective. In other words, we
get a cosimplicial object $(P^{\bullet}\twoheadrightarrow
R)\in\operatorname{Fun}\left(\Delta,\operatorname{AniPair}_{(A\twoheadrightarrow
A^{\prime\prime})\mathord{/}}\right)$. Let $(D^{\bullet}\twoheadrightarrow
R,\gamma_{D^{\bullet}})\in\operatorname{Fun}\left(\Delta,\operatorname{AniPDPair}_{(A\twoheadrightarrow
A^{\prime\prime},\gamma_{A})\mathord{/}}\right)$ denote the cosimplicial
relative animated PD-envelope, i.e. applying the functor
$\operatorname{AniPair}_{(A\twoheadrightarrow
A^{\prime\prime})\mathord{/}}\rightarrow\operatorname{AniPDPair}_{(A\twoheadrightarrow
A^{\prime\prime},\gamma_{A})\mathord{/}}$ (Definition 4.49) pointwise. This
effectively gives rise to a cosimplicial object
$\Delta\rightarrow\operatorname{Cris}(R/(A\twoheadrightarrow
A^{\prime\prime},\gamma_{A}))^{\operatorname{op}}$. Composing with the Hodge-
filtered presheaf
$\operatorname{Fil}_{H}\mathcal{O}:\operatorname{Cris}(R/(A\twoheadrightarrow
A^{\prime\prime},\gamma_{A}))^{\operatorname{op}}\rightarrow\operatorname{CAlg}(\operatorname{DF}^{\geq
0}(A))$, we get a cosimplicial filtered $\mathbb{E}_{\infty}$-$A$-algebra
$\Delta\rightarrow\operatorname{CAlg}(\operatorname{DF}^{\geq 0}(A))$, the
limit of which computes the cohomology
$\operatorname{Fil}_{H}R\Gamma(\operatorname{Cris}(R/(A\twoheadrightarrow
A^{\prime\prime},\gamma_{A})),\mathcal{O})$. In plain terms, this cosimplicial
filtered $\mathbb{E}_{\infty}$-$A$-algebra is just the PD-filtration of the
cosimplicial animated PD-pair $(D^{\bullet}\twoheadrightarrow
R,\gamma_{D^{\bullet}})$.

For this cosimplicial object, the comparison map constructed above is
concretely given by

(3) $\operatorname{Fil}_{H}\operatorname{dR}_{(D^{\bullet}\twoheadrightarrow
R,\gamma_{D^{\bullet}})/(A\twoheadrightarrow
A^{\prime\prime},\gamma_{A})}\rightarrow\operatorname{Fil}_{\operatorname{PD}}D^{\bullet}$

Now Propositions 4.64 and 4.53 gives us an equivalence

$\operatorname{Fil}_{H}\operatorname{CrysCoh}_{R/(P^{\bullet}\twoheadrightarrow
P^{\bullet}\otimes_{A}^{\mathbb{L}}A^{\prime\prime},\gamma_{P^{\bullet}})}\rightarrow\operatorname{Fil}_{\operatorname{PD}}D^{\bullet}$

which is effectively given by

$\operatorname{Fil}_{H}\operatorname{dR}_{(D^{\bullet}\twoheadrightarrow
R,\gamma_{D^{\bullet}})/(P^{\bullet}\twoheadrightarrow
P^{\bullet}\otimes_{A}^{\mathbb{L}}A^{\prime\prime},\gamma_{P^{\bullet}})}\rightarrow\operatorname{Fil}_{\operatorname{PD}}D^{\bullet}$

by chasing the proof. In other words, (3) could be rewritten as the natural
map

$\operatorname{Fil}_{H}\operatorname{dR}_{(D^{\bullet}\twoheadrightarrow
R,\gamma_{D^{\bullet}})/(A\twoheadrightarrow
A^{\prime\prime},\gamma_{A})}\rightarrow\operatorname{Fil}_{H}\operatorname{dR}_{(D^{\bullet}\twoheadrightarrow
R,\gamma_{D^{\bullet}})/(P^{\bullet}\twoheadrightarrow
P^{\bullet}\otimes_{A}^{\mathbb{L}}A^{\prime\prime},\gamma_{P^{\bullet}})}$

or equivalently, the natural map

$\operatorname{Fil}_{H}\operatorname{CrysCoh}_{R/(A\twoheadrightarrow
A^{\prime\prime},\gamma_{A})}\rightarrow\operatorname{Fil}_{H}\operatorname{CrysCoh}_{R/(P^{\bullet}\twoheadrightarrow
P^{\bullet}\otimes_{A}^{\mathbb{L}}A^{\prime\prime},\gamma_{P^{\bullet}})}$

It remains to show that this cosimplicial map gives rise to an equivalence
after taking the limit, i.e., the totalization, and passing to associated
graded pieces. We isolate the remaining part into Lemma 4.77. $\Box$

Before proving Lemma 4.77, we isolate an important observation in the previous
proof into a proposition:

######  Proposition 4.70.

For every crystalline context $((A\twoheadrightarrow
A^{\prime\prime},\gamma_{A}),A^{\prime\prime}\rightarrow
R)\in\operatorname{CrysCon}$, the followings are equivalent:

  1. (1)

The comparison map in Proposition 4.66 is an equivalence.

  2. (2)

There exists a polynomial $A$-algebra $P$ (of possibly infinitely many
variables) along with a surjection $P\twoheadrightarrow R$ of $A$-algebras,
and letting $P^{\bullet}\rightarrow R$ denote the Čech conerve of
$P\rightarrow R$ in the $\infty$-category
$\left(\operatorname{Ani}(\operatorname{Ring})_{A\mathord{/}}\right)_{/R}$ as
in the proof of Proposition 4.66, then the natural maps

(4) $\operatorname{Fil}_{H}\operatorname{CrysCoh}_{R/(A\twoheadrightarrow
A^{\prime\prime},\gamma_{A})}\rightarrow\operatorname{Fil}_{H}\operatorname{CrysCoh}_{R/(P^{\bullet}\twoheadrightarrow
P^{\bullet}\otimes_{A}^{\mathbb{L}}A^{\prime\prime},\gamma_{P^{\bullet}})}$

form a limit diagram in $\operatorname{CAlg}(\operatorname{DF}^{\geq 0}(A))$.

  3. (3)

For all polynomial $A$-algebras $P$ (of possibly infinitely many variables)
along with a surjection $P\twoheadrightarrow R$ of $A$-algebras, and letting
$P^{\bullet}\rightarrow R$ denote the Čech conerve of $P\rightarrow R$ in the
$\infty$-category
$\left(\operatorname{Ani}(\operatorname{Ring})_{A\mathord{/}}\right)_{/R}$,
then the natural maps (4) form a limit diagram in
$\operatorname{CAlg}(\operatorname{DF}^{\geq 0}(A))$.

  4. (4)

(After proving Lemma 4.77) There exists a (or equivalently, for every)
polynomial $A$-algebra $P$ (of possibly infinitely many variables) along with
a surjection $P\twoheadrightarrow R$ of $A$-algebras, and letting
$P^{\bullet}\rightarrow R$ denote the Čech conerve of $P\rightarrow R$ in the
$\infty$-category
$\left(\operatorname{Ani}(\operatorname{Ring})_{A\mathord{/}}\right)_{/R}$ as
in the proof of Proposition 4.66, then the natural maps

$\operatorname{CrysCoh}_{R/(A\twoheadrightarrow
A^{\prime\prime},\gamma_{A})}\rightarrow\operatorname{CrysCoh}_{R/(P^{\bullet}\twoheadrightarrow
P^{\bullet}\otimes_{A}^{\mathbb{L}}A^{\prime\prime},\gamma_{P^{\bullet}})}$

form a limit diagram in $\operatorname{CAlg}(D(A))$.

In order to deal with associated graded pieces of the Hodge filtration, we
need a variant of the Katz-Oda filtration in [GL20, Cons 3.12]. We need an
auxiliary construction:

######  Definition 4.71.

The cotangent complex functor
$\mathbb{L}_{\cdot/\cdot}:\operatorname{dRCon}\rightarrow\operatorname{Ani}(\operatorname{Mod})$
is defined to be the left derived functor (Proposition A.10) of the functor
$\operatorname{dRCon}^{0}\rightarrow\operatorname{Ani}(\operatorname{Mod}),((A,I,\gamma_{A})\rightarrow(B,J,\gamma_{B}))\mapsto(B,\Omega_{(B,J)/(A,I)}^{1})$.

The proof of Lemma 4.10 leads to

######  Lemma 4.72.

The composite functor
$\operatorname{Fun}(\Delta^{1},\operatorname{Ani}(\operatorname{Ring}))\rightarrow\operatorname{dRCon}\rightarrow\operatorname{Ani}(\operatorname{Mod})$
is equivalent to the classical cotangent complex functor.

We now introduce the “stupid” filtration $\operatorname{Fil}_{B}$ on the
Hodge-filtered derived de Rham cohomology
$\operatorname{Fil}_{H}\operatorname{dR}_{\cdot/\cdot}$. For each
$((A,I,\gamma_{A})\rightarrow(B,J,\gamma_{B}))\in\operatorname{dRCon}^{0}$,
consider the filtration $(\Omega_{(B,J,\gamma_{B})/(A,I,\gamma_{A})}^{\geq
n},\mathrm{d})_{n\in(\mathbb{N},\geq)}$ of the Hodge-filtered CDGA, which
gives rise to a bifiltered $\mathbb{E}_{\infty}$-$\mathbb{Z}$-algebra. By
Proposition A.10, we get a functor
$\operatorname{CAlg}(\operatorname{Fun}((\mathbb{N},\geq)\times(\mathbb{N},\geq),D(\mathbb{Z}))),((A\twoheadrightarrow
A^{\prime\prime},\gamma_{A})\rightarrow(B\twoheadrightarrow
B^{\prime\prime},\gamma_{B}))\mapsto\operatorname{Fil}_{B}\operatorname{Fil}_{H}\operatorname{dR}_{(B\twoheadrightarrow
B^{\prime\prime})/(A\twoheadrightarrow A^{\prime\prime})}$.

######  Warning 4.73.

Unlike the Hodge filtration, the “stupid” filtration does not descend to
$\operatorname{CrysCon}$, that is to say, it depends on the choice of $B$ in
question.

We now analyze the associated graded pieces with respect to the “stupid”
filtration:

######  Lemma 4.74.

Let $((A\twoheadrightarrow
A^{\prime\prime},\gamma_{A})\rightarrow(B\twoheadrightarrow
B^{\prime\prime},\gamma_{B}))\in\operatorname{dRCon}$ be a de Rham context.
Then associated graded pieces
$\operatorname{gr}_{B}^{i}\operatorname{Fil}_{H}\operatorname{dR}_{(B\twoheadrightarrow
B^{\prime\prime})/(A\twoheadrightarrow A^{\prime\prime})}$ could be
functorially identified with
$\operatorname{ins}^{i}\left(\bigwedge_{B}^{i}\mathbb{L}_{(B\twoheadrightarrow
B^{\prime\prime})/(A\twoheadrightarrow
A^{\prime\prime})}[-i]\right)\otimes_{B}^{\mathbb{L}}\operatorname{Fil}_{\operatorname{PD}}B$
as a $\operatorname{Fil}_{\operatorname{PD}}B$-module in
$\operatorname{DF}^{\geq 0}(B)$ (where
$\cdot\otimes_{B}^{\mathbb{L}}\operatorname{Fil}_{\operatorname{PD}}B$ is the
base change from $\operatorname{DF}^{\geq 0}(B)$ to the $\infty$-category of
$\operatorname{Fil}_{\operatorname{PD}}B$-modules). Furthermore,
$\operatorname{Fil}_{B}^{i}\operatorname{gr}_{H}^{j}\operatorname{dR}_{(B\twoheadrightarrow
B^{\prime\prime})/(A\twoheadrightarrow A^{\prime\prime})}\simeq 0$ when
$i\char 62\relax j$.

###### Proof.

By Proposition A.10, it suffices to check the equivalences on
$\operatorname{dRCon}^{0}$, which follows from definitions. ∎

We are now ready to introduce the Katz-Oda filtration:

######  Definition 4.75 (cf. [GL20, Cons 3.12]).

Let $(A\twoheadrightarrow
A^{\prime\prime},\gamma_{A})\rightarrow(B\twoheadrightarrow
B^{\prime\prime},\gamma_{B})$ be a map of animated PD-pairs and
$B^{\prime\prime}\rightarrow R$ a map of animated rings. The Katz-Oda
filtration on the Hodge-filtered derived crystalline cohomology
$\operatorname{Fil}_{H}\operatorname{CrysCoh}_{R/(A\twoheadrightarrow
A^{\prime\prime})}$ rewritten as
$\operatorname{Fil}_{H}\operatorname{CrysCoh}_{R/(A\twoheadrightarrow
A^{\prime\prime})}\otimes_{\operatorname{Fil}_{H}\operatorname{dR}_{(B\twoheadrightarrow
B^{\prime\prime})/(A\twoheadrightarrow
A^{\prime\prime})}}^{\mathbb{L}}\operatorname{Fil}_{H}\operatorname{dR}_{(B\twoheadrightarrow
B^{\prime\prime})/(A\twoheadrightarrow A^{\prime\prime})}$ is induced by the
“stupid” filtration on
$\operatorname{Fil}_{H}\operatorname{dR}_{(B\twoheadrightarrow
B^{\prime\prime})/(A\twoheadrightarrow A^{\prime\prime})}$.

We now have

######  Lemma 4.76 (cf. [GL20, Lem 3.13]).

Let $(A\twoheadrightarrow
A^{\prime\prime},\gamma_{A})\rightarrow(B\twoheadrightarrow
B^{\prime\prime},\gamma_{B})$ be a map of animated PD-pairs and
$B^{\prime\prime}\rightarrow R$ a map of animated rings. Then

  1. (1)

The associated graded pieces
$\operatorname{gr}_{\operatorname{KO}}^{i}\operatorname{Fil}_{H}\operatorname{CrysCoh}_{R/(A\twoheadrightarrow
A^{\prime\prime})}$ are functorially equivalent to

$\operatorname{Fil}_{H}\operatorname{CrysCoh}_{R/(B\twoheadrightarrow
B^{\prime\prime})}\otimes_{\operatorname{Fil}_{\operatorname{PD}}B}^{\mathbb{L}}\left(\operatorname{ins}^{i}\left(\bigwedge_{B}^{i}\mathbb{L}_{(B\twoheadrightarrow
B^{\prime\prime})/(A\twoheadrightarrow
A^{\prime\prime})}[-i]\right)\otimes_{B}^{\mathbb{L}}\operatorname{Fil}_{\operatorname{PD}}B\right)$

as $\operatorname{Fil}_{\operatorname{PD}}B$-modules in
$\operatorname{DF}^{\geq 0}(\mathbb{Z})$ for all $i\in\mathbb{N}$, where the
functor $\operatorname{ins}^{i}$ is defined in Section 2.4.

  2. (2)

The induced Katz-Oda filtration on
$\operatorname{gr}_{H}^{\ast}\operatorname{CrysCoh}_{R/(A\twoheadrightarrow
A^{\prime\prime})}$ is complete. In fact, for $i\char 62\relax j$, we have
$\operatorname{Fil}_{\operatorname{KO}}^{i}\operatorname{gr}_{H}^{j}\operatorname{CrysCoh}_{R/(A\twoheadrightarrow
A^{\prime\prime})}\simeq 0$.

###### Proof.

We have seen (Corollary 4.31) that the canonical map

$\operatorname{Fil}_{H}\operatorname{CrysCoh}_{R/(A\twoheadrightarrow
A^{\prime\prime})}\otimes_{\operatorname{Fil}_{H}\operatorname{dR}_{(B\twoheadrightarrow
B^{\prime\prime})/(A\twoheadrightarrow
A^{\prime\prime})}}^{\mathbb{L}}\operatorname{Fil}_{\operatorname{PD}}B\rightarrow\operatorname{Fil}_{H}\operatorname{CrysCoh}_{R/(B\twoheadrightarrow
B^{\prime\prime})}$

is an equivalence. Then both follow from Lemma 4.74. ∎

The convergence of Katz-Oda filtration on associated graded pieces is the key
to Lemma 4.77.

######  Lemma 4.77.

In Proposition 4.70, the maps (4) form a limit diagram after passing to the
associated graded pieces, i.e. after composition with the functor
$\operatorname{CAlg}(\operatorname{DF}^{\geq
0}(A))\rightarrow\operatorname{CAlg}(\operatorname{Gr}^{\geq 0}(A))$.
Furthermore, if the $\pi_{0}(A^{\prime\prime})$-algebra $\pi_{0}(R)$ is of
finite type, then the maps (4) form a limit diagram.

###### Proof.

Note that the map (4) is the canonical map
$\operatorname{Fil}_{H}\operatorname{CrysCoh}_{R/(A\twoheadrightarrow
A^{\prime\prime})}\rightarrow\operatorname{gr}_{\operatorname{KO}}^{0,(\nu)}\operatorname{Fil}_{H}\operatorname{CrysCoh}_{R/(A\twoheadrightarrow
A^{\prime\prime})}$, the Katz-Oda filtration with respect to the cosimplicial
system $((A\twoheadrightarrow
A^{\prime\prime},\gamma_{A})\rightarrow(P^{\nu}\rightarrow
P^{\nu}\otimes_{A}^{\mathbb{L}}A^{\prime\prime},\gamma_{P^{\nu}}),P^{\nu}\rightarrow
R)$. By the completeness in Lemma 4.76, for the equivalence on associated
graded pieces, it suffices to show that, for every $i\in\mathbb{N}_{\char
62\relax 0}$, the totalization
$\lim_{\nu\in\boldsymbol{\Delta}}\operatorname{gr}_{\operatorname{KO}}^{i,(\nu)}\operatorname{Fil}_{H}\operatorname{CrysCoh}_{R/(A\twoheadrightarrow
A^{\prime\prime})}$ is contractible.

The key observation is that
$\operatorname{ins}^{i}\left(\bigwedge^{i}\mathbb{L}_{(P^{\bullet}\rightarrow
P^{\bullet}\otimes_{A}^{\mathbb{L}}A^{\prime\prime})/(A\twoheadrightarrow
A^{\prime\prime})}[-i]\right)$ is homotopy equivalent to $0$ as a cosimplicial
$B^{\bullet}$-module spectrum [Bha12b, Lem 2.6] when $i\char 62\relax 0$ (this
is of course false when $i=0$). It follows that the cosimplicial object
$\operatorname{gr}_{\operatorname{KO}}^{i,(\bullet)}\operatorname{Fil}_{H}\operatorname{CrysCoh}_{R/(A\twoheadrightarrow
A^{\prime\prime})}$ is homotopy equivalent to $0$ as
$\operatorname{Fil}_{\operatorname{PD}}B$-modules by [Sta21, Tag 07KQ] and
Lemma 4.76.

Finally, if $\pi_{0}(R)$ is a finitely generated
$\pi_{0}(A^{\prime\prime})$-algebra, we could pick a polynomial $A$-algebra
$P$ of finitely many variables along with a surjection $P\twoheadrightarrow
R$. In this case, the Katz-Oda filtration is finite and the above argument
works. ∎

######  Warning 4.78.

One should be careful about homotopy equivalences. In an earlier draft of this
article, we came up with the following “proof”: the Hodge-filtered derived de
Rham cohomology
$\operatorname{Fil}_{H}\operatorname{CrysCoh}_{R/(A\twoheadrightarrow
A^{\prime\prime})}$ could be rewritten as

$\operatorname{Fil}_{H}\operatorname{CrysCoh}_{R/(A\twoheadrightarrow
A^{\prime\prime})}\otimes_{\operatorname{Fil}_{H}\operatorname{dR}_{(P^{\bullet}\twoheadrightarrow
P^{\bullet}\otimes_{A}^{\mathbb{L}}A^{\prime\prime})/(A\twoheadrightarrow
A^{\prime\prime})}}^{\mathbb{L}}\operatorname{Fil}_{H}\operatorname{dR}_{(P^{\bullet}\twoheadrightarrow
P^{\bullet}\otimes_{A}^{\mathbb{L}}A^{\prime\prime})/(A\twoheadrightarrow
A^{\prime\prime})}$

and since the map $A\rightarrow P^{\bullet}$ is a homotopy equivalence as
$A$-algebras, the map
$\operatorname{Fil}_{H}\operatorname{dR}_{(P^{\bullet}\twoheadrightarrow
P^{\bullet}\otimes_{A}^{\mathbb{L}}A^{\prime\prime})/(A\twoheadrightarrow
A^{\prime\prime})}\rightarrow\operatorname{Fil}_{\operatorname{PD}}P^{\bullet}$
is also a homotopy equivalence “therefore” the constant cosimplicial algebra
$\operatorname{Fil}_{H}\operatorname{CrysCoh}_{R/(A\twoheadrightarrow
A^{\prime\prime})}$ is homotopy equivalent to
$\operatorname{Fil}_{H}\operatorname{CrysCoh}_{R/(A\twoheadrightarrow
A^{\prime\prime})}\otimes_{\operatorname{Fil}_{H}\operatorname{dR}_{(P^{\bullet}\twoheadrightarrow
P^{\bullet}\otimes_{A}^{\mathbb{L}}A^{\prime\prime})/(A\twoheadrightarrow
A^{\prime\prime})}}^{\mathbb{L}}\operatorname{Fil}_{\operatorname{PD}}P^{\bullet}\simeq\operatorname{Fil}_{H}\operatorname{CrysCoh}_{R/(P^{\bullet}\twoheadrightarrow
P^{\bullet}\otimes_{A}^{\mathbb{L}}A^{\prime\prime})}$ therefore the
conditions in Proposition 4.70.

This argument is incorrect: when playing with homotopy equivalences, one
cannot replace the base cosimplicial algebra by a homotopy equivalent algebra
without justification. In fact, the last homotopy equivalence obtained above
is also incorrect: if it were the case, we consider the special case that
$(A\twoheadrightarrow A^{\prime\prime},\gamma_{A})$ is given by
$(\operatorname{id}_{A}:A\rightarrow A,0)$, and
$\operatorname{CrysCoh}_{R/P^{\bullet}}$ is just the animated PD-envelope of
$P^{\bullet}\twoheadrightarrow R$ (see the proof of Proposition 4.66). We
inspect the homotopy equivalence of cosimplicial objects that we assumed:

$\operatorname{dR}_{R/A}\simeq^{\operatorname{HoEq}}\operatorname{dR}_{R/P^{\bullet}}$

when $A$ is a static $\mathbb{F}_{p}$-algebra and $R$ is a smooth $A$-algebra
such that $\operatorname{dR}_{R/A}$ is not static, the map
$P^{\bullet}\twoheadrightarrow R$ is Koszul regular and the derived de Rham
cohomology $\operatorname{dR}_{R/P^{\bullet}}$ is simply the PD-envelope,
therefore static. Applying $\pi_{i}$ to the homotopy equivalence, where $i\neq
0$ is so chosen that $\pi_{i}(\operatorname{dR}_{R/A})\neq 0$, we get a
contradiction.

In view of this warning, our proof of Lemma 4.77 tells us that the associated
graded pieces with respect to the Katz-Oda filtration are homotopy equivalent,
but the homotopy equivalences could not be glued, even after forgetting all
the richer structures to the underlying $\infty$-category $D(\mathbb{Z})$.

When the $\pi_{0}(A)$-algebra $\pi_{0}(R)$ is not of finite type, we can still
prove that the comparison map is an equivalence with mild smoothness of
$A^{\prime\prime}\rightarrow R$ (Proposition 4.87). We start with another
sufficient condition in characteristic $p$ which is essentially a variant of
[LL20, Prop 2.17] by Proposition 4.70.

######  Lemma 4.79.

Let $((A\twoheadrightarrow
A^{\prime\prime},\gamma_{A}),A^{\prime\prime}\rightarrow
R)\in\operatorname{CrysCon}_{\mathbb{F}_{p}}$. Suppose that

  1. (1)

The cotangent complex $\mathbb{L}_{R/A^{\prime\prime}}\in D_{\geq 0}(R)$ has
$\operatorname{Tor}$-amplitude in $[0,1]$.

  2. (2)

The derived Frobenius twist $\varphi_{A\twoheadrightarrow
A^{\prime\prime}}^{\ast}(R)$ (see Lemma 4.36) is bounded above, i.e.
$\pi_{i}(\varphi_{A\twoheadrightarrow A^{\prime\prime}}^{\ast}(R))\cong 0$ for
$i\gg 0$.

Then the comparison map in Proposition 4.66 is an equivalence.

###### Proof.

Our proof is also adapted from [LL20, Prop 2.17]. By Propositions 4.70 and
4.77, it suffices to that the natural maps

(5) $\operatorname{CrysCoh}_{R/(A\twoheadrightarrow
A^{\prime\prime},\gamma_{A})}\longrightarrow\operatorname{CrysCoh}_{R/(P^{\bullet}\twoheadrightarrow
P^{\bullet}\otimes_{A}^{\mathbb{L}}A^{\prime\prime},\gamma_{P^{\bullet}})}$

form a limit diagram in $\operatorname{CAlg}(D(A))$. We endow both sides with
conjugate filtration (Definition 4.41), and show that this forms in fact a
limit diagram in $\operatorname{CAlg}(\operatorname{DF}^{\leq 0}(A))$.

We show that, after passing to associated graded pieces with respect to the
conjugate filtration, the maps (5) form a limit diagram, which implies that
the natural maps (5) form limit diagrams after passing to finite level of
quotients, and then we control the convergence to deduce the result. To show
the result for associated graded pieces, by Proposition 4.46, it suffices to
show that the maps

(6) $\varphi_{A\twoheadrightarrow
A^{\prime\prime}}^{\ast}\left(\bigwedge_{R}^{\star}\mathbb{L}_{R/A^{\prime\prime}}\right)\left[-\mathord{\star}\right]\longrightarrow\varphi_{P^{\bullet}\twoheadrightarrow
P^{\bullet}\otimes_{A}^{\mathbb{L}}A^{\prime\prime}}^{\ast}\left(\bigwedge_{R}^{\star}\mathbb{L}_{R/(P^{\bullet}\otimes_{A}^{\mathbb{L}}A^{\prime\prime})}\right)\left[-\mathord{\star}\right]$

form a limit diagram in $\operatorname{Gr}^{\geq 0}(D(A))$.

Let $R_{1}:=\varphi_{A\twoheadrightarrow A^{\prime\prime}}^{\ast}(R)$. Note
that the Frobenius map $\varphi_{P^{\bullet}\twoheadrightarrow
P^{\bullet}\otimes_{A}^{\mathbb{L}}A^{\prime\prime}}$ factors as
$P^{\bullet}\otimes_{A}^{\mathbb{L}}A^{\prime\prime}\rightarrow\varphi_{A}^{\ast}(P^{\bullet})\rightarrow
P^{\bullet}$ where the second map is the Frobenius map of $P^{\bullet}$
relative to $A$. Then the maps (6) could be rewritten as the maps

$\bigwedge_{R_{1}}^{\star}\mathbb{L}_{R_{1}/A}\left[-\mathord{\star}\right]\longrightarrow\left(\bigwedge_{R_{1}}^{\star}\mathbb{L}_{R_{1}/\varphi_{A}^{\ast}(P^{\bullet})}\right)\left[-\mathord{\star}\right]\otimes_{\varphi_{A}^{\ast}(P^{\bullet})}^{\mathbb{L}}P^{\bullet}$

or equivalently, the maps

$\operatorname{gr}_{H}^{\star}\operatorname{dR}_{R_{1}/A}\longrightarrow\operatorname{gr}_{H}^{\star}\operatorname{dR}_{R_{1}/\varphi_{A}^{\ast}(P^{\bullet})}\otimes_{\varphi_{A}^{\ast}(P^{\bullet})}^{\mathbb{L}}P^{\bullet}$

by an inverse application of Lemma 4.74 (recall that for derived de Rham
cohomology of animated rings, the “stupid” filtration coincides with the Hodge
filtration). We again consider the Katz-Oda filtration associated to the
cosimplicial system $A\rightarrow\varphi_{A}^{\ast}(P^{\bullet})\rightarrow
R_{1}$ (Lemma 4.76) and by completeness, we could pass to associated graded
pieces for $i=0$:

(7)
$\operatorname{gr}_{H}^{\star}\operatorname{dR}_{R_{1}/\varphi_{A}^{\ast}(P^{\bullet})}\longrightarrow\operatorname{gr}_{H}^{\star}\operatorname{dR}_{R_{1}/\varphi_{A}^{\ast}(P^{\bullet})}\otimes_{\varphi_{A}^{\ast}(P^{\bullet})}^{\mathbb{L}}P^{\bullet}$

and $i\in\mathbb{N}_{\char 62\relax 0}$:

$\operatorname{gr}_{H}^{\star}\operatorname{dR}_{R_{1}/\varphi_{A}^{\ast}(P^{\bullet})}\otimes_{\varphi_{A}^{\ast}(P^{\bullet})}^{\mathbb{L}}\left(\bigwedge_{\varphi_{A}^{\ast}(P^{\bullet})}^{i}\mathbb{L}_{\varphi_{A}^{\ast}(P^{\bullet})/A}[-i]\right)\longrightarrow
0$

As in Lemma 4.77, the later maps constitute a homotopy equivalence by [Bha12b,
Lem 2.6] and [Sta21, Tag 07KQ], therefore constitutes a limit diagram by Lemma
4.69. On the other hand, by Lemma 4.82, the maps (7) constitute a limit
diagram.

Now we control the convergence. Again by Lemma 4.74, we rewrite the maps (7)
as the maps

$\bigwedge_{R_{1}}^{\star}\mathbb{L}_{R_{1}/\varphi_{A}^{\ast}(P^{\bullet})}\left[-\mathord{\star}\right]\longrightarrow\left(\bigwedge_{R_{1}}^{\star}\mathbb{L}_{R_{1}/\varphi_{A}^{\ast}(P^{\bullet})}\left[-\mathord{\star}\right]\right)\otimes_{\varphi_{A}^{\ast}(P^{\bullet})}^{\mathbb{L}}P^{\bullet}$

Now consider the transitivity sequence

$\mathbb{L}_{\varphi_{A}^{\ast}(P^{\bullet})/A}\otimes_{\varphi_{A}^{\ast}(P^{\bullet})}^{\mathbb{L}}R_{1}\longrightarrow\mathbb{L}_{R_{1}/A}\longrightarrow\mathbb{L}_{R_{1}/\varphi_{A}^{\ast}(P^{\bullet})}$

For every static $R_{1}$-module $M$, we get the fiber sequence

$\mathbb{L}_{\varphi_{A}^{\ast}(P^{\bullet})/A}\otimes_{\varphi_{A}^{\ast}(P^{\bullet})}^{\mathbb{L}}M\longrightarrow\mathbb{L}_{R_{1}/A}\otimes_{R_{1}}^{\mathbb{L}}M\longrightarrow\mathbb{L}_{R_{1}/\varphi_{A}^{\ast}(P^{\bullet})}\otimes_{R_{1}}^{\mathbb{L}}M$

Since $\mathbb{L}_{R/A^{\prime\prime}}\in D_{\geq 0}(R)$ has
$\operatorname{Tor}$-amplitude in $[0,1]$, so does $\mathbb{L}_{R_{1}/A}\in
D_{\geq 0}(R_{1})$, therefore
$\pi_{j}(\mathbb{L}_{R_{1}/A}\otimes_{R_{1}}^{\mathbb{L}}M)\cong 0$ for $j\neq
0,1$. Note that $\mathbb{L}_{\varphi_{A}^{\ast}(P^{\bullet})/A}$ is a flat
$\varphi_{A}^{\ast}(P^{\bullet})$-module. It follows that
$\pi_{j}(\mathbb{L}_{R_{1}/\varphi_{A}^{\ast}(P^{\bullet})}\otimes_{R_{1}}^{\mathbb{L}}M)\cong
0$ for $j\neq 0,1$. Furthermore, since
$\varphi_{A}^{\ast}(P^{\bullet})\rightarrow R_{1}$ is surjective,
$\pi_{0}(\mathbb{L}_{R_{1}/\varphi_{A}^{\ast}(P^{\bullet})}\otimes_{R_{1}}^{\mathbb{L}}M)\cong
0$. It follows that $\mathbb{L}_{R_{1}/\varphi_{A}^{\ast}(P^{\bullet})}[-1]$
is a flat $R_{1}$-module, and so is
$\bigwedge_{R_{1}}^{\star}\mathbb{L}_{R_{1}/\varphi_{A}^{\ast}(P^{\bullet})}\left[-\mathord{\star}\right]\simeq\Gamma_{R_{1}}^{\star}(\mathbb{L}_{R_{1}/\varphi_{A}^{\ast}(P^{\bullet})}[-1])$.
By assumption, $R_{1}$ is bounded above, therefore so is
$\bigwedge_{R_{1}}^{\star}\mathbb{L}_{R_{1}/\varphi_{A}^{\ast}(P^{\bullet})}\left[-\mathord{\star}\right]$.

It remains to show that the associated graded pieces are uniformly bounded
above, which implies that (5) form a limit diagram, by Lemma 4.84 and that the
conjugate filtration is exhaustive (Lemma 4.42). Suppose that the homotopy
groups of $R_{1}$ are concentrated in the range $[a,b]$, then the associated
graded pieces of the target could be rewritten as
$\Gamma_{R_{1}}^{\star}(\mathbb{L}_{R_{1}/\varphi_{A}^{\ast}(P^{\bullet})}[-1])\otimes_{\varphi_{A}^{\ast}(P^{\bullet})}^{\mathbb{L}}P^{\bullet}$,
where
$\Gamma_{R_{1}}^{\star}(\mathbb{L}_{R_{1}/\varphi_{A}^{\ast}(P^{\bullet})}[-1])$
is a flat $R_{1}$-module therefore the homotopy groups of it is also
concentrated in the range $[a,b]$. Since the relative Frobenius
$\varphi_{A}^{\ast}(P^{\bullet})\rightarrow P^{\bullet}$ is flat, we get
$\pi_{j}(\Gamma_{R_{1}}^{\star}(\mathbb{L}_{R_{1}/\varphi_{A}^{\ast}(P^{\bullet})}[-1])\otimes_{\varphi_{A}^{\ast}(P^{\bullet})}^{\mathbb{L}}P^{\bullet})\cong\pi_{j}(\Gamma_{R_{1}}^{\star}(\mathbb{L}_{R_{1}/\varphi_{A}^{\ast}(P^{\bullet})}[-1]))\otimes_{\pi_{0}(\varphi_{A}^{\ast}(P^{\bullet}))}\pi_{0}(P^{\bullet})\cong
0$ for $j\not\in[a,b]$. ∎

We need the following lemmas:

######  Lemma 4.80.

Let $\mathcal{C}$ be an $\infty$-category which admits finite coproducts. Let
$\varnothing$ denote the initial object of $\mathcal{C}$, and let $X,Y$ be two
objects of $\mathcal{C}$. Then for any two maps
$g_{0},g_{1}\in\operatorname{Hom}_{\mathcal{C}}(X,Y)$, the induced maps
$X^{\bullet}\rightrightarrows Y^{\bullet}$ of Čech conerves $X^{\bullet}$ of
$X$ (i.e. of $\varnothing\rightarrow X$) and $Y^{\bullet}$ of $Y$ (i.e. of
$\varnothing\rightarrow Y$) are homotopic. More precisely, there exists a
simplicial homotopy from $g_{0}^{\bullet}$ to $g_{1}^{\bullet}$ which is
functorial in $g_{0}$ and $g_{1}$. In particular, if $X=Y$ and $g_{0}=g_{1}$,
then the simplicial homotopy is constant.

###### Proof.

We start with the special case that $\mathcal{C}$ is a $1$-category. We define
the simplicial homotopy
$h:\delta^{\ast}(X^{\bullet})\rightarrow\delta^{\ast}(Y^{\bullet})$ as
follows: for every
$(\alpha_{n}:[n]\rightarrow[1])\in\boldsymbol{\Delta}_{/[1]}$, we note that
$(X^{\bullet})(\alpha_{n})=X^{n}=X\amalg\cdots\amalg X$ and
$(Y^{\bullet})(\alpha_{n})=Y^{n}=Y\amalg\cdots\amalg Y$, and we set
$h_{\alpha_{n}}=\coprod_{i=0}^{n}g_{\alpha_{n}(i)}:X^{n}\rightarrow Y^{n}$. By
construction,
$i_{0}^{\ast}(h)^{\nu}=h_{[\nu]\xrightarrow{0}[1]}=\coprod_{i=0}^{n}g_{0}=g_{0}^{\nu}$
and $i_{1}^{\ast}(h)^{\nu}=g_{1}^{\nu}$.

We need to check that this is a map of functors. For every map
$\psi:(\alpha_{n}:[n]\rightarrow[1])\rightarrow(\alpha_{m}:[m]\rightarrow[1])$
in $\boldsymbol{\Delta}_{/[1]}$, we need to check that the diagram

commutes, where the vertical maps $\psi_{\ast}:X^{n}\rightarrow X^{m}$ and
$\psi_{\ast}:Y^{n}\rightarrow Y^{m}$ are induced by $\psi$. For this end, let
$j_{i}:X\rightarrow X^{n}$ be the $i$-th canonical map for $0\leq i\leq n$.

Then the composite $h_{\alpha_{n}}\circ j_{i}:X\rightarrow X^{n}\rightarrow
Y^{n}$ could be rewritten as the composite $j_{i}\circ
g_{a_{n}(i)}:X\rightarrow Y\rightarrow Y^{n}$, and the composite
$\psi_{\ast}\circ h_{\alpha_{n}}\circ j_{i}:X\rightarrow Y^{m}$ is equivalent
to the composite $j_{\psi(i)}\circ g_{\alpha_{n}(i)}:X\rightarrow Y\rightarrow
Y^{m}$. Similarly, the composite $\psi_{\ast}\circ j_{i}:X\rightarrow
X^{n}\rightarrow X^{m}$ is equivalent to the $\psi(i)$-th canonical map
$j_{\psi(i)}:X\rightarrow X^{m}$, and the composite
$h_{\alpha_{m}}\circ\psi_{\ast}\circ j_{i}$ could be identified with the
composite $j_{\psi(i)}\circ g_{\alpha_{m}(\psi(i))}:X\rightarrow Y\rightarrow
Y^{m}$.

Since $\alpha_{m}(\psi(i))=\alpha_{n}(i)$, it follows that $\psi_{\ast}\circ
h_{\alpha_{n}}\circ j_{i}=h_{\alpha_{m}}\circ\psi_{\ast}\circ j_{i}$ for every
$0\leq i\leq n$. It follows that $\psi_{\ast}\circ
h_{\alpha_{n}}=h_{\alpha_{m}}\circ\psi_{\ast}$. The other claims for the
$1$-category $\mathcal{C}$ follow directly from the construction.

Now we claim that the result for $\infty$-categories follows from that for
$1$-categories. The point is that there exists a universal
$1$-category282828This is informed to us by Denis Nardin. $\mathcal{C}_{0}$
along with two objects $X_{0},Y_{0}\in\mathcal{C}$ and two maps
$X_{0}\rightrightarrows Y_{0}$, which admits all finite products, such that
for every $\infty$-category $\mathcal{C}$ as in the assumption of this lemma,
there exists an essentially unique functor
$\mathcal{C}_{0}\rightarrow\mathcal{C}$ which preserves finite coproducts: let
$K$ be the diagram $\bullet\rightrightarrows\bullet$, and then take the
presheaf $\infty$-category
$\mathcal{P}(K)=\operatorname{Fun}(K^{\operatorname{op}},\mathcal{S})$. Then
we can take $\mathcal{C}_{0}$ to be the full subcategory of $\mathcal{P}(K)$
spanned by finite coproducts of the two vertices of $K$. ∎

######  Corollary 4.81.

Let $\mathcal{C}$ be an $\infty$-category with finite coproducts, and two
objects $X,Y$ in $\mathcal{C}$. Let $i:X\rightarrow Y$ be a map which admits a
left inverse $r:Y\rightarrow X$. Then there is a “strong deformation retract”,
i.e. a simplicial homotopy from $\operatorname{id}_{Y^{\bullet}}$ to
$i^{\bullet}\circ r^{\bullet}$, which restricts to a constant simplicial
homotopy of $X^{\bullet}$ along $i^{\bullet}:X^{\bullet}\rightarrow
Y^{\bullet}$, where $X^{\bullet}$ (resp. $Y^{\bullet}$) is the Čech conerve of
$X$ (resp. $Y$), and $i^{\bullet}:X^{\bullet}\rightarrow Y^{\bullet}$ and
$r^{\bullet}:Y^{\bullet}\rightarrow X^{\bullet}$ are induced simplicial maps.

###### Proof.

We apply Lemma 4.80 to $\operatorname{id}_{Y},i\circ
r\in\operatorname{Hom}_{\mathcal{C}}(Y,Y)$, getting the desired simplicial
homotopy. To see the later statement, it suffices to inspect the commutative
diagram

$\begin{array}[]{ccc}Y&\underset{i\circ
r}{\overset{\operatorname{id}_{Y}}{\mathrel{\begin{subarray}{c}\longrightarrow\\\\[-2.58334pt]
\longrightarrow\end{subarray}}}}&Y\\\
{\mbox{\rotatebox[origin={c}]{90.0}{$\longrightarrow$}}}&&{\mbox{\rotatebox[origin={c}]{90.0}{$\longrightarrow$}}}\\\
X&\underset{\operatorname{id}_{X}}{\overset{\operatorname{id}_{X}}{\mathrel{\begin{subarray}{c}\longrightarrow\\\\[-2.58334pt]
\longrightarrow\end{subarray}}}}&X\end{array}$

and invoke the functoriality. ∎

######  Lemma 4.82.

Let $A\in\operatorname{CAlg}^{\operatorname{cn}}$ be a connective
$\mathbb{E}_{\infty}$-ring and let $B\rightarrow C$ be a faithfully flat map
of connective $\mathbb{E}_{\infty}$-$A$-algebras. Let $B^{\bullet}$ (resp.
$C^{\bullet}$) denote the Čech conerve of the map $A\rightarrow B$ (resp.
$A\rightarrow C$). Then for any cosimplicial $B^{\bullet}$-module
$N^{\bullet}$, the natural cosimplicial map

$N^{\bullet}\longrightarrow
N^{\bullet}\otimes_{B^{\bullet}}^{\mathbb{L}}C^{\bullet}$

induces an equivalence after totalization
$\lim_{\bullet\in\boldsymbol{\Delta}}$in $D(A)$, where the cosimplicial map
$B^{\bullet}\rightarrow C^{\bullet}$ is induced by $B\rightarrow C$.

###### Proof.

Let $D^{\bullet,\bullet}$ denote the cosimplicial Čech conerve of
$B^{\bullet}\rightarrow C^{\bullet}$ (each $D^{\nu,\bullet}$ is the Čech
conerve of $B^{\nu}\rightarrow C^{\nu}$), which is a bicosimplicial object in
$\operatorname{CAlg}_{A}$. We note that there is a unique cosimplicial map
$B^{\bullet}\rightarrow D^{\bullet,\mu}$ for all
$[\mu]\in\boldsymbol{\Delta}$. Consider the bicosimplicial object
$M^{\bullet,\bullet}$:

$\displaystyle\boldsymbol{\Delta}^{2}$ $\displaystyle\longrightarrow$
$\displaystyle D(A)$ $\displaystyle([\nu],[\mu])$ $\displaystyle\longmapsto$
$\displaystyle N^{\nu}\otimes_{B^{\nu}}^{\mathbb{L}}D^{\nu,\mu}$

and its limit $I:=\lim_{([\nu],[\mu])}M^{\nu,\mu}$. The map which we need to
show to be an equivalence factors as
$\lim_{[\nu]}N^{\nu}\rightarrow\lim_{([\nu],[\mu])}M^{\nu,\mu}\rightarrow\lim_{[\nu]}M^{\nu,0}\simeq\lim_{[\nu]}N^{\nu}\otimes_{B^{\nu}}^{\mathbb{L}}C^{\nu}$.
It suffices to show that both maps are equivalences.

For the first map, in fact, for every $[\nu]\in\boldsymbol{\Delta}$, the map
$N^{\nu}\rightarrow\lim_{[\mu]\in\boldsymbol{\Delta}}M^{\nu,\mu}$ is an
equivalence by faithfully flat descent.

For the second map, since
$\boldsymbol{\Delta}_{\operatorname{inj}}^{\operatorname{op}}\rightarrow\boldsymbol{\Delta}^{\operatorname{op}}$
is cofinal [Lur09, Lem 6.5.3.7] where
$\boldsymbol{\Delta}_{\operatorname{inj}}\subseteq\boldsymbol{\Delta}$ is the
(non-full) subcategory with strictly increasing maps $[m]\rightarrow[n]$, we
can replace $\lim_{\boldsymbol{\Delta}}(\cdot)$ by
$\lim_{\boldsymbol{\Delta}_{\operatorname{inj}}}(\cdot)$. By [Lur09, Cor
4.4.4.10], it suffices to show that, for every injective map
$[\mu_{1}]\rightarrow[\mu_{2}]$ in $\boldsymbol{\Delta}$, the induced map
$\lim_{[\nu]}M^{\nu,\mu_{1}}\rightarrow\lim_{[\nu]}M^{\nu,\mu_{2}}$ is an
equivalence. Every injective map $[\mu_{1}]\rightarrow[\mu_{2}]$ admits a
retract in $\boldsymbol{\Delta}$, therefore by Corollary 4.81, the induced map
$D^{\bullet,\mu_{1}}\rightarrow D^{\bullet,\mu_{2}}$ is a homotopy equivalence
of cosimplicial $\mathbb{E}_{\infty}$-$B^{\bullet}$-algebras, therefore
$M^{\bullet,\mu_{1}}\rightarrow M^{\bullet,\mu_{2}}$ is a homotopy equivalence
of cosimplicial $A$-modules by [Sta21, Tag 07KQ]. The result then follows from
Lemma 4.69. ∎

######  Remark 4.83.

When $A$ is a static $\mathbb{F}_{p}$-algebra, $B$ is a polynomial $A$-algebra
and $C=B\otimes_{A,\varphi_{A}}^{\mathbb{L}}A$ is the Frobenius twist of $B$,
we recover [BS19, Lem 5.4].

######  Lemma 4.84.

Let $(M_{i}^{\bullet})_{i\in(\mathbb{Z}_{\leq
0},\geq)}\in\operatorname{Fun}(\boldsymbol{\Delta}\times(\mathbb{Z}_{\leq
0},\geq),\operatorname{Sp})$ be a cosimplicial filtered spectra. Suppose that
it is uniformly bounded above, i.e. there exists $N\in\mathbb{N}$ such that
for every $i\in\mathbb{Z}_{\leq 0}$ and $\nu\in\boldsymbol{\Delta}$, we have
$\pi_{j}(M_{i}^{\nu})\cong 0$ for all $j\char 62\relax N$. Let
$M^{\bullet}:=\operatorname{colim}_{i\rightarrow-\infty}M_{i}^{\bullet}$. Then
the canonical map
$\operatorname{colim}_{i\rightarrow-\infty}\lim_{\nu\in\boldsymbol{\Delta}}M_{i}^{\nu}\rightarrow\lim_{\nu\in\boldsymbol{\Delta}}M^{\nu}$
is an equivalence.

###### Proof.

We could rewrite $\lim_{\nu\in\boldsymbol{\Delta}}$ as
$\lim_{n\rightarrow\infty}\lim_{\nu\in\boldsymbol{\Delta}_{\leq[n]}}$.
Furthermore, the functor
$\boldsymbol{\Delta}_{\leq[n]}^{\operatorname{inj}}\rightarrow\boldsymbol{\Delta}_{\leq[n]}$
is right cofinal, therefore we can replace
$\lim_{\nu\in\boldsymbol{\Delta}_{\leq[n]}}$ by
$\lim_{\nu\in\boldsymbol{\Delta}_{\leq[n]}^{\operatorname{inj}}}$ which is a
finite limit, therefore commutes with
$\operatorname{colim}_{i\rightarrow-\infty}$. For any cosimplicial spectrum
$X^{\bullet}$, there is a canonical map
$\lim_{\nu\in\boldsymbol{\Delta}}X^{\nu}\rightarrow\lim_{\nu\in\boldsymbol{\Delta}_{\leq[n]}}X^{\nu}$.
If $X^{\bullet}$ is assumed to be uniformly bounded above, then the
coconnectivity of
$\operatorname{fib}(\lim_{\nu\in\boldsymbol{\Delta}}X^{\nu}\rightarrow\lim_{\nu\in\boldsymbol{\Delta}_{\leq[n]}}X^{\nu})$
tends to $-\infty$ as $n\rightarrow\infty$ by [Lur17, Cor 1.2.4.18]. The
result then follows. ∎

For the integral version, we need to introduce the following concept of
smoothness:

######  Definition 4.85 (cf. [BMS19, Def 4.9]).

We say that a map $R\rightarrow S$ of animated rings is quasisyntomic if it is
flat and the cotangent complex $\mathbb{L}_{S/R}$ has
$\operatorname{Tor}$-amplitude in $[0,1]$.

######  Example 4.86.

Any smooth map, or more generally, any syntomic map of static rings is
quasisyntomic.

We now phrase the integral comparison:

######  Proposition 4.87.

Let $((A\twoheadrightarrow
A^{\prime\prime},\gamma_{A}),A^{\prime\prime}\rightarrow
R)\in\operatorname{CrysCon}$ such that $A$ is bounded above (that is,
$\pi_{n}(A)\cong 0$ for $n\gg 0$) and the map $A^{\prime\prime}\rightarrow R$
is quasisyntomic. Then the comparison map in Proposition 4.66 is an
equivalence.

###### Proof.

We again appeal to Proposition 4.70. It suffices to show that the map

$\operatorname{CrysCoh}_{R/(A\twoheadrightarrow
A^{\prime\prime},\gamma_{A})}\longrightarrow\lim_{\nu\in\boldsymbol{\Delta}}\operatorname{CrysCoh}_{R/(P^{\nu}\twoheadrightarrow
P^{\nu}\otimes_{A}^{\mathbb{L}}A^{\prime\prime},\gamma_{P^{\nu}})}$

is an equivalence of $\mathbb{Z}$-module spectra (since the forgetful functor
is conservative), which could be checked by base change along
$\mathbb{Z}\rightarrow\mathbb{Z}/p$ for all prime numbers
$p\in\mathbb{N}_{\char 62\relax 0}$ and along
$\mathbb{Z}\rightarrow\mathbb{Q}$. The latter follows from Lemma 4.11 and that
the map $A\rightarrow P$ is faithfully flat therefore the canonical map
$A\rightarrow\lim_{\nu\in\boldsymbol{\Delta}}P^{\nu}$ is an equivalence (in
fact, this is induced by a homotopy equivalence of cosimplicial objects, but
we do not need this). For every prime number $p$, by base change property
(Lemma 4.35) and Lemma 4.79, where the flatness of
$A^{\prime\prime}\rightarrow R$ implies the flatness of
$A/^{\mathbb{L}}p\rightarrow\varphi_{A/^{\mathbb{L}}p\twoheadrightarrow
A^{\prime\prime}/^{\mathbb{L}}p}^{\ast}(R/^{\mathbb{L}}p)$, therefore the
Frobenius twist in question is bounded above. ∎

Finally, we want to compare the cohomology of the affine crystalline site and
the classical crystalline cohomology. We first describe a non-complete variant
of the classical affine crystalline site, which we will name after static
affine crystalline site.

######  Definition 4.88.

Let $(A,I,\gamma_{A})\in\operatorname{PDPair}$ be a PD-pair and let
$A/I\rightarrow R$ be a map of rings. Note that $((A\twoheadrightarrow
A/I,\gamma_{A}),A/I\rightarrow R)\in\operatorname{CrysCon}$ is a crystalline
context. The static affine crystalline site
$\operatorname{Cris}^{\operatorname{st}}(R/(A,I,\gamma_{A}))$ is the full
subcategory of $\operatorname{Cris}(R/(A\twoheadrightarrow A/I,\gamma_{A}))$
spanned by those $(B\twoheadrightarrow B/J,\gamma_{B})$ along with a map
$R\rightarrow B/J$, i.e., the animated PD-pair in question is given by a PD-
pair, equipped with the indiscrete topology.

We note that the structure presheaf $\mathcal{O}$ on
$\operatorname{Cris}(R/(A\twoheadrightarrow A/I,\gamma_{A}))$ restricts to a
presheaf $\operatorname{Cris}^{\operatorname{st}}(R/(A,I,\gamma_{A}))$, still
called the structure presheaf, which is canonically equipped with PD-
filtration, of which the cohomology is called the cohomology of the static
crystalline site (resp. Hodge-filtered cohomology of the static crystalline
site), denoted by
$R\Gamma(\operatorname{Cris}^{\operatorname{st}}(R/(A,I,\gamma_{A})),\mathcal{O})$
(resp.
$\operatorname{Fil}_{H}R\Gamma(\operatorname{Cris}^{\operatorname{st}}(R/(A,I,\gamma_{A})),\mathcal{O})$.
By definition, there is a comparison map
$\operatorname{Fil}_{H}R\Gamma(\operatorname{Cris}(R/(A\twoheadrightarrow
A/I,\gamma_{A})),\mathcal{O})\rightarrow\operatorname{Fil}_{H}R\Gamma(\operatorname{Cris}^{\operatorname{st}}(R/(A,I,\gamma_{A})),\mathcal{O})$
of filtered $\mathbb{E}_{\infty}$-$A$-algebras.

######  Warning 4.89.

Here the PD-filtration is that for animated PD-envelope, although we are
considering PD-pairs. However, when $I=0$, thanks to Proposition 3.77, we can
consider the classical PD-envelope instead.

Now the cohomology of the affine crystalline site coincides with the classical
version:

######  Proposition 4.90.

Let $(A,I,\gamma_{A})\in\operatorname{PDPair}$ be a PD-pair and
$A/I\rightarrow R$ a quasisyntomic map of rings ($R$ is static by flatness).
Then the comparison map

$\operatorname{Fil}_{H}R\Gamma(\operatorname{Cris}(R/(A\twoheadrightarrow
A/I,\gamma_{A})),\mathcal{O})\rightarrow\operatorname{Fil}_{H}R\Gamma(\operatorname{Cris}^{\operatorname{st}}(R/(A,I,\gamma_{A})),\mathcal{O})$

of filtered $\mathbb{E}_{\infty}$-$A$-algebras constructed above is an
equivalence.

###### Proof.

We adapt the Čech-Alexander computation in Proposition 4.66. We pick a
polynomial $A$-algebra $P$ (of possibly infinitely many variables) along with
a surjection $P\twoheadrightarrow R$. Let $P^{\bullet}\rightarrow R$ denote
the Čech conerve of the object $P\rightarrow R$ in
$(\operatorname{Alg}_{A})_{/R}$. Concretely, $P^{\nu}=P^{\otimes_{A}(\nu+1)}$.
Note that since $A\rightarrow P$ is flat, the classical tensor product
coincides with the derived tensor product, therefore the cosimplicial pair
$P^{\bullet}\rightarrow R$ coincides with the cosimplicial animated pair in
the proof of Proposition 4.66, and then
$\operatorname{Fil}_{H}R\Gamma(\operatorname{Cris}^{\operatorname{st}}(R/(A,I,\gamma_{A})),\mathcal{O})$
is computed by the classical PD-envelope of $P^{\bullet}\twoheadrightarrow R$
with respect to $(A,I,\gamma_{A})$, equipped with the PD-filtration.

Let $(D^{\bullet}\twoheadrightarrow R,\gamma_{D^{\bullet}})$ denote the
cosimplicial animated PD-envelope of $(P^{\bullet}\twoheadrightarrow R)$
relative to $(A,I,\gamma_{A})$. It suffices to show that
$(D^{\nu}\twoheadrightarrow R,\gamma_{D^{\bullet}})$ is given by a PD-pair for
all $\nu\in\boldsymbol{\Delta}$, or equivalently, the underlying animated ring
$D^{\nu}$ is static, by virtue of Propositions 3.32 and 3.37, which follows
from Lemma 4.91 below. ∎

######  Lemma 4.91.

Let $(A\twoheadrightarrow A^{\prime\prime},\gamma_{A})$ be an animated PD-
pair, $A^{\prime\prime}\rightarrow R$ a quasisyntomic map of animated rings,
$P$ a polynomial $A$-algebra (of possibly infinitely many variables) and
$P\twoheadrightarrow R$ a surjection of $A$-algebras. Let
$(D\twoheadrightarrow R,\gamma_{D})$ denote the animated PD-envelope of
$P\twoheadrightarrow R$ relative to $(A\twoheadrightarrow
A^{\prime\prime},\gamma_{A})$. Then $D$ is a flat $A$-module.

###### Proof.

This is a “quasi” variant of “flatness of PD-envelope” [BS19, Lem 2.42]. By
Lemma 3.71, it suffices to show that
$D\otimes_{\mathbb{Z}}^{\mathbb{L}}\mathbb{Q}$ is a flat
$A\otimes_{\mathbb{Z}}^{\mathbb{L}}\mathbb{Q}$-module, and for every prime
$p\in\mathbb{N}$, $D/^{\mathbb{L}}p$ is a flat $A/^{\mathbb{L}}p$-module.

By Lemma 4.52, the map
$P\otimes_{\mathbb{Z}}^{\mathbb{L}}\mathbb{Q}\rightarrow
D\otimes_{\mathbb{Z}}^{\mathbb{L}}\mathbb{Q}$ is an equivalence. Since
$A\rightarrow P$ is flat, so is the map
$A\otimes_{\mathbb{Z}}^{\mathbb{L}}\mathbb{Q}\rightarrow
D\otimes_{\mathbb{Z}}^{\mathbb{L}}\mathbb{Q}$.

For every prime $p\in\mathbb{N}$, by base change property (a relative version
of Lemma 3.46, with a similar proof), $D_{0}:=D/^{\mathbb{L}}p$ is the
animated PD-envelope of $P/^{\mathbb{L}}p\twoheadrightarrow R/^{\mathbb{L}}p$
relative to the animated PD-pair $(A\twoheadrightarrow
A^{\prime\prime},\gamma_{A})$. To simplify notations, we let
$P_{0}:=P/^{\mathbb{L}}p,R_{0}:=R/^{\mathbb{L}}p,A_{0}:=A/^{\mathbb{L}}p,A_{0}^{\prime\prime}:=A^{\prime\prime}/^{\mathbb{L}}p$.
Since $A\rightarrow R$ is quasisyntomic, so is $A_{0}\rightarrow R_{0}$.
Consider the transitivity sequence

$\mathbb{L}_{P_{0}/A_{0}}\otimes_{P_{0}}^{\mathbb{L}}R_{0}\longrightarrow\mathbb{L}_{R_{0}/A_{0}}\longrightarrow\mathbb{L}_{R_{0}/P_{0}}$

For every static $R_{0}$-module $M$, we get a fiber sequence

$\mathbb{L}_{P_{0}/A_{0}}\otimes_{P_{0}}^{\mathbb{L}}M\longrightarrow\mathbb{L}_{R_{0}/A_{0}}\otimes_{R_{0}}^{\mathbb{L}}M\longrightarrow\mathbb{L}_{R_{0}/P_{0}}\otimes_{R_{0}}^{\mathbb{L}}M$

Since $P_{0}$ is a polynomial $A_{0}$-algebra, $\mathbb{L}_{P_{0}/A_{0}}$ is a
flat $P_{0}$-module. The map $A_{0}\rightarrow R_{0}$ is quasisyntomic,
therefore
$\pi_{\ast}(\mathbb{L}_{R_{0}/A_{0}}\otimes_{R_{0}}^{\mathbb{L}}M)\cong 0$ for
$\mathord{\ast}\neq 0,1$. It follows that
$\pi_{\ast}(\mathbb{L}_{R_{0}/P_{0}}\otimes_{R_{0}}^{\mathbb{L}}M)\cong 0$ for
$\mathord{\ast}\neq 0,1$. Furthermore, since $P_{0}\rightarrow R_{0}$ is
surjective,
$\pi_{0}(\mathbb{L}_{R_{0}/P_{0}}\otimes_{R_{0}}^{\mathbb{L}}M)\cong 0$. It
follows that $P_{0}\twoheadrightarrow R_{0}$ is a quasiregular animated pair.
By Corollary 4.60, $D_{0}$ is a flat $\varphi_{P_{0}\twoheadrightarrow
P_{0}\otimes_{A_{0}}^{\mathbb{L}}A_{0}^{\prime\prime}}^{\ast}(R_{0})$-module
where $\varphi_{P_{0}\twoheadrightarrow
P_{0}\otimes_{A_{0}}^{\mathbb{L}}A_{0}^{\prime\prime}}:P_{0}\otimes_{A_{0}}^{\mathbb{L}}A_{0}^{\prime\prime}\rightarrow
P_{0}$ is the Frobenius map (Lemma 4.36). It remains to see that the composite
map $A_{0}\rightarrow P_{0}\rightarrow\varphi_{P_{0}\twoheadrightarrow
P_{0}\otimes_{A_{0}}^{\mathbb{L}}A_{0}^{\prime\prime}}^{\ast}(R_{0})=R_{0}\otimes_{P_{0}\otimes_{A_{0}}^{\mathbb{L}}A_{0}^{\prime\prime}}^{\mathbb{L}}P_{0}$
is flat, where the second map is the “map into the second factor”.

We note that the Frobenius $\varphi_{P_{0}\twoheadrightarrow
P_{0}\otimes_{A_{0}}^{\mathbb{L}}A_{0}^{\prime\prime}}$ factors as
$P_{0}\otimes_{A_{0}}^{\mathbb{L}}A_{0}^{\prime\prime}\rightarrow\varphi_{A_{0}}^{\ast}(P_{0})\rightarrow
P_{0}$ where the second map is the Frobenius of $P_{0}$ relative to $A_{0}$.
Let $R_{1}$ denote
$R_{0}\otimes_{A_{0}^{\prime\prime},\varphi_{A_{0}\twoheadrightarrow
A_{0}^{\prime\prime}}}^{\mathbb{L}}A_{0}$. Since
$A_{0}^{\prime\prime}\rightarrow R_{0}$ is flat, so is $A_{0}\rightarrow
R_{1}$, and we have

$\varphi_{P_{0}\twoheadrightarrow
P_{0}\otimes_{A_{0}}^{\mathbb{L}}A_{0}^{\prime\prime}}^{\ast}(R_{0})\simeq
R_{1}\otimes_{\varphi_{A_{0}}^{\ast}(P_{0})}^{\mathbb{L}}P_{0}$

as a pushout of $A_{0}$-algebras. The relative Frobenius
$\varphi_{A_{0}}^{\ast}(P_{0})\rightarrow P_{0}$ is flat, therefore the map
$R_{1}\rightarrow
R_{1}\otimes_{\varphi_{A_{0}}^{\ast}(P_{0})}^{\mathbb{L}}P_{0}$. The result
then follows since flatness is stable under composition. ∎

######  Remark 4.92.

If we examine the proof of Lemma 4.91 closely, we see that, instead of being a
polynomial, what we really need to impose on the map $A\rightarrow P$ is that
the map is quasismooth (i.e. it is flat and $\mathbb{L}_{P/A}$ is a flat
$P$-module), and for every prime $p\in\mathbb{N}$, the Frobenius of
$P/^{\mathbb{L}}p$ relative to $A/^{\mathbb{L}}p$ is flat.

##  5\. Animated prismatic structures

We fix a prime $p\in\mathbb{N}$. In this section, we will develop the theory
of animated $\delta$-rings, that of animated $\delta$-pairs, a non-complete
theory of prisms and prove a variant of the Hodge-Tate comparison, from which
we deduce a result about “flat covers of the final object”. Almost every ring
that we will discuss is a $\mathbb{Z}_{(p)}$-algebra, we will simply denote
$\operatorname{AniPair}_{\mathbb{Z}_{(p)}}$ by $\operatorname{AniPair}$ and
$\operatorname{AniPDPair}_{\mathbb{Z}_{(p)}}$ by $\operatorname{AniPDPair}$.

###  5.1. Animated $\delta$-rings and $\delta$-pairs

In this section, we will define animated $\delta$-rings and animated
$\delta$-pairs and discuss the interaction between the $\delta$-structure and
the PD-structure. Recall that

######  Definition 5.1 ([BS19, Def 2.1]).

A $\delta$-ring is a pair $(R,\delta)$ where $R$ is a
$\mathbb{Z}_{(p)}$-algebra and $\delta:R\rightarrow R$ is an endomorphism of
the underlying set $R$ such that

  1. (1)

$\delta(x+y)=\delta(x)+\delta(y)-P(x,y)$ for all $x,y\in R$ where
$P(X,Y)\in\mathbb{Z}[X,Y]$ is the polynomial

$\frac{(X+Y)^{p}-X^{p}-Y^{p}}{p}:=\sum_{j=1}^{p-1}\frac{1}{p}\binom{p}{j}X^{p-j}Y^{j}$

  2. (2)

$\delta(xy)=x^{p}\delta(y)+y^{p}\delta(x)+p\delta(x)\delta(y)$.

  3. (3)

$\delta(1)=0$.

A map $f:(R,\delta)\rightarrow(S,\delta)$ of $\delta$-rings is a map
$f:R\rightarrow S$ of rings such that $f\circ\delta=\delta\circ g$ as maps of
sets. These form the $1$-category of $\delta$-rings, denoted by
$\operatorname{Ring}_{\delta}$.

######  Remark 5.2 ([BS19, Rem 2.2]).

Given a $\delta$-ring $(R,\delta)$, we write $\varphi:R\rightarrow R$ for the
map $x\mapsto x^{p}+p\delta(x)$. Then $\varphi$ is a ring endomorphism of $R$
which lifts the Frobenius map $R/p\rightarrow R/p$, i.e. $\varphi(x)\equiv
x^{p}\hskip 1.99997pt\pmod{p}$ for every $x\in R$.

The $1$-category $\operatorname{Ring}_{\delta}$ admits an initial object
$\mathbb{Z}_{(p)}$ [BS19, Ex 2.6], and more generally, all small colimits and
small limits, and the forgetful functor
$\operatorname{Ring}_{\delta}\rightarrow\operatorname{Alg}_{\mathbb{Z}_{(p)}}$
preserves them [BS19, Rem 2.7]. The forgetful functor
$\operatorname{Ring}_{\delta}\rightarrow\operatorname{Set}$ admits a left
adjoint $\operatorname{Set}\rightarrow\operatorname{Ring}_{\delta}$, which
sends a set $S$ to the free $\delta$-ring generated by $S$, denoted by
$\mathbb{Z}_{(p)}\\{S\\}$. Indeed, when $S=\\{x\\}$ is a singleton, it is
given by the free $\delta$-ring $\mathbb{Z}_{(p)}\\{x\\}$ of which the
underlying $\mathbb{Z}_{(p)}$-algebra is isomorphic to the polynomial
$\mathbb{Z}_{(p)}$-algebra
$\mathbb{Z}_{(p)}[x,\delta(x),\delta^{2}(x),\ldots]$ [BS19, Lem 2.11], and the
general case follows by taking the coproduct of $S$-copies of
$\mathbb{Z}_{(p)}\\{x\\}$. It then follows from Corollary 2.3 that

######  Lemma 5.3.

The $1$-category $\operatorname{Ring}_{\delta}$ is $1$-projectively generated,
therefore presentable.

By the adjoint functor theorem, the forgetful functor
$\operatorname{Ring}_{\delta}\rightarrow\operatorname{Alg}_{\mathbb{Z}_{(p)}}$
admits a left adjoint. A further application of Corollary 2.3 leads to

######  Lemma 5.4.

There is a pair
$\operatorname{Ani}(\operatorname{Alg}_{\mathbb{Z}_{(p)}})\rightleftarrows\operatorname{Ani}(\operatorname{Ring}_{\delta})$
of adjoint functors, being the animation of the pair
$\operatorname{Alg}_{\mathbb{Z}_{(p)}}\rightleftarrows\operatorname{Ring}_{\delta}$
of adjoint functors. We will call the functor
$\operatorname{Ani}(\operatorname{Ring}_{\delta})\rightarrow\operatorname{Ani}(\operatorname{Alg}_{\mathbb{Z}_{(p)}})$
the free animated $\delta$-ring functor292929The non-animated version was
called the “$\delta$-envelope” in [GLQ20, Def 1.1].. The functor
$\operatorname{Ani}(\operatorname{Ring}_{\delta})\rightarrow\operatorname{Ani}(\operatorname{Alg}_{\mathbb{Z}_{(p)}})$,
called the forgetful functor, is conservative and preserves small colimits
(and as a right adjoint, it preserves small limits as well).

######  Definition 5.5.

The $\infty$-category of animated $\delta$-rings is defined to be the
animation $\operatorname{Ani}(\operatorname{Ring}_{\delta})$, of which an
object is called an animated $\delta$-ring, formally denoted by $(R,\delta)$
where $R$ is the image of $(R,\delta)$ under the forgetful functor
$\operatorname{Ani}(\operatorname{Ring}_{\delta})\rightarrow\operatorname{Ani}(\operatorname{Alg}_{\mathbb{Z}_{p}})$,
or simply by $R$ when the $\delta$-structure is unambiguously obvious.

Concretely, a set of compact projective generators for
$\operatorname{Ani}(\operatorname{Ring}_{\delta})$ is given by free
$\delta$-rings generated by a finite set, which spans a full subcategory
$\operatorname{Ring}_{\delta}^{0}\subseteq\operatorname{Ring}_{\delta}$.
Recall that
$\operatorname{Ring}_{\delta}\hookrightarrow\operatorname{Ani}(\operatorname{Ring}_{\delta})$
is a full subcategory (Remark A.35). Now we characterize this full subcategory
in terms of the underlying animated ring:

######  Lemma 5.6.

Let $(R,\delta)\in\operatorname{Ani}(\operatorname{Ring}_{\delta})$ be an
animated $\delta$-ring. Then the followings are equivalent:

  1. (1)

The animated $\delta$-ring
$(R,\delta)\in\operatorname{Ani}(\operatorname{Ring}_{\delta})$ is
$n$-truncated.

  2. (2)

The underlying animated ring
$R\in\operatorname{Ani}(\operatorname{Alg}_{\mathbb{Z}_{(p)}})$ is
$n$-truncated.

  3. (3)

For every $m\in\mathbb{N}_{\char 62\relax n}$, the homotopy group $\pi_{m}(R)$
vanishes.

###### Proof.

The equivalence of parts 2 and 3 is [Lur18, Prop 25.1.3.3]. On the other hand,
part 1 is equivalent to say that, for every free $\delta$-ring $F$ generated
by a finite set, the mapping anima
$\operatorname{Map}_{\operatorname{Ani}(\operatorname{Ring}_{\delta})}(F,R)$
is $n$-truncated by [Lur09, Rem 5.5.8.26]. Since any such $F$ is a finite
coproduct of $\mathbb{Z}_{(p)}\\{x\\}$, it is equivalent to
$\operatorname{Map}_{\operatorname{Ani}(\operatorname{Ring}_{\delta})}(\mathbb{Z}_{(p)}\\{x\\},R)$
being $n$-truncated, which is equivalent to part 3 since

$\operatorname{Map}_{\operatorname{Ani}(\operatorname{Ring}_{\delta})}(\mathbb{Z}_{(p)}\\{x\\},R)\simeq\operatorname{Map}_{\mathcal{S}}(\\{x\\},R)\simeq
R$

∎

Now we define the Frobenius map on animated $\delta$-rings. We note that the
identity functor
$\operatorname{id}:\operatorname{Ani}(\operatorname{Ring}_{\delta})\rightarrow\operatorname{Ani}(\operatorname{Ring}_{\delta})$
is the animation of the identity functor
$\operatorname{id}:\operatorname{Ring}_{\delta}\rightarrow\operatorname{Ring}_{\delta}$.

######  Definition 5.7.

The Frobenius endomorphism is the endomorphism of the identity functor
$\operatorname{id}:\operatorname{Ani}(\operatorname{Ring}_{\delta})\rightarrow\operatorname{Ani}(\operatorname{Ring}_{\delta})$
defined to be the animation of the Frobenius endomorphism (described in Remark
5.2) of the identity functor
$\operatorname{id}:\operatorname{Ring}_{\delta}\rightarrow\operatorname{Ring}_{\delta}$.

Recall that a $\delta$-pair is the datum $(A,I)$ of a $\delta$-ring $A$ along
with an ideal $I\subseteq A$ [BS19, Def 3.2]. Similar to animated pairs, we
have an “animated version” of $\delta$-pairs:

######  Definition 5.8.

The $\infty$-category of animated $\delta$-pairs
$\operatorname{AniPair}_{\delta}$ is defined to be the fiber product
$\operatorname{Ani}(\operatorname{Ring}_{\delta})\times_{\operatorname{Ani}(\operatorname{Alg}_{\mathbb{Z}_{(p)}})}\operatorname{AniPair}$
where the functor
$\operatorname{Ani}(\operatorname{Ring}_{\delta})\rightarrow\operatorname{Ani}(\operatorname{Alg}_{\mathbb{Z}_{(p)}})$
is the forgetful functor and the functor
$\operatorname{AniPair}\rightarrow\operatorname{Ani}(\operatorname{Alg}_{\mathbb{Z}_{(p)}})$
is the evaluation $(A\twoheadrightarrow A^{\prime\prime})\mapsto A$. An
animated $\delta$-pair is an object in $\operatorname{AniPair}_{\delta}$ which
we will denote by $((A,\delta),A\twoheadrightarrow A^{\prime\prime})$, or
simply by $A\twoheadrightarrow A^{\prime\prime}$ when there is no ambiguity.

It follows from Lemma 5.4 and [Lur09, Lem 5.4.5.5] which characterizes
colimits in the fiber products, that

######  Lemma 5.9.

The $\infty$-category $\operatorname{AniPair}_{\delta}$ is cocomplete, and the
forgetful functor
$\operatorname{AniPair}_{\delta}\rightarrow\operatorname{AniPair}$ is
conservative and preserves small colimits.

Explicitly, an animated $\delta$-pair is given by an animated $\delta$-ring
$(A,\delta)$ along with a surjection $A\twoheadrightarrow A^{\prime\prime}$ of
animated $\mathbb{Z}_{(p)}$-algebras. Since
$\operatorname{Pair}\subseteq\operatorname{AniPair}$ is a full subcategory
(Proposition 3.17) and so is
$\operatorname{Ring}_{\delta}\subseteq\operatorname{Ani}(\operatorname{Ring}_{\delta})$
(Remark A.35), the $1$-category of $\delta$-pairs is a full subcategory of the
$\infty$-category of animated $\delta$-pairs. Similar to the $\infty$-category
of animated pairs, we have

######  Lemma 5.10.

The forgetful functor
$\operatorname{AniPair}_{\delta}\rightarrow\operatorname{AniPair}$ admits a
left adjoint, and the $\infty$-category $\operatorname{AniPair}_{\delta}$ is
projectively generated.

###### Proof.

The left adjoint
$\operatorname{AniPair}\rightarrow\operatorname{AniPair}_{\delta}$ concretely
given by $(A\twoheadrightarrow
A^{\prime\prime})\mapsto((A^{\delta},\delta),(A^{\delta}\twoheadrightarrow
A^{\prime\prime}\otimes_{A}^{\mathbb{L}}A^{\delta}))$ where $A^{\delta}$ is
the image of $A\in\operatorname{Ani}(\operatorname{Alg}_{\mathbb{Z}_{(p)}})$
under the free animated $\delta$-ring functor
$\operatorname{Ani}(\operatorname{Alg}_{\mathbb{Z}_{(p)}})\rightarrow\operatorname{Ani}(\operatorname{Ring}_{\delta})$.
Now the result follows from Corollaries 2.3, 5.3 and 5.4. ∎

Concretely, a set of compact projective generators for
$\operatorname{AniPair}_{\delta}$ is given by the set
$\\{(\mathbb{Z}_{(p)}\\{X,Y\\},(Y))|X,Y\in\operatorname{Fin}\\}$ of
$\delta$-pairs, which spans a full subcategory
$\operatorname{AniPair}_{\delta}^{0}\subseteq\operatorname{AniPair}_{\delta}$.
Now we turn to the PD-structure. Recall that

######  Lemma 5.11 ([BS19, Lem 2.11]).

The Frobenius endomorphism
$\varphi_{\mathbb{Z}_{(p)}\\{x\\}}:\mathbb{Z}_{(p)}\\{x\\}\rightarrow\mathbb{Z}_{(p)}\\{x\\}$
on the free $\delta$-ring $\mathbb{Z}_{(p)}\\{x\\}$, which is in fact induced
by $x\mapsto\varphi(x)=x^{p}+p\delta(x)$, is faithfully flat. The same holds
for free $\delta$-rings generated by arbitrary sets (not-necessarily finite).

We remark that, thanks to Lemma 3.71, it is not necessary to pass to the
polynomial ring of finitely many variables to invoke the fiberwise criterion
of flatness.

We now relate $\delta$-structure to divided powers. Note that, for any
$p$-torsion free $\mathbb{Z}_{(p)}$-algebra $A$, any element $y\in A$ and any
$n\in\mathbb{N}$, we have

$\frac{y^{n}}{n!}\in\frac{y^{n}}{p^{v_{p}(n!)}}\operatorname{GL}_{1}(\mathbb{Z}_{(p)})$

In particular, $y^{p}/p!$ (resp. $y^{p^{2}}/(p^{2})!$) differs
multiplicatively from $y^{p}/p$ (resp. $y^{p^{2}}/p^{p+1}$) by a unit. When
$A$ is a $p$-torsion free $\delta$-ring, we have $\varphi(y)=y^{p}+p\delta(y)$
and $y^{p}/p!\in A[p^{-1}]$ belongs to $A$ if and only if $\varphi(y)$ is
divisible by $p$.

Now we define the animated $\delta$-ring
$\mathbb{Z}_{(p)}\\{x,\varphi(x)/p\\}$ to be the pushout of the diagram

in the $\infty$-category $\operatorname{Ani}(\operatorname{Ring}_{\delta})$.
Since the Frobenius map
$\varphi:\mathbb{Z}_{(p)}\\{y\\}\rightarrow\mathbb{Z}_{(p)}\\{x\\}$ is
faithfully flat, so is the map
$\mathbb{Z}_{(p)}\\{z\\}\rightarrow\mathbb{Z}_{(p)}\\{x,\varphi(x)/p\\}$. It
follows that $\mathbb{Z}_{(p)}\\{x,\varphi(x)/p\\}$ is static and $p$-torsion-
free by Remark 3.63, therefore it is a $\delta$-ring by Lemma 5.6 (this is
essentially [BS19, Lem 2.36]). We need another characterization of the
underlying ring of $\mathbb{Z}_{(p)}\\{x,\varphi(x)/p\\}$:

######  Lemma 5.12 ([BS19, Lem 2.36]).

There is a natural isomorphism

$D_{\mathbb{Z}_{(p)}\\{x\\}}(x)\longrightarrow\mathbb{Z}_{(p)}\\{x,\varphi(x)/p\\}$

of $p$-torsion-free $\mathbb{Z}_{(p)}$-algebras.

This map transfers the surjective map
$D_{\mathbb{Z}_{(p)}\\{x\\}}(x)\twoheadrightarrow\mathbb{Z}_{(p)}\\{x\\}/(x)$
to a surjective map
$\mathbb{Z}_{(p)}\\{x,\varphi(x)/p\\}\twoheadrightarrow\mathbb{Z}_{(p)}\\{x\\}/(x)$,
the existence of which does not seem to be a priori clear (which is implicitly
involved in [BS19, Lem 2.35]).

Note that since $x\in D_{\mathbb{Z}_{(p)}\\{x\\}}(x)$ is a non-zero-divisor,
the map from the animated PD-envelope of $(\mathbb{Z}_{(p)}\\{x\\},(x))$ to
the classical PD-envelope is an equivalence, by base change of
$(\mathbb{Z}_{(p)}[x],(x))$ along the flat map
$\mathbb{Z}_{(p)}[x]\rightarrow\mathbb{Z}_{(p)}\\{x\\}\simeq\mathbb{Z}_{(p)}[x,\delta(x),\delta^{2}(x),\ldots]$,
or alternatively by Proposition 3.72. We could replace $x$ by a finite number
of variables, which leads to

######  Corollary 5.13.

There exists a canonical $\delta$-pair structure on the animated PD-envelope
of every $\delta$-pair
$(\mathbb{Z}_{(p)}\\{X,Y\\},(Y))\in\operatorname{AniPair}_{\delta}^{0}$. More
formally, there exists a canonical functor
$\operatorname{AniPair}_{\delta}^{0}\rightarrow\operatorname{AniPair}_{\delta}$
which fits into a commutative diagram

of $\infty$-categories.

###### Proof.

The functoriality of
$\operatorname{AniPair}_{\delta}^{0}\rightarrow\operatorname{AniPair}_{\delta}$
needs explanation: a map
$(\mathbb{Z}_{(p)}\\{X,Y\\},(Y))\rightarrow(\mathbb{Z}_{(p)}\\{X^{\prime},Y^{\prime}\\},(Y^{\prime}))$
of $\delta$-pairs induces a map
$(\mathbb{Q}\\{X,Y\\},(Y))\rightarrow(\mathbb{Q}\\{X^{\prime},Y^{\prime}\\},(Y^{\prime}))$
of pairs after inverting $p$ which is “Frobenius”-equivariant, where
$\mathbb{Q}\\{X,Y\\}:=\mathbb{Z}_{(p)}\\{X,Y\\}[p^{-1}]$. A careful
$v_{p}$-analysis implies that this map restricts to a map
$\mathbb{Z}_{(p)}\\{X,Y,\varphi(Y)/p\\}\rightarrow\mathbb{Z}_{(p)}\\{X^{\prime},Y^{\prime},\varphi(Y^{\prime})/p\\}$
of $\mathbb{Z}_{(p)}$-subalgebras, which gives rise to the functoriality. ∎

It follows from Propositions A.10, 3.34, 5.9 and 5.13 that

######  Corollary 5.14.

There exists a canonical animated $\delta$-pair structure on the animated PD-
envelope of every animated $\delta$-pair. More formally, there exists a
canonical functor
$\operatorname{AniPair}_{\delta}\rightarrow\operatorname{AniPair}_{\delta}$
which fits into a commutative diagram

of $\infty$-categories. Moreover, the functor
$\operatorname{AniPair}_{\delta}\rightarrow\operatorname{AniPair}_{\delta}$
preserves small colimits.

We give an analysis of the conjugate filtration on the PD-envelope of
$(\mathbb{F}_{p}\\{x\\},(x))$ where
$\mathbb{F}_{p}\\{x\\}:=\mathbb{Z}_{(p)}\\{x\\}/^{\mathbb{L}}p$, which is the
base change of the PD-envelope of $(\mathbb{Z}_{(p)}\\{x\\},(x))$ along
$\mathbb{Z}_{(p)}\rightarrow\mathbb{F}_{p}$. Recall that

  1. (1)

The (animate) PD-envelope $D_{\mathbb{F}_{p}[x]}(x)$ is a free
$\mathbb{F}_{p}[x]/(x^{p})$-module generated by the set
$\\{\gamma_{kp}(x)|k\in\mathbb{N}\\}$ of divided powers of $x$.

  2. (2)

For $i\in\mathbb{N}_{\geq 0}$, the $(-i)$-th piece of the conjugate filtration
of $D_{\mathbb{F}_{p}[x]}(x)$ is generated by $\\{\gamma_{kp}(x)|k\leq i\\}$
as an $\mathbb{F}_{p}[x]/(x^{p})$-submodule.

By the base change property (Lemma 3.46), we have

  1. (1)

The (animate) PD-envelope $D_{\mathbb{F}_{p}\\{x\\}}(x)$ is a free
$\mathbb{F}_{p}\\{x\\}/(x^{p})$-module generated by the set
$\\{\gamma_{kp}(x)|k\in\mathbb{N}\\}$.

  2. (2)

For $i\in\mathbb{N}_{\geq 0}$, the $(-i)$-th piece of the conjugate filtration
of $D_{\mathbb{F}_{p}\\{x\\}}(x)$ is generated by $\\{\gamma_{kp}(x)|k\leq
i\\}$ as an $\mathbb{F}_{p}\\{x\\}/(x^{p})$-submodule.

We follow the argument of [BS19, Lem 2.35]: for every
$y\in\mathbb{Z}_{(p)}\\{x\\}$ with $y^{p}/p\in\mathbb{Z}_{(p)}\\{x\\}$, we
have

(8) $\displaystyle\delta\left(\frac{y^{p}}{p}\right)$ $\displaystyle=$
$\displaystyle\frac{1}{p}\left(\frac{\varphi(y)^{p}}{p}-\left(\frac{y^{p}}{p}\right)^{p}\right)$
$\displaystyle=$
$\displaystyle\frac{(y^{p}+p\delta(y))^{p}}{p^{2}}-\frac{y^{p^{2}}}{p^{p+1}}$
$\displaystyle=$
$\displaystyle\frac{1}{p^{2}}\left(y^{p^{2}}+p^{2}y^{p(p-1)}\delta(y)+\sum_{k=0}^{p-2}\binom{p}{k}y^{kp}(p\delta(y))^{p-k}\right)-\frac{y^{p^{2}}}{p^{p+1}}$
$\displaystyle=$
$\displaystyle\frac{p^{p-1}-1}{p^{p+1}}y^{p^{2}}+y^{p(p-1)}\delta(y)+\sum_{k=0}^{p-2}p^{p-2-k}\binom{p}{k}y^{kp}\delta(y)^{p-k}$

Letting $z=x^{p}/p$, it follows from
$p^{p-1}-1\in\operatorname{GL}_{1}(\mathbb{Z}_{(p)})$ that

  1. (1)

The set
$\\{z^{a_{0}}\delta(z)^{a_{1}}(\delta^{2}(z))^{a_{2}}\cdots(\delta^{r}(z))^{a_{r}}|r\in\mathbb{N},0\leq
a_{0},a_{1},\ldots,a_{r}<p\\}$ forms a basis of the free
$\mathbb{F}_{p}\\{x\\}/(x^{p})$-module
$\mathbb{Z}_{(p)}\\{x,\varphi(x)/p\\}/^{\mathbb{L}}p\simeq
D_{\mathbb{F}_{p}\\{x\\}}(x)$.

  2. (2)

For every $i\in\mathbb{N}$, the $(-i)$-th piece of the conjugate filtration of
$D_{\mathbb{F}_{p}\\{x\\}}(x)$ is generated by
$\\{z^{a_{0}}\delta(z)^{a_{1}}(\delta^{2}(z))^{a_{2}}\cdots(\delta^{r}(z))^{a_{r}}|0\leq
a_{0},a_{1},\ldots,a_{r}<p,a_{0}+a_{1}p+a_{2}p^{2}+\cdots+a_{r}p^{r}\leq
i\\}$.

######  Remark 5.15.

In a bit more imprecise terms, $\delta^{k}(z)$ differs from
$\gamma_{p^{k}}(x)$ up to a unit, modulo “lower terms”.

This generalizes to multivariable case with the same argument:

######  Lemma 5.16.

Let
$(A,I):=(\mathbb{Z}_{(p)}\\{X,Y\\},(Y))\in\operatorname{AniPair}_{\delta}^{0}$
be a $\delta$-pair and let $(B,J,\gamma)$ be the (animated) PD-envelope of
$(\mathbb{F}_{p}\\{X,Y\\},(Y))$. Let $Y=\\{y_{1},y_{2},\ldots\\}$ and
$z_{j}:=\varphi(y_{j})/p$. Then

  1. (1)

The $\varphi_{A}^{\ast}(A/I)$-module $B$ is freely generated by the subset
$\left\\{\prod_{j,k}(\delta^{k}(z_{j}))^{a_{j,k}}|0\leq
a_{j,k}<p\right\\}\subseteq B$.

  2. (2)

For every $i\in\mathbb{N}$, the $(-i)$-th piece of the conjugate filtration of
$B$ is generated by $\left\\{\prod_{j,k}(\delta^{k}(z_{j}))^{a_{j,k}}|0\leq
a_{j,k}<p,\sum_{j,k}a_{j,k}p^{k}\leq i\right\\}$ as a
$\varphi_{A}^{\ast}(A/I)$-submodule.

###  5.2. Oriented prisms

In this subsection, we will study animated $\delta$-rings viewed as “non-
complete oriented prisms”. Recall that a orientable prism is a $\delta$-pair
$(A,I)$ such that the ideal $I\subseteq A$ is principal, the $\delta$-ring $A$
is $I$-torsion free, derived $(p,I)$-complete, and $p\in I+\varphi(I)A$ [BS19,
Def 3.2]. For technical reasons, we will study the “non-complete” analogues
where the completeness and the torsion-freeness are dropped.

We fix a $\delta$-ring $A$ along with a chosen non-zero-divisor $d\in A$. In
practice, we are only interested in the special case that
$A=\mathbb{Z}_{(p)}\\{d\\}$ and some variants like
$A=\mathbb{Z}_{(p)}\\{d,\delta(d)^{-1}\\}$. We denote by
$\operatorname{Ring}_{\delta,A}$ the $1$-category
$(\operatorname{Ring}_{\delta})_{A\mathord{/}}$ of $\delta$-$A$-algebras. It
follows from Lemma 2.10 that

######  Lemma 5.17.

The $1$-category $\operatorname{Ring}_{\delta,A}$ is $1$-projectively
generated, therefore presentable. A set of compact $1$-projective generators
is given by
$\\{A\\{X\\}:=A\otimes_{\mathbb{Z}_{(p)}}^{\mathbb{L}}\mathbb{Z}_{(p)}\\{X\\}|X\in\operatorname{Fin}\\}$,
which spans a full subcategory of $\operatorname{Ring}_{\delta,A}$ denoted by
$\operatorname{Ring}_{\delta,A}^{0}$.

######  Definition 5.18.

Let $B$ be an animated $\delta$-ring. The $\infty$-category of animated
$\delta$-$B$-algebras is defined to be the undercategory
$\operatorname{Ani}(\operatorname{Ring}_{\delta})_{B\mathord{/}}$. When $B$ is
static, it is equivalent to the animation
$\operatorname{Ani}(\operatorname{Ring}_{\delta,B})$ by Corollary 2.14.

By Lemma 4.47, we get an adjunction
$\operatorname{Alg}_{A}\rightleftarrows\operatorname{Ring}_{\delta,A}$, where
the forgetful functor
$\operatorname{Ring}_{\delta,A}\rightarrow\operatorname{Alg}_{A}$ preserves
all small colimits (and as a right adjoint, it preserves small limits as
well). It follows from Corollary 2.3 that

######  Lemma 5.19.

There is a pair
$\operatorname{Ani}(\operatorname{Alg}_{A})\rightleftarrows\operatorname{Ani}(\operatorname{Ring}_{\delta,A})$
of adjoint functors, being the animation of the pair
$\operatorname{Alg}_{A}\rightleftarrows\operatorname{Ring}_{\delta,A}$ of
adjoint functors. We will call the functor
$\operatorname{Ani}(\operatorname{Ring}_{\delta,A})\rightarrow\operatorname{Ani}(\operatorname{Alg}_{A})$
the free animated $\delta$-$A$-algebra functor. The functor
$\operatorname{Ani}(\operatorname{Ring}_{\delta,A})\rightarrow\operatorname{Ani}(\operatorname{Alg}_{A})$,
called the forgetful functor, is conservative and preserves small colimits
(and as a right adjoint, it preserves small limits as well).

######  Definition 5.20.

The $\infty$-category of animated $\delta$-$A$-pairs
$\operatorname{AniPair}_{\delta,A}$ is defined to be the undercategory
$(\operatorname{AniPair}_{\delta})_{(\operatorname{id}_{A}:A\rightarrow
A)\mathord{/}}$, which is equivalent to the fiber product
$\operatorname{Ani}(\operatorname{Ring}_{\delta,A})\times_{\operatorname{Ani}(\operatorname{Alg}_{A})}\operatorname{AniPair}_{A}$
by [Lur09, Lem 5.4.5.4].

The set $\\{(A\\{X,Y\\},(Y))|X,Y\in\operatorname{Fin}\\}$ form a set of
compact projective generators for $\operatorname{AniPair}_{\delta,A}$ by by
Lemma 2.10, which spans a full subcategory
$\operatorname{AniPair}_{\delta,A}^{0}\subseteq\operatorname{AniPair}_{\delta,A}$.
It follows from Lemmas 4.47 and 5.10 that

######  Lemma 5.21.

The forgetful functor
$\operatorname{AniPair}_{\delta,A}\rightarrow\operatorname{AniPair}_{A}$
admits a left adjoint.

There is a canonical functor
$\operatorname{Ani}(\operatorname{Ring}_{\delta,A})\rightarrow\operatorname{AniPair}_{\delta,A}$
given by $B\mapsto(B\twoheadrightarrow B/^{\mathbb{L}}d)$. We observe that

######  Lemma 5.22.

The functor
$\operatorname{Ani}(\operatorname{Ring}_{\delta,A})\rightarrow\operatorname{AniPair}_{\delta,A},B\mapsto(B\twoheadrightarrow
B/^{\mathbb{L}}d)$ admits a left adjoint
$\operatorname{AniPair}_{\delta,A}\rightarrow\operatorname{Ani}(\operatorname{Ring}_{\delta,A})$,
given by the left derived functor (Proposition A.10) of
$\operatorname{AniPair}_{\delta,A}^{0}\rightarrow\operatorname{Ani}(\operatorname{Ring}_{\delta,A}),(A\\{X,Y\\},(Y))\mapsto
A\\{X,Y/d\\}$ where $A\\{X,Y/d\\}$ is an abbreviation for the free
$\delta$-$A$-algebra
$A\\{x_{1},x_{2},\ldots,y_{1}/d,y_{2}/d,\ldots\\}$303030These generators
$y_{i}/d$ are in fact formal variables $z_{i}$. This notation indicates that
the counit map $(A\\{X,Y\\},(Y))\rightarrow(A\\{X,Y/d\\}\twoheadrightarrow
A\\{X,Y/d\\}/^{\mathbb{L}}d)$ is induced by $x_{i}\mapsto x_{i}$ and
$y_{i}\mapsto z_{i}d$..

###### Proof.

Let $G$ denote the functor
$\operatorname{Ani}(\operatorname{Ring}_{\delta,A})\rightarrow\operatorname{AniPair}_{\delta,A},B\mapsto(B\twoheadrightarrow
B/^{\mathbb{L}}d)$. Then we have a functor
$F:\operatorname{AniPair}_{\delta}\rightarrow\operatorname{Fun}(\operatorname{Ani}(\operatorname{Ring}_{\delta}),\mathcal{S})^{\operatorname{op}}$
which preserves small colimits and sends $(B\twoheadrightarrow
B^{\prime\prime})\in\operatorname{AniPair}_{\delta,A}^{0}$ to the functor
$\operatorname{Map}_{\operatorname{AniPair}_{\delta,A}}(B\twoheadrightarrow
B^{\prime\prime},G(\cdot))$. By Proposition A.10, it is the left derived
functor of its restriction to the full subcategory
$\operatorname{AniPair}_{\delta,A}^{0}\subseteq\operatorname{AniPair}_{\delta,A}$.

We now show that, for every
$(A\\{X,Y\\},(Y))\in\operatorname{AniPair}_{\delta,A}^{0}$, the functor
$F((A\\{X,Y\\},(Y)))$ is equivalent to the functor
$\operatorname{Map}_{\operatorname{Ani}(\operatorname{Ring}_{\delta,A})}(A\\{X,Y/d\\},\cdot)$.
In other words, the essential image of
$F|_{\operatorname{AniPair}_{\delta,A}^{0}}$ lies in the full subcategory
$\operatorname{Ring}_{\delta,A}^{0}\hookrightarrow\operatorname{Ani}(\operatorname{Ring}_{\delta,A})\hookrightarrow\operatorname{Fun}(\operatorname{Ani}(\operatorname{Ring}_{\delta,A}),\mathcal{S})^{\operatorname{op}}$.
By adjunctions
$\operatorname{Fun}((\Delta^{1})^{\operatorname{op}},D(\mathbb{Z})_{\geq
0})\rightleftarrows\operatorname{AniPair}_{A}\rightleftarrows\operatorname{AniPair}_{\delta,A}$
(Definitions 3.15 and 5.21), we have

$\displaystyle F(A\\{X,Y\\},(Y))(B)$ $\displaystyle\simeq$
$\displaystyle\operatorname{Map}_{\operatorname{AniPair}_{A}}(A[X,Y]\twoheadrightarrow
A[X],B\twoheadrightarrow B/^{\mathbb{L}}d)$ $\displaystyle\simeq$
$\displaystyle\operatorname{Map}_{\operatorname{Fun}((\Delta^{1})^{\operatorname{op}},D(\mathbb{Z})_{\geq
0})}\left(X\mathbb{Z}\oplus Y\mathbb{Z}\leftarrow
Y\mathbb{Z},B\xleftarrow{d}B\right)$ $\displaystyle\simeq$ $\displaystyle
B^{\operatorname{Card}(Y)}\times B^{\operatorname{Card}(X)}$
$\displaystyle\simeq$
$\displaystyle\operatorname{Map}_{\operatorname{Ani}(\operatorname{Ring}_{\delta,A})}(A\\{X,Y/d\\},B)$

which are functorial in
$B\in\operatorname{Ani}(\operatorname{Ring}_{\delta,A})$ (note that naively
speaking, the “values” of $Y/d$ correspond to the “preimages” of $Y$ under the
map $B\xleftarrow{d}B$, therefore the formal notation $Y/d$).

Since the Yoneda embedding
$\operatorname{Ani}(\operatorname{Ring}_{\delta,A})\hookrightarrow\operatorname{Fun}(\operatorname{Ani}(\operatorname{Ring}_{\delta,A}),\mathcal{S})^{\operatorname{op}}$
is stable under small colimits, it follows that the essential image of $F$
lies in $\operatorname{Ani}(\operatorname{Ring}_{\delta,A})$, which proves
that $G$ admits a left adjoint
$L:\operatorname{AniPair}_{\delta,A}\rightarrow\operatorname{Ani}(\operatorname{Ring}_{\delta,A})$,
and that $L(A\\{X,Y/d\\})\simeq A\\{X,Y/d\\}$.

We still need to show that $L|_{\operatorname{AniPair}_{\delta,A}^{0}}$
coincides with the functor defined in the obvious way. We have shown this
objectwise, and since $L(\operatorname{AniPair}_{\delta,A}^{0})$ lies in the
full subcategory
$\operatorname{Ring}_{\delta,A}^{0}\hookrightarrow\operatorname{Ani}(\operatorname{Ring}_{\delta,A})$
which is a $1$-category, we only need to show that the image of morphisms
coincide with the “obvious” choice, i.e. without higher categorical
complication. This can be checked putting different $d$-torsion-free
$\delta$-$A$-algebras $B\in\operatorname{Ring}_{\delta,A}$ into the functorial
isomorphism

$\operatorname{Hom}_{\operatorname{Pair}_{\delta,A}}((A\\{X,Y\\},(Y)),(B,(d)))\cong\operatorname{Hom}_{\operatorname{Ring}_{\delta,A}}(A\\{X,Y/d\\},B)$

given by the adjunction. ∎

Now we introduce a variant of Definition 5.20:

######  Definition 5.23.

The $\infty$-category of animated $\delta$-$(A,d)$-pairs
$\operatorname{AniPair}_{\delta,(A,d)}$ is defined to be the undercategory
$(\operatorname{AniPair}_{\delta,A})_{(A\twoheadrightarrow
A/^{\mathbb{L}}d)\mathord{/}}$, which is equivalent to the fiber product
$\operatorname{Ani}(\operatorname{Ring}_{\delta,A})\times_{\operatorname{Ani}(\operatorname{Alg}_{A})}\operatorname{AniPair}_{(A\twoheadrightarrow
A/^{\mathbb{L}}d)\mathord{/}}$ by [Lur09, Lem 5.4.5.4].

By Lemma 2.10, we have

######  Lemma 5.24.

The $\infty$-category $\operatorname{AniPair}_{\delta,(A,d)}$ is projectively
generated. A set of compact projective generators is given by
$\\{(A\\{X,Y\\},(d,Y))|X,Y\in\operatorname{Fin}\\}$, which spans a full
subcategory of $\operatorname{AniPair}_{\delta,(A,d)}$ denoted by
$\operatorname{AniPair}_{\delta,(A,d)}^{0}$.

Note that $A$ is initial in $\operatorname{AniPair}_{\delta,A}$. It follows
from Lemmas 4.47 and 5.22 that

######  Corollary 5.25.

The functor
$\operatorname{Ani}(\operatorname{Ring}_{\delta,A})\rightarrow\operatorname{AniPair}_{\delta,(A,d)},B\mapsto(B\twoheadrightarrow
B/^{\mathbb{L}}d)$ admits a left adjoint
$\operatorname{AniPair}_{\delta,(A,d)}\rightarrow\operatorname{Ani}(\operatorname{Ring}_{\delta,A})$,
which will be denoted by $\operatorname{PrismEnv}$313131This is understood as
a “non-complete” prismatic envelope when $p$ lies in the Jacobson radical
$\operatorname{Rad}(A)$ and $d\in A$ is weakly distinguished (Definition
5.28)., given by the left derived functor (Proposition A.10) of
$\operatorname{AniPair}_{\delta,(A,d)}^{0}\rightarrow\operatorname{Ani}(\operatorname{Ring}_{\delta,A}),(A\\{X,Y\\},(d,Y))\mapsto
A\\{X,Y/d\\}$.

Furthermore, for every
$B\in\operatorname{Ani}(\operatorname{Ring}_{\delta,A})$, by unrolling the
definitions, the counit map $\operatorname{PrismEnv}(B\twoheadrightarrow
B/^{\mathbb{L}}d)\rightarrow B$ is an equivalence, therefore

######  Lemma 5.26.

The functor
$\operatorname{Ani}(\operatorname{Ring}_{\delta,A})\rightarrow\operatorname{AniPair}_{\delta,(A,d)}$
is in fact fully faithful, the image of which is a reflective subcategory
(Definition 2.44).

The following concept is not strictly necessary, but it would help us to
understand when we need to “divide by $d$”:

######  Definition 5.27.

Let $A$ be a $\delta$-ring and $d\in A$ a non-zero-divisor. Let $M\in
D(A/^{\mathbb{L}}d)$ be a $A/^{\mathbb{L}}d$-module spectrum. For every
$n\in\mathbb{Z}$, the $n$-th Breuil-Kisin twist of $M$ with respect to
$(A,d)$, denoted by $M\\{n\\}$, is defined to be
$M\otimes_{A/^{\mathbb{L}}d}^{\mathbb{L}}(dA/d^{2}A)^{\otimes_{A/^{\mathbb{L}}d}^{\mathbb{L}}n}$.

Note that when $d\in A$ is a non-zero-divisor, the $A/^{\mathbb{L}}d$-module
$d^{n}A/d^{n+1}A$ is a free of rank $1$, therefore equivalent to
$A/^{\mathbb{L}}d$. The Breuil-Kisin twists are strictly necessary when we
want to generalize to non-orientable prisms. In our case, we understand
$M\\{1\\}$ “formally multiplied by $d$” and $M\\{-1\\}$ “formally divided by
$d$”, just as the formal notations $y_{i}/d$ in Lemma 5.22.

Finally, we introduce a variant of the concept of distinguished elements
[BS19, Def 2.19]:

######  Definition 5.28.

Let $A$ be a $\delta$-ring. We say that an element $d\in A$ is weakly
distinguished if the ideal $(d,\delta(d))$ is the unital ideal $A$, or
equivalently, $\delta(d)$ is invertible in $A/d$.

######  Remark 5.29.

Let $A$ be a $\delta$-ring and $d\in\operatorname{Rad}(A)$ an element in the
Jacobson radical. Then $d$ is weakly distinguished if and only if it is
distinguished.

The following lemma is a motivation for the introduction of weakly
distinguished elements:

######  Lemma 5.30 (cf. [BS19, Lem 2.23]).

Let $A$ be a $\delta$-ring, $I=(d)\subseteq A$ a principal ideal. Then for any
invertible element $u\in\operatorname{GL}_{1}(A)$, the principal ideals
$\delta(d)(A/I)$ and $\delta(ud)(A/I)$ are the same. In particular, when $I$
is generated by a non-zero-divisor, the principal ideal $\delta(d)(A/I)$ does
not depend on the choice of the generator $d\in I$.

###### Proof.

We have
$\delta(ud)=\varphi(u)\delta(d)+\delta(u)d^{p}\equiv\varphi(u)\delta(d)\hskip
1.99997pt\pmod{ud}$. Since $u$ is invertible, so is $\varphi(u)$, and the
result follows. ∎

######  Corollary 5.31.

Let $A$ be a $\delta$-ring, $I\subseteq A$ a principal ideal generated by a
non-zero-divisor. Then the followings are equivalent:

  1. (1)

There exists a weakly distinguished generator $d$ of $I$.

  2. (2)

Every generator $d$ of $I$ is weakly distinguished.

######  Remark 5.32 (Bhatt).

We have a variant of Corollary 5.31 which does not involve non-zero-divisors,
by replacing a principal ideal $I$ by the equivalence classes of maps
$A\rightarrow A$ of $A$-modules, and the proof of Lemma 5.30 implies that the
concept of “weakly distinguished” is invariant under this equivalence. More
generally, we can consider the equivalence classes of an invertible $A$-module
$I$ along with a map $I\rightarrow A$, and define the concept of such a map
$I\rightarrow A$ being weakly distinguished when $p\in\operatorname{Rad}(A)$.
This generalizes to animated $\delta$-rings.

Recall that an animated ring $A$ is $p$-local if the element $p\in\pi_{0}(A)$
lies in the Jacobson radical323232The Jacobson radical $\operatorname{Rad}(A)$
of a ring $A$ is defined to be the subset (and a fortiori, the ideal) of
elements $x\in A$ such that for every $a\in A$, the element $1+ax$ is
invertible in $A$. $\operatorname{Rad}(\pi_{0}(A))$.

######  Lemma 5.33.

Let $A$ be a $p$-local $\delta$-ring and $d\in A$ a weakly distinguished
element. Then for every $n\in\mathbb{N}$, $\varphi^{n}(\delta(d))$ is
invertible in $A/d$.

###### Proof.

By induction, it suffices to show that, for every $u\in A$ of which the image
in $A/d$ is invertible, then so is the image of $\varphi(u)$ in $A/d$. It
follows from the identity $\varphi(u)=u^{p}+p\delta(u)$, since the image of
$u^{p}$ in $A/d$ invertible, and $p\in\operatorname{Rad}(A/d)$. ∎

###  5.3. Conjugate filtration