In this subsection, we will introduce the conjugate filtration on “non-
complete prismatic envelopes”, which plays a similar role as the conjugate
filtrations on animated PD-envelopes and derived crystalline cohomology. Let
$A$ be a $p$-local $\delta$-ring, $d\in A$ a weakly distinguished non-zero-
divisor. To simplify the presentation, we mostly concentrate on the “single
variable” case: $\operatorname{PrismEnv}(A\\{y\\},(d,y))/^{\mathbb{L}}d\simeq
A\\{y/d\\}/^{\mathbb{L}}d$ as an $A\\{y\\}/^{\mathbb{L}}(d,y)$-algebra (or
module).

First, note that the identity

(9) $\displaystyle\delta(u^{p})$ $\displaystyle=$
$\displaystyle(\varphi(u^{p})-u^{p^{2}})/p$ $\displaystyle=$
$\displaystyle((u^{p}+p\delta(u))^{p}-u^{p^{2}})/p$ $\displaystyle=$
$\displaystyle\sum_{k=1}^{p}\binom{p}{k}u^{p(p-k)}p^{k-1}\delta(u)^{k}$

holds in the free $\delta$-ring $\mathbb{Z}_{(p)}\\{u\\}$, therefore it is an
identity in any $\delta$-ring.

We now compute $\delta^{n}(y)$ in terms of $\delta^{n}(z)$ where $y=zd$ in the
free $\delta$-$A$-algebra $A\\{y\\}$:

$\displaystyle\delta(y)$ $\displaystyle=$ $\displaystyle\delta(zd)$
$\displaystyle=$ $\displaystyle\delta(z)\varphi(d)+z^{p}\delta(d)$
$\displaystyle\delta^{2}(y)$ $\displaystyle=$
$\displaystyle\delta(\delta(z)\varphi(d)+z^{p}\delta(d))$ $\displaystyle=$
$\displaystyle\delta(\delta(z)\varphi(d))+\delta(z^{p}\delta(d))-\underbrace{\sum_{k=1}^{p-1}\frac{1}{p}\binom{p}{k}(\delta(z)\varphi(d))^{p-k}(z^{p}\delta(d))^{k}}_{=:R_{2}}$
$\displaystyle=$
$\displaystyle\delta^{2}(z)\varphi^{2}(d)+\delta(z)^{p}\delta(\varphi(d))+\delta(z^{p})\delta(\varphi(d))+z^{p^{2}}\delta^{2}(d)-R_{2}$
$\displaystyle=$
$\displaystyle\delta^{2}(z)\varphi^{2}(d)+\varphi(\delta(d))(1+p^{p-1})\delta(z)^{p}+\sum_{k=1}^{p-1}\cdots+z^{p^{2}}\delta^{2}(d)-R_{2}$

where we used the fact that $\varphi\circ\delta=\delta\circ\varphi$ and (9)
(which leads to the summand $\sum_{k=1}^{p-1}\cdots$), and in general, we have

######  Lemma 5.34.

Let $A\\{z\\}$ be the free $\delta$-$A$-algebra and $y:=zd$. For every
$n\in\mathbb{N}$, there exists a unique polynomial $P_{n}\in
A[X_{0},\ldots,X_{n-1}]$ with $\deg_{X_{n-1}}P_{n}\leq p$ such that

$\delta^{n}(y)=\delta^{n}(z)\varphi^{n}(d)+P_{n}(z,\delta(z),\ldots,\delta^{n-1}(z))$

Moreover, there exists a unique $Q_{n}\in A[X_{0},\ldots,X_{n-1}]$ with
$\deg_{X_{n-1}}Q_{n}<p$ such that
$P_{n}=a_{n}\varphi^{n-1}(\delta(d))X_{n-1}^{p}+Q_{n}$ where $a_{n}$ are
partial sums $\sum_{k=0}^{n-1}p^{k(p-1)}$ of the geometric progression
$(p^{k(p-1)})_{k\in\mathbb{N}}$. Note that
$a_{n}\in\operatorname{GL}_{1}(\mathbb{Z}_{(p)})$ for $n\char 62\relax 0$. On
the other hand, if we endow $X_{i}$ with degree $p^{i}$, then $P_{n}$ is
homogeneous of degree $p^{n}$.

###### Proof.

The uniqueness follows from the freeness. We prove the existence inductively
on $n\in\mathbb{N}$. When $n=0$, this is obvious. Now let
$n\in\mathbb{N}_{\char 62\relax 0}$, and assume that this is true for every
$m<n$, Now we have

$\displaystyle\delta^{n}(y)$ $\displaystyle=$
$\displaystyle\delta(\delta^{n-1}(y))$ $\displaystyle=$
$\displaystyle\delta(\delta^{n-1}(z)\varphi^{n-1}(d)+P_{n-1}(z,\delta(z),\ldots,\delta^{n-2}(z)))$
$\displaystyle=$
$\displaystyle\delta(\delta^{n-1}(z)\varphi^{n-1}(d))+\delta(P_{n-1}(z,\delta(z),\ldots,\delta^{n-2}(z)))-R_{n}$

where
$\delta(\delta^{n-1}(z)\varphi^{n-1}(d))=\delta^{n}(z)\varphi^{n}(d)+(\delta^{n-1}(z))^{p}\varphi^{n-1}(\delta(d))$
and

$R_{n}:=\sum_{k=1}^{p-1}\frac{1}{p}\binom{p}{k}(\delta^{n-1}(z)\varphi^{n-1}(d))^{p-k}(P_{n-1}(z,\delta(z),\ldots,\delta^{n-2}(z)))^{k}$

Note that the “degree” of $\delta^{n-1}(z)$ in $R_{n}$ is strictly less than
$p$. Let $b_{n-1}=a_{n-1}\varphi^{n-2}(\delta(d))$, we have

$\displaystyle\delta(P_{n-1}(z,\delta(z),\ldots))$ $\displaystyle=$
$\displaystyle\delta(b_{n-1}(\delta^{n-2}(z))^{p}+Q_{n-1}(z,\delta(z),\ldots))$
$\displaystyle=$
$\displaystyle\delta(b_{n-1}(\delta^{n-2}(z))^{p})+\delta(Q_{n-1}(z,\delta(z),\ldots))-\underbrace{\sum_{k=1}^{p-1}\cdots}_{=:R_{n}^{\prime}}$
$\displaystyle=$
$\displaystyle\varphi(b_{n-1})\delta((\delta^{n-2}(z))^{p})+\delta(Q_{n-1}(\cdots))+\delta(b_{n-1})(\delta^{n-2}(z))^{p^{2}}$
$\displaystyle-R_{n}^{\prime}$

and only $\varphi(b_{n-1})\delta((\delta^{n-2}(z))^{p})$ has contribution on
$\delta^{n-1}(z)^{p}$, and

$\delta((\delta^{n-2}(z))^{p})=\sum_{k=1}^{p}\binom{p}{k}(\delta^{n-2}(z))^{p(p-k)}p^{k-1}(\delta^{n-1}(z))^{k}$

has contribution on $\delta^{n-1}(z)^{p}$ only at $k=p$, i.e.
$p^{p-1}\delta^{n-1}(z)^{p}$. Note that
$\varphi(b_{n-1})=a_{n-1}\varphi^{n-1}(\delta(d))$, the result then follows. ∎

We now rewrite $A\\{y\\}\rightarrow A\\{z\\},y\mapsto zd$ as the sequential
composite (i.e. the $A\\{y\\}$-algebra $A\\{z\\}$ is equivalent to the
sequential colimit of)

(10) $A\\{y\\}\longrightarrow
A\\{y\\}\otimes_{B_{0}}^{\mathbb{L}}C_{0}\longrightarrow
A\\{y\\}\otimes_{B_{1}}^{\mathbb{L}}C_{1}\longrightarrow\cdots$

where $A_{n}:=A[z,\ldots,\delta^{n-1}(z)],B_{n}:=A_{n}[\delta^{n}(y)]$ and
$C_{n}:=A_{n}[\delta^{n}(z)]$ are polynomial algebras, and the map
$B_{n}\rightarrow C_{n}$ is given by the evaluation
$\delta^{n}(y)\mapsto\delta^{n}(z)\varphi^{n}(d)+P_{n}(z,\delta(z),\ldots,\delta^{n-1}(z))$
by Lemma 5.34. Thus $B_{n}\rightarrow C_{n}$ could be written as the composite
(where we replace $\delta^{n}(y)$ by $u$ and $\delta^{n}(z)$ by $v$)

(11) $B_{n}=A_{n}[u]\rightarrow
A_{n}[u,v]/(u-\varphi^{n}(d)v-P_{n}(z,\delta(z),\ldots,\delta^{n-1}(z)))\cong
A_{n}[v]=C_{n}$

In other words, $B_{n}\rightarrow C_{n}$ is essentially formally
adjoining333333Note that this is true although $\varphi^{n}(d)$ is not
necessarily a non-zero-divisor.
$(\delta^{n}(y)-P_{n}(z,\delta(z),\ldots,\delta^{n-1}(z)))/\varphi^{n}(d)$ to
$B_{n}$ as an (animated) $A$-algebra, and the $A\\{y\\}$-algebra $A\\{z\\}$ is
obtained by formally adjoining
$(\delta^{n}(y)-P_{n}(z,\delta(z),\ldots,\delta^{n-1}(z)))/\varphi^{n}(d)$
iteratively from $A\\{y\\}$.

The conjugate filtration on $A\\{y/d\\}/^{\mathbb{L}}d$ is given by
$\operatorname{Fil}^{-i}_{\operatorname{conj}}(A\\{y/d\\}/^{\mathbb{L}}d)$
being the $A\\{y\\}/^{\mathbb{L}}(y,d)$-submodule of
$A\\{y/d\\}/^{\mathbb{L}}d$ spanned by
$\\{(y/d)^{a_{0}}\delta(y/d)^{a_{1}}(\delta^{2}(y/d))^{a_{2}}\cdots(\delta^{r}(y/d))^{a_{r}}|r\in\mathbb{N},0\leq
a_{0},a_{1},\ldots,a_{r}<p\\}$. Passing to the multivariable version, we get:

######  Lemma 5.35.

Let $A$ be a $p$-local $\delta$-ring and $d\in A$ a weakly distinguished non-
zero-divisor. Then there exists a canonical functor
$\operatorname{Fil}_{\operatorname{conj}}^{\ast}(\operatorname{PrismEnv}(\cdot)/^{\mathbb{L}}d):\operatorname{AniPair}_{\delta,(A,d)}^{0}\rightarrow\operatorname{CAlg}(\operatorname{DF}^{\leq
0}(A/^{\mathbb{L}}d))$ which preserves finite coproducts, along with a
functorial map
$\operatorname{Fil}_{\operatorname{conj}}^{\ast}(\operatorname{PrismEnv}(B,J)/^{\mathbb{L}}d)\rightarrow\operatorname{PrismEnv}(B,J)/^{\mathbb{L}}d$,
understood as the conjugate filtration on
$\operatorname{PrismEnv}(B,J)/^{\mathbb{L}}d$, such that

  1. (1)

The conjugate filtration is exhaustive, that is to say, the induced map
$\operatorname{Fil}_{\operatorname{conj}}^{-\infty}(\operatorname{PrismEnv}(B,J)/^{\mathbb{L}}d)\rightarrow\operatorname{PrismEnv}(B,J)/^{\mathbb{L}}d$
is an equivalence in $D(A)$.

  2. (2)

The filtration
$\operatorname{Fil}_{\operatorname{conj}}^{\ast}(\operatorname{PrismEnv}(A\\{y\\}\twoheadrightarrow
A\\{y\\}/^{\mathbb{L}}(y,d))/^{\mathbb{L}}d)$ coincides with the filtration
$\operatorname{Fil}_{\operatorname{conj}}^{\ast}(A\\{y/d\\}/^{\mathbb{L}}d)$
constructed above.

  3. (3)

The maps
$\operatorname{Fil}_{\operatorname{conj}}^{-i}(\operatorname{PrismEnv}(A\\{x\\}\twoheadrightarrow
A\\{x\\}/^{\mathbb{L}}d)/^{\mathbb{L}}d)\rightarrow\operatorname{PrismEnv}(A\\{x\\}\twoheadrightarrow
A\\{x\\}/^{\mathbb{L}}d)/^{\mathbb{L}}d\simeq A\\{x\\}/^{\mathbb{L}}d$ are
equivalences for all $i\in\mathbb{N}$, that is to say, the conjugate
filtration on $(A/^{\mathbb{L}}d)\\{x\\}$ is “constant343434More precisely, it
is constant after restriction to $\mathbb{Z}_{\leq 0}$, but this restriction
is expected as the conjugate filtration is non-positive.”.

###### Proof.

The conjugate filtration on each object $\operatorname{PrismEnv}(B,J)$ for
$(B,J)\in\operatorname{AniPair}_{\delta,(A,d)}^{0}$ is completely determined
by these properties and that the functor preserves finite coproducts, since
every $(B,J)$ could be written as a coproduct of $A\\{x\\}\twoheadrightarrow
A\\{x\\}/^{\mathbb{L}}d$ and $A\\{y\\}\twoheadrightarrow
A\\{y\\}/^{\mathbb{L}}(y,d)$. Concretely,
$\operatorname{Fil}_{\operatorname{conj}}^{-i}(A\\{X,Y/d\\}/^{\mathbb{L}}d)$
are generated, as an $A\\{X,Y\\}/(d,Y)$-submodule, by “standard monomials”
$\prod_{(r,y)\in E}\delta^{r}(y/d)$ “of total degree$\leq i$” where
$E\subseteq\mathbb{N}\times Y$ is a finite subset and the element
$\delta^{r}(y/d)$ is of degree $p^{r}$ for $r\in\mathbb{N}$ and $y\in Y$. One
verifies that this indeed gives rise to a functor.

Alternatively, if we further assume that $d$ is weakly transversal (Definition
5.39), then we can invoke Lemma 5.42 to reduce significantly the computations.
∎

######  Definition 5.36.

Let $A$ be a $p$-local $\delta$-ring and $d\in A$ a weakly distinguished non-
zero-divisor. Then the conjugate filtration on
$\operatorname{PrismEnv}(B\twoheadrightarrow B^{\prime\prime})/^{\mathbb{L}}d$
for $(B\twoheadrightarrow
B^{\prime\prime})\in\operatorname{AniPair}_{\delta,(A,d)}$ is given by the
left derived functor (Proposition A.10)
$\operatorname{AniPair}_{\delta,(A,d)}\rightarrow\operatorname{CAlg}(\operatorname{DF}^{\leq
0}(A/^{\mathbb{L}}d))$ of the functor
$\operatorname{AniPair}_{\delta,(A,d)}^{0}\rightarrow\operatorname{CAlg}(\operatorname{DF}^{\leq
0}(A/^{\mathbb{L}}d))$ in Lemma 5.35.

It follows from Lemma 2.43 that

######  Lemma 5.37.

Let $A$ be a $p$-local $\delta$-ring and $d\in A$ a weakly distinguished non-
zero-divisor. Then the conjugate filtration on
$\operatorname{PrismEnv}(B\twoheadrightarrow B^{\prime\prime})/^{\mathbb{L}}d$
for every $(B\twoheadrightarrow
B^{\prime\prime})\in\operatorname{AniPair}_{\delta,(A,d)}$ is exhaustive, i.e.
$\operatorname{Fil}^{-\infty}\operatorname{PrismEnv}(B\twoheadrightarrow
B^{\prime\prime})/^{\mathbb{L}}d\rightarrow\operatorname{PrismEnv}(B\twoheadrightarrow
B^{\prime\prime})/^{\mathbb{L}}d$ is an equivalence.

We now analyze the “denominators” $\varphi^{n}(d)$ when $A$ is $p$-local and
$d$ is weakly distinguished:

######  Lemma 5.38 (cf. [AL19b, Lem 3.5]).

Let $A$ be a $p$-local $\delta$-ring and $d\in A$ a weakly distinguished
element. Then for every $n\in\mathbb{N}_{\char 62\relax 0}$, there exists a
unit $u\in\operatorname{GL}_{1}(A/d)$ such that $\varphi^{n}(d)\equiv pu\hskip
1.99997pt\pmod{d}$.

###### Proof.

We will construct inductively on $n\in\mathbb{N}_{\char 62\relax 0}$ a
sequence $(u_{n})_{n}\in A^{\mathbb{N}_{\char 62\relax 0}}$ such that for
every $n\in\mathbb{N}_{\char 62\relax 0}$, the image of $u_{n}$ in $A/d$ is
invertible, and $\varphi^{n}(d)-d^{p^{n}}=pu_{n}$. We take $u_{1}=\delta(d)$,
and suppose that $u_{m}$ are already constructed for $1\leq m<n$, then

$\displaystyle\varphi^{n}(d)$ $\displaystyle=$
$\displaystyle\varphi^{n-1}(\varphi(d))$ $\displaystyle=$
$\displaystyle\varphi^{n-1}(d^{p}+p\delta(d))$ $\displaystyle=$
$\displaystyle(\varphi^{n-1}(d))^{p}+p\varphi(\delta(d))$ $\displaystyle=$
$\displaystyle(d^{p^{n-1}}+pu_{n-1})^{p}+p\varphi(\delta(d))$ $\displaystyle=$
$\displaystyle
d^{p^{n}}+p\left(\varphi(\delta(d))+\sum_{k=1}^{p}\binom{p}{k}d^{p^{n-1}(p-k)}p^{k-1}u_{n-1}^{k}\right)$

We pick
$u_{n}=\delta(d)+\sum_{k=1}^{p}\binom{p}{k}d^{p^{n-1}(p-k)}p^{k-1}u_{n-1}^{k}$.
Note that the second summand $\sum_{k=1}^{p}\cdots$ is canonically divisible
by $p$ (separating the cases $k=0$ and $k\char 62\relax 1$), thus
$u_{n}\equiv\delta(d)\hskip 1.99997pt\pmod{p}$ of which the image in $A/(p,d)$
is invertible. The result then follows from the fact that
$p\in\operatorname{Rad}(A/d)$. ∎

We introduce the following temporary terminology:

######  Definition 5.39.

Let $A$ be a $\delta$-ring. We say that an element $d\in A$ is weakly
transversal if it is weakly distinguished and the sequence $(d,p)$ is regular
in $A$, that is to say, $d$ is a non-zero-divisor and $A/d$ is $p$-torsion-
free.

Recall that for a ring $A$, the Zariski localization of $A$ along an ideal
$I\subseteq A$ is defined to be the localization of $A$ at the multiplicative
set $1+I$. The image of $I$ in $(1+I)^{-1}A$ lies in the Jacobson radical.

######  Example 5.40.

The element $d$ in the $p$-local $\delta$-ring
$\mathbb{Z}_{(p)}\\{d,\delta(d)^{-1}\\}_{(p)}$ is weakly transversal. In fact,
this special case suffices for our applications.

Now we assume that $d\in A$ is weakly transversal. In the “single variable”
case $A\\{y\\}/^{\mathbb{L}}(d,y)\rightarrow A\\{y/d\\}/^{\mathbb{L}}d$, by
Lemmas 5.38 and 5.34, the sequence $(z,\delta(z),\delta^{2}(z),\ldots)$ forms
a system similar to that of divided $p^{r}$-powers
$(\gamma_{p^{r}})_{r\in\mathbb{N}}$ up to a multiplication of a unit after
modulo $d$:

$\displaystyle p\delta(z)$ $\displaystyle\equiv$ $\displaystyle-
a_{1}\delta(d)z^{p}\hskip 1.99997pt\pmod{B}$ $\displaystyle p\delta^{2}(z)$
$\displaystyle\equiv$ $\displaystyle-
a_{2}\varphi(\delta(d))\delta(z)^{p}\hskip 1.99997pt\pmod{B[\delta(z)]}$
$\displaystyle p\delta^{3}(z)$ $\displaystyle\equiv$ $\displaystyle-
a_{3}\varphi^{2}(\delta(d))\delta^{2}(z)^{p}\hskip
1.99997pt\pmod{B[\delta(z),\delta^{2}(z)]}$

where $B:=A\\{y\\}/^{\mathbb{L}}(d,y)$ and
$a_{n}\varphi^{n-1}(\delta(d))\in\operatorname{GL}_{1}(A/d)$ (cf. Remark
5.15). We now translate this observation to an analysis of the conjugate
filtration, which seems hard to attack directly. We look at the maps
$B_{0}/^{\mathbb{L}}(d,y)\rightarrow C_{0}/^{\mathbb{L}}d$ and
$B_{n}/^{\mathbb{L}}d\rightarrow C_{n}/^{\mathbb{L}}d$ for
$n\in\mathbb{N}_{\char 62\relax 0}$ induced by the map (11). We first note
that the map $B_{0}/^{\mathbb{L}}(d,y)\rightarrow C_{0}/^{\mathbb{L}}d$ is the
polynomial algebra in single variable $z$.

If we further (derived) modulo $p$, we see that
$B_{n}/^{\mathbb{L}}(d,p)\rightarrow C_{n}/^{\mathbb{L}}(d,p)$ for
$n\in\mathbb{N}_{\char 62\relax 0}$ is killing a polynomial
$\delta^{n}(y)-P_{n}(z,\delta(z),\ldots,\delta^{n-1}(z))$ monic in
$\delta^{n-1}(z)$ of degree $p$, and then adjoining a formal variable
$\delta^{n}(z)$. In view of (10), we see that the map
$A\\{y\\}/^{\mathbb{L}}(d,p,y)\xrightarrow{y\mapsto
zd}A\\{z\\}/^{\mathbb{L}}(d,p)$ is the composition of consecutively adjoining
a root of a monic polynomial of degree $p$, and consequently, as a
$A\\{y\\}/^{\mathbb{L}}(d,p,y)$-module, $A\\{z\\}/^{\mathbb{L}}(d,p)$ is
freely generated by
$\\{z^{a_{0}}\delta(z)^{a_{1}}(\delta^{2}(z))^{a_{2}}\cdots(\delta^{r}(z))^{a_{r}}|r\in\mathbb{N},0\leq
a_{0},a_{1},\ldots,a_{r}<p\\}$.

On the other hand, if we invert $p$, we see that, for every
$n\in\mathbb{N}_{\char 62\relax 0}$, the maps
$(B_{n}/^{\mathbb{L}}d)[p^{-1}]\rightarrow(C_{n}/^{\mathbb{L}}d)[p^{-1}]$ are
equivalences, therefore
$(A\\{y\\}/^{\mathbb{L}}(d,y))[p^{-1}]\rightarrow(A\\{z\\}/^{\mathbb{L}}d)[p^{-1}]$
is the polynomial algebra in one variable $z$.

The $\operatorname{mod}p$ conjugate filtration
$\operatorname{Fil}_{\operatorname{conj}}^{-i}(A\\{z\\}/^{\mathbb{L}}(d,p))/^{\mathbb{L}}p$
is then freely generated by
$\\{z^{a_{0}}\delta(z)^{a_{1}}(\delta^{2}(z))^{a_{2}}\cdots(\delta^{r}(z))^{a_{r}}|r\in\mathbb{N},0\leq
a_{0},a_{1},\ldots,a_{r}<p,a_{0}+pa_{1}+\cdots+p^{r}a_{r}\leq i\\}$. On the
other hand, the rationalized conjugate filtration
$\operatorname{Fil}_{\operatorname{conj}}^{-i}(A\\{z\\}/^{\mathbb{L}}(d,p))[p^{-1}]$
is given by the $(A\\{y\\}/^{\mathbb{L}}(d,y))[p^{-1}]$-polynomials in $z$ of
degree$\leq i$. This follows from the following lemma, which can be
established by induction on $n$:

######  Lemma 5.41.

In the rationalized free $\delta$-ring
$\mathbb{Z}_{(p)}\\{x\\}[p^{-1}]\cong\mathbb{Q}[x,\varphi(x),\varphi^{2}(x),\ldots]$,
for every $n\in\mathbb{N}$, the image of
$\delta^{n}(x)\in\mathbb{Z}_{(p)}\\{x\\}$ in
$\mathbb{Q}[x,\varphi(x),\varphi^{2}(x),\ldots]$ is given by a polynomial
$D_{n}(x,\varphi(x),\ldots,\varphi^{n}(x))$ such that $\deg_{x}D_{n}=p^{n}$
with leading term $(-p^{-1})^{1+p+\cdots+p^{n}}x^{p^{n}}$ for all
$n\in\mathbb{N}$.

We summarize the “multi-variable” version as follows:

######  Lemma 5.42.

Let $A$ be a $p$-local $\delta$-ring and $d\in A$ a weakly transversal
element. Let $(A\\{X,Y\\},(d,Y))\in\operatorname{AniPair}_{\delta,(A,d)}^{0}$.
Then

  1. (1)

The generator $\left\\{\prod_{(r,y)\in E}\delta^{r}(y/d)\right\\}_{E}$ for
$\operatorname{Fil}_{\operatorname{conj}}^{-i}(A\\{X,Y/d\\}/^{\mathbb{L}}d)$
as an $A\\{X,Y\\}/^{\mathbb{L}}(d,Y)$-submodule, “of total degree$\leq i$”
where $E\subseteq\mathbb{N}\times Y$ is a finite subset and the element
$\delta^{r}(y/d)$ is of degree $p^{r}$, becomes an basis after (derived)
modulo $p$. This also holds for $i=+\infty$.

  2. (2)

The $(-i)$-th piece of the rationalized conjugate filtration
$\operatorname{Fil}_{\operatorname{conj}}^{-i}(A\\{X,Y/d\\}/^{\mathbb{L}}d)[p^{-1}]\subseteq(A\\{X,Y/d\\}/^{\mathbb{L}}d)[p^{-1}]$
is given by the $A\\{X,Y\\}/^{\mathbb{L}}(d,Y)$-polynomials in variables $Y/d$
of total degree$\leq i$. This also holds for $i=+\infty$.

Furthermore, an element $x\in A\\{X,Y/d\\}/^{\mathbb{L}}d$ belongs to the
$(-i)$-th piece of the conjugate filtration
$\operatorname{Fil}_{\operatorname{conj}}^{-i}(A\\{X,Y/d\\}/^{\mathbb{L}}d)$
if and only if so does it after (derived) modulo $p$ and after
rationalization.

######  Remark 5.43.

In some vague terms, in Lemma 5.42, the derived modulo $p$ is about
“controlling the denominators”, and the rationalization is about “controlling
the degree”.

Recall that for every $(B,J)\in\operatorname{AniPair}_{\delta,(A,d)}^{0}$,
there exists a canonical map $B/^{\mathbb{L}}d\simeq
B\otimes_{A}^{\mathbb{L}}(A/^{\mathbb{L}}d)\rightarrow B/J$ which is in fact
surjective. Then we have the following “multivariable” version:

######  Lemma 5.44.

Let $A$ be a $p$-local $\delta$-ring and $d\in A$ a weakly transversal
element. For every
$(A\\{X,Y\\},(d,Y))=:(B,J)\in\operatorname{AniPair}_{\delta,(A,d)}^{0}$, let
$K:=\ker(B/^{\mathbb{L}}d\rightarrow B/J)$. Note that $K/K^{2}$ is naturally a
$B/J$-module. Then there exists a comparison map

$\Gamma_{B/J}^{\ast}((K/K^{2})\\{-1\\})\longrightarrow\operatorname{gr}_{\operatorname{conj}}^{-\mathord{\ast}}(\operatorname{PrismEnv}(B,J)/^{\mathbb{L}}d)$

of graded $B/J$-algebras induced by
$[\gamma_{n}(z_{i})]\mapsto\prod_{j=0}^{r}\left(\frac{\delta^{j}(z_{i})}{-a_{j}\varphi^{j}(\delta(d))}\right)^{n_{j}}$
where $Y=\\{y_{1},\ldots\\}$, $z_{i}=y_{i}/p$ and $n=\sum_{j=0}^{r}n_{j}p^{j}$
is the $p$-adic expansion of $n$. The comparison map is functorial in
$(B,J)\in\operatorname{AniPair}_{\delta,(A,d)}^{0}$.

###### Proof.

The comparison map is induced by
$\left[\gamma_{n}\left(\frac{y}{d}\right)\right]\mapsto\prod_{j=0}^{r}\left(\frac{\delta^{j}(y/d)}{-a_{j}\varphi^{j}(\delta(d))}\right)^{n_{j}}$
for every $y$ in the ideal $(d,Y)$. To see that this is well-defined, the most
nontrivial part is to show that this vanishes when $y\in(d,Y^{2})$. By the
multiplicity of the conjugate filtration, we can assume that $n_{r}=1$ and
$n_{j}=0$ for $j\neq r$, and it suffices to analyze $\delta^{r}(y/d)$ when
$y\in(d,Y^{2})$, which can be reduced to the special case that $y=y_{1}y_{2}$
where $y_{1},y_{2}\in Y$.

By Lemma 5.41, the element $\delta^{r}(y_{1}y_{2}/d)\in
A[p^{-1}][X,Y/d,\varphi(Y),\varphi^{2}(Y/d),\ldots]$ is a polynomial in
$y_{1}y_{2}/d=y_{1}z_{2},\varphi(y_{1}y_{2}),\ldots,\varphi^{r}(y_{1}y_{2})$.
The crucial point is that $y_{1}y_{2}/d=(y_{1}/d)(y_{2}/d)d=0$ in
$A\\{X,Y/d\\}/^{\mathbb{L}}d$, therefore after rationalization,
$\delta^{r}(y_{1}y_{2}/d)$ lies in
$\operatorname{Fil}_{\operatorname{conj}}^{0}(\operatorname{PrismEnv}(B,J)/^{\mathbb{L}}d)[p^{-1}]$.

By Lemma 5.34,
$\delta^{r}(y_{1}z_{2})=\delta^{r}(z_{2})\varphi^{r}(y_{1})+P_{r}(z_{2},\ldots,\delta^{r-1}(z_{2}))$
where $P_{r}$ is an $A\\{y_{1}\\}$-polynomial. Note that
$\varphi^{r}(y_{1})=\varphi^{r}(z_{1}d)=\varphi^{r}(z_{1})\varphi^{r}(d)\equiv
0\hskip 1.99997pt\pmod{(d,p)}$ by Lemma 5.38. Since $P_{r}$ is homogeneous of
degree $p^{r}$ when $\deg(\delta^{j}(z_{2}))=p^{j}$, it follows that for every
monomial $\prod_{j}T_{j}^{n_{j}}$ of $P_{r}$, there exists a $j$ such that
$n_{j}\geq p$, but then $\delta^{j}(z_{2})^{n_{j}}$ is a linear combination of
basis elements in Lemma 5.42 which shows that
$\prod_{j}(\delta^{j}(z_{2}))^{n_{j}}\in\operatorname{Fil}_{\operatorname{conj}}^{-(p^{r}-1)}(\operatorname{PrismEnv}(B,J)/^{\mathbb{L}}d)/^{\mathbb{L}}p$.
The result then follows from the last part of Lemma 5.42. ∎

It again follows from Lemma 5.42, via derived modulo $p$ and rationalization,
that

######  Lemma 5.45.

Let $A$ be a $p$-local $\delta$-ring and $d\in A$ a weakly transversal
element. For every $(B,J)\in\operatorname{AniPair}_{\delta,(A,d)}^{0}$, the
comparison map in Lemma 5.44 is an equivalence.

After such a long march, let us harvest the Hodge-Tate comparison, which is a
prismatic analogue of Corollary 3.60. Note that for every
$(B\twoheadrightarrow
B^{\prime\prime})\in\operatorname{AniPair}_{\delta,(A,d)}$, note that the
commutative diagram

$\begin{array}[]{ccc}A&\longrightarrow&B\\\
{\mbox{\rotatebox[origin={c}]{-90.0}{$\longrightarrow$}}}&&{\mbox{\rotatebox[origin={c}]{-90.0}{$\longrightarrow$}}}\\\
A/^{\mathbb{L}}d&\longrightarrow&B^{\prime\prime}\end{array}$

induces a natural map $B/^{\mathbb{L}}d\simeq
B\otimes_{A}^{\mathbb{L}}(A/^{\mathbb{L}}d)\rightarrow B^{\prime\prime}$ which
is surjective, that is to say, $B/^{\mathbb{L}}d\twoheadrightarrow
B^{\prime\prime}$ is an animated pair. It then follows from Lemmas 5.45 and
A.10 that

######  Theorem 5.46 (Hodge-Tate).

Let $A$ be a $p$-local $\delta$-ring and $d\in A$ a weakly transversal
element. Then for every animated $\delta$-$(A,d)$-pair $(B\twoheadrightarrow
B^{\prime\prime})\in\operatorname{AniPair}_{\delta,(A,d)}$, there exists a
canonical equivalence

$\Gamma_{B^{\prime\prime}}^{i}(\operatorname{gr}^{1}(\mathbb{L}\operatorname{AdFil}(B/^{\mathbb{L}}d\twoheadrightarrow
B^{\prime\prime}))\\{-1\\})\longrightarrow\operatorname{gr}_{\operatorname{conj}}^{-i}(\operatorname{PrismEnv}(B\twoheadrightarrow
B^{\prime\prime})/^{\mathbb{L}}d)$

which is functorial in $(B\twoheadrightarrow
B^{\prime\prime})\in\operatorname{AniPair}_{\delta,(A,d)}$, where
$\mathbb{L}\operatorname{AdFil}$ is the adic filtration functor defined before
Corollary 3.54.

Let $R$ be an $\mathbb{E}_{1}$-ring. Recall that a right $R$-module $M$ is
faithfully flat if it is flat (Definition 3.62) and $\pi_{0}(M)$ is a
faithfully flat right $\pi_{0}(R)$-module. A map $R\rightarrow S$ of
$\mathbb{E}_{\infty}$-rings is faithfully flat if $S$ is faithfully flat as an
$R$-module. There is a useful characterization of faithfully flat algebras:

######  Lemma 5.47 ([Lur04, Lemma 5.5]).

Let $f:R\rightarrow S$ be a map of static (commutative) rings. Then $f$ is
faithfully flat if and only if $f$ is flat, injective and that
$\operatorname{coker}(f)$ taken in the category of $R$-modules is flat.

######  Lemma 5.48.

Let $f:R\rightarrow S$ a map of $\mathbb{E}_{\infty}$-rings. If $f$ is
faithfully flat, then $\operatorname{cofib}(f)$ taken in the $\infty$-category
of $R$-module spectra is flat. The converse is true if $R$ is supposed to be
connective.

###### Proof.

Assume first that $f$ is faithfully flat. Let
$M:=\operatorname{coker}(\pi_{0}(R)\rightarrow\pi_{0}(S))$. By Lemma 5.47, the
map $\pi_{0}(R)\rightarrow\pi_{0}(S)$ is injective and the $\pi_{0}(R)$-module
$M$ is flat, Then for every $n\in\mathbb{Z}$, we have the exact sequence

$\operatorname{Tor}_{1}^{\pi_{0}(R)}(\pi_{n}(R),M)\rightarrow\pi_{n}(R)\rightarrow\pi_{n}(R)\otimes_{\pi_{0}(R)}\pi_{0}(S)\rightarrow\pi_{n}(R)\otimes_{\pi_{0}(R)}M\rightarrow
0$

which implies that the map
$\pi_{n}(R)\rightarrow\pi_{n}(R)\otimes_{\pi_{0}(R)}\pi_{0}(S)$ is injective.
Since $f$ is flat, the canonical map
$\pi_{n}(R)\otimes_{\pi_{0}(R)}\pi_{0}(S)\rightarrow\pi_{n}(S)$ is an
isomorphism, therefore the map $\pi_{n}(R)\rightarrow\pi_{n}(S)$ is injective.
Then the long exact sequence associated to the fiber sequence $R\rightarrow
S\rightarrow\operatorname{cofib}(f)$ splits into short exact sequences

$0\longrightarrow\pi_{n}(R)\longrightarrow\pi_{n}(S)\longrightarrow\pi_{n}(\operatorname{cofib}(f))\longrightarrow
0$

which implies that the canonical map
$M\rightarrow\pi_{0}(\operatorname{cofib}(f))$ is an isomorphism. Furthermore,
we have a morphism of short exact sequences

By the short five lemma, the map
$\pi_{n}(R)\otimes_{\pi_{0}(R)}M\rightarrow\pi_{n}(\operatorname{cofib}(f))$
is an isomorphism, therefore $\operatorname{cofib}(f)$ is flat.

Now we assume that $R$ is connective and that $\operatorname{cofib}(f)$ is
flat. By definition, $\operatorname{cofib}(f)$ is connective and so is $S$ by
the fiber sequence $R\rightarrow S\rightarrow\operatorname{cofib}(f)$. For
every static $R$-module $M$, we have the fiber sequence

$M\longrightarrow M\otimes_{R}^{\mathbb{L}}S\longrightarrow
M\otimes_{R}^{\mathbb{L}}\operatorname{cofib}(f)$

By flatness of $\operatorname{cofib}(f)$ and [Lur17, Prop 7.2.2.13],
$M\otimes_{R}^{\mathbb{L}}\operatorname{cofib}(f)$ is static, therefore so is
$M\otimes_{R}^{\mathbb{L}}S$. It then follows from [Lur17, Thm 7.2.2.15] that
$S$ is a flat $R$-module. It remains to show that the map
$\pi_{0}(R)\rightarrow\pi_{0}(S)$ is faithfully flat. By Lemma 5.47, it
suffices to show that $\pi_{0}(R)\rightarrow\pi_{0}(S)$ is injective and
$\operatorname{coker}(\pi_{0}(R)\rightarrow\pi_{0}(S))$ is flat. The first
follows from the connectivity of $\operatorname{cofib}(f)$, and the later
follows from the isomorphism
$\operatorname{coker}(\pi_{0}(R)\rightarrow\pi_{0}(S))\cong\pi_{0}(\operatorname{cofib}(f))$
and the flatness of $\operatorname{cofib}(f)$. ∎

Now we have a prismatic analogue of Corollary 3.66, with a similar argument:

######  Proposition 5.49.

Let $A$ be a $p$-local $\delta$-ring and $d\in A$ a weakly transversal
element. Let $(B\twoheadrightarrow
B^{\prime\prime})\in\operatorname{AniPair}_{\delta,(A,d)}$ be an animated
$\delta$-$(A,d)$-pair such that the canonical animated pair
$B/^{\mathbb{L}}d\twoheadrightarrow B^{\prime\prime}$ is quasiregular. Then
the unit map
$B^{\prime\prime}\rightarrow\operatorname{PrismEnv}(B\twoheadrightarrow
B^{\prime\prime})/^{\mathbb{L}}d$ is faithfully flat.

###### Proof.

By Theorem 5.46 and the quasiregularity of $B/^{\mathbb{L}}d\twoheadrightarrow
B^{\prime\prime}$, for every $i\in\mathbb{N}$, the $B^{\prime\prime}$-module
$\operatorname{gr}_{\operatorname{conj}}^{-i}(\operatorname{PrismEnv}(B\twoheadrightarrow
B^{\prime\prime})/^{\mathbb{L}}d)$ is flat. By Lemma 3.64, for every
$i\in\mathbb{N}_{\char 62\relax 0}$,
$\operatorname{cofib}(B^{\prime\prime}\rightarrow\operatorname{Fil}_{\operatorname{conj}}^{-i}(\operatorname{PrismEnv}(B\twoheadrightarrow
B^{\prime\prime})/^{\mathbb{L}}d))$ is flat. Since the conjugate filtration is
exhaustive (Lemma 5.35) and the collection of flat modules is stable under
filtered colimits [Lur17, Lem 7.2.2.14(1)], we get
$\operatorname{cofib}(B^{\prime\prime}\rightarrow\operatorname{PrismEnv}(B\twoheadrightarrow
B^{\prime\prime})/^{\mathbb{L}}d)$ is a flat $B^{\prime\prime}$-module. Then
the result follows from Lemma 5.48. ∎

######  Remark 5.50.

In Proposition 5.49, if we further assume that $B^{\prime\prime}$ is static,
then so is $\operatorname{PrismEnv}(B\twoheadrightarrow
B^{\prime\prime})/^{\mathbb{L}}d$. This does not imply that
$\operatorname{PrismEnv}(B\twoheadrightarrow B^{\prime\prime})$ is static.
However, it implies that, after taking $d$-completion,
$\operatorname{PrismEnv}(B\twoheadrightarrow B^{\prime\prime})$ becomes static
which should be understood as a “static $d$-completed envelope”.

######  Remark 5.51.

There is a $p$-completed analogue of Proposition 5.49: suppose that the
animated pair $B/^{\mathbb{L}}d\twoheadrightarrow B^{\prime\prime}$ is
$p$-completely353535“$p$-complete” concepts are usually applied to
$p$-complete objects. However, this is not necessary because we can always
derived $p$-complete a non-complete object. quasiregular, that is to say, the
shifted cotangent complex
$\mathbb{L}_{B^{\prime\prime}/(B/^{\mathbb{L}}d)}[-1]$ is a $p$-completely
flat $B^{\prime\prime}$-module, then the same proof shows that the unit map
$B^{\prime\prime}\rightarrow\operatorname{PrismEnv}(B\twoheadrightarrow
B^{\prime\prime})/^{\mathbb{L}}d$ is $p$-completely faithfully flat (i.e., it
becomes faithfully flat after derived modulo $p$).

In particular, if if $(B,d)$ is a bounded oriented prism [BS19, Def 3.2] and
that $B^{\prime\prime}$ is static and has bounded $p$-power torsion, then the
$p$-completion of $\operatorname{PrismEnv}(B\twoheadrightarrow
B^{\prime\prime})/^{\mathbb{L}}d$ is static. Moreover, by [BS19, Lem
3.7(2,3)], the $(p,d)$-completion of
$C:=\operatorname{PrismEnv}(B\twoheadrightarrow B^{\prime\prime})$ is static
and thus it follows from a formal argument that $(C_{(p,d)}^{\wedge},d)$ is
the prismatic envelope of the $\delta$-pair $B\twoheadrightarrow
B^{\prime\prime}$ as long as it is $d$-torsion free. In other words, we
generalize [BS19, Prop 3.13] by weakening regularity to quasiregularity.

As we will explain in the introduction, we prefer to deal with completeness
(and complete flatness) in the general framework of condensed (or more
precisely, solid) mathematics.

We record a simple corollary which furnishes a quite general class of “flat
covers of the final object” in the affine prismatic site (similar to
Definition 4.65) which will be studied in a future work. For this, we need the
following definition:

######  Definition 5.52.

Let $A$ be a $\delta$-ring, $d\in A$ an element and $B$ an animated
$\delta$-$A$-algebra. The $\infty$-category of $\delta$-$(B,d)$-pairs, denoted
by $\operatorname{AniPair}_{\delta,(B,d)}$, is defined to be the undercategory
$(\operatorname{AniPair}_{\delta,(A,d)})_{(B\twoheadrightarrow
B/^{\mathbb{L}}d)\mathord{/}}$.

Let $A$ be a $p$-local $\delta$-ring and $d\in A$ a weakly distinguished non-
zero-divisor. Let $B$ be an animated $\delta$-$A$-algebra, and $R$ an animated
$B/^{\mathbb{L}}d$-algebra. Similar to Definition 4.65, we can consider the
category of animated $\delta$-$B$-algebras $C$ along with a map363636Unlike
the crystalline case, here we do not assume that the map $R\rightarrow
C/^{\mathbb{L}}d$ is an equivalence. $R\rightarrow C/^{\mathbb{L}}d$ of
animated $B/^{\mathbb{L}}d$-algebras, which we will denoted by $R\rightarrow
C/^{\mathbb{L}}d\twoheadleftarrow C$, depicted by the commutative diagram

More formally, this is the fiber product
$\operatorname{Ani}(\operatorname{Ring}_{\delta})_{B\mathord{/}}\times_{\operatorname{Ani}(\operatorname{Ring})_{(B/^{\mathbb{L}}d)\mathord{/}}}\operatorname{Ani}(\operatorname{Ring})_{R\mathord{/}}$
of $\infty$-categories, the opposite category of which will be denoted by
${{\mathbbl{\Delta}}}(R/(B,d))$373737In [BS19], they used the notation
$(R/A)_{{{\mathbbl{\Delta}}}}$. However, this notation is usually devoted to
topoi (such as $X_{\operatorname{et}}$ and $X_{\operatorname{cris}}$). We
therefore adopt the traditional notation for sites.. In a future work, we will
show that the $\infty$-category ${{\mathbbl{\Delta}}}(R/(B,d))$ admits a
Grothendieck topology given by flat covers.

Now let $P$ be an animated $\delta$-$B$-algebra along with a surjection
$P\twoheadrightarrow R$ of animated $B$-algebras such that the cotangent
complex $\mathbb{L}_{P/B}/^{\mathbb{L}}d$ is a flat $P/^{\mathbb{L}}d$-module.

######  Remark 5.53.

We note that such $P$ exists in abundance. For example, this happens when $R$
is a smooth $B/^{\mathbb{L}}d$-algebra which admits a smooth $B$-lift $P$ with
a $\delta$-structure compatible with that on $B$, or $P$ is a polynomial
$B$-algebra $B[x_{i}]$ (of possibly infinitely many variables) with
$\delta(x_{i})=0$ along with a surjection $P\twoheadrightarrow R$ of animated
$B$-algebras.

Then the animated pair $P\twoheadrightarrow R$ admits a canonical animated
$\delta$-$(B,d)$-pair structure, and thus the animated $\delta$-ring
$\operatorname{PrismEnv}(P\twoheadrightarrow R)$ gives rise to an object of
${{\mathbbl{\Delta}}}(R/(B,d))$ (by abuse of notation, we will still denote by
$\operatorname{PrismEnv}(P\twoheadrightarrow R)$ the object of
${{\mathbbl{\Delta}}}(R/(B,d))$).

######  Remark 5.54.

By Lemma 5.26, when $P\twoheadrightarrow R$ is “already” a non-completed prism
in the sense that the induced map $P/^{\mathbb{L}}d\rightarrow R$ is an
equivalence, the non-completed prismatic envelope
$\operatorname{PrismEnv}(P\twoheadrightarrow R)$ is equivalent to $P$ itself.

For any object $(R\rightarrow C/^{\mathbb{L}}d\twoheadleftarrow
C)\in{{\mathbbl{\Delta}}}(R/(B,d))$, by unrolling the definitions, the product
of $(R\rightarrow C/^{\mathbb{L}}d\twoheadleftarrow C)$ and
$\operatorname{PrismEnv}(P\twoheadrightarrow R)$ in
${{\mathbbl{\Delta}}}(R/(B,d))$ is given by
$\operatorname{PrismEnv}(P\otimes_{B}^{\mathbb{L}}C\twoheadrightarrow R)$. We
have therefore a map
$C/^{\mathbb{L}}d\rightarrow\operatorname{PrismEnv}(P\otimes_{B}^{\mathbb{L}}C\twoheadrightarrow
C/^{\mathbb{L}}d)/^{\mathbb{L}}d$ of animated $R$-algebras. The following
proposition is essentially equivalent to the “flat cover of the final object”,
cf. [Cha20, Prop 1.1.2]383838This characterization was already implicit in the
Faltings’s proof of “independence of the choice of the framing”..

######  Proposition 5.55.

Let $A$ be a $p$-local $\delta$-ring and $d\in A$ a weakly transversal
element. Let $B$ be an animated $\delta$-$A$-algebra and $P\twoheadrightarrow
R$ an animated $\delta$-$(B,d)$-pair such that the cotangent complex
$\mathbb{L}_{P/B}/^{\mathbb{L}}d$ is a flat $P/^{\mathbb{L}}d$-module. Then
for every $(R\rightarrow C/^{\mathbb{L}}d\twoheadleftarrow
C)\in{{\mathbbl{\Delta}}}(R/(B,d))$, the map
$C/^{\mathbb{L}}d\rightarrow\operatorname{PrismEnv}(P\otimes_{B}^{\mathbb{L}}C\twoheadrightarrow
C/^{\mathbb{L}}d)/^{\mathbb{L}}d$ is faithfully flat.

###### Proof.

By Proposition 5.49, it suffices to show that the map
$(P/^{\mathbb{L}}d)\otimes_{B/^{\mathbb{L}}d}^{\mathbb{L}}(C/^{\mathbb{L}}d)\rightarrow
C/^{\mathbb{L}}d$ is quasiregular. To simplify the notations, let
$P^{\prime\prime}:=P/^{\mathbb{L}}d$, $B^{\prime\prime}:=B/^{\mathbb{L}}d$ and
$C^{\prime\prime}:=C/^{\mathbb{L}}d$. We have the transitivity sequence

$\mathbb{L}_{(P^{\prime\prime}\otimes_{B^{\prime\prime}}^{\mathbb{L}}C^{\prime\prime})/C^{\prime\prime}}\otimes_{P^{\prime\prime}\otimes_{B^{\prime\prime}}^{\mathbb{L}}C^{\prime\prime}}^{\mathbb{L}}C^{\prime\prime}\rightarrow\mathbb{L}_{C^{\prime\prime}/C^{\prime\prime}}\simeq
0\rightarrow\mathbb{L}_{C^{\prime\prime}/(P^{\prime\prime}\otimes_{B^{\prime\prime}}^{\mathbb{L}}C^{\prime\prime})}$

associated to the maps $C^{\prime\prime}\rightarrow
P^{\prime\prime}\otimes_{B^{\prime\prime}}^{\mathbb{L}}C^{\prime\prime}\rightarrow
C^{\prime\prime}$ whose composite is $\operatorname{id}_{C^{\prime\prime}}$.
Note that
$\mathbb{L}_{(P^{\prime\prime}\otimes_{B^{\prime\prime}}^{\mathbb{L}}C^{\prime\prime})/C^{\prime\prime}}\simeq\mathbb{L}_{P^{\prime\prime}/B^{\prime\prime}}\otimes_{B^{\prime\prime}}^{\mathbb{L}}C^{\prime\prime}$
is a flat
$P^{\prime\prime}\otimes_{B^{\prime\prime}}^{\mathbb{L}}C^{\prime\prime}$-module.
It follows that
$\mathbb{L}_{C^{\prime\prime}/(P^{\prime\prime}\otimes_{B^{\prime\prime}}^{\mathbb{L}}C^{\prime\prime})}[-1]$
is a flat $C^{\prime\prime}$-module. ∎

We first learned the possibility of such kind of result from [MT, Prop 3.4]
(which is closely related to [Cha20, Prop 1.1.2]). Later we came up with an
argument which is essentially equivalent to the proof of Proposition 5.55, but
the foundation was lacking then, therefore the current article could be
understood as paving the way to this proof. Now we want to point out that,
with minor modifications, this proof would imply [MT, Prop 3.4] and the
relevant technical lemmas in the recent works by Y. Tian and by A. Ogus
[Ogu21] announced in Illusie conference. Furthermore, when the proper
foundation is laid, the same proof would lead to a flat cover of the final
object in the absolute prismatic site, and in particular, it would recover
[AL19a, Lem 5.2.8]. We now show this implication.

As in Remark 5.51, we assume that $(B,d)$ is a bounded oriented prism, $R$ is
derived $p$-complete and the map $B/^{\mathbb{L}}d\rightarrow R$ is a
$p$-completely quasisyntomic (i.e. the map $B/^{\mathbb{L}}d\rightarrow R$ is
$p$-completely flat and the cotangent complex
$\mathbb{L}_{R/(B/^{\mathbb{L}}d)}$ has $p$-complete
$\operatorname{Tor}$-amplitude in $[0,1]$ as an $R$-module spectrum). Then by
[BMS19, Lem 4.7], $R$ is static and has bounded $p$-power torsion. Let $P$ be
a derived $(p,d)$-complete animated $\delta$-$B$-algebra which is
$(p,d)$-completely quasismooth (i.e. the map $B\rightarrow P$ is
$(p,d)$-completely flat and the cotangent complex $\mathbb{L}_{P/B}$ is a
$(p,d)$-completely flat $B$-module). Then by [BS19, Lem 3.7(2,3)], $P$ is
static and for every $n\in\mathbb{N}$, the multiplication map
$d^{n}:P\rightarrow P$ is injective and $P/d^{n}$ has bounded $p$-power
torsion.

Now suppose that we are given a surjection $P\twoheadrightarrow R$ of
$B$-algebras. Then by Remark 5.51, the derived $(p,d)$-completion of
$\operatorname{PrismEnv}(P\twoheadrightarrow R)$ is static and the prism
defined by this $(p,d)$-completed algebra is the prismatic envelope in the
sense of [BS19, Prop 3.13], where the $d$-torsion-freeness follows from the
complete flatness of $B\rightarrow P$ and [BS19, Lem 3.7(2)]. Moreover, since
both $B/^{\mathbb{L}}d\rightarrow R$ and
$R\rightarrow\operatorname{PrismEnv}(P\twoheadrightarrow R)/^{\mathbb{L}}d$
are $p$-completely flat, the map
$B\rightarrow\operatorname{PrismEnv}(P\twoheadrightarrow R)$ is
$(p,d)$-completely flat (this in fact generalizes the flatness in [BS19, Prop
3.13]). The proof of Proposition 5.55 shows that

######  Proposition 5.56.

Let $(B,d)$ be a bounded oriented prism, $R$ a derived $p$-complete and
$p$-completely quasisyntomic $B/d$-algebra. Let $P$ be a derived
$(p,d)$-complete animated $\delta$-$B$-algebra which is $(p,d)$-completely
quasismooth over $B$, equipped with a surjection $P\twoheadrightarrow R$ of
$B$-algebras. Then the $(p,d)$-completion of
$\operatorname{PrismEnv}(P\twoheadrightarrow R)$ is static which gives rise to
a bounded prism $(C,d)$ in the prismatic site393939It is the non-animated but
$(p,d)$-completed version of our ${{\mathbbl{\Delta}}}(R/(B,d))$. defined in
[BS19, Def 4.1] of $R$ relative to the base prism $(B,d)$. Furthermore,
$(C,d)$ is a flat cover of the final object in this site.

This implies virtually all the similar technical cover results for relative
prismatic site mentioned above, cf. Remark 5.54.

######  Remark 5.57.

For the absolute prismatic site, the proof also works in the special case of
[AL19a, Lem 5.2.8], but we are not aware of a statement as general as
Proposition 5.56.

##  Appendix A Animations and projectively generated categories

In this appendix, we recollect basic category-theoretic facts about animations
[CS19] and projectively generated categories needed in the text.

###  A.1. Projectively generated $\infty$-categories

We recall that $\mathcal{S}$ is the $\infty$-category of animae (see Section
1). In very few places, we will encounter large animae of which the
$\infty$-category will be denoted by $\hat{\mathcal{S}}$. One way to
distinguish small objects and large objects is to fix a Grothendieck universe.

######  Definition A.1 ([Lur09, Rem 5.5.8.20]).

Let $\mathcal{C}$ be a cocomplete $\infty$-category and $X\in\mathcal{C}$ an
object. We say that $X$ is compact and projective, or that $X$ is a compact
projective object, if the functor
$\mathcal{C}\rightarrow\hat{\mathcal{S}},Y\mapsto\operatorname{Map}_{\mathcal{C}}(X,Y)$
corepresented by $X$ commutes with filtered colimits and geometric
realizations.

######  Remark A.2.

Here we need $\hat{\mathcal{S}}$ in lieu of $\mathcal{S}$ because the
$\infty$-category $\mathcal{C}$ is not necessarily locally small. In practice,
the $\infty$-categories that we encounter, e.g. projectively generated
$\infty$-categories, are a fortiori locally small, but not necessarily a
priori locally small.

######  Definition A.3 ([Lur09, Def 5.5.8.23]).

Let $\mathcal{C}$ be a cocomplete $\infty$-category and
$S\subseteq\mathcal{C}$ a (small) collection of objects of $\mathcal{C}$. We
say that $S$ is a set of compact projective generators for $\mathcal{C}$ if
the following conditions are satisfied:

  1. (1)

Each element of $S$ is a compact projective object of $\mathcal{C}$.

  2. (2)

The full subcategory of $\mathcal{C}$ spanned by finite coproducts of elements
of $S$ is essentially small.

  3. (3)

The set $S$ generates $\mathcal{C}$ under small colimits.

We say that an $\infty$-category $\mathcal{C}$ is projectively generated if it
is cocomplete and there exists a set $S$ of compact projective generators for
$\mathcal{C}$.

######  Remark A.4.

Let $\mathcal{C}$ be a cocomplete $\infty$-category and
$\mathcal{C}_{0}\subseteq\mathcal{C}$ an essentially small full subcategory.
Then we will abuse the terminology by saying that $\mathcal{C}_{0}$ is a set
of compact projective generators for $\mathcal{C}$ if a skeleton of
$\mathcal{C}_{0}$ is a set of compact generators for $\mathcal{C}$.

A closely related concept is that of sifted colimits which is based on the
following definition:

######  Definition A.5 ([Lur09, Def 5.5.8.1]).

A simplicial set $K$ is called sifted if it satisfies the following
conditions:

  1. (1)

The simplicial set $K$ is nonempty.

  2. (2)

The diagonal map $K\rightarrow K\times K$ is cofinal.

Projective generated $\infty$-categories are essentially determined by a set
of compact projective generators. More precisely, we have the following:

######  Notation A.6 ([Lur09, Def 5.5.8.8]).

Let $\mathcal{C}$ be a small $\infty$-category which admits finite coproducts.
We let $\mathcal{P}_{\Sigma}(\mathcal{C})$ denote the full subcategory of
$\mathcal{P}(\mathcal{C}):=\operatorname{Fun}(\mathcal{C}^{\operatorname{op}},\mathcal{S})$
spanned by those functors
$\mathcal{C}^{\operatorname{op}}\rightarrow\mathcal{S}$ which preserves finite
products.

######  Proposition A.7.

Let $\mathcal{C}$ be a small $\infty$-category which admits finite coproducts.
Then

  1. (1)

The $\infty$-category $\mathcal{P}_{\Sigma}(\mathcal{C})$ is an accessible
localization of $\mathcal{P}(\mathcal{C})$, therefore presentable.

  2. (2)

The Yoneda embedding $j:\mathcal{C}\rightarrow\mathcal{P}(\mathcal{C})$
factors through $\mathcal{P}_{\Sigma}(\mathcal{C})$. Moreover, the induced
functor $\mathcal{C}\rightarrow\mathcal{P}_{\Sigma}(\mathcal{C})$ preserves
finite coproducts.

  3. (3)

Let $\mathcal{D}$ be a presentable $\infty$-category and let
$\mathcal{P}(\mathcal{C})\underset{G}{\overset{F}{\mathrel{\begin{subarray}{c}\longrightarrow\\\\[-2.58334pt]
\longleftarrow\end{subarray}}}}\mathcal{D}$ be a pair of adjoint functors.
Then $G$ factors through $\mathcal{P}_{\Sigma}(\mathcal{C})$ if and only if
$f=F\circ j:\mathcal{C}\rightarrow\mathcal{D}$ preserves finite coproducts.

  4. (4)

The full subcategory
$\mathcal{P}_{\Sigma}(\mathcal{C})\subseteq\mathcal{P}(\mathcal{C})$ is stable
under sifted colimits.

We recall that, for a small $\infty$-category $\mathcal{C}$,
$\operatorname{Ind}(\mathcal{C})\subseteq\mathcal{P}(\mathcal{C})$ is the full
subcategory generated under filtered colimits by the essential image of the
Yoneda embedding $\mathcal{C}\rightarrow\mathcal{P}(\mathcal{C})$, [Lur09,
Prop 5.3.5.3 & Cor 5.3.5.4]. It follows from [Lur09, Prop 5.3.5.11] that

######  Lemma A.8.

Let $\mathcal{C}$ be a small $\infty$-category which admits finite coproducts.
Then the fully faithful embedding
$\mathcal{C}\hookrightarrow\mathcal{P}_{\Sigma}(\mathcal{C})$ extends uniquely
to a functor
$\operatorname{Ind}(\mathcal{C})\rightarrow\mathcal{P}_{\Sigma}(\mathcal{C})$
which preserves filtered colimit. This functor
$\operatorname{Ind}(\mathcal{C})\rightarrow\mathcal{P}_{\Sigma}(\mathcal{C})$
is fully faithful.

######  Lemma A.9.

Let $\mathcal{C}$ be a small $\infty$-category which admits finite coproducts.
Then the $\infty$-category $\mathcal{P}_{\Sigma}(\mathcal{C})$ is projectively
generated for which $\mathcal{C}\subseteq\mathcal{P}_{\Sigma}(\mathcal{C})$ is
a set of projective generators. In fact, for any
$X\in\mathcal{P}_{\Sigma}(\mathcal{C})$, there exists a simplicial object
$U_{\bullet}:\boldsymbol{\Delta}^{\operatorname{op}}\rightarrow\operatorname{Ind}(\mathcal{C})$
whose colimit is $X$.

###### Proof.

First, since
$\mathcal{P}_{\Sigma}(\mathcal{C})\subseteq\mathcal{P}(\mathcal{C})$ is a
accessible localization, $\mathcal{P}_{\Sigma}(\mathcal{C})$ is presentable
[Lur09, Rem 5.5.1.6] therefore cocomplete. Since
$\mathcal{P}_{\Sigma}(\mathcal{C})\subseteq\mathcal{P}(\mathcal{C})$ is stable
under sifted colimits (Proposition A.7), the objects of $\mathcal{C}$ are
compact and projective. The last statement then follows from [Lur09, Lem
5.5.8.14]. ∎

######  Proposition A.10 ([Lur09, Prop 5.5.8.15]).

Let $\mathcal{C}$ be a small $\infty$-category which admits finite coproducts
and let $\mathcal{D}$ be an $\infty$-category which admits filtered colimits
and geometric realizations. Let
$\operatorname{Fun}_{\Sigma}(\mathcal{P}_{\Sigma}(\mathcal{C}),\mathcal{D})$
denote the full subcategory spanned by those functors
$\mathcal{P}_{\Sigma}(\mathcal{C})\rightarrow\mathcal{D}$ which preserve
filtered colimits and geometric realizations. Then

  1. (1)

Composition with the Yoneda embedding
$j:\mathcal{C}\rightarrow\mathcal{P}_{\Sigma}(\mathcal{C})$ induces an
equivalence
$\theta:\operatorname{Fun}_{\Sigma}(\mathcal{P}_{\Sigma}(\mathcal{C}),\mathcal{D})\rightarrow\operatorname{Fun}(\mathcal{C},\mathcal{D})$
of categories. The inverse $\theta^{-1}$ is given by the left Kan extension
along $j$. In this case, we will call $\theta^{-1}(f)$ the left derived
functor of $f\in\operatorname{Fun}(\mathcal{C},\mathcal{D})$.

  2. (2)

Any functor
$g\in\operatorname{Fun}_{\Sigma}(\mathcal{P}_{\Sigma}(\mathcal{C}),\mathcal{D})$
preserves sifted colimits.

  3. (3)

Assume that $\mathcal{D}$ admits finite coproducts. A functor
$g\in\operatorname{Fun}_{\Sigma}(\mathcal{P}_{\Sigma}(\mathcal{C}),\mathcal{D})$
preserves small colimits if and only if $g\circ j$ preserves finite
coproducts.

######  Proposition A.11 ([Lur09, Prop 5.5.8.22]).

Let $\mathcal{C}$ be a small $\infty$-category which admits finite coproducts,
$\mathcal{D}$ an $\infty$-category which admits filtered colimits and
geometric realizations, and
$F:\mathcal{P}_{\Sigma}(\mathcal{C})\rightarrow\mathcal{D}$ a left derived
functor of $f=F\circ j:\mathcal{C}\rightarrow\mathcal{D}$, where
$j:\mathcal{C}\rightarrow\mathcal{P}_{\Sigma}(\mathcal{D})$ denotes the Yoneda
embedding. Consider the following conditions:

  1. (1)

The functor $f$ is fully faithful.

  2. (2)

The essential image of $f$ consists of compact projective objects of
$\mathcal{D}$.

  3. (3)

The $\infty$-category $\mathcal{D}$ is generated by the essential image of $f$
under filtered colimits and geometric realizations.

If 1 and 2 are satisfied, then $F$ is fully faithful. Moreover, $F$ is an
equivalence if and only if 1, 2 and 3 are satisfied.

######  Proposition A.12 ([Lur09, Prop 5.5.8.25]).

Let $\mathcal{C}$ be a projectively generated $\infty$-category with a set $S$
of compact projective generators for $\mathcal{C}$. Then

  1. (1)

Let $\mathcal{C}^{0}\subseteq\mathcal{C}$ be the full subcategory spanned by
finite coproducts of the objects in $S$. Then $\mathcal{C}^{0}$ is essentially
small, and the left derived functor
$F:\mathcal{P}_{\Sigma}(\mathcal{C}^{0})\rightarrow\mathcal{C}$ is an
equivalence of $\infty$-categories. In particular, $\mathcal{C}$ is a
compactly generated presentable $\infty$-category.

  2. (2)

Let $C\in\mathcal{C}$ be an object. The following conditions are equivalent:

    1. (a)

The object $C\in\mathcal{C}$ is compact and projective.

    2. (b)

The functor $\mathcal{C}\rightarrow\mathcal{S}$ corepresented by $C$ preserves
sifted colimits.

    3. (c)

There exists an object $C^{\prime}\in\mathcal{C}^{0}$ such that $C$ is a
retract of $C^{\prime}$.

###### Proof.

We explain more details of the first point than [Lur09, Prop 5.5.8.25]. It
follows from definitions that $\mathcal{C}^{0}$ is essentially small. Then it
follows from Proposition A.11 that the left derived functor
$F:\mathcal{P}_{\Sigma}(\mathcal{C}^{0})\rightarrow\mathcal{C}$ is fully
faithful. Since $\mathcal{C}^{0}\subseteq\mathcal{C}$ is stable under finite
coproducts taken in $\mathcal{C}$, the embedding
$\mathcal{C}^{0}\hookrightarrow\mathcal{C}$ preserves finite coproducts. It
follows from Proposition A.10 that $F$ preserves small colimits, thus the
essential image of $F$ is stable under small colimits. By assumption, $S$
generates $\mathcal{C}$ under small colimits, therefore $F$ is essentially
surjective. ∎

######  Corollary A.13.

Let $\mathcal{C}$ be a projectively generated $\infty$-category and let
$\mathcal{D}$ be an $\infty$-category which admits filtered colimits and
geometric realizations. If a functor $\mathcal{C}\rightarrow\mathcal{D}$
preserves filtered colimits and geometric realizations, then it also preserve
sifted colimits.

######  Warning A.14.

Filtered colimits and geometric realizations are sifted colimits, therefore if
a functor preserves sifted colimits, then it also preserves filtered colimits
and geometric realizations. However, the converse is in general false.

The following proposition is extremely useful to detect projectively generated
$\infty$-categories:

######  Proposition A.15 ([Lur17, Cor 4.7.3.18]).

Given a pair
$\mathcal{C}\underset{G}{\overset{F}{\mathrel{\begin{subarray}{c}\longrightarrow\\\\[-2.58334pt]
\longleftarrow\end{subarray}}}}\mathcal{D}$ of adjoint functors between
$\infty$-categories. Assume that

  1. (1)

The $\infty$-category $\mathcal{D}$ admits filtered colimits and geometric
realizations, and the functor $G$ preserves filtered colimits and geometric
realizations.

  2. (2)

The $\infty$-category $\mathcal{C}$ is projectively generated.

  3. (3)

The functor $G$ is conservative.

Then

  1. (1)

The $\infty$-category $\mathcal{D}$ is projectively generated.

  2. (2)

An object $D\in\mathcal{D}$ is compact and projective if and only if there
exists a compact projective object $C\in\mathcal{C}$ such that $D$ is a
retract of $F(C)$.

  3. (3)

The functor $G$ preserves all sifted colimits.

###  A.2. Projectively generated $n$-categories

In this subsection, we will briefly describe the $n$-categorical analogue of
Section A.1. We say that an anima $X$ is $n$-truncated for
$n\in\mathbb{N}_{\geq 0}$ if the homotopy groups $\pi_{i}(X,x)=0$ for every
point $x\in X$ and every $i\in\mathbb{N}_{\char 62\relax n}$, and
$(-1)$-truncated if $X$ is either empty or contractible, and $(-2)$-truncated
if $X=\varnothing$. An $\infty$-category $\mathcal{C}$ is an $n$-category
[Lur09, Prop 2.3.4.18] if for every pair
$(X,Y)\in\mathcal{C}\times\mathcal{C}$ of objects, the mapping anima
$\operatorname{Map}_{\mathcal{C}}(X,Y)$ is $(n-1)$-truncated. We will denote
by $\mathcal{S}_{\leq n}$ the $\infty$-category of $n$-truncated animae, and
by $\hat{\mathcal{S}}_{\leq n}$ the $\infty$-category of large $n$-truncated
animae.

######  Remark A.16.

$1$-categories are just categories in the classical category theory. If we
define $\infty$-categories as quasicategories as in [Lur09], this
identification is given by the nerve construction. Since in our texts,
categories often mean $\infty$-categories, we usually add “$1$-” to avoid
possible ambiguities.

In fact, for the text, we only need results for $n=1$ (and $n=\infty$ in some
sense), but the generalization to general $n\in\mathbb{N}_{\char 62\relax 0}$
is quite cost-free.

######  Proposition A.17 ([Lur09, Cor 2.3.4.8]).

Let $\mathcal{C}$ be an $n$-category and $K$ a simplicial set. Then
$\operatorname{Fun}(K,\mathcal{C})$ is an $n$-category.

######  Definition A.18.

Let $\mathcal{C}$ be a cocomplete $n$-category and $X\in\mathcal{C}$ an
object. We say that $X$ is compact and $n$-projective, or that $X$ is a
compact $n$-projective object, if the functor
$\mathcal{C}\rightarrow\hat{\mathcal{S}}_{\leq
n-1},Y\mapsto\operatorname{Map}_{\mathcal{C}}(X,Y)$ corepresented by $X$
commutes with filtered colimits and geometric realizations.

######  Remark A.19.

In fact, an object $X\in\mathcal{C}$ is called $n$-projective if and only if
the functor $\mathcal{C}\rightarrow\hat{\mathcal{S}}_{\leq
n-1},Y\mapsto\operatorname{Map}_{\mathcal{C}}(X,Y)$ corepresented by $X$
commutes with geometric realizations. In particular, when $\mathcal{C}$ is an
abelian $1$-category, an object $X\in\mathcal{C}$ is $1$-projective if and
only if it is a “projective object” of the abelian $1$-category $\mathcal{C}$.

######  Remark A.20.

Let $\mathcal{C}$ be a cocomplete $n$-category and $X\in\mathcal{C}$ a compact
$n$-projective object. In general, $X$ is not a compact projective object of
$\mathcal{C}$ as an $\infty$-category. In fact, the inclusion
$\hat{\mathcal{S}}_{\leq n-1}\rightarrow\hat{\mathcal{S}}$ does not commute
with geometric realizations. That is to say, for general simplicial objects
$Y_{\bullet}:\boldsymbol{\Delta}^{\operatorname{op}}\rightarrow\mathcal{C}$,
the geometric realization
$|\operatorname{Map}_{\mathcal{C}}(X,Y_{\bullet})|_{\bullet\in\boldsymbol{\Delta}^{\operatorname{op}}}$
is not in general $(n-1)$-truncated.

######  Remark A.21.

There is another way to characterize geometric realizations in an $n$-category
$\mathcal{C}$. In fact, the fully faithful embedding
$\boldsymbol{\Delta}_{\leq[n]}^{\operatorname{op}}\hookrightarrow\boldsymbol{\Delta}^{\operatorname{op}}$
is “$n$-cofinal”, therefore the geometric realization of a simplicial object
$\boldsymbol{\Delta}^{\operatorname{op}}\rightarrow\mathcal{C}$ exists if and
only if colimit of the composite functor
$\boldsymbol{\Delta}_{\leq[n]}^{\operatorname{op}}\hookrightarrow\boldsymbol{\Delta}^{\operatorname{op}}\rightarrow\mathcal{C}$
exists, and the two colimits are equivalent. Furthermore, for any diagram
$\boldsymbol{\Delta}_{\leq[n]}^{\operatorname{op}}\rightarrow\mathcal{C}$, the
left Kan extension along
$\boldsymbol{\Delta}_{\leq[n]}^{\operatorname{op}}\hookrightarrow\boldsymbol{\Delta}^{\operatorname{op}}$
always exists. Thus for a cocomplete $n$-category $\mathcal{C}$, an object
$X\in\mathcal{C}$ is $n$-projective if and only if the functor
$\operatorname{Map}_{\mathcal{C}}(X,\cdot)$ corepresented by $X$ preserves
$\boldsymbol{\Delta}_{\leq[n]}^{\operatorname{op}}$-indexed colimits. See
[Nar16] and the proof of [Lur17, Lem 1.3.3.10].

######  Definition A.22.

Let $\mathcal{C}$ be a cocomplete $n$-category and $S\subseteq\mathcal{C}$ a
(small) collection of objects of $\mathcal{C}$. We say that $S$ is a set of
compact $n$-projective generators for $\mathcal{C}$ if the following
conditions are satisfied:

  1. (1)

Each element of $S$ is a compact $n$-projective object of $\mathcal{C}$.

  2. (2)

The full subcategory of $\mathcal{C}$ spanned by finite coproducts of elements
of $S$ is essentially small.

  3. (3)

The set $S$ generates $\mathcal{C}$ under small colimits.

We say that an $n$-category $\mathcal{C}$ is $n$-projectively generated if it
is cocomplete and there exists a set $S$ of compact $n$-projective generators
for $\mathcal{C}$.

Similar to $\infty$-categories, we have

######  Notation A.23.

Let $\mathcal{C}$ be a small $n$-category which admits finite coproducts. We
let $\mathcal{P}_{\Sigma,n}(\mathcal{C})$ denote the full subcategory of
$\mathcal{P}_{n}(\mathcal{C}):=\operatorname{Fun}(\mathcal{C}^{\operatorname{op}},\mathcal{S}_{\leq
n-1})$ spanned by those functors
$\mathcal{C}^{\operatorname{op}}\rightarrow\mathcal{S}_{\leq n-1}$ which
preserves finite products.

######  Proposition A.24.

Let $\mathcal{C}$ be a small $n$-category which admits finite coproducts. Then

  1. (1)

The $\infty$-category $\mathcal{P}_{\Sigma,n}(\mathcal{C})$ is an accessible
localization of $\mathcal{P}_{n}(\mathcal{C})$, therefore presentable.

  2. (2)

The Yoneda embedding $j:\mathcal{C}\rightarrow\mathcal{P}_{n}(\mathcal{C})$
factors through $\mathcal{P}_{\Sigma,n}(\mathcal{C})$. Moreover, the induced
functor $\mathcal{C}\rightarrow\mathcal{P}_{\Sigma,n}(\mathcal{C})$ preserves
finite coproducts.

  3. (3)

Let $\mathcal{D}$ be a presentable $n$-category and let
$\mathcal{P}(\mathcal{C})\underset{G}{\overset{F}{\mathrel{\begin{subarray}{c}\longrightarrow\\\\[-2.58334pt]
\longleftarrow\end{subarray}}}}\mathcal{D}$ be a pair of adjoint functors.
Then $G$ factors through $\mathcal{P}_{\Sigma,n}(\mathcal{C})$ if and only if
$f=F\circ j:\mathcal{C}\rightarrow\mathcal{D}$ preserves finite coproducts.

  4. (4)

The full subcategory
$\mathcal{P}_{\Sigma,n}(\mathcal{C})\subseteq\mathcal{P}_{n}(\mathcal{C})$ is
stable under sifted colimits404040We do not introduce $n$-sifted diagrams, so
a priori it is a sifted diagram defined in [Lur09, Def 5.5.8.1]. However, here
one can replace sifted diagrams by $n$-sifted diagram. See Remark A.21. .

######  Lemma A.25.

Let $\mathcal{C}$ be a small $n$-category which admits finite coproducts. Then
the fully faithful embedding
$\mathcal{C}\hookrightarrow\mathcal{P}_{\Sigma,n}(\mathcal{C})$ extends
uniquely to a functor
$\operatorname{Ind}(\mathcal{C})\rightarrow\mathcal{P}_{\Sigma,n}(\mathcal{C})$
which preserves filtered colimit. This functor
$\operatorname{Ind}(\mathcal{C})\rightarrow\mathcal{P}_{\Sigma,n}(\mathcal{C})$
is fully faithful.

######  Lemma A.26.

Let $\mathcal{C}$ be a small $n$-category which admits finite coproducts. Then
the $n$-category $\mathcal{P}_{\Sigma,n}(\mathcal{C})$ is $n$-projectively
generated for which $\mathcal{C}\subseteq\mathcal{P}_{\Sigma,n}(\mathcal{C})$
is a set of $n$-projective generators. In fact, for any
$X\in\mathcal{P}_{\Sigma,n}(\mathcal{C})$, there exists a simplicial object
$U_{\bullet}:\boldsymbol{\Delta}^{\operatorname{op}}\rightarrow\operatorname{Ind}(\mathcal{C})$
(or equivalently, a diagram $\boldsymbol{\Delta}_{\leq
n}^{\operatorname{op}}\rightarrow\operatorname{Ind}(\mathcal{C})$ by Remark
A.21) whose colimit is $X$.

######  Proposition A.27.

Let $\mathcal{C}$ be a small $n$-category which admits finite coproducts and
let $\mathcal{D}$ be an $n$-category which admits filtered colimits and
geometric realizations. Let
$\operatorname{Fun}_{\Sigma}(\mathcal{P}_{\Sigma,n}(\mathcal{C}),\mathcal{D})$
denote the full subcategory spanned by those functors
$\mathcal{P}_{\Sigma,n}(\mathcal{C})\rightarrow\mathcal{D}$ which preserve
filtered colimits and geometric realizations. Then

  1. (1)

Composition with the Yoneda embedding
$j:\mathcal{C}\rightarrow\mathcal{P}_{\Sigma,n}(\mathcal{C})$ induces an
equivalence
$\theta:\operatorname{Fun}_{\Sigma}(\mathcal{P}_{\Sigma,n}(\mathcal{C}),\mathcal{D})\rightarrow\operatorname{Fun}(\mathcal{C},\mathcal{D})$
of categories. The inverse $\theta^{-1}$ is given by the left Kan extension
along $j$. In this case, we will call $\theta^{-1}(f)$ the left derived
functor of $f\in\operatorname{Fun}(\mathcal{C},\mathcal{D})$.

  2. (2)

Any functor
$g\in\operatorname{Fun}_{\Sigma}(\mathcal{P}_{\Sigma,n}(\mathcal{C}),\mathcal{D})$
preserves sifted colimits.

  3. (3)

Assume that $\mathcal{D}$ admits finite coproducts. A functor
$g\in\operatorname{Fun}_{\Sigma}(\mathcal{P}_{\Sigma,n}(\mathcal{C}),\mathcal{D})$
preserves small colimits if and only if $g\circ j$ preserves finite
coproducts.

######  Proposition A.28.

Let $\mathcal{C}$ be a small $n$-category which admits finite coproducts,
$\mathcal{D}$ an $n$-category which admits filtered colimits and geometric
realizations, and
$F:\mathcal{P}_{\Sigma,n}(\mathcal{C})\rightarrow\mathcal{D}$ a left derived
functor of $f=F\circ j:\mathcal{C}\rightarrow\mathcal{D}$, where
$j:\mathcal{C}\rightarrow\mathcal{P}_{\Sigma,n}(\mathcal{D})$ denotes the
Yoneda embedding. Consider the following conditions:

  1. (1)

The functor $f$ is fully faithful.

  2. (2)

The essential image of $f$ consists of compact $n$-projective objects of
$\mathcal{D}$.

  3. (3)

The $n$-category $\mathcal{D}$ is generated by the essential image of $f$
under filtered colimits and geometric realizations.

If 1 and 2 are satisfied, then $F$ is fully faithful. Moreover, $F$ is an
equivalence if and only if 1, 2 and 3 are satisfied.

######  Proposition A.29.

Let $\mathcal{C}$ be a $n$-projectively generated $n$-category with a set $S$
of compact $n$-projective generators for $\mathcal{C}$. Then

  1. (1)

Let $\mathcal{C}^{0}\subseteq\mathcal{C}$ be the full subcategory spanned by
finite coproducts of the objects in $S$. Then $\mathcal{C}^{0}$ is essentially
small, and the left derived functor
$F:\mathcal{P}_{\Sigma,n}(\mathcal{C}^{0})\rightarrow\mathcal{C}$ is an
equivalence of $n$-categories. In particular, $\mathcal{C}$ is a compactly
generated presentable $n$-category.

  2. (2)

Let $C\in\mathcal{C}$ be an object. The following conditions are equivalent:

    1. (a)

The object $C\in\mathcal{C}$ is compact and $n$-projective.

    2. (b)

The functor $\mathcal{C}\rightarrow\mathcal{S}_{\leq n-1}$ corepresented by
$C$ preserves sifted colimits.

    3. (c)

There exists an object $C^{\prime}\in\mathcal{C}^{0}$ such that $C$ is a
retract of $C^{\prime}$.

######  Corollary A.30.

Let $\mathcal{C}$ be a projectively generated $n$-category and let
$\mathcal{D}$ be an $n$-category which admits filtered colimits and geometric
realizations. If a functor $\mathcal{C}\rightarrow\mathcal{D}$ preserves
filtered colimits and geometric realizations, then it also preserve sifted
colimits.

As in the case of $\infty$-categories, the following proposition is essential:

######  Proposition A.31 ([Lur17, Cor 4.7.3.18]).

Given a pair
$\mathcal{C}\underset{G}{\overset{F}{\mathrel{\begin{subarray}{c}\longrightarrow\\\\[-2.58334pt]
\longleftarrow\end{subarray}}}}\mathcal{D}$ of adjoint functors between
$n$-categories. Assume that

  1. (1)

The $n$-category $\mathcal{D}$ admits filtered colimits and geometric
realizations, and the functor $G$ preserves filtered colimits and geometric
realizations.

  2. (2)

The $n$-category $\mathcal{C}$ is $n$-projectively generated.

  3. (3)

The functor $G$ is conservative.

Then

  1. (1)

The $n$-category $\mathcal{D}$ is $n$-projectively generated.

  2. (2)

An object $D\in\mathcal{D}$ is compact and $n$-projective if and only if there
exists a compact $n$-projective object $C\in\mathcal{C}$ such that $D$ is a
retract of $F(C)$.

  3. (3)

The functor $G$ preserves all sifted colimits.

###  A.3. Animation of $n$-projectively generated $n$-categories

In this subsection, we describe a procedure, called animation, introduced in
[CS19, §5.1], to produce a projectively generated $\infty$-category from an
$n$-projectively generated $n$-category. Roughly speaking, this projectively
generated $\infty$-category is determined by a set of compact $n$-projective
generators for the $n$-category in question.

######  Definition A.32.

Let $\mathcal{C}$ be an $n$-projectively generated $n$-category. We choose a
set $S\subseteq\mathcal{C}$ of compact $n$-projective generators for
$\mathcal{C}$. Let $\mathcal{C}^{0}\subseteq\mathcal{C}$ be the full
subcategory spanned by finite coproducts of the objects in $S$. Then the
animation of $\mathcal{C}$, denoted by $\operatorname{Ani}(\mathcal{C})$, is
defined to be the projectively generated $\infty$-category
$\mathcal{P}_{\Sigma}(\mathcal{C}^{0})$.

######  Remark A.33.

The definition of the animation does not depend on the choice of the set of
compact $n$-projective generators. The key is that if $S^{\prime}$ is another
compact $n$-projective generators, then it follows from Proposition A.29 that
every object $X^{\prime}\in S^{\prime}$ is a retract of an object
$X\in\mathcal{C}^{0}$ in Definition A.32. The same applies to the discussions
below.

######  Example A.34.

Let $\operatorname{Ab}$ be the abelian category of abelian groups. Then
$\operatorname{Ani}(\operatorname{Ab})$ coincides with the (connective)
derived category $D_{\geq 0}(\operatorname{Ab})$.

######  Remark A.35.

In the context of Definition A.32, we have
$\mathcal{C}\simeq\mathcal{P}_{\Sigma,n}(\mathcal{C}^{0})$ by Proposition A.29
and
$\operatorname{Ani}(\mathcal{C})\simeq\mathcal{P}_{\Sigma}(\mathcal{C}^{0})$.
It follows that the $n$-category $\mathcal{C}$ could be identified with
$n$-truncated objects in $\operatorname{Ani}(\mathcal{C})$. In particular,
there exists a left adjoint $\tau_{\leq
n-1}:\operatorname{Ani}(\mathcal{C})\rightarrow\mathcal{C}$ to the fully
faithful embedding
$\mathcal{C}\hookrightarrow\operatorname{Ani}(\mathcal{C})$, cf. [Lur09, Rem
5.5.8.26].

We now discuss the animation of functors.

######  Definition A.36 ([CS19, §5.1.4]).

Let $\mathcal{C},\mathcal{D}$ be two $n$-projectively generated $n$-categories
and $F:\mathcal{C}\rightarrow\mathcal{D}$ a functor. Then the animation of the
functor $F$, denoted by
$\operatorname{Ani}(F):\operatorname{Ani}(\mathcal{C})\rightarrow\operatorname{Ani}(\mathcal{D})$,
is defined as follows:

We choose a set $S\subseteq\mathcal{C}$ of compact $n$-projective generators
for $\mathcal{C}$. Let $\mathcal{C}^{0}\subseteq\mathcal{C}$ be the full
subcategory spanned by finite coproducts of the objects in $S$. Then the
functor $F:\mathcal{C}\rightarrow\mathcal{D}$ gives rise to the composite
$\mathcal{C}^{0}\rightarrow\mathcal{C}\rightarrow\mathcal{D}\rightarrow\operatorname{Ani}(\mathcal{D})$.
We define
$\operatorname{Ani}(F):\operatorname{Ani}(\mathcal{C})\rightarrow\operatorname{Ani}(\mathcal{D})$
to be the left derived functor (in Proposition A.10) of
$\mathcal{C}^{0}\rightarrow\operatorname{Ani}(\mathcal{D})$.

######  Example A.37.

Let $F:\operatorname{Ab}\rightarrow\operatorname{Ab}$ be an additive functor.
Then the animation
$\operatorname{Ani}(F):\operatorname{Ani}(\operatorname{Ab})\rightarrow\operatorname{Ani}(\operatorname{Ab})$
coincides with the left derived functor $\mathbb{L}F:D_{\geq
0}(\operatorname{Ab})\rightarrow D_{\geq 0}(\operatorname{Ab})$ in homological
algebra.

It follows from Propositions A.27, A.29, A.10 and A.12 that

######  Corollary A.38.

In Definition A.36, if $F$ preserves sifted colimits (cf. Corollary A.30),
then so does $\operatorname{Ani}(F)$. Furthermore, if $F$ preserves small
colimits, then so does $\operatorname{Ani}(F)$.

In homological algebra, there is a natural comparison map
$H_{0}\circ\mathbb{L}F\rightarrow F\circ H_{0}$, which becomes an equivalence
when $F$ is assumed to be right exact. Now we study the animated analogue. In
the context of Definition A.36, the composite functor
$\operatorname{Ani}(\mathcal{C})\xrightarrow{\tau_{\leq
n-1}}\mathcal{C}\xrightarrow{F}\mathcal{D}\overset{j_{\mathcal{D}}}{\hookrightarrow}\operatorname{Ani}(\mathcal{D})$
is an extension of the composite functor
$\mathcal{C}\xrightarrow{F}\mathcal{D}\hookrightarrow\operatorname{Ani}(\mathcal{D})$.
Since
$\operatorname{Ani}(F):\operatorname{Ani}(\mathcal{C})\rightarrow\operatorname{Ani}(\mathcal{D})$
is the left Kan extension, there exists an essentially unique map
$\operatorname{Ani}(F)\rightarrow j_{\mathcal{D}}\circ F\circ\tau_{\leq n-1}$
of functors
$\operatorname{Ani}(\mathcal{C})\rightrightarrows\operatorname{Ani}(\mathcal{D})$.
By adjunction, we get a canonical map $\tau_{\leq
n-1}\circ\operatorname{Ani}(F)\rightarrow F\circ\tau_{\leq n-1}$ of functors
$\operatorname{Ani}(\mathcal{C})\rightrightarrows\mathcal{D}$.

######  Lemma A.39 ([CS19, §5.1.4]).

In Definition A.36, suppose that the functor
$F:\mathcal{C}\rightarrow\mathcal{D}$ (between $n$-categories) preserves
sifted colimits. Then the map $\tau_{\leq
n-1}\circ\operatorname{Ani}(F)\rightarrow F\circ\tau_{\leq n-1}$ of functors
constructed above is an equivalence of functors.

###### Proof.

First, note that the map $\tau_{\leq n-1}\circ\operatorname{Ani}(F)\rightarrow
F\circ\tau_{\leq n-1}$ of functors
$\operatorname{Ani}(\mathcal{C})\rightrightarrows\mathcal{D}$ is an
equivalence of functors after composing with the inclusion
$\mathcal{C}^{0}\hookrightarrow\operatorname{Ani}(\mathcal{C})$. We claim that
both functors $\tau_{\leq n-1}\circ\operatorname{Ani}(F)$ and
$F\circ\tau_{\leq n-1}$ preserve sifted colimits, thus belonging to
$\operatorname{Fun}_{\Sigma}(\operatorname{Ani}(\mathcal{C}),\mathcal{D})$
which becomes an equivalence after mapped along
$\operatorname{Fun}_{\Sigma}(\operatorname{Ani}(\mathcal{C}),\mathcal{D})\rightarrow\operatorname{Fun}(\mathcal{C},\mathcal{D})$,
and hence by Proposition A.10, the constructed map of functors is an
equivalence.

In fact, since $\tau_{\leq n-1}$ is a left adjoint, therefore commutes with
small colimits, which implies that $\tau_{\leq n-1}\circ\operatorname{Ani}(F)$
commutes with sifted colimits. On the other hand,
$F:\mathcal{C}\rightarrow\mathcal{D}$ is a functor which preserves sifted
colimits, therefore also preserves sifted colimits since
$\mathcal{C},\mathcal{D}$ are $n$-categories. Thus $F\circ\tau_{\leq n-1}$
also preserves sifted colimits. ∎

In homological algebra, leftly deriving functors is not compatible with
compositions, therefore neither is animation of functors in general. However,
recall that with some acyclicity conditions [Sta21, Tag 015M], there is a
compatibility of leftly deriving functors and compositions. Here is such a
condition in the world of animations:

######  Proposition A.40 ([CS19, Prop 5.1.5]).

Let $\mathcal{C},\mathcal{D},\mathcal{E}$ be three $n$-projectively generated
$n$-categories and
$F:\mathcal{C}\rightarrow\mathcal{D},G:\mathcal{D}\rightarrow\mathcal{E}$ two
functors preserving sifted colimits (cf. Corollary A.30). Then

  1. (1)

There is a natural transformation from the composite
$\operatorname{Ani}(G)\circ\operatorname{Ani}(F)$ to
$\operatorname{Ani}(G\circ F)$ (In fact, for this, we only need that $G$
preserves sifted colimits).

  2. (2)

Let $\mathcal{C}^{0}\subseteq\mathcal{C}$ and
$\mathcal{D}^{0}\subseteq\mathcal{D}$ be full subcategories determined by a
choice of set of compact $n$-projective generators as in Definition A.32. If
either $F(\mathcal{C}^{0})\subseteq\operatorname{Ind}(\mathcal{D}^{0})$ in
$\mathcal{D}$ or
$(\operatorname{Ani}(G))(F(\mathcal{C}^{0}))\subseteq\mathcal{E}$ in
$\operatorname{Ani}(\mathcal{E})$, then the natural transformation
$\operatorname{Ani}(G)\circ\operatorname{Ani}(F)\rightarrow\operatorname{Ani}(G\circ
F)$ is an equivalence.

##  Appendix B Stable symmetric monoidal $\infty$-categories

In this section, we recall some basic facts about stable symmetric monoidal
$\infty$-categories that we need in our text. Roughly speaking, a symmetric
monoidal $\infty$-category $(\mathcal{C},\otimes)$ is an $\infty$-category
with an operation
$\cdot\otimes\cdot:\mathcal{C}\times\mathcal{C}\rightarrow\mathcal{C}$ which
is commutative and associative “up to coherent homotopies”. The precise
definition is the following:

######  Definition B.1 ([Lur17, Def 2.0.0.7 & Rem 2.1.2.20]).

A symmetric monoidal $\infty$-category is a coCartesian fibration of
simplicial sets $p:\mathcal{C}^{\otimes}\rightarrow\operatorname{Fin}_{\ast}$
with the following property:

For each $n\in\mathbb{N}_{\geq 0}$, the maps $(\rho^{i}:\langle
n\rangle\rightarrow\langle 1\rangle)_{1\leq i\leq n}$ induce functors
$\rho_{!}^{i}:\mathcal{C}_{\langle
n\rangle}^{\otimes}\rightarrow\mathcal{C}_{\langle 1\rangle}^{\otimes}$ which
determines an equivalence $\mathcal{C}_{\langle
n\rangle}^{\otimes}\xrightarrow{\simeq}(\mathcal{C}_{\langle
1\rangle}^{\otimes})^{n}$.

In this case, we will refer to the fiber $\mathcal{C}:=\mathcal{C}_{\langle
1\rangle}^{\otimes}$ as the underlying $\infty$-category of
$\mathcal{C}^{\otimes}$. The active morphism $\beta:\langle
2\rangle\rightarrow\langle 1\rangle,0\mapsto 0,1\mapsto 1,2\mapsto 1$ induces
the tensor bifunctor
$\cdot\otimes\cdot:\mathcal{C}\times\mathcal{C}\rightarrow\mathcal{C}$. We
will abuse terminology by referring to $\mathcal{C}$ or
$(\mathcal{C},\otimes)$ as a symmetric monoidal $\infty$-category.

Most symmetric monoidal $\infty$-categories that we consider are presentable
and the tensor product behaves well with colimits:

######  Definition B.2 ([Nik16, §2]).

A symmetric monoidal $\infty$-category $(\mathcal{C},\otimes)$ is called
presentable symmetric monoidal if the underlying $\infty$-category
$\mathcal{C}$ is presentable and the tensor bifunctor
$\cdot\otimes\cdot:\mathcal{C}\times\mathcal{C}\rightarrow\mathcal{C}$
preserves small colimits separately in each variable.

In most cases in which we are interested, the underlying $\infty$-category
$\mathcal{C}$ is stable, which satisfies some compatibility conditions
formulated as follows:

######  Definition B.3 ([Nik16, Def 4.1]).

A symmetric monoidal $\infty$-category $(\mathcal{C},\otimes)$ is called
stable symmetric monoidal if the underlying $\infty$-category $\mathcal{C}$ is
stable and the tensor bifunctor
$\cdot\otimes\cdot:\mathcal{C}\times\mathcal{C}\rightarrow\mathcal{C}$
preserves finite colimits separately in each variable.

######  Corollary B.4.

A presentable symmetric monoidal $\infty$-category $(\mathcal{C},\otimes)$ is
stable symmetric monoidal if and only if the underlying $\infty$-category
$\mathcal{C}$ is stable.

We usually need to impose a $t$-structure which is compatible with the
symmetric monoidal structure:

######  Definition B.5.

Let $(\mathcal{C},\otimes)$ be a stable symmetric monoidal $\infty$-category.
A symmetric monoidal $t$-structure on the underlying stable $\infty$-category
$\mathcal{C}$ is a $t$-structure $(\mathcal{C}_{\geq 0},\mathcal{C}_{\leq 0})$
such that $\boldsymbol{1}_{\mathcal{C}}\in\mathcal{C}_{\geq 0}$ and
$\mathcal{C}_{\geq 0}\in\mathcal{C}$ is closed under tensor product: $X\otimes
Y\in\mathcal{C}_{\geq 0}$ for all $X,Y\in\mathcal{C}_{\geq 0}$. In this case,
$(\mathcal{C}_{\geq 0},\otimes)$ is a symmetric monoidal $\infty$-category and
the inclusion $\mathcal{C}_{\geq 0}\hookrightarrow\mathcal{C}$ is symmetric
monoidal.

## References

  * [AL19a] Johannes Anschütz and Arthur-César Le Bras.  Prismatic Dieudonné theory.  ArXiv e-prints, jul 2019.
  * [AL19b] Johannes Anschütz and Arthur-César Le Bras.  The $p$-completed cyclotomic trace in degree $2$.  ArXiv e-prints, page 0, jul 2019.
  * [Bei87] A. A. Beilinson.  On the derived category of perverse sheaves.  In $K$-theory, arithmetic and geometry (Moscow, 1984–1986), volume 1289 of Lecture Notes in Math., pages 27–41. Springer, Berlin, 1987.
  * [Ber74] Pierre Berthelot.  Cohomologie cristalline des schémas de caractéristique $p\char 62\relax 0$.  Lecture Notes in Mathematics, Vol. 407\. Springer-Verlag, Berlin-New York, 1974.
  * [Bha12a] B. Bhatt.  $p$-adic derived de Rham cohomology.  ArXiv e-prints, apr 2012.
  * [Bha12b] B. Bhatt.  Completions and derived de Rham cohomology.  ArXiv e-prints, jul 2012.
  * [Bha18] Bhatt.  Prismatic cohomology.  http://www-personal.umich.edu/~bhattb/teaching/prismatic-columbia/, 2018\.
  * [BMS19] Bhargav Bhatt, Matthew Morrow, and Peter Scholze.  Topological Hochschild homology and integral $p$-adic Hodge theory.  Publications Mathématiques. Institut de Hautes Études Scientifiques, 129:199–310, 2019.
  * [BO78] Pierre Berthelot and Arthur Ogus.  Notes on crystalline cohomology.  Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1978\.
  * [BS19] Bhargav Bhatt and Peter Scholze.  Prisms and Prismatic Cohomology.  ArXiv e-prints, May 2019.
  * [Cha20] Andre Chatzistamatiou.  $Q$-crystals and $q$-connections.  ArXiv e-prints, oct 2020.
  * [CS19] Kestutis Cesnavicius and Peter Scholze.  Purity for flat cohomology.  ArXiv e-prints, dec 2019.
  * [GL20] Haoyang Guo and Shizhang Li.  Period sheaves via derived de Rham cohomology.  ArXiv e-prints, aug 2020.
  * [GLQ20] Michel Gros, Bernard Le Stum, and Adolfo Quirós.  Twisted differential operators and $q$-crystals.  ArXiv e-prints, apr 2020.
  * [GP18] Owen Gwilliam and Dmitri Pavlov.  Enhancing the filtered derived category.  Journal of Pure and Applied Algebra, 222(11):3621–3674, 2018.
  * [H+98] J. van der Hoeven et al.  GNU TeXmacs.  https://www.texmacs.org, 1998.
  * [Hov14] Mark Hovey.  Smith ideals of structured ring spectra.  ArXiv e-prints, jan 2014.
  * [Ill72] Luc Illusie.  Complexe cotangent et déformations. II.  Lecture Notes in Mathematics, Vol. 283. Springer-Verlag, Berlin-New York, 1972.
  * [Kat70] Nicholas M. Katz.  Nilpotent connections and the monodromy theorem: Applications of a result of Turrittin.  Institut des Hautes Études Scientifiques. Publications Mathématiques, (39):175–232, 1970.
  * [LL20] Shizhang Li and Tong Liu.  Comparison of prismatic cohomology and derived de Rham cohomology.  ArXiv e-prints, dec 2020.
  * [Lur04] Jacob Lurie.  Tannaka Duality for Geometric Stacks.  ArXiv Mathematics e-prints, page 0, dec 2004.
  * [Lur09] Jacob Lurie.  Higher topos theory, volume 170 of Annals of Mathematics Studies.  Princeton University Press, Princeton, NJ, 2009.
  * [Lur15] Jacob Lurie.  Rotation invariance in algebraic K-theory.  https://www.math.ias.edu/~lurie/papers/Waldhaus.pdf, sep 2015\.
  * [Lur17] Jacob Lurie.  Higher Algebra.  https://www.math.ias.edu/~lurie/papers/HA.pdf, sep 2017\.
  * [Lur18] Jacob Lurie.  Spectral Algebraic Geometry.  https://www.math.ias.edu/~lurie/papers/SAG-rootfile.pdf, feb 2018.
  * [Lur20] Jacob Lurie.  Kerodon.  https://kerodon.net, 2020.
  * [MT] Matthew Morrow and Takeshi Tsuji.  Generalised representations as q-connections in integral $p$-adic Hodge theory.  ArXiv e-prints.
  * [Nar16] Denis Nardin.  Difficulties with descent data as homotopy limit of image of Čech nerve.  MathOverflow, apr 2016.  https://mathoverflow.net/a/236600/.
  * [Nik16] Thomas Nikolaus.  Stable $\infty$-Operads and the multiplicative Yoneda lemma.  ArXiv e-prints, aug 2016.
  * [Ogu21] Arthur Ogus.  Prismatic and $p$-de Rham cohomology: comparison theorems and antecedents.  Illusie conference: https://server.mcm.ac.cn/~zheng/LI/Ogus.pdf, jun 2021\.
  * [Qui67] Daniel G. Quillen.  Homotopical algebra.  Lecture Notes in Mathematics, No. 43. Springer-Verlag, Berlin-New York, 1967.
  * [Rak20] Arpon Raksit.  Hochschild homology and the derived de Rham complex revisited.  ArXiv e-prints, jul 2020.
  * [Sch19] Peter Scholze.  Lectures on Analytic Geometry.  https://www.math.uni-bonn.de/people/scholze/Analytic.pdf, oct 2019.
  * [Sta21] StacksProjectAuthors.  The Stacks project.  https://stacks.math.columbia.edu, 2021.
  * [WY17a] David White and Donald Yau.  Arrow Categories of Monoidal Model Categories.  ArXiv e-prints, mar 2017.
  * [WY17b] David White and Donald Yau.  Smith Ideals of Operadic Algebras in Monoidal Model Categories.  ArXiv e-prints, mar 2017.