# Non-decomposability of the de Rham complex and non-semisimplicity of the Sen
operator

Alexander Petrov  Max Planck Institute for Mathematics, Bonn, Germany
<EMAIL_ADDRESS>
###### Abstract.

We describe the obstruction to decomposing in degrees $\leq p$ the de Rham
complex of a smooth variety over a perfect field $k$ of characteristic $p$
that lifts over $W_{2}(k)$, and show that there exist liftable smooth
projective varieties of dimension $p+1$ whose Hodge-to-de Rham spectral
sequence does not degenerate at the first page. We also describe the action of
the Sen operator on the de Rham complex in degrees $\leq p$ and give examples
of varieties with a non-semisimple Sen operator.

Our methods rely on the commutative algebra structure on de Rham and Hodge-
Tate cohomology, and are inspired by the properties of Steenrod operations on
cohomology of cosimplicial commutative algebras. The example of a non-
degenerate Hodge-to-de Rham spectral sequence relies on a non-vanishing result
on cohomology of groups of Lie type. We give applications to other situations
such as describing extensions in the canonical filtration on de Rham, Hodge,
and étale cohomology of an abelian variety equipped with a group action. We
also show that the de Rham complex of a smooth variety over $k$ is formal as
an $E_{\infty}$-algebra if and only if the variety lifts to $W_{2}(k)$
together with its Frobenius endomorphism.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries on homotopical algebra
  3. 3 Symmetric power $S^{p}$, class $\alpha$, and Steenrod operations
  4. 4 Extensions in complexes underlying derived commutative algebras
  5. 5 Applications to de Rham and Hodge-Tate cohomology
  6. 6 Preliminaries on the Sen operator
  7. 7 Sen operator via descent from semiperfectoid rings
  8. 8 Sen operator of a fibration in terms of Kodaira-Spencer class
  9. 9 Cohomology of abelian varieties
  10. 10 Liftable variety with non-degenerate conjugate spectral sequence
  11. 11 Extensions in higher degrees
  12. 12 Non-vanishing in rational group cohomology
  13. 13 From algebraic cohomology to cohomology of the group of ${\mathbb F}_{q}$-points
  14. 14 Cohomology of $SL_{p}$ over rings of integers

##  1\. Introduction

###  1.1. The main results.

A celebrated theorem of Deligne and Illusie provides an analog of Hodge
decomposition on de Rham cohomology in degrees $<p$ of varieties over a
perfect field $k$ of positive characteristic $p$ that lift to $W_{2}(k)$. In
[DI87] they proved that for a smooth variety over $X_{0}$ over $k$ that admits
a lift to a smooth scheme $X_{1}$ over the ring $W_{2}(k)$ of length $2$ Witt
vectors, the canonical truncation of the de Rham complex
$\mathrm{dR}_{X_{0}/k}:=(F_{X_{0}/k*}\Omega^{\bullet}_{X_{0}/k},d)$ in degrees
$\leq p-1$ decomposes in the derived category $D(X_{0}^{(1)})$ of
${\mathcal{O}}_{X^{(1)}_{0}}$-modules:

(1.1) $\tau^{\leq
p-1}\mathrm{dR}_{X_{0}/k}\simeq\bigoplus\limits_{i=0}^{p-1}\Omega^{i}_{X^{(1)}_{0}/k}[-i]$

Passing to cohomology, this gives decompositions for $n<p$:

(1.2)
$H^{n}_{\mathrm{dR}}(X_{0}/k)\simeq\bigoplus\limits_{i+j=n}H^{j}(X_{0},\Omega^{i}_{X_{0}/k})^{(1)}$

In this paper we investigate whether the de Rham complex of a variety liftable
to $W_{2}(k)$ decomposes in further degrees. We prove that in general it does
not decompose, and decomposition (1.2) might not exist, answering a question
of Deligne and Illusie [DI87, Remarque 2.6(iii)]:

######  Theorem 1.1 (Corollary 10.7).

There exists a smooth projective variety $X_{0}$ over $k$ of dimension $p+1$
that lifts to $W(k)$ such that
$\dim_{k}H^{p}_{\mathrm{dR}}(X_{0}/k)<\displaystyle\sum\limits_{i+j=p}\dim_{k}H^{j}(X_{0},\Omega^{i}_{X_{0}/k})$.
In particular, the Hodge-to-de Rham spectral sequence of $X_{0}$ does not
degenerate at the first page, and $\mathrm{dR}_{X_{0}/k}$ is not quasi-
isomorphic to $\bigoplus\limits_{i\geq 0}\Omega^{i}_{X^{(1)}_{0}/k}[-i]$.

Moreover, we compute the obstruction to decomposing the truncation $\tau^{\leq
p}\mathrm{dR}_{X_{0}/k}$ of the de Rham complex in degrees $\leq p$ in terms
of other invariants of the variety $X_{0}$ and its lift $X_{1}$. Given that
$\tau^{\leq p-1}\mathrm{dR}_{X_{0}/k}$ decomposes, the next truncation fits
into an exact triangle in the derived category $D(X^{(1)}_{0})$:

(1.3) $\bigoplus\limits_{i=0}^{p-1}\Omega^{i}_{X^{(1)}_{0}/k}[-i]\to\tau^{\leq
p}\mathrm{dR}_{X_{0}/k}\to\Omega^{p}_{X^{(1)}_{0}/k}[-p]$

and hence gives an extension class

$e_{X_{1},p}\in H^{p+1}(X^{(1)}_{0},\Lambda^{p}T_{X^{(1)}_{0}/k})\oplus
H^{p}(X^{(1)}_{0},\Lambda^{p}T_{X^{(1)}_{0}/k}\otimes\Omega^{1}_{X^{(1)}_{0}/k})\oplus\ldots$

that vanishes if and only if (1.3) splits in $D(X^{(1)}_{0})$. We compute this
class:

######  Theorem 1.2 (Theorem 5.12 if $X_{1}$ lifts to $W(k)$, Corollary 7.5 in
general).

All components of $e_{X_{1},p}$ except for the one in
$H^{p+1}(X^{(1)}_{0},\Lambda^{p}T_{X^{(1)}_{0}/k})$ vanish, and that component
is equal to

(1.4) $\mathrm{Bock}_{X^{(1)}_{1}}(\operatorname{ob}_{F,X_{1}}\cdot\hskip
2.0pt\alpha(\Omega^{1}_{X^{(1)}_{0}}))\in
H^{p+1}(X^{(1)}_{0},\Lambda^{p}T_{X^{(1)}_{0}/k})$

where

  * •

$\mathrm{Bock}_{X^{(1)}_{1}}:H^{p}(X^{(1)}_{0},\Lambda^{p}T_{X^{(1)}_{0}/k})\to
H^{p+1}(X^{(1)}_{0},\Lambda^{p}T_{X^{(1)}_{0}/k})$ is the Bockstein
homomorphism, that is the connecting morphism arising from the short exact
sequence of sheaves on $X_{1}$:
$0\to\Lambda^{p}T_{X^{(1)}_{0}/k}\to\Lambda^{p}T_{X^{(1)}_{1}/W_{2}(k)}\to\Lambda^{p}T_{X^{(1)}_{0}/k}\to
0$.

  * •

$\operatorname{ob}_{F,X_{1}}\in
H^{1}(X^{(1)}_{0},F_{X_{0}^{(1)}}^{*}T_{X^{(1)}_{0}})$ is the obstruction to
lifting the Frobenius morphism $F_{X_{0}}:X_{0}\to X_{0}$ to an endomorphism
of $X_{1}$,

  * •

$\alpha(\Omega^{1}_{X^{(1)}_{0}})\in
H^{p-1}(X^{(1)}_{0},F_{X^{(1)}_{0}}^{*}\Omega^{1}_{X^{(1)}_{0}}\otimes(\Lambda^{p}\Omega^{1}_{X^{(1)}_{0}})^{\vee})$
is a certain ‘characteristic class’ of the cotangent bundle, defined in
Definition 3.4.

The class $\alpha(E)\in
H^{p-1}(X_{0},F_{X_{0}}^{*}E\otimes(\Lambda^{p}E)^{\vee})$ is defined for any
vector bundle $E$ on $X_{0}$, and it can be made especially explicit when
$p=2$, in which case it is the class of the extension

(1.5) $0\to F_{X_{0}}^{*}E\to S^{2}E\xrightarrow{e_{1}\cdot e_{2}\mapsto
e_{1}\wedge e_{2}}\Lambda^{2}E\to 0.$

The fact that $e_{X_{1},p}$ has only one potentially non-zero component is a
shadow of an additional ${\mathbb Z}/p$-grading on the de Rham complex
$\mathrm{dR}_{X_{0}/k}$ in the presence of a lift $X_{1}$, discovered by
Drinfeld through his work on the prismatization [Dri21]. More generally,
Bhatt-Lurie [BL22a] and Li-Mondal [LM21] showed that a lift of $X_{0}$ to a
smooth $W_{2}(k)$-scheme $X_{1}$ induces an endomorphism $\Theta_{X_{1}}$ of
$\mathrm{dR}_{X_{0}/k}$ in the derived category $D(X^{(1)}_{0})$, called the
Sen operator.

The action of $\Theta_{X_{1}}$ on the $i$th cohomology sheaf
$H^{i}(\mathrm{dR}_{X_{0}/k})\simeq\Omega^{i}_{X^{(1)}_{0}/k}$ is given by
multiplication by $(-i)$. Therefore generalized eigenspaces for this operator
form a decomposition

(1.6)
$\mathrm{dR}_{X_{0}/k}\simeq\bigoplus\limits_{i=0}^{p-1}\mathrm{dR}_{X_{0}/k,i}$

such that the object $\mathrm{dR}_{X_{0}/k,i}$ has non-zero cohomology only in
degrees congruent to $i$ modulo $p$. In particular, $\tau^{\leq
p}\mathrm{dR}_{X_{0}/k,0}$ is an extension of $\Omega^{p}_{X^{(1)}_{0}/k}[-p]$
by ${\mathcal{O}}_{X^{(1)}_{0}}$ that splits off as a direct summand from the
extension (1.3), corroborating the fact that the components of $e_{X_{1},p}$
in $H^{p+1-i}(X_{0},\Lambda^{p}T_{X_{0}}\otimes\Omega^{i}_{X_{0}})$ vanish for
$i>0$. It was observed earlier by Achinger and Suh [AS20] that these
components vanish for $i>1$, for a purely homological-algebraic reason which
is related to our method.

The Sen operator $\Theta_{X_{1}}$ not only induces the decomposition (1.6) but
also equips each direct summand $\mathrm{dR}_{X_{0}/k,i}$ with an additional
structure of the nilpotent endomorphism $\Theta_{X_{1}}+i$. Our methods allow
to describe the action of $\Theta_{X_{1}}$ on $\tau^{\leq
p}\mathrm{dR}_{X_{0},0}$. Since $\tau^{\leq p}\mathrm{dR}_{X_{0}/k,0}$ has
only two non-zero cohomology sheaves and $\Theta_{X_{1}}$ acts on them by
zero, it naturally defines a map
$\Omega^{p}_{X^{(1)}_{0}}[-p]\to{\mathcal{O}}_{X^{(1)}_{0}}$ in the derived
category $D(X^{(1)}_{0})$ whose cohomology class we denote by
$c_{X^{(1)}_{1},p}\in H^{p}(X^{(1)}_{0},\Lambda^{p}T_{X^{(1)}_{0}})$. This
class can be described as:

######  Theorem 1.3 (Theorem 7.1).

For a smooth $W_{2}(k)$-scheme $X_{1}$ with the special fiber
$X_{0}=X_{1}\times_{W_{2}(k)}k$ we have

(1.7) $c_{X_{1},p}=\operatorname{ob}_{F,X_{1}}\cdot\alpha(\Omega^{1}_{X_{0}})$

where the right hand side is the product of classes
$\operatorname{ob}_{F,X_{1}}\in H^{1}(X_{0},F_{X_{0}}^{*}T_{X_{0}/k})$ and
$\alpha(\Omega^{1}_{X_{0}/k})\in
H^{p-1}(X_{0},F_{X_{0}}^{*}\Omega^{1}_{X_{0}/k}\otimes\Lambda^{p}T_{X_{0}/k})$

It is not a coincidence that $e_{X_{1},p}$ is the result of applying the
Bockstein homomorphism to $c_{X_{1},p}$ – this follows from the basic
properties of the action of the Sen operator on the diffracted Hodge
cohomology of $X_{1}$, as we prove in Lemma 6.12. In particular, if
$\tau^{\leq p}\mathrm{dR}_{X_{0}/k}$ is not decomposable then the Sen operator
on $\tau^{\leq p}\mathrm{dR}_{X_{0}/k}$ must be non-semi-simple. Thus Theorem
1.1 also provides an example of a liftable variety with a non-semisimple Sen
operator on its de Rham cohomology, answering a question of Bhatt.

In fact, the non-vanishing of $c_{X_{1},p}$ is a more frequent phenomenon than
that of $e_{X_{1},p}$ as we demonstrate by the following:

######  Corollary 1.4 (Proposition 8.4).

There exists a smooth projective variety $X_{0}$ of dimension $p$ over $k$
equipped with a lift $X_{1}$ over $W_{2}(k)$ such that the Sen operator
$\Theta_{X_{1}}$ on $\mathrm{dR}_{X_{0}/k}$ is not semisimple.

The examples we construct admit a smooth proper morphism to a curve, and for
such varieties the class $\alpha(\Omega^{1}_{X_{0}})$ can be expressed in
terms of the Kodaira-Spencer class of this fibration, giving a more tangible
reformulation (Theorem 8.1) of Theorem 1.3.

We give two different, but similar in spirit, proofs of Theorem 1.2, both of
which use crucially the multiplicative structure on de Rham and Hodge-Tate
complexes.

###  1.2. Proof of Theorem 1.2.

The first proof, for which we additionally need to assume that $X_{0}$ lifts
to a smooth (formal) scheme $X$ over $W(k)$, takes place in homotopical
algebra rather than algebraic geometry. To prove Theorem 1.2 in full
generality, in a situation when only a lift over $W_{2}(k)$ exists, we appeal
to Theorem 1.3, which implies it by Corollary 7.5.

To describe the idea, recall the structure of the proof of [DI87]. Given the
lift $X_{1}$ over $W_{2}(k)$, they produce a map
$s:\Omega^{1}_{X^{(1)}_{0}}[-1]\to\mathrm{dR}_{X_{0}/k}$ in the derived
category $D(X^{(1)}_{0})$ inducing an isomorphism on cohomology sheaves in
degree $1$. Using that $\mathrm{dR}_{X_{0}/k}$ can be represented by a
cosimplicial commutative algebra in quasi-coherent sheaves on $X_{0}$ via the
Čech resolution, $s$ gives rise to maps

(1.8) $s_{i}:S^{i}(\Omega^{1}_{X^{(1)}_{0}/k}[-1])\to\mathrm{dR}_{X_{0}/k}$

for all $i\geq 0$ where $S^{i}$ denotes the derived symmetric power functor in
the sense of [Ill71]. When $i<p$, the derived symmetric power
$S^{i}(\Omega^{1}_{X^{(1)}_{0}/k}[-1])$ is identified with
$\Omega^{i}_{X^{(1)}_{0}/k}[-i]$, and maps $s_{i}$ produce a quasi-isomorphism
$\bigoplus
s_{i}:\bigoplus\limits_{i=0}^{p-1}\Omega^{i}_{X^{(1)}_{0}/k}[-i]\simeq\tau^{\leq
p-1}\mathrm{dR}_{X_{0}/k}$.

This method of decomposing $\mathrm{dR}_{X_{0}/k}$ stops working for $i=p$,
because $S^{p}(\Omega^{1}_{X^{(1)}_{0}/k}[-1])$ is now different from
$\Omega^{p}_{X^{(1)}_{0}/k}[-p]$. It has non-zero cohomology sheaves in
several degrees, and we analyze this object in detail in Section 3.
Specifically, there is a natural map
$N_{p}:S^{p}(\Omega^{1}_{X^{(1)}_{0}/k}[-1])\to\Omega^{p}_{X^{(1)}_{0}/k}[-p]$
but the map $s_{p}$ need not factor through it. In Lemma 3.10 we compute the
map $s_{p}$ on the fiber of $N_{p}$, and thus relate the obstruction to
splitting $\tau^{\leq p}\mathrm{dR}_{X_{0}/k}$ to the obstruction to the
existence of a section of the map $N_{p}$, the latter being precisely the
class $\alpha(\Omega^{1}_{X_{0}})$.

This argument is not specific to the de Rham complex and applies more
generally to derived commutative algebras whose cohomology algebra is freely
generated by $H^{1}$. In the body of the paper we work with derived
commutative algebras in the sense of Mathew, but here we state the main
algebraic result for the more classical notion of cosimplicial commutative
algebras:

######  Theorem 1.5 (Theorem 4.1).

Let $A$ be a cosimplicial commutative algebra in quasi-coherent sheaves on a
flat ${\mathbb Z}_{(p)}$-scheme $X$. Suppose that
$H^{0}(A)={\mathcal{O}}_{X}$, the sheaf $H^{1}(A)$ is a locally free sheaf of
${\mathcal{O}}_{X}$-modules, and the multiplication on the cohomology of $A$
induces an isomorphism $\Lambda^{\bullet}H^{1}(A)\simeq H^{\bullet}(A)$. If
there exists a map $s:H^{1}(A)[-1]\to A$ in $D(X)$ inducing an isomorphism on
cohomology in degree $1$, then

  1. (1)

$\tau^{\leq p-1}A$ is quasi-isomorphic to
$\bigoplus\limits_{i=0}^{p-1}H^{i}(A)[-i].$

  2. (2)

The connecting map $H^{p}(A)\to(\tau^{\leq p-1}A)[p+1]$ corresponding to the
fiber sequence $\tau^{\leq p-1}A\to\tau^{\leq p}A\to H^{p}(A)[-p]$ can be
described as the composition

(1.9)
$H^{p}(A)=\Lambda^{p}H^{1}(A)\to\Lambda^{p}H^{1}(A)/p\xrightarrow{\alpha(H^{1}(A)/p)}F_{X_{0}}^{*}(H^{1}(A)/p)[p-1]\xrightarrow{F_{X_{0}}^{*}s[p]}(\tau^{\leq
1}F^{*}_{X_{0}}(A/p))[p]\\\ \xrightarrow{\varphi_{A/p}}(\tau^{\leq
1}A/p)[p]\xrightarrow{\tau^{\leq 1}\mathrm{Bock}_{A}}(\tau^{\leq
1}A)[p+1]\xrightarrow{\oplus}(\tau^{\leq p-1}A)[p+1]$

Here $\varphi_{A/p}$ is the Frobenius morphism of the cosimplicial commutative
algebra $A/p$, and $\mathrm{Bock}_{A}:A/p\to A[1]$ is the morphism
corresponding to the fiber sequence $A\xrightarrow{p}A\to A/p$ on $X$.

The fact that the map
$s_{p}:S^{p}(\Omega^{1}_{X^{(1)}_{0}}[-1])\to\mathrm{dR}_{X_{0}/k}$ need not
factor through $\Omega^{p}_{X^{(1)}_{0}}[-p]$ is an incarnation of the
classical phenomenon responsible for the existence of non-trivial Steenrod
operations on mod $p$ cohomology of topological spaces. If the map
$s:\Omega^{1}_{X^{(1)}_{0}/k}[-1]\to\mathrm{dR}_{X_{0}/k}$ in the derived
category $D(X^{(1)}_{0})$ was represented by a genuine map between these
complexes, all the maps $s_{i}$ would have to factor through
$\Omega^{i}_{X^{(1)}_{0}/k}[-i]$ because of the commutative DG algebra
structure on the de Rham complex. However, $s$ can generally only be
represented by a chain-level map into a model of $\mathrm{dR}_{X_{0}/k}$ where
multiplication is commutative up to a set of coherent homotopies, rather than
commutative on the nose.

Theorem 1.5 draws inspiration from Steenrod’s construction [Ste53], [May70] of
cohomology operations from homology classes of symmetric groups, and our
method of proof implies the following statement about operations on cohomology
of cosimplicial commutative ${\mathbb F}_{p}$-algebras, defined by Priddy
[Pri73]. The appearance of Frobenius and Bockstein homomorphisms in the
description of the degree $1$ Steenrod operation is directly related (Lemma
3.10 being the common reason for both) to their appearance in (1.9).

######  Proposition 1.6 (Proposition 3.14).

For a cosimplicial commutative ${\mathbb F}_{p}$-algebra $B$,

  1. (1)

the degree zero Steenrod operation $P^{0}:H^{i}(B)\to H^{i}(B)$ is equal to
the Frobenius endomorphism $\varphi_{B}$ of $B$,

  2. (2)

the degree $1$ Steenrod operation $P^{1}:H^{i}(B)\to H^{i+1}(B)$ is equal to
Serre’s Witt vector Bockstein homomorphism induced by the exact sequence
$B\xrightarrow{V}W_{2}(B)\to B$. In particular, for any cosimplicial
commutative flat ${\mathbb Z}/p^{2}$-algebra $\widetilde{B}$ lifting $B$,
$P^{1}$ can be described as the composition
$H^{i}(B)\xrightarrow{\varphi_{B}}H^{i}(B)\xrightarrow{\mathrm{Bock}_{\widetilde{B}}}H^{i+1}(B)$.

We apply Theorem 1.5 to the diffracted Hodge complex
$A:=\Omega^{\operatorname{\not{D}}}_{X}$ of [BL22a] (it coincides with the
Hodge-Tate cohomology of [BS22] over an appropriately chosen prism), and
deduce Theorem 1.2 by reducing modulo $p$, using that $\mathrm{dR}_{X_{0}/k}$
is the reduction of $\Omega^{\operatorname{\not{D}}}_{X^{(1)}}$. We thus along
the way obtain in Theorem 5.9 a similar description of the extension class
corresponding to $\tau^{\leq p}\Omega^{\operatorname{\not{D}}}_{X}$ as well.
Theorem 1.5 also applies to a number of cohomology theories evaluated on
abelian varieties: in Section 9 we use it to study extensions in the canonical
filtration on coherent (Proposition 9.1), de Rham (Proposition 9.3), and étale
cohomology (Proposition 9.4) of an abelian scheme being acted on by a group.
The results on coherent and de Rham equivariant cohomology of abelian schemes
play the key role in our example of a liftable variety with a non-degenerate
conjugate spectral sequence.

A related approach to decomposability of the de Rham complex has been
previously used by Beuchot [Beu19], who used a description of cohomology
sheaves of $(\Omega^{1}_{X^{(1)}_{0}}[-1]^{\otimes p})_{hS_{p}}$ to prove
decomposability of the truncation $\tau^{\leq p}\mathrm{dR}_{X_{0}/k}$ for a
liftable variety $X_{0}$ under the assumption that certain cohomology groups
of $X_{0}$ vanish. The consequence of Theorem 1.5 for the de Rham complex was
also obtained independently by Robert Burklund, by a similar method.

At the moment it is unclear how to generalize Theorem 1.5, and therefore
Theorem 1.2, to describe extensions in degrees $>p$. We make some preliminary
observations on this in Section 11.

Computations of the map
$S^{p}(\Omega^{1}_{X^{(1)}_{0}}[-1])\to\mathrm{dR}_{X_{0}/k}$ that we perform
for the proof of Theorem 1.2 also allow us to determine when the de Rham
complex can be decomposed compatibly with the algebra structure. The following
result is to be contrasted with the formality result of [DGMS75] for smooth
projective varieties in characteristic zero:

######  Proposition 1.7 (Proposition 5.14).

For a smooth variety $X_{0}$ over $k$ the de Rham complex
$\mathrm{dR}_{X_{0}/k}$ is quasi-isomorphic to $\bigoplus\limits_{i\geq
0}\Omega^{i}_{X^{(1)}_{0}/k}[-i]$ as an $E_{\infty}$-algebra in
$D(X^{(1)}_{0})$ if and only if $X_{0}$ admits a lift over $W_{2}(k)$ together
with its Frobenius endomorphism.

The ‘if’ direction was proven in [DI87] and is the starting point of their
proof of the decomposition (1.1).

###  1.3. Liftable variety with a non-degenerate conjugate spectral sequence

It is rather non-obvious that the formula (1.4) ever takes non-zero values.
Our example for Theorem 1.1 is an approximation (in the sense of [ABM21]) of
the classifying stack of a finite flat non-commutative group scheme $G$ over
$W(k)$ defined as follows. Let $E$ be an elliptic curve over $W(k)$ whose
reduction is supersingular. The group scheme $G$ is

(1.10) $G:=SL_{p}({\mathbb F}_{p^{2}})\ltimes(E[p]\otimes_{{\mathbb
F}_{p}}{\mathbb F}_{p^{2}}^{\oplus p}).$

Here $E[p]\otimes_{{\mathbb F}_{p}}{\mathbb F}_{p^{2}}^{\oplus p}$ is the
product of $2p$ copies of the $p$-torsion group scheme $E[p]$, and the
discrete group $SL_{p}({\mathbb F}_{p^{2}})$ acts on it via the tautological
representation on ${\mathbb F}_{p^{2}}^{\oplus p}$.

We do not apply formula (1.4) to the stack $BG$ directly, but rather study the
conjugate spectral sequence of an auxiliary quotient stack by applying Theorem
1.5 to equivariant Hodge and de Rham cohomology of the abelian scheme
$E^{\times d}$ for a certain $d$, with respect to a certain discrete group.

First, we show that the class $\alpha(-)$ does not vanish in the universal
example:

######  Proposition 1.8 (Proposition 12.1).

Denote by $V$ the tautological $p$-dimensional representation of the algebraic
group $GL_{p,k}$. The class
$\alpha(V)\in\operatorname{{Ext}}^{p-1}_{GL_{p,k}}(\Lambda^{p}V,V^{(1)})$ is
non-zero.

For the proof, we find an explicit representation for $\alpha(V)$ as a Yoneda
extension and apply Kempf’s vanishing theorem to compare spectral sequences
associated to bête and canonical filtrations on this explicit complex. We also
prove an enhancement of this non-vanishing: the class $\alpha(V)$ remains non-
zero after being restricted to the discrete group $GL_{p}({\mathbb
F}_{p^{r}})$ whenever $r>1$ (Proposition 13.1), and that the Bockstein
homomorphism applied to $\alpha(V)$ gives a non-zero class in the cohomology
of the special linear group $SL_{p}({\mathcal{O}}_{F})$ of the ring of
integers in an appropriately chosen number field $F$ (Proposition 14.1). These
enhancements use the technique of Cline-Parshall-Scott-van der Kallen
[CPSvdK77] for comparing the cohomology a reductive group with that of its
group of ${\mathbb F}_{q}$-points.

With these non-vanishing results in hand, we apply Theorem 1.5 to de Rham and
coherent cohomology of an abelian scheme being acted on by a group. Our
example in Theorem 1.1 stems from the following potential discrepancy between
de Rham and coherent cohomology:

######  Proposition 1.9 (Proposition 10.3).

There exists an abelian scheme $A$ over $W(k)$ with an action of a discrete
group $\Gamma$ such that the truncation $\tau^{\leq
p}\mathrm{R}\Gamma(A_{0},{\mathcal{O}})$ of the cohomology of the structure
sheaf of the special fiber $A_{0}=A\times_{W(k)}k$ is $\Gamma$-equivariantly
decomposable, while its de Rham cohomology $\tau^{\leq
p}\mathrm{R}\Gamma_{\mathrm{dR}}(A_{0}/k)$ is not.

The ultimate reason for different behavior of de Rham and coherent cohomology
in Proposition 1.9 is that the Frobenius morphism appearing in (1.9) is zero
on $H^{1}(A_{0},{\mathcal{O}})$, but is non-zero on
$H^{1}_{\mathrm{dR}}(A_{0}/k)$. We moreover arrange that the stack
$[A_{0}/\Gamma]$ has a non-degenerate conjugate spectral sequence. We then
construct a map $[A_{0}/\Gamma]\to BG$ and deduce that the conjugate spectral
sequence of $BG$ does not degenerate either, as desired.

The dependence of the extension class (1.9) on the Frobenius action yields the
following criterion for supersingularity of an elliptic curve:

######  Proposition 1.10 (Corollary 9.2).

Suppose that $n\geq 2p$. For any elliptic curve $E_{0}$ over $k$, the discrete
group $GL_{n}({\mathbb Z})$ naturally acts on $E_{0}^{\times n}$ and therefore
on $\mathrm{R}\Gamma(E_{0}^{\times n},{\mathcal{O}})$. The complex $\tau^{\leq
p}\mathrm{R}\Gamma(E_{0}^{\times n},{\mathcal{O}})$ is $GL_{n}({\mathbb
Z})$-equivariantly quasi-isomorphic to the direct sum
$\bigoplus\limits_{i=0}^{p}H^{i}(E_{0}^{\times n},{\mathcal{O}})[-i]$ of its
cohomology if and only if $E_{0}$ is supersingular.

###  1.4. Computation of the Sen operator.

Finally, let us describe the idea of our proof of Theorem 1.3. It might be
possible to upgrade Theorem 1.5 to work in the category of sheaves on $X$
equipped with an endomorphism, but we instead evaluate the Sen operator on
$\tau^{\leq p}\mathrm{dR}_{X_{0}/k}$ using the technique of descent to
quasiregular semiperfectoid rings developed by [BMS18]. We set up the
foundation for our computation in Section 6, defining in particular a version
of diffracted Hodge cohomology equipped with a Sen operator relative to the
base ${\mathbb Z}/p^{n}$ for $n\geq 2$ (Theorem 6.10). We then prove formula
(1.7) in Theorem 7.1.

For a quasiregular semiperfectoid $W(k)$-algebra $S$ with mod $p$ reduction
$S_{0}=S/p$ the derived de Rham complex $\mathrm{dR}_{S_{0}/k}$ is
concentrated in degree zero so the Sen operator in question is an endomorphism
of a classical module, rather than an object of the derived world. To compute
it, we consider the natural map

(1.11) $\bigoplus\limits_{i\geq
0}S^{i}(L\Omega^{1}_{S}[-1])\to\Omega^{\operatorname{\not{D}}}_{S}$

from the symmetric algebra on the shifted cotangent complex of $S$, comprised
of the analogs of the maps (1.8).

Since $S$ is quasiregular semiperfectoid, the map (1.11) is an injection of
flat $W(k)$-modules with torsion cokernel. As $\Theta_{S}$ acts on the source
of this map via multiplication by $-i$ on the summand
$S^{i}(L\Omega^{1}_{S}[-1])$, this allows us to pin down exactly how
$\Theta_{S}$ acts on $\Omega^{\operatorname{\not{D}}}_{S}$. A slight variation
of this argument works for a semiperfectoid $W_{2}(k)$-algebra $S_{1}$ lifting
$S_{0}$, even if there is no lift over $W(k)$, as we prove in Lemma 7.2.
Though as a result we only compute the Sen operator on $\mathrm{dR}_{S_{0}}$
rather than $\Omega^{\operatorname{\not{D}}}_{S_{1}/{\mathbb Z}/p^{2}}$.

Note that, as in the proof of Theorem 1.2, we do not directly use here the
construction of the Sen operator via the Cartier-Witt stack, but rather rely
crucially on its basic properties: that it is a derivation with respect to the
multiplication on $\Omega^{\operatorname{\not{D}}}_{X}$ and that it acts by
scalar multiplication on graded quotients of the conjugate filtration.

### Notation.

We use the machinery of $\infty$-categories in the sense of [Lur09]. When
referring to an ordinary category we always view it as an $\infty$-category
via the simplicial nerve construction [Lur09, 1.1.2].

For an object ${\mathcal{F}}\in D(X)$ of the derived category of quasi-
coherent sheaves on a prestack $X$ we denote by
$H^{i}({\mathcal{F}})\in\operatorname{QCoh}(X)$ the cohomology sheaf of
${\mathcal{F}}$ in degree $i$, and by $H^{i}(X,{\mathcal{F}})$ the abelian
group of the degree $i$ cohomology of the derived global sections complex
$\mathrm{R}\Gamma(X,{\mathcal{F}})$. By $\tau^{\leq i}{\mathcal{F}}$ we denote
the canonical truncation: it is an object with a map $\tau^{\leq
i}{\mathcal{F}}\to{\mathcal{F}}$ such that $H^{j}(\tau^{\leq
i}{\mathcal{F}})\simeq H^{j}({\mathcal{F}})$ for $j\leq i$, and
$H^{j}(\tau^{\leq i}{\mathcal{F}})=0$ for $j>i$.

When working in a stable $\infty$-category ${\mathcal{C}}$ linear over a
commutative ring $R$, for objects $M,N\in{\mathcal{C}}$ we denote by
$\mathrm{RHom}_{{\mathcal{C}}}(M,N)\in D(R)$ the corresponding object of
morphisms. The mapping space $\operatorname{Map}_{{\mathcal{C}}}(M,N)$ is
equivalent to the image of the truncation $\tau^{\leq
0}\mathrm{RHom}_{{\mathcal{C}}}(M,N)$ under the forgetful functor
$D(R)\to\operatorname{{Sp}}$ to the $\infty$-category of spaces.

For an algebraic stack $X$ over a ring $R$ we denote by $L\Omega^{i}_{X/R}$
the $i$th exterior power of its cotangent complex, viewed as an object of the
derived category $D(X)$ of quasicoherent sheaves on $X$. If $pR=0$ we denote
by $\mathrm{dR}_{X/R}$ the (derived) de Rham complex of $X$ relative to $R$,
viewed as an object of $D(X\times_{R,F_{R}}R)$, cf. [Bha12, §3]. We sometimes
drop the base $R$ from the notation if it is clear from the context,
especially if $R$ is perfect.

### Acknowledgements.

I am very grateful to Vadim Vologodsky for many conversations on the
surrounding topics throughout several years, I learned most of the techniques
used in this paper from him. I am thankful to Luc Illusie for numerous
stimulating conversations, encouragement, and many very helpful comments and
corrections on the previous versions of this text.

I am also grateful to Bhargav Bhatt, Lukas Brantner, Robert Burklund, Sanath
Devalapurkar, Dmitry Kubrak, Shizhang Li, Shubhodip Mondal, Arthur Ogus,
Alexei Skorobogatov, and Bogdan Zavyalov for helpful conversations,
suggestions, and explanations. Special thanks to Dmitry and Shizhang for
pointing out errors in the previous versions of some of the arguments.

This work was completed during my stay at the Max Planck Institute for
Mathematics in Bonn, I am grateful to MPIM for hospitality. This research was
partially conducted during the period the author served as a Clay Research
Fellow.

##  2\. Preliminaries on homotopical algebra

The key to our approach to the study of the de Rham and Hodge-Tate cohomology
is the fact that the de Rham complex and the diffracted Hodge complex admit a
structure of a commutative algebra, in the appropriate derived sense. In this
expository section we summarize the necessary facts about non-abelian derived
functors, derived commutative algebras in the sense of Mathew, and
cosimplicial commutative algebras. The specific piece of structure that will
be used in Section 4 is a map $S^{p}A\to A$ where $S^{p}$ is the derived
symmetric power functor in the sense of [Ill71]. Such a map is naturally
induced both by the structure of a derived commutative algebra and a
cosimplicial commutative algebra structure on $A$. We choose to work with the
former, but comment in Subsection 2.3 on how to translate the results in the
language of cosimplicial commutative algebras.

Our exposition here definitely does not introduce any original mathematics
but, on the contrary, is aiming to convince the reader that the part of this
machinery relevant for us is powered by a fairly classical piece of
homolog(top)ical algebra.

###  2.1. Non-abelian derived functors.

Let $X$ be an arbitrary prestack, that is a functor
$X:\operatorname{Ring}^{\operatorname{{op}}}\to\operatorname{{Sp}}$ from the
category of ordinary commutative rings to the $\infty$-category of spaces. We
denote by $\operatorname{PreStk}$ the $\infty$-category of prestacks. For a
prestack $X$, we denote by $D(X)$ the derived $\infty$-category of quasi-
coherent sheaves on $X$ defined as the limit
$\lim\limits_{\operatorname{{Spec}}R\to X}D(R)$ of derived categories of $R$
modules, cf. [GR17, Chapter I.3]. We work in this large generality simply
because all actual proofs in this section take place over affine schemes and
the language of prestacks provides a convenient framework for generalizing the
result to more general geometric objects via descent.

By an algebraic stacks we will always mean a 1-Artin stack. In all of our
applications, $X$ will either be a scheme, a $p$-adic formal scheme, or a
global quotient of a scheme by a group scheme. If $X$ is a quasi-compact
scheme with affine diagonal then $D(X)$ is equivalent to the derived category
(in the sense of [Lur17, Definition 1.3.5.8]) of the usual abelian category
$\operatorname{QCoh}(X)$ of quasi-coherent sheaves on $X$. If such an $X$ is
being acted on by an affine flat group scheme $G$ then $D([X/G])$ is
equivalent to the left completion of the derived category of the abelian
category $\operatorname{QCoh}_{G}(X)$ of $G$-equivariant quasicoherent sheaves
on $X$.

For each $n\geq 0$ we have the derived functors
$S^{n},\Gamma^{n},\Lambda^{n}:D(X)\to D(X)$ of symmetric, divided, and
exterior power functors, respectively. These are the $\infty$-categorical
enhancements for the non-abelian derived functors defined by Illusie [Ill71].
We refer to [BM19, Section 3] and [BGMN21, Section 2.3] for a treatment in the
case $X=\operatorname{{Spec}}R$ is an affine scheme, and [KP21, Section A.2]
for the general case. We will now briefly recall the construction of these
functors and their basic properties. Denote by $\operatorname{{Mod}}_{R}$ the
ordinary abelian category of $R$-modules. These functors are uniquely
characterized by the following properties:

  1. (1)

If $X=\operatorname{{Spec}}R$ is affine then for a flat module
$M\in\operatorname{{Mod}}_{R}\subset D(R)$, the values

$S^{n}(M)=(M^{\otimes n})_{S_{n}},\Gamma^{n}(M)=(M^{\otimes n})^{S_{n}},\\\
\Lambda^{n}(M)=(M^{\otimes n})/\langle m_{1}\otimes\ldots\otimes
m_{n}|m_{i}=m_{j}\text{ for some }i\neq j\rangle$

are the usual symmetric, divided, and exterior powers.

  2. (2)

If $X$ is an affine scheme, the functors $S^{n},\Lambda^{n},\Gamma^{n}$
preserve sifted colimits, and are polynomial functors of degree $\leq n$ in
the sense of [BGMN21, Definition 2.11].

  3. (3)

These functors are natural in morphisms of the underlying stacks. That is, the
functors $S^{n},\Gamma^{n},\Lambda^{n}$ can be enhanced to endomorphisms of
the functor
$D(-):\operatorname{PreStk}^{\operatorname{{op}}}\to\operatorname{Cat}_{\infty}$
that sends $X$ to $D(X)$ and a morphism $f:X\to Y$ to the pullback functor
$f^{*}:D(Y)\to D(X)$.

These derived functors are first constructed on affine schemes using the
following:

######  Lemma 2.1 ([BGMN21]).

For a commutative ring $R$, the restriction functor

(2.1) $\operatorname{Func}_{\leq n}(D(R),D(R))\to\operatorname{Func}_{\leq
n}(\operatorname{Proj}_{R}^{\mathrm{f.g}},D(R))$

is an equivalence. Here the source of the functor is the category of
polynomial of degree $\leq n$ endofunctors of the stable $\infty$-category
$D(R)$ that preserve sifted colimits. The target category is the category of
polynomial functors of degree $\leq n$ from the abelian category of finitely
generated projective $R$-modules to the $\infty$-category $D(R)$.

###### Proof.

Statement (2) is a combination of [BGMN21, Theorem 2.19] applied to
${\mathcal{A}}=\operatorname{Proj}(R)^{\mathrm{f.g}}$ with the fact that
$D(R)$ is the ind-completion of
$\mathrm{Stab}({\mathcal{A}})=\operatorname{{Perf}}(R)$. ∎

Denote by $\operatorname{{Mod}}:\operatorname{Ring}\to\operatorname{Cat}$ the
functor from the ordinary category of commutative rings to the $2$-category of
ordinary categories that sends a commutative ring $R$ to the ordinary category
$\operatorname{{Mod}}_{R}$ or $R$ modules, and a morphism of rings $f:R\to
R^{\prime}$ to the functor
$R^{\prime}\otimes_{R}-:\operatorname{{Mod}}_{R}\to\operatorname{{Mod}}_{R^{\prime}}$.
Similarly, denote by $\operatorname{Proj}^{\mathrm{f.g.}}$ the subfunctor of
$\operatorname{{Mod}}$ that sends a ring $R$ to the category
$\operatorname{Proj}^{\mathrm{f.g.}}_{R}$ of finitely generated projective
$R$-modules.

We denote by $\operatorname{{Mor}}_{\leq
n}(\operatorname{Proj}^{\mathrm{f.g.}},\operatorname{{Mod}})\subset\operatorname{{Mor}}(\operatorname{Proj}^{\mathrm{f.g.}},\operatorname{{Mod}})$
the full subcategory of the ordinary $1$-category of natural transformations
$\operatorname{Proj}^{\mathrm{f.g.}}\to\operatorname{{Mod}}$ spanned by
transformations that are given by polynomial functors
$\operatorname{Proj}^{\mathrm{f.g.}}_{R}\to\operatorname{{Mod}}_{R}$ of degree
$\leq n$, for every ring $R$. We have the following construction principle for
polynomial functors on arbitrary prestacks

######  Lemma 2.2.

For every prestack $X$ there is a functor

(2.2) $\Sigma_{X}:\operatorname{{Mor}}_{\leq
n}(\operatorname{Proj}^{\mathrm{f.g.}},\operatorname{{Mod}})\to\operatorname{Func}(D(X),D(X))$

satisfying the following property:

For any morphism $f:\operatorname{{Spec}}R\to X$ from an affine scheme, and a
polynomial functor $T\in\operatorname{{Mor}}_{\leq
n}(\operatorname{Proj}^{\mathrm{f.g.}},\operatorname{{Mod}})$ there is an
equivalence of functors $f^{*}\circ\Sigma_{X}(T)\simeq
T^{\mathrm{derived}}_{R}\circ f^{*}$ from $D(X)$ to $D(R)$, where
$T^{\mathrm{derived}}_{R}$ is the result of applying the inverse of the
equivalence (2.1) to
$T_{R}:\operatorname{Proj}_{R}^{\mathrm{f.g.}}\to\operatorname{{Mod}}_{R}$.

Moreover, if $X$ is an algebraic stack flat over a ring $A$, then $\Sigma_{X}$
factors through the category $\operatorname{{Mor}}_{\leq
n}(\operatorname{Proj}^{\mathrm{f.g.}}|_{A-flat},\operatorname{{Mod}}|_{A-flat})$
of morphisms between these functors restricted to the category of flat
$A$-algebras.

###### Proof.

For a general $X$ the category $D(X)$ is, by definition, equivalent to
$\lim\limits_{\operatorname{{Spec}}R\to X}D(R)$, where the limit is taken over
all affine schemes mapping to $X$. Therefore
$\operatorname{Func}(D(X),D(X))\simeq\lim\limits_{\operatorname{{Spec}}R\to
X}\operatorname{Func}(D(X),D(R))$ and we define $\Sigma_{X}(T)$ as the object
of this limit given by $T_{R}^{\mathrm{derived}}\circ
f^{*}\in\operatorname{Func}(D(X),D(R))$ for every map
$f:\operatorname{{Spec}}R\to X$.

The last assertion follows from the fact that if $X$ is a flat algebraic stack
over $A$, then in the above limit we may restrict to morphisms
$\operatorname{{Spec}}R\to X$ for which $R$ is flat over $A$, by [GR17,
Proposition 3.1.4.2]. ∎

######  Definition 2.3.

For a prestack $X$, we define functors $S^{n},\Gamma^{n},\Lambda^{n}:D(X)\to
D(X)$ as images under $\Sigma_{X}$ of the corresponding polynomial functors on
projective modules over rings. We will denote these functors by
$S^{n}_{X},\Gamma^{n}_{X},\Lambda^{n}_{X}$ if the base is not clear from the
context.

######  Remark 2.4.

We can give an explicit recipe for computing the derived functors
$S^{n},\Gamma^{n},\Lambda^{n}$ that we state in a special case, for
simplicity. Suppose that $X$ is a quasi-compact separated scheme and $M\in
D^{-}(X)$ is an object with cohomology bounded from above, that can be
represented by a complex $M^{0}\to M^{1}\to\ldots$ concentrated in degrees
$\geq 0$ of flat quasicoherent sheaves. Then $S^{n}(M)$ (and likewise for the
other functors $\Lambda^{n},\Gamma^{n}$) is the totalization of the
cosimplicial diagram

(2.3)
${S^{n}(\operatorname{DK}(M)^{0})}$${S^{n}(\operatorname{DK}(M)^{1})}$${\ldots}$

where $\operatorname{DK}(M)$ is the cosimplicial sheaf associated to the
complex $M$ by the Dold-Kan equivalence. The miracle is that the resulting
object $S^{n}(M)$ does not depend on the choice of a resolution, up to a
quasi-isomorphism.

We will use Lemma 2.2 frequently for homotopy-coherent computations with
polynomial functors. Note that specifying an object of
$\operatorname{{Mor}}_{\leq
n}(\operatorname{Proj}^{\mathrm{f.g.}},\operatorname{{Mod}})$ amounts to a
manageable collection of data: we need to give a functor
$T_{R}:\operatorname{Proj}^{\mathrm{f.g.}}_{R}\to\operatorname{{Mod}}_{R}$ for
every ring $R$, construct equivalences $R^{\prime}\otimes_{R}T_{R}(-)\simeq
T_{R^{\prime}}(R^{\prime}\otimes_{R}-)$ for all maps of rings $R\to
R^{\prime}$, and then check (rather than provide any additional constructions)
that these equivalences are associative with respect to the composition of
maps of rings. Let us illustrate this technique by constructing certain maps
between the symmetric and divided power functors.

For a projective module $M$ over a ring $R$ there are natural maps between
$S^{n}M=(M^{\otimes n})_{S_{n}}$ and $\Gamma^{n}M=(M^{\otimes n})^{S_{n}}$:

(2.4) $r_{n}:\Gamma^{n}M\hookrightarrow M^{\otimes n}\twoheadrightarrow
S^{n}M\qquad N_{n}=\sum\limits_{\sigma\in S_{n}}\sigma:S^{n}M\to\Gamma^{n}M$

The compositions $N_{n}\circ r_{n}$ and $r_{n}\circ N_{n}$ are equal to
$n!\cdot\mathrm{Id}_{\Gamma^{n}M}$ and $n!\cdot\mathrm{Id}_{S^{n}M}$,
respectively. Since maps $r_{n}$ and $N_{n}$ are compatible with base change
along arbitrary maps of rings $R\to R^{\prime}$, they define morphism between
functors $S^{n}$ and $\Gamma^{n}$ in the category $\operatorname{{Mor}}_{\leq
n}(\operatorname{Proj}^{\mathrm{f.g.}},\operatorname{{Mod}})$. Therefore Lemma
2.2 provides us with maps

(2.5) $r_{n}:\Gamma^{n}\to S^{n}\qquad N_{n}:S^{n}\to\Gamma^{n}$

of endofucntors of the category $D(X)$ for every prestack $X$.

######  Lemma 2.5.

For an ${\mathbb F}_{p}$-algebra $R_{0}$ and a projective $R_{0}$-module $M$
the map $r_{p}:\Gamma^{p}M\to S^{p}M$ naturally factors as

(2.6)
$\Gamma^{p}M\xrightarrow{\psi_{M}}F^{*}_{R_{0}}M\xrightarrow{\Delta_{M}}S^{p}M$

where the first arrow is a surjection and the second one is an injection.

###### Proof.

We start by constructing the map
$\Delta_{M}:M\otimes_{R_{0},F_{R_{0}}}R_{0}=F^{*}_{R_{0}}M\to S^{p}M$. Define
$\Delta_{M}(m\otimes r)=r\cdot m^{p}$, this is a well-defined additive
$R_{0}$-linear map. To check that $r_{p}$ factors through $\Delta_{M}$ we may
work locally on $\operatorname{{Spec}}R_{0}$ and thus assume that $M$ is a
free $R_{0}$-module. Denote a basis for $M$ by $e_{1},\ldots,e_{n}$. Then a
basis for the module $\Gamma^{p}M\subset M^{\otimes p}$ is given by the
elements $e_{I}:=\sum\limits_{\sigma\in
S_{p}/\operatorname{Stab}_{I}}e_{i_{1}}\otimes\ldots\otimes e_{i_{p}}$ where
$I=(i_{1},\ldots,i_{p})$ runs through $S_{p}$-orbits on
$\\{1,\ldots,n\\}^{\times p}$. The map $r_{p}$ sends $e_{I}$ to
$[S_{p}:\operatorname{Stab}_{I}]\cdot e_{i_{1}}\cdot\ldots\cdot e_{i_{p}}$. If
the stabilizer of $I$ inside $S_{p}$ has index coprime to $p$, then
$\operatorname{Stab}_{I}$ contains a long cycle of length $p$ which forces all
$i_{1},\ldots,i_{p}$ to be equal. Hence only basis elements of the form
$e_{i}^{\otimes p}\in\Gamma^{p}M$ are not killed by $r_{p}$, which gives the
desired factorization. ∎

By Lemma 2.2, having constructed maps $\psi_{M},\Delta_{M}$ for projective
modules in a functorial fashion, we get maps

(2.7) $\Delta_{M}:F_{X_{0}}^{*}M\to S^{p}M\qquad\psi_{M}:\Gamma^{p}M\to
F_{X_{0}}^{*}M$

for all objects $M\in D(X_{0})$ for an arbitrary ${\mathbb F}_{p}$-prestack
$X_{0}$. The composition $\Delta_{M}\circ\psi_{M}$ is naturally homotopic to
$r_{p}:\Gamma^{p}M\to S^{p}M$, by construction.

A useful computation tool for us will be the following ‘décalage’
isomorphisms:

######  Lemma 2.6 ([Ill71, Proposition 4.3.2.1],[KP21, Proposition A.2.49]).

There are natural equivalences for $M\in D(X)$

(2.8)
$S^{n}(M[1])\simeq(\Lambda^{n}M)[n]\quad\Gamma^{n}(M[-1])\simeq(\Lambda^{n}M)[-n]$

###  2.2. Derived commutative algebras.

The precise version of the notion of a commutative algebra that we will need
is derived commutative algebras, in the sense of Mathew. In this section we
collect the necessary material about this notion, largely following [Rak20,
Section 4]. Let us stress again that this section is purely expository and
contains no original results.

Recall the following point of view on ordinary commutative algebras over a
field $k$, afforded by the Barr-Beck theorem. The endofunctor
$S^{\bullet}=\bigoplus\limits_{n\geq 0}S^{n}$ of the category of $k$-vector
spaces has a structure of a monad, and the category of $k$-algebras is
equivalent to the category of modules over the monad $S^{\bullet}$ in the
category of $k$-vector spaces. The idea of derived commutative algebras is to
replicate this definition in the derived world by using the appropriate
derived version of the symmetric algebra monad.

######  Construction 2.7 ([Rak20, Construction 4.2.19]).

For a ring $R$ consider the endofunctor $S^{\bullet}:=\bigoplus\limits_{n\geq
0}S^{n}:\operatorname{Proj}_{R}\to\operatorname{Proj}_{R}$ of the ordinary
category of projective $R$-modules. It has a monad structure induced by the
maps $S^{i}(S^{j}M)\twoheadrightarrow S^{i\cdot j}M$ for every projective
module $M$. It naturally extends to a monad, in the sense of [Lur17,
Definition 4.7.0.1], on the category $D(X)$ for every prestack $X$ which we
denote as

(2.9) $S^{\bullet}:=\bigoplus\limits_{n\geq 0}S^{n}:D(X)\to D(X)$

and call it the derived symmetric algebra monad.

We can now define derived commutative algebras on $X$:

######  Definition 2.8.

The category $\operatorname{DAlg}(X)$ of derived commutative algebras on $X$
is the category of modules over the derived symmetric algebra monad
$S^{\bullet}$ in $D(X)$.

By construction, there is a functor
$S^{\bullet}:D(X)\to\operatorname{DAlg}(X)$ left adjoint to the forgetful
functor $\operatorname{DAlg}(X)\to D(X)$. For a derived commutative algebra
$A\in\operatorname{DAlg}(X)$ we will usually denote the underlying object in
$D(X)$ by the same symbol $A$. The structure of a derived commutative algebra,
in particular, gives a map

(2.10) $m:A\otimes_{{\mathcal{O}}_{X}}A\to S^{2}A\to A$

which induces a graded commutative product operation
$H^{i}(A)\otimes_{{\mathcal{O}}_{X}}H^{j}(A)\to H^{i+j}(A)$ on cohomology
sheaves.

######  Definition 2.9.

For an object $M$, we call $S^{\bullet}(M)\in\operatorname{DAlg}(X)$ the free
derived commutative algebra on $M$.

######  Remark 2.10.

For every $M\in D(X)$ there are natural maps $(M^{\otimes n})_{hS_{n}}\to
S^{n}(M)$ which give rise to a map from the monad
$M\mapsto\bigoplus\limits_{n\geq 0}(M^{\otimes n})_{hS_{n}}$ to $S^{\bullet}$.
It is in general far from equivalence, we make a precise comparison between
their values in a special case in Lemma 3.8. Since modules over the the former
monad are the $E_{\infty}$-algebras on $X$, we get a functor
$\operatorname{DAlg}(X)\to\operatorname{Alg}_{E_{\infty}}D(X)$.

Derived commutative algebras in characteristic $p$ have natural Frobenius
endomorphisms:

######  Lemma 2.11.

Suppose that $X$ is a prestack over ${\mathbb F}_{p}$. For all
$A\in\operatorname{DAlg}(X)$ there is a natural morphism
$\varphi_{A}:F_{X}^{*}A\to A$ in $D(X)$ that is equal to the linearization of
the usual Frobenius endomorphism when $A$ is a flat ordinary commutative
algebra on an affine scheme $X$.

######  Remark 2.12.

It should be possible to enhance $\varphi_{A}$ to a morphism of derived
commutative algebras, but we do not pursue this here.

###### Proof.

We define $\varphi_{A}$ as the composition
$F_{X}^{*}A\xrightarrow{\Delta_{A}}S^{p}A\xrightarrow{m_{A}}A$ where $m_{A}$
is a part of the $S^{\bullet}$-module structure on $A$, and $\Delta_{A}$ is
the morphism defined in (2.7). Compatibility with the usual Frobenius follows
from the defining formula of $\Delta_{A}$ given in the proof of Lemma 2.5. ∎

One of the main motivations for introducing the notion of derived commutative
algebras is that many cohomological invariants arising in geometry are
naturally equipped with the structure of a derived commutative algebra.

######  Proposition 2.13.

If $f:X\to Y$ is a morphism of prestacks then the functor $Rf_{*}:D(X)\to
D(Y)$ can be naturally enhanced to a functor
$Rf^{\mathrm{alg}}_{*}:\operatorname{DAlg}(X)\to\operatorname{DAlg}(Y)$.

###### Proof.

Recall that $Rf_{*}:D(X)\to D(Y)$ is defined as the right adjoint functor to
the pullback functor $f^{*}:D(Y)\to D(X)$. The equivalences $f^{*}\circ
S^{n}_{Y}\simeq S^{n}_{X}\circ f^{*}$ induce a colimit-preserving functor
$f^{*}_{\mathrm{alg}}:\operatorname{DAlg}(Y)\to\operatorname{DAlg}(X)$ given
by $f^{*}$ on the underlying objects of $D(-)$. Since $\operatorname{DAlg}(Y)$
is presentable (e.g. by [Rak20, Proposition 4.1.10]), this functor admits a
right adjoint
$Rf_{*}^{\mathrm{alg}}:\operatorname{DAlg}(X)\to\operatorname{DAlg}(Y)$, by
the adjoint functor theorem [Lur09, Corollary 5.5.2.9(1)].

It remains to check that $Rf_{*}^{\mathrm{alg}}$ defined this way induces
$Rf_{*}$ on the underlying sheaves. By construction, the compositions
$D(Y)\xrightarrow{f^{*}}D(X)\xrightarrow{S^{\bullet}_{X}}\operatorname{DAlg}(X)$
and
$D(Y)\xrightarrow{S^{\bullet}_{Y}}\operatorname{DAlg}(Y)\xrightarrow{f^{*}_{\mathrm{alg}}}\operatorname{DAlg}(X)$
are equivalent. Their right adjoints are $\operatorname{DAlg}(X)\to
D(X)\xrightarrow{Rf_{*}}D(Y)$ and
$\operatorname{DAlg}(X)\xrightarrow{Rf_{*}^{\mathrm{alg}}}\operatorname{DAlg}(Y)\xrightarrow{}D(Y)$,
and their equivalence is precisely the desired compatibility between $Rf_{*}$
and $Rf_{*}^{\mathrm{alg}}$. ∎

In particular, $Rf_{*}{\mathcal{O}}_{X}\in D(Y)$ is naturally equipped with a
derived commutative algebra structure. The functor $Rf^{\mathrm{alg}}_{*}$ is
compatible with Frobenius endomorphisms:

######  Lemma 2.14.

Suppose that $Y$ is a prestack over ${\mathbb F}_{p}$ and $f:X\to Y$ is a
morphism of prestacks. For $A\in\operatorname{DAlg}(X)$ there is a natural
equivalence between the composition $F_{Y}^{*}Rf_{*}A\to
Rf_{*}F_{X}^{*}A\xrightarrow{Rf_{*}\varphi_{A}}Rf_{*}A$ and the map
$F_{Y}^{*}Rf_{*}A\xrightarrow{\varphi_{Rf_{*}^{\mathrm{alg}}A}}Rf_{*}A$ in
$D(Y)$. Here $F_{Y}^{*}Rf_{*}A\to Rf_{*}F_{X}^{*}A$ is the base change map
corresponding to the equality $F_{Y}\circ f=f\circ F_{X}$.

###### Proof.

By adjunction between $f^{*}$ and $Rf_{*}$, and the definition of
$\varphi_{A}$ and $\varphi_{Rf_{*}^{\mathrm{alg}}A}$, our task is equivalent
to identifying the composition

(2.11) $f^{*}F_{Y}^{*}Rf_{*}A\to
f^{*}Rf_{*}F_{X}^{*}A\xrightarrow{f^{*}Rf_{*}\Delta_{A}}f^{*}Rf_{*}S^{p}_{X}A\xrightarrow{f^{*}Rf_{*}m_{A}}f^{*}Rf_{*}A\to
A$

with the composition

(2.12)
$f^{*}F_{Y}^{*}Rf_{*}A\xrightarrow{f^{*}\Delta_{Rf_{*}A}}f^{*}S^{p}_{Y}Rf_{*}A\xrightarrow{f^{*}m_{Rf_{*}^{\mathrm{alg}}A}}f^{*}Rf_{*}A\to
A$

The map $S^{p}_{Y}Rf_{*}A\xrightarrow{m_{Rf_{*}^{\mathrm{alg}}A}}Rf_{*}A$ is
adjoint to the map $f^{*}S^{p}_{Y}Rf_{*}A\simeq S^{p}_{X}f^{*}Rf_{*}A\to
S^{p}_{X}A\xrightarrow{m_{A}}A$, which allows us to rewrite (2.12) as

(2.13) $f^{*}F_{Y}^{*}Rf_{*}A\xrightarrow{}f^{*}S^{p}_{Y}Rf_{*}A\simeq
S^{p}_{X}f^{*}Rf_{*}A\to S^{p}_{X}A\xrightarrow{m_{A}}A$

For any object $M\in D(Y)$ the map
$f^{*}F_{Y}^{*}M\xrightarrow{f^{*}\Delta_{M}}f^{*}S^{p}_{Y}M\simeq
S^{p}_{X}f^{*}M$ can be identified with the composition $f^{*}F_{Y}^{*}M\simeq
F_{X}^{*}f^{*}M\xrightarrow{\Delta_{f^{*}M}}S^{p}_{X}f^{*}M$ where the first
equivalence arises from the fact that $f$ intertwines the Frobenius
endomorphisms of $X$ and $Y$. Applying this to $M=Rf_{*}A$ allows us to
identify (2.11) with (2.13), as desired. ∎

We can use Lemma 2.13 to construct examples of derived commutative algebras:

######  Definition 2.15.

For a finite locally free sheaf $M$ (concentrated in degree $0$) on a scheme
$X$ we define the free divided power algebra on $M[-1]$ denote by
$\Gamma^{\bullet}(M[-1])\in\operatorname{DAlg}(X)$ as
$R\pi_{*}^{\mathrm{alg}}{\mathcal{O}}_{B_{X}M^{\vee\\#}}$ where
$\pi:B_{X}M^{\vee\\#}\to X$ is the relative classifying stack of the divided
power group scheme $M^{\vee\\#}$ on $X$ associated to $M^{\vee}$.

This terminology is justified by the fact that the underlying
$E_{\infty}$-algebra of $\Gamma^{\bullet}(M[-1])$ is identified with
$\bigoplus\limits_{i\geq 0}\Lambda^{i}M[-i]$, e.g. by [BL22b, Lemma 7.8]. If
$X$ is an ${\mathbb F}_{p}$-scheme, by Lemma 2.14 the Frobenius map
$\varphi^{*}_{\Gamma^{\bullet}(M[-1])}:F_{X}^{*}\Gamma^{\bullet}(M[-1])\to\Gamma^{\bullet}(M[-1])$
factors as
$F_{X}^{*}\Gamma^{\bullet}(M[-1])\to{\mathcal{O}}_{X}\to\Gamma^{\bullet}(M[-1])$
because the Frobenius endomorphism of $M^{\vee\\#}$ factors through the
identity section.

Cohomology of a sheaf of rings on a site can be equipped with a derived
commutative algebra structure:

######  Lemma 2.16.

Let ${\mathcal{C}}$ be a site and ${\mathcal{F}}$ be a sheaf of (ordinary)
commutative algebras over a ring $R$ on ${\mathcal{C}}$. Then for any object
$X\in{\mathcal{C}}$ the complex $\mathrm{R}\Gamma(X,{\mathcal{F}})\in D(R)$ is
naturally endowed with a structure of a derived commutative $R$-algebra.

###### Proof.

By [Sta23, 01GZ] we can compute $\mathrm{R}\Gamma(X,{\mathcal{F}})$ as a
filtered colimit over all hypercovers $U_{\bullet}\to X$ of Čech cohomology
with respect to $U_{\bullet}$:

(2.14)
$\mathrm{R}\Gamma(X,{\mathcal{F}})\simeq\operatorname*{colim}\limits_{U_{\bullet}\to
X}\lim\limits_{n}{\mathcal{F}}(U_{n})$

Each ${\mathcal{F}}(U_{n})$ is a commutative $R$-algebra, which we view as an
object of $\operatorname{DAlg}(R)$. Since the forgetful functor
$\operatorname{DAlg}(R)\to D(R)$ commutes with limits and filtered colimits,
this endows $\mathrm{R}\Gamma(X,{\mathcal{F}})$ with the structure of a
derived commutative $R$-algebra. ∎

Applying this construction to étale cohomology with coefficients in ${\mathbb
F}_{p}$ produces derived commutative algebras with the special property that
the Frobenius endomorphism is homotopic to identity:

######  Lemma 2.17.

If $X$ is a scheme then the Frobenius endomorphism
$\varphi_{\mathrm{R}\Gamma_{\text{et}}(X,{\mathbb F}_{p})}$ of the derived
commutative ${\mathbb F}_{p}$-algebra $\mathrm{R}\Gamma_{\text{et}}(X,{\mathbb
F}_{p})$ is naturally homotopic to the identity morphism.

###### Proof.

In the formula (2.14) each ${\mathbb
F}_{p}(U_{n})=\operatorname{Func}(\pi_{0}(U_{n}),{\mathbb F}_{p})$ is the
algebra of ${\mathbb F}_{p}$-valued functions on a set, and its Frobenius
endomorphism is the identity, so the lemma follows. ∎

When studying the Sen operator, we will use that it is compatible with the
derived commutative algebra structure on the diffracted Hodge cohomology (to
be defined in Section 5). Specifically, we need the following notion

######  Definition 2.18.

For a prestack $X$ and a derived commutative algebra
$A\in\operatorname{DAlg}(X)$, a derivation $f:A\to A$ is a map in $D(X)$ such
that the map $\mathrm{Id}_{A}+\varepsilon\cdot f:A\otimes{\mathbb
Z}[\varepsilon]/\varepsilon^{2}\to A\otimes{\mathbb
Z}[\varepsilon]/\varepsilon^{2}$ in $D(X\times{\mathbb
Z}[\varepsilon]/\varepsilon^{2})$ is equipped with the structure of a map in
$\operatorname{DAlg}(X\times{\mathbb Z}[\varepsilon]/\varepsilon^{2})$.

###  2.3. Cosimplicial commutative algebras.

For the duration of this subsection assume that $X$ is a scheme. In all of our
main applications we will in fact be presented with a cosimplicial commutative
algebra in the ordinary category $\operatorname{QCoh}(X)$. We denote by
$\operatorname{CAlg}^{\Delta}_{X}$ the ordinary category of cosimplicial
commutative algebras in the abelian symmetric monoidal category
$\operatorname{QCoh}(X)$. In this subsection we make some remarks on the
relation between $\operatorname{CAlg}^{\Delta}_{X}$ and
$\operatorname{DAlg}(X)$. These facts are not used in any of our main results,
but the reader who feels more comfortable with
$\operatorname{CAlg}^{\Delta}_{X}$ than with $\operatorname{DAlg}(X)$ is
encouraged to specialize the results in Section 4 to a situation where the
algebra $A$ is a cosimplicial commutative algbera, using this subsection as a
dictionary.

In general, a cosimplicial commutative algebra gives rise to a derived
commutative algebra:

######  Lemma 2.19.

Let $X$ be a scheme. There is a natural functor
${\mathcal{R}}_{X}:\operatorname{CAlg}^{\Delta}(X)\to\operatorname{DAlg}(X)$
from the ordinary category of cosimplicial commutative algebras in quasi-
coherent sheaves on $X$ to the category of derived commutative algebras on
$X$. This functor is compatible with the cosimplicial totalization functor on
the level of underlying complexes.

######  Remark 2.20.

For all of our main results we will work with derived commutative algebras
concentrated in degrees $\geq 0$. In a forthcoming work, Mathew and Mondal
will show that for any commutative base ring $R$ the $\infty$-category of
derived commutative $R$-algebras concentrated in degrees $\geq 0$ is
equivalent to the $\infty$-category of cosimplicial commutative $R$-algebras,
cf. [MR23, Remark 2.1.8].

###### Proof of Lemma 2.19.

First of all, there is a functor from ordinary commutative algebras in
$\operatorname{QCoh}(X)$ to $\operatorname{DAlg}(X)$. Indeed, for an ordinary
algebra $A$ there is a natural map
$S^{n}A\to(A^{\otimes_{{\mathcal{O}}_{X}}n})_{S_{n}}$ where the tensor product
and coinvariants are taken in the non-derived sense. Hence the commutative
multiplication on $A\in\operatorname{QCoh}(X)$ endows it with a structure of a
monad over $S^{\bullet}$ in $D(X)$.

Now, if $A=\left[{\leavevmode\hbox to69.32pt{\vbox
to20.86pt{\pgfpicture\makeatletter\hbox{\hskip
34.66109pt\lower-11.87703pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{\offinterlineskip{}{}{{{}}{{}}}{{{}}}{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-34.66109pt}{-8.97859pt}\pgfsys@invoke{
}\hbox{\vbox{\halign{\pgf@matrix@init@row\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding&&\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding\cr\hfil\hskip
9.45555pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-5.15001pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${A^{0}}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
9.45555pt\hfil&\hfil\hskip 37.20552pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-8.90001pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${A^{1}\ldots}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
13.20555pt\hfil\cr}}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}{{{{}}}{{}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} { {}{}{}}{}{ {}{}{}}{}{ {}{}{}} {{{{{}}{
{}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{}{{{}{}}}\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{}}{}{}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{{}}{}{{{{}}}{{{}}}}{{}}{{{}}{{}}}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{}}{}{}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{{{}}{{}}}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{}}{}{}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{{}}{}{{}} {}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-15.54999pt}{-3.68pt}\pgfsys@lineto{7.65002pt}{-3.68pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{7.85pt}{-3.68pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{
{}{}{}}{}{ {}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{}{{{}{}}}\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{}}{}{}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{{}}{}{{{{}}}{{{}}}}{{}}{{{}}{{}}}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{}}{}{}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{{{}}{{}}}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{}}{}{}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{{}}{}{{}} {}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-15.54999pt}{-9.27719pt}\pgfsys@lineto{7.65002pt}{-9.27719pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{7.85pt}{-9.27719pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\right]$ is a
cosimplicial commutative algebra in $\operatorname{QCoh}(X)$, we define
${\mathcal{R}}_{X}(A)$ as
$\lim\limits_{[n]\in\Delta}{\mathcal{R}}_{X}(A^{n})$, where each
${\mathcal{R}}_{X}(A^{n})$ was defined in the previous paragraph. This indeed
induces the totalization on underlying sheaves because the forgetful functor
$\operatorname{DAlg}(X)\to D(X)$ commutes with limits. ∎

######  Example 2.21.

If $f:X\to\operatorname{{Spec}}R$ is a morphism from a separated scheme to the
spectrum of a ring $R$, the object
$\mathrm{R}\Gamma(X,{\mathcal{O}}_{X})=f_{*}({\mathcal{O}}_{X})\in D(R)$ is
endowed with a structure of a cosimplicial commutative algebra using the Čech
construction associated to a cover $X=\bigcup\limits_{i}U_{i}$:

(2.15) $\mathrm{R}\Gamma(X,{\mathcal{O}}_{X})\simeq\left[{\leavevmode\hbox
to181.3pt{\vbox to30.52pt{\pgfpicture\makeatletter\hbox{\hskip
90.65114pt\lower-18.10681pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{\offinterlineskip{}{}{{{}}{{}}{{}}}{{{}}}{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-90.65114pt}{-12.40977pt}\pgfsys@invoke{
}\hbox{\vbox{\halign{\pgf@matrix@init@row\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding&&\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding\cr\hfil\hskip
22.69386pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-18.38832pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{${\prod\limits_{i}{\mathcal{O}}(U_{i})}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
22.69386pt\hfil&\hfil\hskip 59.90173pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-31.59622pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{${\prod\limits_{i<j}{\mathcal{O}}(U_{i}\cap U_{j})}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
35.90176pt\hfil&\hfil\hskip 32.05551pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.75pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${\ldots}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
8.05554pt\hfil\cr}}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}{{{{}}}{{}}{{}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} { {}{}{}}{}{ {}{}{}}{}{ {}{}{}} {{{{{}}{
{}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{}{{{}{}}}\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{}}{}{}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{{}}{}{{{{}}}{{{}}}}{{}}{{{}}{{}}}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{}}{}{}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{{{}}{{}}}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{}}{}{}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{{}}{}{{}} {}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-45.06342pt}{-7.11118pt}\pgfsys@lineto{-21.8634pt}{-7.11118pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-21.66342pt}{-7.11118pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{
{}{}{}}{}{ {}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{}{{{}{}}}\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{}}{}{}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{{}}{}{{{{}}}{{{}}}}{{}}{{{}}{{}}}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{}}{}{}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{{{}}{{}}}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{}}{}{}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{{}}{}{{}} {}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-45.06342pt}{-12.70837pt}\pgfsys@lineto{-21.8634pt}{-12.70837pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-21.66342pt}{-12.70837pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{
{}{}{}}{}{ {}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{}{{{}{}}}\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{}}{}{}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{{}}{}{{{{}}}{{{}}}}{{}}{{{}}{{}}}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{}}{}{}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{{{}}{{}}}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{}}{}{}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{{}}{}{{}} {}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{50.74008pt}{-4.31258pt}\pgfsys@lineto{73.9401pt}{-4.31258pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{74.14008pt}{-4.31258pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{
{}{}{}}{}{ {}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{}{{{}{}}}\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{}}{}{}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{{}}{}{{{{}}}{{{}}}}{{}}{{{}}{{}}}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{}}{}{}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{{{}}{{}}}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{}}{}{}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{{}}{}{{}} {}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{50.74008pt}{-15.50697pt}\pgfsys@lineto{73.9401pt}{-15.50697pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{74.14008pt}{-15.50697pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{
{}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{50.74008pt}{-9.90977pt}\pgfsys@lineto{73.9401pt}{-9.90977pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{74.14008pt}{-9.90977pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\right]$

The result of applying ${\mathcal{R}}_{R}$ to this cosimplicial algebra is
naturally equivalent to $f_{*}^{\mathrm{alg}}{\mathcal{O}}_{X}$ from Lemma
2.13.

If $X$ is a scheme over ${\mathbb F}_{p}$, for every cosimplicial commutative
algebra $A\in\operatorname{CAlg}^{\Delta}_{X}$ there is a natural map
$\varphi^{\mathrm{cosimp}}_{A}:F_{X}^{*}A\to A$ of algebras induced by the
term-wise Frobenius endomorphism. It follows from the construction of the
functor ${\mathcal{R}}_{X}$ that $\varphi_{A}^{\mathrm{cosimp}}$ coincides
with the Frobenius map arising from the derived commutative algebra structure:

######  Lemma 2.22.

For an ${\mathbb F}_{p}$-scheme $X$, and a term-wise flat cosimplicial
commutative algebra $A\in\operatorname{CAlg}^{\Delta}_{X}$ there is a natural
homotopy between ${\mathcal{R}}_{X}(\varphi_{A}^{\mathrm{cosimp}})$ and
$\varphi_{A}$ in $D(X)$.

###### Proof.

If $A$ is an ordinary commutative algebra, this was established along with
defining $\varphi_{A}$ in Lemma 2.11. The case of an arbitrary $A$ is obtained
by passing to the limit over the cosimplicial category $\Delta$. ∎

For a locally free sheaf $M$ on a scheme $X$ we can represent the free divided
power algebra $\Gamma^{\bullet}(M[-1])\in\operatorname{DAlg}(X)$ of Definition
2.15 by a cosimplicial commutative algebra. Denote by
$\operatorname{DK}(M[-1])$ the cosimplicial object in the category of locally
free sheaves on $X$, obtained by applying the Dold-Kan correspondence to the
complex $M[-1]$.

######  Lemma 2.23.

The derived commutative algebra $\Gamma^{\bullet}(M[-1])$ is equivalent to
${\mathcal{R}}_{X}(\Gamma^{\bullet}_{\mathrm{naive}}(\operatorname{DK}(M[-1])))$
where $\Gamma^{\bullet}_{\mathrm{naive}}(\operatorname{DK}(M[-1]))$ is the
result of applying term-wise the free divided power algebra functor to the
cosimplicial sheaf $\operatorname{DK}(M[-1])$.

###### Proof.

The derived pushforward of the structure sheaf along the map
$B_{X}M^{\vee\\#}\to X$ is equivalent, as a derived commutative algebra, to
the totalization of the cosimplicial diagram

(2.16)
${{\mathcal{O}}_{X}}$${\pi_{*}{\mathcal{O}}_{M^{\vee\\#}}}$${\pi_{*}{\mathcal{O}}_{M^{\vee\\#}\times_{X}M^{\vee\\#}}\ldots}$

obtained by applying the functor $\pi_{*}{\mathcal{O}}_{(-)}$ to the bar-
resolution associated to the group scheme $\pi:M^{\vee\\#}\to X$ over $X$. The
$n$th term of the diagram (2.16) is the commutative algebra
$\pi_{*}{\mathcal{O}}_{M^{\vee\\#\times_{X}n}}\simeq\Gamma^{\bullet}_{X}(M^{\oplus
n})$ concentrated in degree $0$, so the cosimplicial commutative algebra
defined by (2.16) is indeed equivalent to the result of applying
$\Gamma^{\bullet}_{\mathrm{naive}}$ to $\operatorname{DK}(M[-1])$. ∎

###  2.4. Bockstein morphisms.

For a stable ${\mathbb Z}$-linear $\infty$-category ${\mathcal{C}}$, any
object $M\in{\mathcal{C}}$ gives rise to a natural fiber sequence

(2.17) $M\xrightarrow{p}M\to M/p$

We will denote the corresponding connecting map by $\mathrm{Bock}_{M}:M/p\to
M[1]$ and refer to it as the Bockstein morphism corresponding to $M$. Note
that $\mathrm{Bock}_{M[1]}$ is naturally homotopic to
$(-\mathrm{Bock}_{M}[1])$.

Similarly, for a ${\mathbb Z}/p^{n}$-linear stable $\infty$-category
${\mathcal{C}}$ for any object $M\in{\mathcal{C}}$ we have a fiber sequence

(2.18) $M\otimes_{{\mathbb Z}/p^{n}}{\mathbb Z}/p^{n-1}\to M\to
M\otimes_{{\mathbb Z}/p^{n}}{\mathbb Z}/p$

inducing the connecting map $\mathrm{Bock}_{M}:M\otimes{\mathbb Z}/p\to
M\otimes{\mathbb Z}/p^{n-1}[1]$. These constructions over ${\mathbb Z}$ and
${\mathbb Z}/p^{n}$ are compatible in the sense that $\mathrm{Bock}_{M/p^{n}}$
is the composition $M/p\xrightarrow{\mathrm{Bock}_{M}}M[1]\to M/p^{n-1}[1]$.

##  3\. Symmetric power $S^{p}$, class $\alpha$, and Steenrod operations

In this section we study in detail the derived functor $S^{p}$ of $p$th
symmetric power by comparing it with the divided power functor $\Gamma^{p}$.
For complexes of vector spaces over a field $k$ the values of $S^{n}$ can be
described non-canonically using the computations of $S^{n}(k[-i])$ done by
Priddy [Pri73], but we crucially need to understand $S^{p}M$ as a complex of
sheaves, rather that its separate cohomology sheaves.

###  3.1. Symmetric powers vs. divided powers.

Let $X$ be an arbitrary prestack. Recall that in the previous section for an
object $M\in D(X)$ we defined natural morphisms

(3.1) $N_{n}:S^{n}M\to\Gamma^{n}M\quad r_{n}:\Gamma^{n}M\to S^{n}M$

Denote by $T_{n}(M)$ the cofiber
$\operatorname{cofib}(N_{n}:S^{n}M\to\Gamma^{n}M)$ of the norm map. The
functor $T_{n}$, especially for $n=p$, has been extensively studied in the
literature. References close to our point of view are works of Friedlander-
Suslin [FS97, Section 4], and Kaledin [Kal18, 6.3]. We have the following
classical results about $T_{n}$:

######  Lemma 3.1 ([FS97, Lemma 4.12],[Kal18, Lemma 6.9]).

  1. (1)

If $X$ is a prestack over ${\mathbb Z}[\frac{1}{n!}]$ then $T_{n}(M)\simeq 0$
for every $M\in D(X)$.

  2. (2)

If $R$ is a flat ${\mathbb Z}_{p}$-algebra and $M$ is a flat $R$-module, then
$N_{p}:S^{p}M\to\Gamma^{p}M$ is an injection and its cokernel $T_{p}(M)$ is
naturally isomorphic to $F^{*}_{R/p}(M/p)$ as an $R$-module.

  3. (3)

If $X$ is a flat algebraic stack over ${\mathbb Z}_{p}$ then there is a
natural equivalence $T_{p}(M)\simeq i_{*}F_{X_{0}}^{*}i^{*}M$, where
$i:X_{0}=X\times_{{\mathbb Z}_{p}}{\mathbb F}_{p}\to X$ is the closed
immersion of the special fiber, and $F_{X_{0}}:X_{0}\to X_{0}$ is the absolute
Frobenius morphism.

  4. (4)

If $X_{0}$ is a prestack over ${\mathbb F}_{p}$ then $T_{p}(M_{0})$ fits into
a natural fiber sequence

(3.2) $F_{X_{0}}^{*}M_{0}[1]\to T_{p}(M_{0})\to F_{X_{0}}^{*}M_{0}.$

######  Remark 3.2.

  1. (1)

In the setting of (3), if $M_{0}\simeq i^{*}M\in D(X_{0})$ is the reduction of
an object $M\in D(X)$ then $T_{p}(M_{0})$ can be naturally (in $M$) described
as

(3.3) $T_{p}(M_{0})\simeq i^{*}T_{p}(M)\simeq i^{*}i_{*}F_{X_{0}}^{*}M_{0}$

Under this identification, the fiber sequence in (3.2) is identified with the
sequence induced by the fiber sequence of functors $\mathrm{Id}[1]\to
i^{*}i_{*}\to\mathrm{Id}$ from $D(X_{0})$ to $D(X_{0})$.

  2. (2)

This will not be used in any of the proofs but let us remark that one can
describe the extension (3.2) completely, even in the absence of a lift of
$X_{0}$ to ${\mathbb Z}_{p}$ together with the object $M$. It is proven in
[PVV18, Theorem 6] that in the setting of (4), at least if $X$ is a scheme,
$T_{p}(M)$ can be upgraded to an object of the derived category of crystals of
quasi-coherent crystals of ${\mathcal{O}}$-modules
$D(\operatorname{{Cris}}(X_{0}/{\mathbb F}_{p}))$ on the scheme $X_{0}$ such
that (3.2) is an extension of crystals where $F^{*}_{X_{0}}M$ is endowed with
a crystal structure using the canonical connection. Moreover, homotopy classes
of splittings of (3.2) in $D(\operatorname{{Cris}}(X_{0}/{\mathbb F}_{p}))$
are in bijection with lifts of $F_{X_{0}}^{*}M$ to an object in
$D(\operatorname{{Cris}}(X_{0}/({\mathbb Z}/p^{2})))$. If $X_{0}$ is equipped
with a lift $X_{1}$ over ${\mathbb Z}/p^{2}$ then splittings of (3.2) in
$D(X_{0})$ are in bijection with lifts of $F_{X_{0}}^{*}M$ to an object of
$D(X_{1})$.

###### Proof.

1) We have $r_{n}\circ N_{n}=n!\cdot\ \mathrm{Id}_{S^{n}M}$, $N_{n}\circ
r_{n}=n!\cdot\mathrm{Id}_{\Gamma^{n}M}$. Therefore $N_{n}$ is an isomorphism
if $n!$ is invertible, so $T^{n}$ vanishes on prestacks over ${\mathbb
Z}[\frac{1}{n!}]$.

2) By the previous part, the map $N_{p}:S^{p}M\to\Gamma^{p}M$ becomes an
isomorphism after inverting $p$. Since the module $S^{p}M$ is $p$-torsion
free, the map $N_{p}$ is injective.

We will now construct a natural isomorphism $\alpha$ between the module
$F^{*}_{R/p}(M/p)=M/p\otimes_{R/p,F_{R/p}}R/p$ and the cokernel of $N_{p}$. To
an element $m\otimes r\in M/p\otimes_{R/p,F_{R/p}}R/p$ assign the element
$\alpha(m\otimes r):=\widetilde{r}\cdot\widetilde{m}^{\otimes
p}\in\operatorname{coker}N_{p}$ where $\widetilde{r}\in R$ and
$\widetilde{m}\in M$ are arbitrary lifts of $r$ and $m$, respectively.

To see that $\alpha$ gives a well-defined map
$F_{R/p}^{*}(M/p)\to\operatorname{coker}N_{p}$ we need to check that
$\widetilde{r}\cdot\widetilde{m}^{\otimes p}\in\operatorname{coker}N_{p}$ does
not depend on the choices of the lifts $\widetilde{r},\widetilde{m}$ and that
$(\widetilde{m}_{1}+\widetilde{m}_{2})^{\otimes p}=\widetilde{m}_{1}^{\otimes
p}+\widetilde{m}_{2}^{\otimes p}\in\operatorname{coker}N_{p}$. The first claim
follows from the fact that
$p\cdot\Gamma_{R}^{p}(M)\subset\operatorname{{Im}}N_{p}$, and the additivity
is demonstrated by the formula

(3.4) $(\widetilde{m}_{1}+\widetilde{m}_{2})^{\otimes
p}-\widetilde{m}_{1}^{\otimes p}-\widetilde{m}_{2}^{\otimes
p}=N_{p}\left(\sum\limits_{i=1}^{p-1}\frac{1}{i!(p-i)!}\widetilde{m}_{1}^{i}\widetilde{m}_{2}^{p-i}\right)$

Finally, to show that $\alpha:F_{R/p}^{*}(M/p)\to\operatorname{coker}(N_{p})$
is an isomorphism, we may assume that $M$ is a free $R$-module, because both
functors $M\mapsto\operatorname{coker}(N_{p}:S^{p}M\to\Gamma^{p}M)$ and
$M\mapsto F_{R/p}^{*}(M/p)$ commute with filtered colimits and every flat
module can be represented as a filtered colimit of free modules. Let
$\\{e_{i}\\}_{i\in I}$ be a basis of $M$ over $R$. Then
$\\{N_{p}(e_{i_{1}}\ldots e_{i_{p}})\\}_{(i_{1},\ldots,i_{p})\in
I^{p}\setminus I}\cup\\{e_{i}^{\otimes p}\\}_{i\in I}$ is an $R$-basis for
$\Gamma_{R}^{p}(M)$ so $\\{e_{i}^{\otimes p}\\}_{i\in I}$ is an $R/p$-basis
for $\operatorname{coker}N_{p}$, as desired.

3) Part (2) produced a natural short exact sequence $S^{p}M\to\Gamma^{p}M\to
F^{*}_{R/p}(M/p)$ of $R$-modules for every projective module $M$ over a flat
${\mathbb Z}_{p}$-algebra $R$. The functor $M\mapsto F^{*}_{R/p}(M/p)$ from
$\operatorname{Proj}^{\mathrm{f.g.}}_{R}$ to $\operatorname{{Mod}}_{R}$ is
polynomial (in fact, linear), and the formation of this short exact sequence
is compatible with base change along arbitrary maps $R\to R^{\prime}$ of flat
${\mathbb Z}_{p}$-algebras, so Lemma 2.2 produces a fiber sequence
$S^{p}M\to\Gamma^{p}M\to i_{*}F^{*}_{X_{0}}i^{*}M$ for every $M\in D(X)$ which
gives the desired identification.

4) We will establish such a fiber sequence when $M_{0}$ is a flat sheaf on an
affine scheme $X_{0}=\operatorname{{Spec}}R$ and the general case will follow
formally as in (3). If $M_{0}$ is a flat $R$-module where $R$ is an ${\mathbb
F}_{p}$-algebra, then $T_{p}(M_{0})$ is represented by the complex
$S^{p}M_{0}\xrightarrow{N_{p}}\Gamma^{p}M_{0}$ concentrated in degrees
$[-1,0]$. In Lemma 2.5 we defined maps $\psi_{M_{0}}:\Gamma^{p}M_{0}\to
F_{R_{0}}^{*}M_{0}$ and $\Delta_{M_{0}}:F_{R_{0}}^{*}M_{0}\to S^{p}M_{0}$ that
give rise to a sequence

(3.5) $0\to
F^{*}_{R_{0}}M_{0}\xrightarrow{\Delta_{M_{0}}}S^{p}_{R_{0}}M_{0}\xrightarrow{N_{p}}\Gamma^{p}M_{0}\xrightarrow{\psi_{M_{0}}}F^{*}_{R_{0}}M_{0}\to
0$

that one checks to be exact by a direct calculation with a basis of $M_{0}$ in
case it is free, as in part (2). This exact sequence gives rise to the desired
fiber sequence (3.2) by Lemma 2.2. ∎

Non-decomposability of the de Rham complex will arise from the fact that the
map $N_{p}:S^{p}M\to\Gamma^{p}M$ does not have a section in general. By
definition, for an object $M\in D(X)$ on an arbitrary prestack $X$ there is a
natural fiber sequence

(3.6)
$T_{p}(M)[-1]\xrightarrow{\gamma_{M}}S^{p}M\xrightarrow{N_{p}}\Gamma^{p}M$

It will be slightly more natural for us to start with an arbitrary object
$E\in D(X)$, and apply (3.6) to $M=E[-1]$ to get a fiber sequence

(3.7) $T_{p}(E[-1])[-1]\to S^{p}(E[-1])\to(\Lambda^{p}E)[-p]$

where we used the décalage identification
$\Gamma^{p}(E[-1])\simeq(\Lambda^{p}E)[-p]$ from Lemma 2.8 to rewrite the
third term. If $X$ is an algebraic stack flat over ${\mathbb Z}_{p}$ then, by
Lemma 3.1(3) this fiber sequence takes the form

(3.8) $F_{X_{0}}^{*}(E/p)[-2]\to S^{p}(E[-1])\to(\Lambda^{p}E)[-p]$

Here $F_{X_{0}}^{*}(E/p)$ is an abbreviation for $i_{*}F_{X_{0}}^{*}i^{*}E$,
we will prefer using this notation in what follows. Let us record the
description of the cohomology sheaves of $S^{p}(E[-1])$ in the case $E$ is a
locally free sheaf, for $p>2$, the analogous result for $p=2$ will be
established in Corollary 3.7:

######  Lemma 3.3.

Suppose that $p>2$.

  1. (1)

If $E$ is a locally free sheaf on a scheme $X$ flat over ${\mathbb Z}_{p}$,
then

(3.9) $H^{2}(S^{p}(E[-1]))\simeq F_{X_{0}}^{*}(E/p),\quad
H^{p}(S^{p}(E[-1]))\simeq\Lambda^{p}E,$

and all other cohomology sheaves of $S^{p}(E[-1])$ are zero.

  2. (2)

If $E$ is a locally free sheaf on a scheme $X_{0}$ over ${\mathbb F}_{p}$,
then

(3.10) $H^{1}(S^{p}(E[-1]))\simeq H^{2}(S^{p}(E[-1]))\simeq
F_{X_{0}}^{*}E,\quad H^{p}(S^{p}(E[-1]))\simeq\Lambda^{p}E$

and all other cohomology sheaves of $S^{p}(E[-1])$ are zero.

###### Proof.

This is immediate from (3.7) and Lemma 3.1(3),(4). ∎

Let now $X_{0}$ be an arbitrary prestack over ${\mathbb F}_{p}$. For an object
$E\in D(X_{0})$ the pushout of the fiber sequence (3.7) along the map
$T_{p}(E[-1])[-1]\to F_{X_{0}}^{*}E[-2]$ from (3.2) defines a fiber sequences

(3.11) $F_{X_{0}}^{*}E[-2]\to
S^{p}(E[-1])\bigsqcup\limits_{T_{p}(E)[-2]}F_{X_{0}}^{*}E[-2]\to(\Lambda^{p}E)[-p]$

In case where $E$ is a locally free sheaf on an ${\mathbb F}_{p}$-scheme
scheme $X_{0}$, the sequence (3.11) is simply the truncation of (3.7) in
degrees $\geq 2$.

######  Definition 3.4.

We will denote by $\alpha(E):\Lambda^{p}E\to F_{X_{0}}^{*}E[p-1]$ the shift by
$[p]$ of the connecting morphism corresponding to the fiber sequence (3.11).

Suppose now that $X$ is a flat algebraic stack over ${\mathbb Z}_{p}$. Denote
by $i:X_{0}\simeq X\times_{{\mathbb Z}_{p}}{\mathbb F}_{p}\hookrightarrow X$
the closed embedding of the special fiber. Let us remark that the information
contained in the extension (3.7) for an object $E\in D(X)$ is completely
captured by $\alpha(i^{*}E)$:

######  Lemma 3.5.

For an object $E\in D(X)$ denote $i^{*}E$ by $E_{0}$. The image of the map
$\alpha(E_{0})$ under the adjunction identification
$\mathrm{RHom}_{X_{0}}(\Lambda^{p}E_{0},F_{X_{0}}^{*}E_{0}[p-1])=\mathrm{RHom}_{X}(\Lambda^{p}E,i_{*}F_{X_{0}}^{*}i^{*}E[p-1])$
is the connecting morphism corresponding to the fiber sequence

(3.12) $T_{p}(E[-1])[-1]\to S^{p}(E[-1])\to(\Lambda^{p}E)[-p]$

where we identify $T_{p}(E[-1])[-1]$ with $i_{*}F_{X_{0}}^{*}i^{*}E[-2]$ via
Lemma 3.1(3).

###### Proof.

In general, for two objects $M\in D(X),N\in D(X_{0})$ the adjunction
identification
$\mathrm{RHom}_{D(X)}(M,i_{*}N)\simeq\mathrm{RHom}_{D(X_{0})}(i^{*}M,N)$ can
be described as sending a map $f:M\to i_{*}N$ to the composition
$i^{*}M\xrightarrow{i^{*}f}i^{*}i_{*}N\to N$ where the second map is the
counit of the adjunction. The fiber sequence $T_{p}(E_{0}[-1])[-1]\to
S^{p}(E_{0}[-1])\to(\Lambda^{p}E_{0})[-p]$ is the result of applying $i^{*}$
to the sequence (3.12), and the map $T_{p}(E_{0}[-1])[-1]\to
F_{X_{0}}^{*}E_{0}[-2]$ used to form the pushout sequence (3.11) is precisely
the counit map $i^{*}i_{*}\to\mathrm{Id}$ evaluated on
$F^{*}_{X_{0}}E_{0}[-2]$ by Remark 3.2, so the lemma follows. ∎

The extension defining $\alpha(E)$ can be described in more classical terms
for $p=2$. For a projective module $M$ over an ${\mathbb F}_{2}$-algebra
$R_{0}$ we have a natural short exact sequence

(3.13) $0\to
F_{R_{0}}^{*}M\xrightarrow{\Delta_{M}}S^{2}M\xrightarrow{j}\Lambda^{2}M\to 0$

Here $\Delta_{M}$ is the map defined in (2.7), it sends an element $m\otimes
1\in M\otimes_{R_{0},F_{R_{0}}}R_{0}$ to $m\cdot m\in S^{2}M$, and $j$ sends
$m_{1}\cdot m_{2}\in S^{2}M$ to $m_{1}\wedge m_{2}\in\Lambda^{2}M$. By Lemma
2.2 this gives rise to a fiber sequence

(3.14)
$F^{*}_{X_{0}}E\xrightarrow{\Delta_{E}}S^{2}E\xrightarrow{j}\Lambda^{2}E$

for every object $E\in D(X_{0})$ on any ${\mathbb F}_{2}$-prestack $X_{0}$.

######  Lemma 3.6.

When $p=2$, and $E$ is an object of $D(X_{0})$, the fiber sequence (3.11) is
naturally equivalent to the shift by $[-2]$ of the sequence (3.14).

###### Proof.

Consider the shift of the fiber sequence (3.11) by $[2]$:

(3.15) $F_{X_{0}}^{*}E\to
S^{2}(E[-1])[2]\bigsqcup\limits_{T_{2}(E)}F_{X_{0}}^{*}E\to\Lambda^{2}E$

When $X_{0}=\operatorname{{Spec}}R_{0}$ is an affine scheme and $E$
corresponds to a projective $R_{0}$-module, first and third terms of (3.15)
are concentrated in degree $0$, so this fiber sequence is an exact sequence of
$R_{0}$-modules. We will functorially identify it with the sequence (3.13),
which will prove the Lemma in general thanks to Lemma 2.2.

First, note that the middle term of (3.15) is isomorphic to the cohomology
module $H^{2}(S^{2}(E[-1]))$. On the other hand, the fiber sequence

$F_{X_{0}}^{*}E[-1]\xrightarrow{\Delta_{E[-1]}}S^{2}(E[-1])\xrightarrow{j_{E[-1]}}\Lambda^{2}(E[-1])$

identifies $H^{2}(S^{2}(E[-1]))$ with $H^{2}(\Lambda^{2}E[-1])\simeq S^{2}E$.
This already identifies the middle terms of exact sequences (3.14) and (3.15),
so it remains to check that this identification fits into an identification of
fiber sequences.

To this end, note that for any projective $R_{0}$-module $M$ the norm map
$N_{2}:S^{2}M\to\Gamma^{2}M$ factors as
$S^{2}M\xrightarrow{j}\Lambda^{2}M\xrightarrow{j^{\prime}}\Gamma^{2}M$ where
$j^{\prime}$ sends $m_{1}\wedge m_{2}$ to $m_{1}\otimes m_{2}+m_{2}\otimes
m_{1}\in\Gamma^{2}M\subset M^{\otimes 2}$. Therefore we get such a
factorization for any object $M\in D(X_{0})$. Applying this to $M=E[-1]$ we
learn that the map $S^{2}(E[-1])\to\Gamma^{2}(E[-1])\simeq\Lambda^{2}E$
factors through $j_{E[-1]}$ which proves the lemma. ∎

We can deduce the computation of cohomology sheaves of $S^{p}(E[-1])$ when
$p=2$, complementing Lemma 3.3(2):

######  Corollary 3.7.

For a locally free sheaf $E$ on an ${\mathbb F}_{2}$-scheme $X_{0}$ we have

(3.16) $H^{1}(S^{2}(E[-1]))\simeq F^{*}_{X_{0}}E\quad
H^{2}(S^{2}(E[-1]))\simeq S^{2}E$

and all other cohomology sheaves are zero.

For an arbitrary $p$, we can relate $S^{p}(E[-1])$ to the homotopy
coinvariants $(E[-1]^{\otimes p})_{hS_{p}}$ of the symmetric group $S_{p}$
acting on $E[-1]^{\otimes p}$ by permutations. This identification will be
used in our proof of non-vanishing of the class $\alpha$ in Section 12.

######  Lemma 3.8.

For a locally free sheaf $E$ on an ${\mathbb F}_{p}$-scheme $X_{0}$ there is a
natural equivalence $S^{p}(E[-1])\simeq\tau^{\geq 1}(E[-1]^{\otimes
p})_{hS_{p}}$

###### Proof.

First, let us recall the values of the cohomology sheaves of $(E[-1]^{\otimes
p})_{hS_{p}}$:

######  Lemma 3.9.

If $p>2$ then

(3.17) $H^{i}((E[-1]^{\otimes
p})_{hS_{p}})\simeq\begin{cases}\Lambda^{p}E,i=p\\\ F_{X_{0}}^{*}E,i\equiv
1\text{ or }2\bmod p-1\text{, and }i\leq 2\\\ 0\text{ otherwise}\end{cases}$

If $p=2$ then

(3.18) $H^{i}((E[-1]^{\otimes 2})_{hS_{2}})\simeq\begin{cases}S^{2}E,i=2\\\
F_{X_{0}}^{*}E,i<2\\\ 0,i>2\end{cases}$

###### Proof.

We will give the argument in the case $p>2$ and the case $p=2$ is proven
analogously. Note that the $S_{p}$-equivariant object $E[-1]^{\otimes p}$ is
equivalent to $\mathrm{sgn}\otimes E^{\otimes p}[-p]$ where $S_{p}$ acts on
$E^{\otimes p}$ via permutation of factors, and $\mathrm{sgn}$ is the sign
character. In particular, $H^{p}((E[-1]^{\otimes p})_{hS_{p}})$ is isomorphic
to $\Lambda^{p}E$ by definition of the exterior power.

To identify other cohomology sheaves, we will compare homology of the
symmetric group with that of a cyclic group. Denote by $C_{p}\subset S_{p}$ a
cyclic subgroup of order $p$. There is a natural map $(E[-1]^{\otimes
p})_{hC_{p}}\to(E[-1]^{\otimes p})_{hS_{p}}$. The coinvariants
$(E[-1]^{\otimes p})_{hC_{p}}$ for the cyclic group can be represented by the
following two-periodic complex

(3.19) $\ldots\xrightarrow{1-\sigma}E^{\otimes
p}\xrightarrow{N_{C_{p}}}E^{\otimes p}\xrightarrow{1-\sigma}E^{\otimes p}$

where $\sigma$ is the endomorphism of $E^{\otimes p}$ given by cyclic
permutation of the factors, and $N_{C_{p}}=\sum\limits_{i=0}^{p-1}\sigma^{i}$.
For all $i<p$ we get a map $F^{*}_{X_{0}}E\to H^{i}((E[-1]^{\otimes
p})_{hC_{p}})$ given by sending a section $e\otimes 1\in
E(U)\otimes_{{\mathcal{O}}_{X_{0}}(U),F_{X_{0}}}{\mathcal{O}}_{X_{0}}(U)$ on
an affine open $U\subset X_{0}$ to $e^{\otimes p}\in E^{\otimes p}(U)$. One
checks on stalks that this map is an isomorphism for all $i<p$. We get

(3.20) $H^{i}((E[-1]^{\otimes p})_{hC_{p}})\simeq\begin{cases}(E^{\otimes
p})_{C_{p}},i=p\\\ F_{X_{0}}^{*}E,i<p\\\ 0\text{ otherwise}\end{cases}$

Since $C_{p}\subset S_{p}$ is a $p$-Sylow subgroup, the map $(E[-1]^{\otimes
p})_{hC_{p}}\to(E[-1]^{\otimes p})_{hS_{p}}$ establishes $(E[-1]^{\otimes
p})_{hS_{p}}$ as a direct summand of $(E[-1]^{\otimes p})_{hC_{p}}$. Hence to
prove the lemma it remains to check that the surjective map
$H^{i}((E[-1]^{\otimes p})_{hC_{p}})\to H^{i}((E[-1]^{\otimes p})_{hS_{p}})$
is an isomorphism for $i\equiv 1,2\bmod p$ and is the zero map for all other
$i<p$. This can be checked on stalks, and $F_{X_{0}}^{*}$ is an additive
functor, so we may assume that $X_{0}=\operatorname{{Spec}}{\mathbb F}_{p}$
and $E$ is a $1$-dimensional ${\mathbb F}_{p}$-vector space. The statement is
now a consequence of classical computations of homology of the symmetric group
with coefficients in the sign character, cf. [Ste62, Chapter V, Proposition
7.8]. ∎

There is a natural map $E[-1]^{\otimes p}\to S^{p}(E[-1])$ which is
$S_{p}$-equivariant with respect to the permutation action on the source and
the trivial action on the target. Hence it induces a natural map
$(E[-1]^{\otimes p})_{hS_{p}}\to S^{p}(E[-1])$. Moreover, this map factors
through $(E[-1]^{\otimes p})_{hS_{p}}\to\tau^{\geq 1}(E[-1]^{\otimes
p})_{hS_{p}}$ because $S^{p}(E[-1])$ is concentrated in degrees $\geq 1$. We
thus obtain a map

(3.21) $\xi_{E}:\tau^{\geq 1}(E[-1]^{\otimes p})_{hS_{p}}\to S^{p}(E[-1])$

between objects of $D(X_{0})$ whose cohomology sheaves are isomorphic, by
(3.17) and Lemma 3.3(2). It remains to check that this particular map induces
isomorphisms on cohomology sheaves.

For cohomology in degree $p$, this is a consequence of the fact that the
composition $E[-1]^{\otimes p}\to S^{p}(E[-1])$ induces (up to a sign) the
surjection $E^{\otimes p}\twoheadrightarrow\Lambda^{p}E$ on $H^{p}$ when
$p>2$, and the surjection $E^{\otimes 2}\twoheadrightarrow S^{2}E$ when $p=2$.
To show that the map $\xi_{E}$ induces isomorphisms on $H^{1}$ and $H^{2}$ it
is enough to consider the case $X_{0}=\operatorname{{Spec}}{\mathbb F}_{p}$
and $E$ being a $1$-dimensional vector space. This follows from the fact that
cohomology classes of $({\mathbb F}_{p}[-1]^{\otimes p})_{hS_{p}}$ are
responsible for Steenrod operations in the cohomology of
$E_{\infty}$-algebras, as proven in Lemma 3.16 below. ∎

###  3.2. $T_{p}$ and commutative algebras.

We will now compute how the defect between $S^{p}$ and $\Gamma^{p}$, given by
$T_{p}$, interacts with the structure of a derived commutative algebra. This
computation plays a central role in our proof of Theorem 4.1.

######  Lemma 3.10.

Let $X$ be an algebraic stack or a formal scheme flat over ${\mathbb Z}_{p}$
with the special fiber $X_{0}:=X\times_{{\mathbb Z}_{p}}{\mathbb F}_{p}$. For
a derived commutative algebra $A\in\operatorname{DAlg}(X)$ the composition
$F_{X_{0}}^{*}(A/p)[-1]\xrightarrow{\gamma_{A}}S^{p}A\xrightarrow{m_{A}}A$ is
naturally homotopic in $D(X)$ to

(3.22)
$F_{X_{0}}^{*}(A/p)[-1]\xrightarrow{\varphi_{A/p}}A/p[-1]\xrightarrow{\mathrm{Bock}_{A}[-1]}A$

###### Proof.

For a projective module $M$ over a flat ${\mathbb Z}_{p}$-algebra $R$ there is
a diagram of $R$-modules

(3.23)
${S^{p}M}$${\Gamma^{p}M}$${F^{*}_{R/p}(M/p)}$${S^{p}M}$${S^{p}M}$${S^{p}_{R/p}(M/p)}$$\scriptstyle{N_{p}}$$\scriptstyle{\operatorname{{id}}_{S^{p}M}}$$\scriptstyle{r_{p}}$$\scriptstyle{\Delta_{M/p}}$$\scriptstyle{p!}$

where both rows are short exact sequences of $R$-modules. It evidently gives
rise to a diagram of polynomial functors that is moreover natural in arbitrary
maps $R\to R^{\prime}$. We can construct from this the following diagram in
$D(X)$ where rows are fiber sequences and vertical maps organize into
morphisms of fiber sequences:

(3.24)
${S^{p}A}$${\Gamma^{p}A}$${F^{*}_{X_{0}}(A/p)}$${S^{p}A}$${S^{p}A}$${S^{p}_{X_{0}}(A/p)}$${A}$${A}$${A/p}$$\scriptstyle{N_{p}}$$\scriptstyle{\Delta_{A/p}}$$\scriptstyle{p!}$$\scriptstyle{m_{A}}$$\scriptstyle{m}$$\scriptstyle{m_{A}}$$\scriptstyle{p!}$

The maps between the first two rows are obtained by applying the functor
$\Sigma_{X}$ of Lemma 2.2 to the diagram (3.23), and evaluating the resulting
maps on the object $A\in D(X)$. The maps between the second and third rows are
obtained by applying the functor $\operatorname{cofib}(p!)$ to the map
$m_{A}:S^{p}A\to A$.

The connecting morphism $F^{*}_{X_{0}}(A/p)\to(S^{p}A)[1]$ corresponding to
the top row of (3.24) is equivalent to $-\gamma_{A}[1]$, by definition of
$\gamma_{A}$ given in (3.6). Commutativity of the diagram implies the lemma
because the connecting morphism $A/p\to A[1]$ of the bottom row is the
negative $(-\mathrm{Bock}_{A})$ of the Bockstein morphism by the identity
$(p-1)!=-1$ in ${\mathbb F}_{p}$, and the composition of the right vertical
column is the map $\varphi_{A/p}:F_{X_{0}}^{*}(A/p)\to A/p$, by the definition
of Frobenius endomorphisms of derived commutative algebras. ∎

######  Remark 3.11.

If $A$ happens to be represented by a term-wise flat cosimplicial commutative
algebra $A^{\bullet}$ in $\operatorname{QCoh}(X)$, for a flat ${\mathbb
Z}_{p}$-scheme $X$, then the diagram (3.24) can be obtained from the strictly
commutative diagram in the ordinary category of cosimplicial sheaves on $X$,
where the rows are term-wise exact:

(3.25)
${S^{p}_{\mathrm{naive}}A^{\bullet}}$${\Gamma_{\mathrm{naive}}^{p}A^{\bullet}}$${F^{*}(A/p)}$${A}$${A}$${A/p}$$\scriptstyle{N}$$\scriptstyle{m}$$\scriptstyle{m}$$\scriptstyle{\varphi^{\mathrm{cosimp}}_{A/p}}$$\scriptstyle{p!}$

where $S^{p}_{\mathrm{naive}}$ and $\Gamma^{p}_{\mathrm{naive}}$ denote the
endofunctors of the ordinary category of cosimplicial objects in
$\operatorname{QCoh}(X)$ induced by the (non-derived) functors $S^{p}$ and
$\Gamma^{p}$.

###  3.3. Steenrod operations on cohomology of cosimplicial algebras.

This subsection is not used in the rest of the paper and contains classical
material, if only presented somewhat differently, but we include it, as
Proposition 3.14 was the original motivation for our approach and Theorem 4.1
should be viewed as its generalization. Let us also mention the result of
Scavia [Sca21, Theorem 1.1(iv),(v)] which is related to and implied by
Proposition 3.14.

Recall that cohomology of derived symmetric powers is related to natural
operations on cohomology of cosimplicial commutative algebras:

######  Lemma 3.12.

Fix a commutative base ring $R$. The $R$-module of natural transformations
between functors
$H^{i},H^{j}:\mathrm{CAlg}_{R}^{\Delta}\to\operatorname{{Mod}}_{R}$ can be
described as

(3.26) $\operatorname{{Hom}}(H^{i},H^{j})\simeq H^{j}(S^{\bullet}(R[-i]))$

where the isomorphism takes a natural transformation $\alpha:H^{i}\to H^{j}$
to the image of the class $1\in R=H^{i}(S^{1}(R[-i]))\subset
H^{i}(S^{\bullet}(R[-i]))$ under $\alpha$ evaluated on the free algebra
$S^{\bullet}(R[-i])$.

######  Remark 3.13.

The resulting module of natural transformations can be computed completely,
cf. [Pri73] for the case $R={\mathbb F}_{p}$.

###### Proof.

We will describe the inverse map. Given a class $c\in
H^{j}(S^{\bullet}(R[-i]))$, for every cosimplicial commutative algebra $A$ we
define a morphism $H^{i}(A)\to H^{j}(A)$ as follows. Let $x:R[-i]\to A$ be a
map representing a cohomology class $[x]\in H^{i}(A)$. Applying the functor
$S^{\bullet}$ to this map of cosimplicial $R$-modules and composing it with
the multiplication on $A$ gives a map
$S^{\bullet}(R[-i])\xrightarrow{S^{\bullet}x}S^{\bullet}A\xrightarrow{m_{A}}A$.
We declare the image of $[x]$ under the natural transformation to be the image
of the class $c$ under $m_{A}\circ S^{\bullet}x$. This produces a map
$H^{j}(S^{\bullet}(R[-i]))\to\operatorname{Func}(H^{i},H^{j})$ inverse to the
map (3.26). ∎

We will now deduce from Lemma 3.10 that certain cohomology operations of
degree $0$ and $1$ are related to Frobenius endomorphism, and Witt vectors
Bockstein homomorphism introduced by Serre in [Ser58, §3]. Recall that for any
object $M\in D(R)$ of the derived category of modules over a ring $R$ we
defined a map $T_{p}(M)[-1]\xrightarrow{\gamma_{M}}S^{p}_{R}M$ in (3.6).

In the case $M={\mathbb Z}_{p}[-i]$ over $R={\mathbb Z}_{p}$ it takes form
${\mathbb F}_{p}[-i-1]\to S^{p}({\mathbb Z}_{p}[-i])$, and we denote by
$P^{1}_{{\mathbb Z}_{p}}\in H^{i+1}(S^{p}({\mathbb Z}_{p}[-i]))$ the resulting
$p$-torsion class. For $M={\mathbb F}_{p}[-i]$ over $R={\mathbb F}_{p}$ the
map $\gamma_{{\mathbb F}_{p}}$ has the form ${\mathbb F}_{p}[-i]\oplus{\mathbb
F}_{p}[-i-1]\to S^{p}_{{\mathbb F}_{p}}({\mathbb F}_{p}[-i])$, and we denote
by $P^{0}_{{\mathbb F}_{p}}\in H^{i}(S^{p}({\mathbb
F}_{p}[-i])),P^{1}_{{\mathbb F}_{p}}\in H^{i+1}(S^{p}_{{\mathbb
F}_{p}}({\mathbb F}_{p}[-i]))$ the resulting classes. Note that
$P^{1}_{{\mathbb F}_{p}}$ is the image of $P^{1}_{{\mathbb Z}_{p}}$ under the
reduction map.

By Lemma 3.12 the class $P^{1}_{{\mathbb Z}_{p}}$ gives rise to a natural
homomorphism

(3.27) $P^{1}_{{\mathbb Z}_{p}}:H^{i}(A)\to H^{i+1}(A)$

for every cosimplicial commutative ${\mathbb Z}_{p}$-algebra $A$. Similarly,
classes $P^{0}_{{\mathbb F}_{p}},P^{1}_{{\mathbb F}_{p}}$ define operations

(3.28) $P^{0}_{{\mathbb F}_{p}}:H^{i}(A_{0})\to H^{i}(A_{0})\quad
P^{1}_{{\mathbb F}_{p}}:H^{i}(A_{0})\to H^{i+1}(A_{0})$

for every cosimplicial commutative ${\mathbb F}_{p}$-algebra $A_{0}$. In the
following result by $W_{2}(A_{0})$ we mean the cosimplicial commutative
${\mathbb Z}/p^{2}$-algebra obtained by applying term-wise the length $2$ Witt
vectors functor to $A_{0}$.

######  Proposition 3.14.

  1. (1)

For a cosimplicial commutative algebra $A$ over ${\mathbb Z}_{p}$ the
operation $P^{1}_{{\mathbb Z}_{p}}:H^{i}(A)\to H^{i+1}(A/p)$ is equal to the
composition

$H^{i}(A)\to
H^{i}(A/p)\xrightarrow{\varphi_{A/p}}H^{i}(A/p)\xrightarrow{\mathrm{Bock}^{i}_{A}}H^{i+1}(A)$

  2. (2)

For a cosimplicial commutative algebra $A_{0}$ over ${\mathbb F}_{p}$ the
operation $P^{0}_{{\mathbb F}_{p}}:H^{i}(A_{0})\to H^{i}(A_{0})$ is equal to
the Frobenius endomorphism of $A_{0}$.

  3. (3)

For a cosimplicial commutative algebra $A_{0}$ over ${\mathbb F}_{p}$ the
operation $P^{1}_{{\mathbb F}_{p}}:H^{i}(A_{0})\to H^{i+1}(A_{0})$ is equal to
the connecting homomorphism induced by the exact sequence
$A_{0}\xrightarrow{V}W_{2}(A_{0})\to A_{0}$.

###### Proof.

Given a class $[x]\in H^{i}(A)$ represented by a map $x:{\mathbb Z}_{p}[-i]\to
A$ we get a commutative diagram

(3.29) ${T_{p}({\mathbb Z}_{p}[-i])[-1]}$${S^{p}{\mathbb
Z}_{p}[-i]}$${T_{p}(A)[-1]}$${S^{p}A}$${A}$$\scriptstyle{T_{p}(x)[-1]}$$\scriptstyle{\gamma_{{\mathbb
Z}_{p}[-i]}}$$\scriptstyle{S^{p}x}$$\scriptstyle{\gamma_{A}}$$\scriptstyle{m_{A}}$

By definition, the value $P^{1}_{{\mathbb Z}_{p}}([x])\in H^{i+1}(A)$ is equal
to the image of the class $1\in{\mathbb F}_{p}\simeq H^{i+1}(T_{p}({\mathbb
Z}_{p}[-i])[-1])$ under the clockwise composition in the diagram (3.29). On
the other hand, the counter-clockwise composition is homotopic to ${\mathbb
F}_{p}[-i-1]\xrightarrow{x[-1]\bmod
p}A/p[-1]\xrightarrow{\varphi_{A/p}[-1]}A/p[-1]\xrightarrow{\mathrm{Bock}_{A}}A$
by Lemma 3.10, and this implies part (1).

For a class $[x]\in H^{i}(A_{0})$ represented by a map $x:{\mathbb
F}_{p}[-i]\to A_{0}$ the value $P^{0}_{{\mathbb F}_{p}}([x])\in H^{i}(A_{0})$
is defined as the image of the unit in ${\mathbb F}_{p}=H^{i}({\mathbb
F}_{p}[-i])$ under the composition ${\mathbb
F}_{p}[-i]\xrightarrow{\Delta_{{\mathbb F}_{p}[-i]}}S^{p}{\mathbb
F}_{p}[-i]\xrightarrow{S^{p}x}S^{p}A_{0}\xrightarrow{m_{A_{0}}}A_{0}$. This
composition is homotopic to ${\mathbb
F}_{p}[-i]\xrightarrow{x}A_{0}\xrightarrow{\Delta_{A_{0}}}S^{p}A_{0}\xrightarrow{m_{A_{0}}}A_{0}$
which implies statement (2) by the definition of Frobenius on $A_{0}$.

By Lemma 3.12, in (3) it is enough to check the claimed formula for
$P^{1}_{{\mathbb F}_{p}}$ on the universal class $1\in
H^{i}(S^{\bullet}{\mathbb F}_{p}[-i])$ for $A_{0}=S^{\bullet}{\mathbb
F}_{p}[-i]$. This cosimplicial ${\mathbb F}_{p}$-algebra lifts to
$A=S^{\bullet}{\mathbb Z}_{p}[-i]$ and the universal class lifts along the map
$H^{i}(A)\to H^{i}(A_{0})$. Therefore we can apply part (1) to get that
$P^{1}_{{\mathbb F}_{p}}(1)$ is equal to the image of $1$ under the
composition
$H^{i}(A_{0})\xrightarrow{\varphi_{A_{0}}}H^{i}(A_{0})\xrightarrow{\mathrm{Bock}_{A/p^{2}}}H^{i+1}(A_{0})$.

To see that this composition is equal to the Witt vector Bockstein
homomorphism, recall that there is a map of cosimplicial rings
$\phi:W_{2}(A_{0})\to A/p^{2}$ term-wise given by
$[a_{0}]+V[a_{1}]\mapsto\widetilde{a}_{0}^{p}+p\widetilde{a}_{1}$ where
$\widetilde{a}_{0},\widetilde{a}_{1}\in A^{i}/p^{2}$ are arbitrary lifts of
$a_{0},a_{1}\in A^{i}_{0}$. This map fits into a commutative diagram where
rows are term-wise exact sequences:

(3.30)
${0}$${A_{0}}$${W_{2}(A_{0})}$${A_{0}}$${0}$${0}$${A_{0}}$${A/p^{2}}$${A_{0}}$${0}$$\scriptstyle{V}$$\scriptstyle{\phi}$$\scriptstyle{\varphi_{A_{0}}}$$\scriptstyle{p}$

The induced map between the associated long exact sequence of cohomology
proves that the connecting homomorphism induced by the top row is equal to
$\mathrm{Bock}_{A/p^{2}}\circ\varphi_{A_{0}}$ which finishes the proof of (3).
∎

###  3.4. Steenrod operations on cohomology of $E_{\infty}$-algebras.

In this expository subsection we recall how power operations on cohomology of
an $E_{\infty}$-algebra is defined, and relate them to the operations on the
cohomology of cosimplicial commutative algebras. All of this material is
contained in [Ste62], [May70], [Pri73].

Let $A$ be an $E_{\infty}$-algebra over a (ordinary) commutative ring $R$.
This structure, in particular, gives a multiplication map $A^{\otimes p}\to A$
in $D(R)$ which is $S_{p}$-equivant where $S_{p}$ acts via permutations of the
factors in $A^{\otimes p}$ and acts trivially on $A$. The multiplication map
therefore factors through a map

(3.31) $m_{A}:(A^{\otimes p})_{hS_{p}}\to A$

This map is used to define operations on cohomology of $A$. Given a cohomology
class $[x]\in H^{i}(A)$ we can represent it by a map $x:R[-i]\to A$ in $D(R)$
and form the composition

(3.32) $(R[-i]^{\otimes p})_{hS_{p}}\xrightarrow{x^{\otimes p}}(A^{\otimes
p})_{hS_{p}}\xrightarrow{m_{A}}A$

We now specialize to the case $R={\mathbb F}_{p}$. The cohomology groups of
$({\mathbb F}_{p}[-i]^{\otimes p})_{hS_{p}}$ are given by

(3.33) $H^{j}(({\mathbb F}_{p}[-i]^{\otimes
p})_{hS_{p}})=\begin{cases}{\mathbb F}_{p},\text{ for }j=pi-(2k+1)(p-1)\text{
or }pi-(2k+1)(p-1)+1\text{ with }k\geq 0\\\ 0,\text{ otherwise}\end{cases}$

if $i$ is odd, and by

(3.34) $H^{j}(({\mathbb F}_{p}[-i]^{\otimes
p})_{hS_{p}})=\begin{cases}{\mathbb F}_{p},\text{ for }j=pi-2k(p-1)\text{ or
}pi-2k(p-1)+1\text{ with }k\geq 0\\\ 0,\text{ otherwise}\end{cases}$

if $i$ is even, see [Ste62, Chapter V, Lemmas 6.1, 6.2 + Proposition 7.8].
Thus for every $m$ of the form $2k(p-1)$ or $2k(p-1)+1$ we can define
$P^{m}([x])\in H^{i+m}(A)$ as the image of a fixed generator of
$H^{i+m}(({\mathbb F}_{p}[-i]^{\otimes p})_{hS_{p}})$ under the map (3.32).
Applying this construction to the singular cohomology of a topological space
gives rise to natural cohomology operations which satisfy special properties
that are false for general $E_{\infty}$-algebras:

######  Theorem 3.15.

For a topological space $X$ and an integer $i\geq 0$ the operations
$P^{0}:H^{i}_{\mathrm{sing}}(X,{\mathbb F}_{p})\to
H^{i}_{\mathrm{sing}}(X,{\mathbb
F}_{p}),P^{1}:H^{i}_{\mathrm{sing}}(X,{\mathbb F}_{p})\to
H^{i+1}_{\mathrm{sing}}(X,{\mathbb F}_{p})$ arising from the
$E_{\infty}$-algebra structure on $C^{\bullet}_{\mathrm{sing}}(X,{\mathbb
F}_{p})$ are described as

  1. (1)

$P^{0}=\mathrm{Id}$

  2. (2)

$P^{1}$ is the Bockstein homomorphism corresponding to the term-wise exact
sequence of complexes $C^{\bullet}_{\mathrm{sing}}(X,{\mathbb F}_{p})\to
C^{\bullet}_{\mathrm{sing}}(X,{\mathbb Z}/p^{2})\to
C^{\bullet}_{\mathrm{sing}}(X,{\mathbb F}_{p})$.

Moreover, all operations $P^{m}$ with $m<0$ are zero.

We can reconcile these operations with the operations on cohomology of
cosimplicial commutative algebras, which will justify denoting these by the
same symbols $P^{i}$. Let $A\in\operatorname{CAlg}^{\Delta}_{{\mathbb F}_{p}}$
be a cosimplicial commutative ${\mathbb F}_{p}$-algebra. We can view it as an
$E_{\infty}$-algebra with the symmetric multiplication factoring through the
cosimplicial symmetric multiplication

(3.35) $(A^{\otimes p})_{hS_{p}}\to S^{p}A\xrightarrow{m_{A}}A$

In particular, for a class $x\in H^{i}(A)$ the map (3.32) factors through the
natural map $({\mathbb F}_{p}[-i]^{\otimes p})_{hS_{p}}\to S^{p}({\mathbb
F}_{p}[-i])$.

######  Lemma 3.16.

For every $i$ the map $({\mathbb F}_{p}[-i]^{\otimes p})_{hS_{p}}\to
S^{p}({\mathbb F}_{p}[-i])$ factors through an equivalence $\tau^{\geq
i}({\mathbb F}_{p}[-i]^{\otimes p})_{hS_{p}}\simeq S^{p}({\mathbb
F}_{p}[-i])$.

###### Proof.

The cohomology groups of the object $S^{p}({\mathbb F}_{p}[-i])$ are
abstractly isomorphic to that of $\tau^{\geq i}({\mathbb F}_{p}[-i]^{\otimes
p})_{hS_{p}}$ by [Pri73, Theorems 4.1.1,4.2.1] so it is enough to show that
the map $\tau^{\geq i}({\mathbb F}_{p}[-i]^{\otimes p})_{hS_{p}}\to
S^{p}{\mathbb F}_{p}[-i]$ induces an injection on cohomology in degrees $\geq
i$.

For this it suffices to provide, for every $m\geq 0$, a topological space $X$
such that the operation $P^{m}:H^{i}_{\mathrm{sing}}(X,{\mathbb F}_{p})\to
H^{i+m}_{\mathrm{sing}}(X,{\mathbb F}_{p})$ is non-zero. The Eilenberg-MacLane
space $X=K({\mathbb F}_{p},i)$ satisfies this condition, because the
operations on cohomology of topological spaces defined from cohomology classes
of Eilenberg-MacLane spaces coincide with those defined using the
$E_{\infty}$-algebra structure, e.g. by the uniqueness of functorial
operations [Ste62, §VIII.3]. ∎

This discussion can be applied to the cosimplicial commutative algebra
$A=C^{\bullet}_{\mathrm{sing}}(X,{\mathbb F}_{p})$ of singular cochains on a
topological space $X$ whose underlying $E_{\infty}$-algebra was referenced in
Theorem 3.15. The fact that operations $P^{m}$ vanish for $m<0$ can thus be
explained by the fact that $S^{p}({\mathbb F}_{p}[-i])$ is concentrated in
degrees $\geq i$. The algebras of the form
$C^{\bullet}_{\mathrm{sing}}(X,{\mathbb F}_{p})$ are special among general
cosimplicial commutative ${\mathbb F}_{p}$-algebras in that their Frobenius
endomorphism is equal to the identity, because $C^{\mathrm{sing}}(X,{\mathbb
F}_{p})$ is defined as the algebra of ${\mathbb F}_{p}$-valued functions on
the simplicial singular set of $X$, cf. [Pri73, 6.1]. In particular, the
relations $P^{0}=\mathrm{Id}$ and $P^{1}=\mathrm{Bock}$ in Theorem 3.15 can be
deduced from our Proposition 3.14.

##  4\. Extensions in complexes underlying derived commutative algebras

In this section we prove our main algebraic result on extensions in the
canonical filtration on the complexes underlying certain derived commutative
algebras. In this section $X$ will be an algebraic stack flat over ${\mathbb
Z}_{p}$, or a formal scheme flat over $\operatorname{{Spf}}{\mathbb Z}_{p}$.
In both regimes we denote by $X_{0}=X\times_{{\mathbb Z}_{p}}{\mathbb F}_{p}$
the special fiber of $X$.

######  Theorem 4.1.

Let $A\in\operatorname{DAlg}^{\geq 0}(X)$ be a derived commutative algebra on
$X$ concentrated in degrees $\geq 0$ such that $H^{0}(A)={\mathcal{O}}_{X}$,
$H^{1}(A)$ a locally free ${\mathcal{O}}_{X}$-module, and multiplication on
cohomology induces an isomorphism
$H^{\bullet}(A)\simeq\Lambda^{\bullet}H^{1}(A)$. Assume further that there is
a morphism $s:H^{1}(A)[-1]\to A$ in $D(X)$ splitting the canonical filtration
on $\tau^{\leq 1}A$.

  1. (1)

There exists a natural equivalence
$\bigoplus\limits_{i=0}^{p-1}H^{i}(A)[-i]\simeq\tau^{\leq p-1}A$ in $D(X)$.

  2. (2)

There is a natural homotopy between the map
$H^{p}(A)\simeq\operatorname{cofib}(\tau^{\leq p-1}A\to\tau^{\leq
p}A)[p]\to(\tau^{\leq p-1}A)[p+1]$ corresponding to the extension $\tau^{\leq
p-1}A\to\tau^{\leq p}A\to H^{p}(A)[-p]$ and the composition

(4.1)
$H^{p}(A)=\Lambda^{p}H^{1}(A)\xrightarrow{\alpha(H^{1}(A)/p)}F_{X_{0}}^{*}H^{1}(A/p)[p-1]\xrightarrow{F_{X_{0}}^{*}s[p]}(\tau^{\leq
1}F_{X_{0}}^{*}(A/p))[p]\xrightarrow{\varphi_{A/p}}\\\ (\tau^{\leq
1}A/p)[p]\xrightarrow{\mathrm{Bock}_{A}}(\tau^{\leq 1}A)[p+1]\to(\tau^{\leq
p-1}A)[p+1]$

###### Proof.

For each $i\geq 0$, consider the map

(4.2)
$s_{i}:S^{i}(H^{1}(A)[-1])\xrightarrow{S^{i}s}S^{i}A\xrightarrow{m_{A}}A$

The composition $H^{1}(A)[-1]^{\otimes i}\to
S^{i}(H^{1}(A)[-1])\xrightarrow{s_{i}}A$ induces the multiplication map
$m:H^{1}(A)^{\otimes i}\to H^{i}(A)$ on $i$-th cohomology and $0$ on all other
cohomology groups. By the assumption that $H^{\bullet}(A)$ is freely generated
by $H^{1}(A)$, the map $m:H^{1}(A)^{\otimes i}\to H^{i}(A)$ identifies
$H^{i}(A)$ with $\Lambda^{i}H^{1}(A)$.

For $i\leq p-1$ we have
$S^{i}(H^{1}(A)[-1])\simeq\Gamma^{i}(H^{1}(A)[-1])\simeq(\Lambda^{i}H^{1}(A))[-i]$,
and the map $H^{1}(A)[-1]^{\otimes i}\to S^{i}(H^{1}(A)[-1])$ is the shift by
$[-i]$ of the natural surjection $H^{1}(A)^{\otimes i}\to\Lambda^{i}H^{1}(A)$.
Therefore $s_{i}$ induces an isomorphism on $H^{i}$, which proves the first
part of the theorem.

For $i=p$ the natural map
$S^{p}(H^{1}(A)[-1])\xrightarrow{N_{p}}\Gamma^{p}(H^{1}(A)[-1])\simeq(\Lambda^{p}H^{1}(A))[-p]$
is not an equivalence anymore. Using the map $s_{p}$, we will relate the
extension $\tau^{\leq p-1}A\to\tau^{\leq p}A\to H^{p}(A)[-p]$ to the extension

(4.3) $F_{X_{0}}^{*}(H^{1}(A)/p)[-2]\to
S^{p}(H^{1}(A)[-1])\to\Lambda^{p}H^{1}(A)[-p]$

constructed in (3.8). To begin, let us compute the map induced by $s_{p}$ on
the cohomology in degree $p$.

Since $S^{p}(H^{1}(A)[-1])$ is concentrated in degrees $\leq p$, the map
$s_{p}$ naturally factors through $\tau^{\leq p}A$. We claim that the
composition

(4.4) $S^{p}(H^{1}(A)[-1])\xrightarrow{s_{p}}\tau^{\leq p}A\to H^{p}(A)[-p]$

is naturally homotopic to the norm map
$N_{p}:S^{p}(H^{1}(A)[-1])\to(\Lambda^{p}H^{1}(A))[-p]$. This composition
factors uniquely through the norm map, because the mapping space
$\operatorname{Map}_{D(X)}(\mathrm{fib}(N_{p}),H^{p}(A)[-p])$ is contractible
as $H^{p}(A)[-p]$ is a locally free sheaf placed in degree $p$, and
$\mathrm{fib}(N_{p})\simeq F^{*}_{X_{0}}H^{1}(A)/p[-2]$ is a $p$-torsion
object concentrated in degrees $\leq 2\leq p$. Therefore the composition (4.4)
has the form
$S^{p}(H^{1}(A)[-1])\xrightarrow{N_{p}}\Lambda^{p}H^{1}(A)[-p]\xrightarrow{\psi}H^{p}(A)[-p]$
for some map $\psi$. To check that $\psi$ is equal to the cup-product map, we
may precompose this composition with the map $H^{1}(A)[-1]^{\otimes p}\to
S^{p}H^{1}(A)[-1]$, and the claim follows from:

######  Lemma 4.2.

Let $E$ be a locally free sheaf on $X$. Under the décalage identification
$\Gamma^{p}(E[-1])\simeq\Lambda^{p}E[-p]$ the composition $E[-1]^{\otimes
p}\to S^{p}(E[-1])\xrightarrow{N_{p}}\Gamma^{p}(E[-1])$ is identified with the
shift by $[-p]$ of the map $E^{\otimes p}\to\Lambda^{p}E$.

###### Proof.

[Ill71, Proposition I.4.3.2.1] shows that décalage equivalences are compatible
with graded algebra structures on symmetric, divided power, and exterior
algebras. Our assertion is a special case of this. ∎

This allows us to fit $s_{p}$ into the following map of fiber sequences for
some map $\beta_{p}^{\leq p-1}$:

(4.5)
${F^{*}_{X_{0}}(H^{1}(A)/p)[-2]}$${S^{p}(H^{1}(A)[-1])}$${(\Lambda^{p}H^{1}(A))[-p]}$${\tau^{\leq
p-1}A}$${\tau^{\leq
p}A}$${H^{p}(A)[-p]}$$\scriptstyle{\gamma_{H^{1}(A)[-1]}}$$\scriptstyle{\beta_{p}^{\leq
p-1}}$$\scriptstyle{N_{p}}$$\scriptstyle{s_{p}}$$\scriptstyle{\sim}$

This diagram implies that the extension class $H^{p}(A)\to(\tau^{\leq
p-1}A)[p+1]$ corresponding to the bottom row can be described as the
composition

(4.6)
$H^{p}(A)\simeq\Lambda^{p}H^{1}(A)\xrightarrow{\alpha(H^{1}(A)/p)}F_{X_{0}}^{*}(H^{1}(A)/p)[p-1]\xrightarrow{\beta_{p}^{\leq
p-1}[p+1]}(\tau^{\leq p-1}A)[p+1]$

Here $\alpha(H^{1}(A)/p)$ is the natural class attached to the vector bundle
$H^{1}(A)/p$ on $X_{0}$ by Definition 3.4, we view it as a map from
$\Lambda^{p}H^{1}(A)$ to $F^{*}_{X_{0}}(H^{1}(A)/p)[p-1]$ via adjunction as in
Lemma 3.5.

To finish the proof of Theorem 4.1, it remains to compute $\beta_{p}^{\leq
p-1}$. Note that $\beta_{p}^{\leq p-1}$ can be naturally recovered from the
composition $\beta_{p}:F^{*}_{X_{0}}H^{1}(A)/p[-2]\xrightarrow{\beta^{\leq
p-1}_{p}}\tau^{\leq p-1}A\to A$ as the truncation $\tau^{\leq p-1}\beta_{p}$,
because the mapping space
$\operatorname{Map}_{D(X)}(F^{*}_{X_{0}}H^{1}(A)/p[-2],\tau^{\geq p}A)$ is
contractible. To identify $\beta_{p}$, consider the following commutative
diagram

(4.7)
${F^{*}_{X_{0}}(H^{1}(A)/p)[-2]}$${S^{p}(H^{1}(A)[-1])}$${F^{*}_{X_{0}}(A/p)[-1]}$${S^{p}A}$${A}$$\scriptstyle{\gamma_{H^{1}(A)[-1]}}$$\scriptstyle{F_{X_{0}}^{*}s[-1]}$$\scriptstyle{\beta_{p}}$$\scriptstyle{S^{p}s}$$\scriptstyle{m}$

The composition of the bottom row was proven in Lemma 3.10 to be homotopic to
$F^{*}_{X_{0}}(A/p)[-1]\xrightarrow{\varphi_{A/p}}A/p[-1]\xrightarrow{\mathrm{Bock}_{A}}A$,
hence $\beta_{p}:F_{X_{0}}^{*}(H^{1}(A)/p)[-2]\to A$ is homotopic to the
composition

(4.8)
$F^{*}_{X_{0}}(H^{1}(A)/p)[-2]\xrightarrow{F_{X_{0}}^{*}s[-1]}F_{X_{0}}^{*}(A/p)[-1]\xrightarrow{\varphi_{A/p}}A/p[-1]\xrightarrow{\mathrm{Bock}_{A}[-1]}A$

that can be factored as

(4.9)
$F^{*}_{X_{0}}(H^{1}(A)/p)[-2]\xrightarrow{F_{X_{0}}^{*}s[-1]}(\tau^{\leq
1}F_{X_{0}}^{*}(A/p))[-1]\xrightarrow{\varphi_{A/p}}\tau^{\leq
1}(A/p)[-1]\xrightarrow{\mathrm{Bock}_{A}[-1]}\tau^{\leq 1}A\to A$

Therefore $\beta^{\leq p-1}_{p}$ is the composition of the first $3$ arrows in
(4.9) followed by the map $\tau^{\leq 1}A\to\tau^{\leq p-1}A$. Plugging this
expression for $\beta_{p}^{\leq p-1}$ into (4.6) finishes the proof of the
second part of the theorem. ∎

Let us record the special form that Theorem 4.1 takes in the case of augmented
algebras.

######  Corollary 4.3.

Let $A\in\operatorname{DAlg}^{\geq 0}(X)$ be a derived commutative algebra
such that $H^{0}(A)\simeq{\mathcal{O}}_{X}$ and the multiplication on
cohomology induces an isomorphism $H^{\bullet}(A)=\Lambda^{\bullet}H^{1}(A)$.
Assume also that $A$ is equipped with a map
$\varepsilon:A\to{\mathcal{O}}_{X}$ of derived commutative algebras that
induces an isomorphism on $H^{0}$. Then

  1. (1)

$\tau^{\leq p-1}A$ naturally decomposes in $D(X)$ as
$\bigoplus\limits_{i=0}^{p-1}H^{i}(A)[-i]$

  2. (2)

The extension class $H^{p}(A)\to\tau^{\leq p-1}A[p+1]$ corresponding to
$\tau^{\leq p}A$ can be described as the composition

(4.10)
$H^{p}(A)\xrightarrow{\alpha(H^{1}(A)/p)}F^{*}(H^{1}(A)/p)[p-1]\xrightarrow{\varphi_{A/p}}(H^{1}(A)/p)[p-1]\xrightarrow{\mathrm{Bock}_{H^{1}(A)}[p-1]}\\\
\to H^{1}(A)[p]\xrightarrow{\oplus}\tau^{\leq p-1}A[p+1]$

###### Proof.

The augmentation map $\varepsilon$ induces a splitting of the fiber sequence
$H^{0}(A)\to\tau^{\leq 1}A\to H^{1}(A)[-1]$. In particular, there exists a map
$s:H^{1}(A)[-1]\to A$ inducing an isomorphism on $H^{1}$, so we are in a
position to apply Theorem 4.1. The formula (4.1) specializes to (4.10) because
under the decompositions $\tau^{\leq 1}A\simeq H^{0}(A)\oplus H^{1}(A)[-1]$
and $\tau^{\leq 1}(A/p)\simeq H^{0}(A/p)\oplus H^{1}(A/p)[-1]$ the Bockstein
and Frobenius morphisms are diagonalized. ∎

###  4.1. Equivariant situation

Suppose that $X$ is a flat ${\mathbb Z}_{p}$-scheme equipped with an action of
a discrete group $G$. Let us explicitly record that Theorem 4.1 applied to the
global quotient $[X/G]$ can be rephrased as a statement about $G$-equivariant
algebras on $X$, where we take the definition of the category of
$G$-equivariant derived commutative algebras on $X$ to be
$\operatorname{DAlg}([X/G])$. We state the result in the augmented setting
because this is the version that will be used in all the applications.

######  Theorem 4.4.

Given a $G$-equivariant augmented derived commutative algebra $A$ on $X$, such
that $H^{0}(A)={\mathcal{O}}_{X}$, the sheaf $H^{1}(A)$is a locally free sheaf
of ${\mathcal{O}}_{X}$-modules, and the multiplication induces isomorphisms
$\Lambda^{\bullet}H^{1}(A)\simeq H^{\bullet}(A)$, we have

  1. (1)

There is an equivalence $\tau^{\leq
p-1}A\simeq\bigoplus\limits_{i=0}^{p-1}H^{i}(A)[-i]$ in $D_{G}(X)$.

  2. (2)

The map $H^{p}(A)\to\tau^{\leq p-1}A[p+1]$ corresponding to the fiber sequence
$\tau^{\leq p-1}A\to\tau^{\leq p}A\to H^{p}(A)[-p]$ in $D_{G}(X)$ can be
described as

(4.11)
$H^{p}(A)\xrightarrow{\alpha(H^{1}(A)/p)}F^{*}(H^{1}(A)/p)[p-1]\xrightarrow{\varphi_{A/p}}(H^{1}(A)/p)[p-1]\xrightarrow{\mathrm{Bock}_{H^{1}(A)}[p-1]}H^{1}(A)[p]\xrightarrow{\oplus}\tau^{\leq
p-1}A[p+1].$

##  5\. Applications to de Rham and Hodge-Tate cohomology

In this section we apply Theorem 4.1 to de Rham and diffracted Hodge
complexes. Let $k$ be a perfect field of characteristic $p>0$. We now work in
a setup where $X$ is a formally smooth formal scheme over $W(k)$, and as
before denote by $X_{0}$ its special fiber $X\times_{W(k)}k$. We denote by
$F_{X_{0}/k}:X_{0}\to X_{0}^{(1)}:=X_{0}\times_{k,\operatorname{Fr}_{p}}k$ the
relative Frobenius morphism, and as before $F_{X_{0}}:X_{0}\to X_{0}$ denotes
the absolute Frobenius morphism. We have the following cohomological
invariants associated to $X$ and $X_{0}$, each equipped with a derived
commutative algebra structure.

  * •

The diffracted Hodge complex $\Omega^{\operatorname{\not{D}}}_{X}\in D(X)$
defined in [BL22a, Notation 4.7.12] whose cohomology algebra
$H^{\bullet}(\Omega^{\operatorname{\not{D}}}_{X})$ is isomorphic to the
algebra $\Omega^{\bullet}_{X}$ of differential forms on $X$. By [BL22b,
Theorem 7.20(2)] it can be identified with the derived pushforward
$R\pi^{\mathrm{HT}}_{*}{\mathcal{O}}_{X^{\operatorname{\not{D}}}}$ of the
structure sheaf along the map $\pi^{\mathrm{HT}}:X^{\operatorname{\not{D}}}\to
X$, hence Lemma 2.13 equips $\Omega^{\operatorname{\not{D}}}_{X}$ with a
structure of a derived commutative algebra in $D(X)$. The Sen operator on
$\Omega^{\operatorname{\not{D}}}_{X}$ induces a decomposition $\tau^{\leq
p-1}\Omega^{\operatorname{\not{D}}}_{X}\simeq\bigoplus\limits_{i=0}^{p-1}\Omega^{i}_{X}[-i]$
in $D(X)$, and, in particular, gives rise to a map
$s:\Omega^{1}_{X}[-1]\to\Omega^{\operatorname{\not{D}}}_{X}$ that induces an
isomorphism on $H^{1}$.

  * •

The de Rham complex
$\mathrm{dR}_{X_{0}/k}=F_{X_{0}/k*}{\mathcal{O}}_{X_{0}}\xrightarrow{d}F_{X_{0}/k*}\Omega^{1}_{X_{0}/k}\xrightarrow{d}\dots$
viewed as an object of $D(X^{(1)}_{0})$. The Cartier isomorphism provides an
identification
$H^{\bullet}(\mathrm{dR}_{X_{0}/k})\simeq\Omega^{\bullet}_{X^{(1)}_{0}/k}$ of
graded algebras. By de Rham comparison, $\mathrm{dR}_{X_{0}/k}$ is naturally
identified with $\Omega^{\operatorname{\not{D}}}_{X^{(1)}}/p$, where
$X^{(1)}:=X\times_{W(k),\operatorname{Fr}_{p}}W(k)$ is the Frobenius-twist of
the formal $W(k)$-scheme $X$. We give $\mathrm{dR}_{X_{0}/k}$ the structure of
an object of $\operatorname{DAlg}(X^{(1)}_{0})$ by identifying it with the
derived pushforward of the structure sheaf along the map
$\pi^{\mathrm{HT}}:(X^{(1)}_{0})^{\operatorname{\not{D}}}\to X_{0}^{(1)}$
using Lemma 2.13.

######  Remark 5.1.

The diffracted Hodge complex can be identified with the Hodge-Tate cohomology
of $X$ relative to an appropriate prism, by [BL22a, Example 4.7.8]. Theorem
4.1 also applies to Hodge-Tate cohomology of smooth formal schemes over
arbitrary prisms, as well as to the decompleted version of the diffracted
Hodge cohomology [BL22a, Construction 4.9.1].

We will apply Theorem 4.1 to $\Omega^{\operatorname{\not{D}}}_{X^{(1)}}$ and
will then compute the extension in the canonical filtration on
$\mathrm{dR}_{X_{0}/k}$ by reducing modulo $p$. To begin, we will relate the
Frobenius endomorphism of the de Rham complex to the obstruction to lifting
Frobenius onto $X\times_{W(k)}W_{2}(k)$.

###  5.1. Obstruction to lifting Frobenius over $W_{2}(k)$.

We prove the results in this subsection without the smoothness assumption, and
only assuming the existence of a flat lift over $W_{2}(k)$, for a future
application in Section 7. For a scheme $Y_{0}$ over $k$ equipped with a lift
$Y_{1}$ over $W_{2}(k)$ we denote by
$\operatorname{ob}_{F,Y_{1}}:F_{Y_{0}/k}^{*}L\Omega^{1}_{Y^{(1)}_{0}/k}\to{\mathcal{O}}_{Y_{0}}[1]$
the obstruction to lifting $F_{Y_{0}/k}:Y_{0}\to Y_{0}^{(1)}$ to a morphism
$Y_{1}\to Y_{1}^{(1)}$, as defined by Illusie [Ill71]. We also denote by
$\mathrm{dR}_{Y_{0}/k}$ the derived de Rham complex of $Y_{0}$ relative to
$k$, viewed as an object of $D(Y_{0}^{(1)})$, cf. [Bha12, §3]. It is equipped
with a filtration $\mathrm{Fil}_{\mathrm{conj}}^{\bullet}$ whose graded pieces
are equivalent to the shifted exterior powers of the cotangent complex:
$\operatorname{{gr}}_{\mathrm{conj}}^{i}\mathrm{dR}_{Y_{0}/k}\simeq
L\Omega^{i}_{Y_{0}^{(1)}/k}[-i]$. Note also that the natural map induces an
equivalence $\mathrm{dR}_{Y_{0}/{\mathbb F}_{p}}\simeq\mathrm{dR}_{Y_{0}/k}$.

Using the relation between the cotangent complex of $Y_{0}$ over $W(k)$ and
the de Rham complex of $Y_{0}$, due to [BS22] and [Ill20] we will prove:

######  Proposition 5.2.

Let $Y_{0}$ be a quasisyntomic scheme over $k$ equipped with a flat lift
$Y_{1}$ over $W_{2}(k)$. Denote by
$s:L\Omega^{1}_{Y^{(1)}_{0}/k}[-1]\to\mathrm{dR}_{Y_{0}/k}$ the splitting of
the conjugate filtration in degree $1$ arising from $Y_{1}$. The composition

(5.1)
$F_{Y_{0}/k}^{*}L\Omega^{1}_{Y^{(1)}_{0}/k}[-1]\xrightarrow{F^{*}_{Y_{0}/k}s}F_{Y_{0}/k}^{*}\mathrm{Fil}_{1}^{\mathrm{conj}}\mathrm{dR}_{Y_{0}/k}\xrightarrow{dF_{Y^{(-1)}_{0}/k}}\mathrm{Fil}^{\mathrm{conj}}_{1}\mathrm{dR}_{Y^{(-1)}_{0}/k}$

is homotopic to the composition
$F_{Y_{0}/k}^{*}L\Omega^{1}_{Y^{(1)}_{0}}[-1]\xrightarrow{\operatorname{ob}_{F,Y_{1}}}{\mathcal{O}}_{Y_{0}}\to\mathrm{Fil}_{1}^{\mathrm{conj}}\mathrm{dR}_{Y^{(-1)}_{0}}$.
Here $Y_{0}^{(-1)}$ is the twist $Y_{0}\times_{k,\operatorname{Fr}_{p}^{-1}}k$
by the inverse of Frobenius and $dF_{Y^{(-1)}_{0}/k}$ is the map induced by
the functoriality of the de Rham complex.

######  Remark 5.3.

For $Y_{0}$ smooth over $k$ this result also follows from the explicit model
for the map $s:\Omega^{1}_{Y_{0}^{(1)}/k}\to\mathrm{dR}_{Y_{0}/k}$ using local
Frobenius lifts, provided by [DI87], cf. [Sri90].

######  Remark 5.4.

Using the isomorphism $Y_{0}^{(1)}\simeq Y_{0}$ of ${\mathbb F}_{p}$-schemes
we may view $\operatorname{ob}_{F,Y_{1}}$ as a map
$F^{*}_{Y_{0}}L\Omega^{1}_{Y_{0}}[-1]\to{\mathcal{O}}_{Y_{0}}[1]$ that
coincides with the obstruction to lifting the absolute Frobenius morphism
$F_{Y_{0}}:Y_{0}\to Y_{0}$ to an endomorphism of $Y_{1}$, we will denote this
map by the same symbol $\operatorname{ob}_{F,Y_{1}}$.

######  Convention 5.5.

We take $\operatorname{ob}_{F,Y_{1}}$ to mean the negative of the obstruction
class defined in [Ill71], to avoid a trailing sign in all of the subsequent
expressions.

We start by recalling from [Ill20] how a lift of $Y_{0}$ over $W_{2}(k)$
provides a decomposition of $L\Omega^{1}_{Y_{0}/W(k)}$. In general, if $Y_{0}$
is a quasisyntomic scheme over $k$ we have the fundamental triangle
corresponding to the morphisms
$Y_{0}\to\operatorname{{Spec}}k\to\operatorname{{Spec}}W(k)$

(5.2)
${\mathcal{O}}_{Y_{0}}[1]\simeq{\mathcal{O}}_{Y_{0}}\otimes_{k}L\Omega^{1}_{k/W(k)}\to
L\Omega^{1}_{Y_{0}/W(k)}\to L\Omega^{1}_{Y_{0}/k}$

The natural map $\mathrm{fib}(L\Omega^{1}_{Y_{0}/W(k)}\to
L\Omega^{1}_{Y_{0}/k})\to\mathrm{fib}(L\Omega^{1}_{Y_{0}/W_{2}(k)}\to
L\Omega^{1}_{Y_{0}/k})$ establishes the source as a direct summand of the
target, hence a flat scheme $Y_{1}$ over $W_{2}(k)$ lifting $Y_{0}$ induces a
splitting of this fiber sequence via the map $di:L\Omega^{1}_{Y_{0}/k}\simeq
i^{*}L\Omega^{1}_{Y_{1}/W_{2}(k)}\to L\Omega^{1}_{Y_{0}/W_{2}(k)}$, where
$i:Y_{0}\hookrightarrow Y_{1}$ is the inclusion of the special fiber, cf.
[Ill20, §4]. We denote by $s^{\prime}_{Y_{1}}:L\Omega^{1}_{Y_{0}/k}\to
L\Omega^{1}_{Y_{0}/W(k)}$ the resulting section of (5.2).

######  Lemma 5.6.

For a flat scheme $Y_{1}$ over $W_{2}(k)$ the obstruction
$\operatorname{ob}_{F,Y_{1}}:F_{Y_{0}/k}^{*}L\Omega^{1}_{Y^{(1)}_{0}/k}\to{\mathcal{O}}_{Y_{0}}[1]$
to lifting $F_{Y_{0}/k}:Y_{0}\to Y^{(1)}_{0}$ to a morphism from $Y_{1}$ to
$Y_{1}^{(1)}$ is homotopic to the composition

(5.3)
$F_{Y_{0}/k}^{*}L\Omega^{1}_{Y^{(1)}_{0}/k}\xrightarrow{F^{*}_{Y_{0}/k}di^{(1)}}F_{Y_{0}/k}^{*}L\Omega^{1}_{Y^{(1)}_{0}/W_{2}(k)}\xrightarrow{dF_{Y_{0}/k}}L\Omega^{1}_{Y_{0}/W_{2}(k)}\to{\mathcal{O}}_{Y_{0}}[1]$

###### Proof.

Let us start by recalling the definition of this obstruction given in [Ill71,
Proposition III.2.2.4]. The scheme $Y_{1}$ viewed as a lift of $Y_{0}$
produces a map $L\Omega^{1}_{Y_{0}/W_{2}(k)}\to{\mathcal{O}}_{Y_{0}}[1]$, and
composing it with $dF_{Y_{0}/k}$ we get a map

(5.4)
$\gamma_{Y_{0}}:F_{Y_{0}/k}^{*}L\Omega^{1}_{Y^{(1)}_{0}/W_{2}(k)}\xrightarrow{dF_{Y_{0}/k}}L\Omega^{1}_{Y_{0}/W_{2}(k)}\to{\mathcal{O}}_{Y_{0}}[1]$

On the other hand, the lift $Y^{(1)}_{1}$ of $Y^{(1)}_{0}$ gives rise to a map
$L\Omega^{1}_{Y^{(1)}_{0}/W_{2}(k)}\to{\mathcal{O}}_{Y^{(1)}_{0}}[1]$, and
pulling it back along $F_{Y_{0}/k}$ we obtain a map

(5.5)
$\gamma_{Y^{(1)}_{0}}:F_{Y_{0}/k}^{*}L\Omega^{1}_{Y^{(1)}_{0}/W_{2}(k)}\to
F^{*}_{Y_{0}/k}{\mathcal{O}}_{Y^{(1)}_{0}}[1]\simeq{\mathcal{O}}_{Y_{0}}[1]$

By definition,
$\operatorname{ob}_{F,Y_{1}}:F_{Y_{0}/k}^{*}L\Omega^{1}_{Y^{(1)}_{0}/k}\to{\mathcal{O}}_{Y_{0}}[1]$
is the unique (up to a contractible space of choices) map such that the
composition $F_{Y_{0}/k}^{*}L\Omega^{1}_{Y^{(1)}_{0}/W_{2}(k)}\to
F_{Y_{0}/k}^{*}L\Omega^{1}_{Y^{(1)}_{0}/k}\xrightarrow{\operatorname{ob}_{F,Y_{1}}}{\mathcal{O}}_{Y_{0}}[1]$
is homotopic to the difference $\gamma_{Y_{0}}-\gamma_{Y_{0}^{(1)}}$ (we have
incorporated Convention 5.5 at this point). Equivalently,
$\operatorname{ob}_{F,Y_{1}}$ is the composition

(5.6)
$F^{*}_{Y_{0}/k}L\Omega^{1}_{Y^{(1)}_{0}/k}\xrightarrow{F_{Y_{0}/k}^{*}di^{(1)}}F_{Y_{0}/k}^{*}L\Omega^{1}_{Y^{(1)}_{0}/W_{2}(k)}\xrightarrow{\gamma_{Y_{0}}-\gamma_{Y^{(1)}_{0}}}{\mathcal{O}}_{Y_{0}}[1]$

However, the composition
$F_{Y_{0}/k}^{*}L\Omega^{1}_{Y^{(1)}_{0}/k}\xrightarrow{F^{*}_{Y_{0}/k}di^{(1)}}F_{Y_{0}/k}^{*}L\Omega^{1}_{Y^{(1)}_{0}/W_{2}(k)}\xrightarrow{\gamma_{Y^{(1)}_{0}}}{\mathcal{O}}_{Y_{0}}[1]$
is zero by construction, so the lemma follows from the definition of
$\gamma_{Y_{0}}$. ∎

###### Proof of Proposition 5.2.