# Divided powers and derived De Rham cohomology

Kirill Magidson  Northwestern University

###### Abstract

We develop the formalism of derived divided power algebras, and revisit the
theory of derived De Rham and derived crystalline cohomology in this
framework. We characterize derived De Rham cohomology of a derived commutative
algebra $A$ over a base $R$, together with the Hodge filtration on it, in
terms of the universal property as the largest filtered divided power
thickening of $A$. We show that our approach recovers the classical De Rham
cohomology in the case of a smooth map $R\rightarrow A$, and therefore in
general, recovers the derived De Rham cohomology in the sense of Illusie.
Along the way, we develop some generalities on square-zero extensions and
derivations in derived algebraic geometry and apply them to give the universal
property of the first Hodge truncation of the derived De Rham cohomology.
Finally, we define derived crystalline cohomology relative to a general
divided power base, show that it satisfies the main properties of the
crystalline cohomology and coincides with the classical crystalline cohomology
in the smooth case.

###### Contents

  1. 1 Introduction and motivation.
    1. 1.1 Crystalline cohomology and De Rham cohomology.
    2. 1.2 Notations and terminology.
  2. 2 Preliminaries.
    1. 2.1 Higher category theory.
    2. 2.2 Non-abelian derived functors.
    3. 2.3 Symmetric sequences and derived pd operads.
    4. 2.4 Recollection on gradings and filtrations.
  3. 3 Filtrations and divided power algebras.
    1. 3.1 Smith ideals.
    2. 3.2 Divided power algebras.
      1. 3.2.1 Non-unital divided power algebras.
      2. 3.2.2 Divided power Smith ideals.
      3. 3.2.3 Filtered divided power algebras.
    3. 3.3 Two-step filtrations and square-zero extensions.
  4. 4 Derived De Rham complex and derived crystalline cohomology.
    1. 4.1 Divided powers and derived De Rham complex.
    2. 4.2 Cosimplicial techniques.
    3. 4.3 Derived crystalline cohomology.
      1. References

##  1 Introduction and motivation.

###  1.1 Crystalline cohomology and De Rham cohomology.

Assume that $A$ is a commutative algebra, smooth over some field $k$. The De
Rham cohomology of $A$ is a sequence of $k$-vector spaces
$H^{n}_{{\mathrm{dR}}}(A/k)$ defined as the cohomology of the De Rham complex
of $A$

$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d_{{\mathrm{dR}}}}$$\textstyle{\Omega^{1}_{A/k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d_{{\mathrm{dR}}}}$$\textstyle{\Omega^{2}_{A/k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\;\;...\;\;,}$

where $\Omega^{n}_{A/k}:=\Lambda^{n}_{A}(\Omega^{1}_{A/k})$ is the $n$-th
exterior power of the $A$-module of Kahler differentials of $A$ relative to
$k$, and $d_{dR}$ is the De Rham differential, i.e. the unique differential
extending the universal $R$-linear derivation of $A$,
$d_{{\mathrm{dR}}}:A\rightarrow\Omega^{1}_{A/k}$ and satisfying the graded
Leibnitz rule. The importance of De Rham cohomology in algebraic geometry
comes from the fact when $k=\mathbb{C}$, De Rham cohomology computes the
singular cohomology of the topological space of complex points of
$\operatorname{Spec}(A)$. However, over a field of positive characteristic, De
Rham cohomology is a less satisfactory invariant, as it consists purely of
torsion, and one can not recover characteristic $0$ information from it.
Instead, the theory of crystalline cohomology comes into play.

The genesis of crystalline cohomology theory started with a deeper
understanding of the De Rham cohomology. In the seminal paper, [Gr]
A.Grothendieck found a remarkable geometric interpretation of De Rham
cohomology as the cohomology of a certain site. Assume first that $A$ is
smooth over a field of characteristic $0$. The infinitesimal site of
$\operatorname{Spec}(A)$ is the category of pairs $(U,Z)$ where $U$ is an open
affine subscheme of $\operatorname{Spec}(A)$ and $Z$ is a nilpotent thickening
of $U$. The infinitesimal site has a structure sheaf $\mathcal{O}$ whose value
on the pair $(U,Z)$ is the algebra of functions $\Gamma(Z,\mathcal{O}_{Z})$.
Grothendieck showed that over a field of characteristic zero, the De Rham
cohomology is equivalent to the cohomology of the structure sheaf of the
infinitesimal site.

Now assume $A$ is defined over $\mathbb{F}_{p}$. Based on the observation that
in characteristic $0$ De Rham cohomology computes the cohomology of the
infinitesimal site, Grothendieck defined a new theory of crystalline
cohomology as the cohomology of the so-called crystalline site. The definition
of the crystalline site is similar to the definition of the infinitesimal
site, in the sense that both sites parametrize certain thickenings of open
affine subscheme of $\operatorname{Spec}(A)$. However, a crucial change is
implemented in the crystalline case: instead of merely nilpotent thickenings,
one now considers divided power thickenings. Let us recall what it means.
Assume $I\subset A$ is an ideal in a ring $A$. A divided power structure on
$I$ is a sequence of maps $\gamma_{n}:I\rightarrow A$ for $n\geq 1$ satisfying
a number of relations ensuring that $\gamma_{n}(x)$ behaves like
$\frac{x^{n}}{n!}$ for any $x\in I$ and $n\geq 1$. Over a field of
characteristic $0$, any ideal has unique divided powers. But over other base
rings, there can be many different divided power structures on an ideal, or
none at all. The crystalline site of $\operatorname{Spec}(A)$ is the category
whose objects are triples $(U,Z,\gamma)$ where $U$ is an open affine subscheme
of $\operatorname{Spec}(A)$, $Z$ is a nilpotent thickening of $U$ over
$\mathbb{Z}_{p}$ with a divided power structure $\gamma$ on the ideal defining
the closed embedding $U\subset Z$, where divided powers are required to be
compatible with the divided powers on the ideal $(p)\subset\mathbb{Z}_{p}$.
The scheme $Z$ is referred to as a “divided power thickening” of $U$. The
crystalline site has a crystalline structure sheaf which takes any triple
$(U,Z,\gamma)$ to the algebra of global functions $\Gamma(Z,\mathcal{O}_{Z})$
on $Z$. The crystalline cohomology $H^{n}_{{\mathrm{crys}}}(A/\mathbb{Z}_{p})$
is the cohomology of the structure sheaf of the crystalline site.

It turns out that the crystalline cohomology defined as the cohomology of the
crystalline site, is closely related to the De Rham cohomology defined in
terms of differential forms. A comparison theorem of Grothendieck says that
after base change to $\mathbb{F}_{p}$, the crystalline cohomology groups are
canonically isomorphic to the De Rham cohomology groups of $A$ over
$\mathbb{F}_{p}$:

$H^{n}_{{\mathrm{crys}}}(A/\mathbb{Z}_{p})\otimes\mathbb{F}_{p}\simeq
H^{n}_{{\mathrm{dR}}}(A/\mathbb{F}_{p}).$ (1.1)

Another statement concerns the presence of a smooth lift of
$\operatorname{Spec}(A)$ to a formally smooth $p$-adic scheme
$\operatorname{Spec}\widetilde{A}$. In this situation, the crystalline
cohomology of $A$ can be computed as $p$-adically completed De Rham cohomology
of the lift:

$H^{n}_{{\mathrm{crys}}}(A/\mathbb{Z}_{p})\simeq
H^{n}_{{\mathrm{dR}}}(\widetilde{A}/\mathbb{Z})_{p}^{\widehat{\;\;}}.$ (1.2)

These statements can seem surprising at first glance. The right side of
Equations 1.1 and 1.2 are defined in very different terms than the left side.
The De Rham cohomology is defined as the cohomology of a concrete commutative
differential graded algebra. On the other hand, the crystalline cohomology is
defined in terms of divided power algebras rather than commutative
differential graded algebras. The equivalences 1.1 and 1.2 suggest that there
should be a close relationship between commutative differential graded
algebras and divided power algebras. This is further affirmed by a classical
computation due to L.Illusie [I2]. The formation of De Rham complex

$(A/R)\longmapsto(\Omega^{\bullet}_{A/R},d_{{\mathrm{dR}}})$

on the category of smooth $R$-algebras can be derived to all $R$-algebras to
obtain the theory of the derived De Rham complex developed by L.Illusie in the
foundational texts [I1, I2]. The derived De Rham complex assigns to any
commutative $R$-algebra $A$ a commutative algebra object
$|\mathbb{L}\Omega^{\bullet}_{A/R}|$ in the derived category $\mathrm{D}(R)$
of $R$-modules. Assume $R\rightarrow A$ is a surjective map and the ideal
$I=\ker(R\rightarrow A)$ is generated by a regular sequence. Then it is shown
in [I2] that there is an equivalence

$|\mathbb{L}\Omega^{\bullet}_{A/R}|\simeq\operatorname{Env}^{\mathrm{pd}}_{R}(I),$
(1.3)

where $\operatorname{Env}^{\mathrm{pd}}_{R}(I)$ is the divided power envelope
of $R$ at the ideal $I$, i.e. the algebra obtained by freely adjoining to $R$
all divided powers of generators of the ideal $I$. The equivalence 1.3 respect
an additional piece of structure present on both sides, namely, a filtration.
For a smooth map $R\rightarrow A$, the De Rham complex
$\Omega_{A/R}^{\bullet}$ has a multiplicative Hodge filtration whose $n$-th
stage is the subcomplex

$\Omega^{\geq n}_{A/R}:=0\rightarrow 0\rightarrow...\rightarrow
0\rightarrow\Omega^{n}_{A/R}\rightarrow\Omega_{A/R}^{n+1}\rightarrow...$

In other words, the subcomplex $\Omega^{\geq
n}_{A/R}\subset\Omega^{\bullet}_{A/R}$ is spanned by differential forms
$a_{0}da_{1}\wedge da_{2}\wedge...\wedge da_{i}\in\Omega_{A/R}^{i}$ of degree
$i\geq n$. Deriving this construction, we obtain a “derived” Hodge filtration
$\mathbb{L}\Omega^{\geq\star}_{A/R}$ and thereby provides us a with functor
$\mathbb{L}\Omega_{-/R}:\operatorname{CAlg}_{R,\heartsuit}\rightarrow\operatorname{Fil}^{\geq
0}\operatorname{CAlg}_{R}$, where the target is the $\infty$-category of
$\mathbb{E}_{\infty}$-algebra objects in the symmetric monoidal
$\infty$-category $\operatorname{Fil}^{\geq 0}\mathrm{Mod}_{R}$ of non-
negatively filtered $R$-modules.

The derived Hodge filtration has a counterpart on the divided power side; for
a surjective map $R\rightarrow A$ with regular ideal $I$, the divided power
envelope $\operatorname{Env}^{\mathrm{pd}}_{R}(I)$ carries a filtration
$\operatorname{Env}^{\mathrm{pd},\geq\star}_{R}(I)$ whose $n$-th stage is
spanned by divided power monomials
$\gamma_{i_{1}}(x_{1})....\gamma_{i_{k}}(x_{k})$ with $i_{1}+...+i_{k}\geq n$
for $x_{i}\in I$.

A classical computation of L.Illusie [I2, Corollary 1.4.4.1] states that the
equivalence 1.3 does in fact refine to a filtered equivalence

$\mathbb{L}\Omega^{\geq\star}_{A/R}\simeq\operatorname{Env}_{R}^{\mathrm{pd},\geq\star}(I).$
(1.4)

This paper is an attempt to develop a formalism in which the equivalence 1.4
becomes obvious by giving a universal property of the derived De Rham
cohomology in the world of divided power algebras. Such a theory ought to make
various crystalline comparisons conceptually simpler. Another expectation from
such a theory, is that it should make more geometrically clear the relation
between derived De Rham cohomology and the stacky approach via the De Rham
stack developed recently by Drinfeld and Bhatt-Lurie in the texts [Dr20],
[BL22(2)], [Bh23].

Let us now outline our approach to derived De Rham cohomology derived via
divided power algebras. Fix a base commutative ring $R$, and let $A$ be a
commutative $R$-algebra. The De Rham complex of $A$ relative to $R$ is a
differential graded $R$-algebra $(\Omega^{\bullet}_{A/R},d_{{\mathrm{dR}}})$
which has the following universal property. Assume $(B^{\bullet},d)$ is
another differential graded $R$-algebra endowed with a commutative $R$-algebra
map $f^{0}:A\rightarrow B^{0}$. Then $f^{0}$ uniquely extends to a
differential graded $R$-algebra map
$f^{\bullet}:(\Omega^{\bullet}_{A/R},d_{{\mathrm{dR}}})\rightarrow(B^{\bullet},d)$.
In other words, the De Rham complex of $A$ relative to $R$ is the universal
commutative differential graded $R$-algebra which has $A$ as its degree $0$
component.

First, let us start by clarifying what we mean by a derived divided power
algebra. Let $\mathrm{DAlg}_{\mathbb{Z}}$ be the $\infty$-category of all
derived rings as defined by A.Raksit [R]. The $\infty$-category
$\mathrm{DAlg}_{\mathbb{Z}}$ is defined as the $\infty$-category of algebras
over a certain monad

$\mathrm{LSym}_{\mathbb{Z}}:\mathrm{Mod}_{\mathbb{Z}}\rightarrow\mathrm{Mod}_{\mathbb{Z}}.$

The monad $\mathrm{LSym}_{\mathbb{Z}}$ is the unique monad which has the
properties that its value on a finitely generated free abelian group on $n$
generators is the polynomial ring $\mathbb{Z}[x_{1},...,x_{n}]$, and that it
preserves sifted colimits and finite totalizations. Recall that the free
polynomial ring on an abelian group $M$ can be defined more conceptually by
the formula

$\operatorname{Sym}_{\mathbb{Z}}(M):=\bigoplus_{n\geq
0}(M^{\otimes^{n}})_{\Sigma_{n}},$

where $(-)_{\Sigma_{n}}$ stands for coinvariants with respect to the symmetric
group, and the symmetric group acts on the tensor power $M^{\otimes^{n}}$ by
permutation.

The notion of a divided power algebra is obtained by replacing coinvariants
with respect to symmetric groups by _invariants_ , i.e. the free divided power
algebra on $M$ is given by the formula

$\Gamma_{\mathbb{Z}}(M):=\bigoplus_{n\geq 0}(M^{\otimes^{n}})^{\Sigma_{n}}.$

A more concrete description of the free divided power ring is as follows. Let
us choose a basis $M=\mathbb{Z}^{n}$. Then $\mathbb{Z}\langle
x_{1},...,x_{n}\rangle:=\Gamma_{\mathbb{Z}}(M)$ is the smallest subring of
$\mathbb{Q}[x_{1},...,x_{n}]$ which contains $\mathbb{Z}[x_{1},...,x_{n}]$ and
all divided powers of the generators $x_{i}$, i.e. elements of the form

$\frac{x_{i}^{d}}{d!}\in\mathbb{Q}[x_{1},...,x_{n}].$

To promote the construction $M\longmapsto\Gamma_{\mathbb{Z}}(M)$ to a monad,
one has to work with non-unital algebras111this has to do with the fact that
outside of characteristic $0$, the unit in a unital ring never supports
divided powers.. The ring $\mathbb{Z}\langle x_{1},...,x_{n}\rangle$ is
augmented, and the augmentation ideal $\mathbb{Z}\langle
x_{1},...,x_{n}\rangle^{+}$ is a non-unital ring, which we call the non-unital
polynomial divided power ring on $n$ generators. We will denote
$\Gamma_{\mathbb{Z}}^{+}(\mathbb{Z}^{n}):=\mathbb{Z}\langle
x_{1},...,x_{n}\rangle$. There exists the non-unital derived divided power
algebra monad on $\mathrm{Mod}_{\mathbb{Z}}$, denoted

$\mathrm{L\Gamma}_{\mathbb{Z}}^{+}:\mathrm{Mod}_{\mathbb{Z}}\rightarrow\mathrm{Mod}_{\mathbb{Z}},$

which similarly to $\mathrm{LSym}_{\mathbb{Z}}$, is defined as the unique
sifted colimit and finite totalization preserving monad whose value on a
finitely generated free abelian group on $n$ generators is
$\Gamma_{\mathbb{Z}}^{+}(\mathbb{Z}^{n})$. We denote

$\mathrm{DAlg}_{\mathbb{Z}}^{\mathrm{nu},\mathrm{pd}}:=\operatorname{Alg}_{\mathrm{L\Gamma}^{+}_{\mathbb{Z}}}(\mathrm{Mod}_{\mathbb{Z}})$

the $\infty$-category of non-unital derived divided power rings. These
definitions can be extended to any ground derived ring $R$, thus giving us the
$\infty$-category $\mathrm{DAlg}_{R}$ of derived $R$-algebras, and the
$\infty$-category $\mathrm{DAlg}_{R}^{\mathrm{nu},\mathrm{pd}}$ of non-unital
derived divided power $R$-algebras.

A global version of the notion of a non-unital derived divided power ring is
the notion of a derived divided power map, or a derived divided power Smith
ideal. Roughly, a derived divided power map $A\rightarrow B$ is the data of a
map of derived rings $A\rightarrow B$ together with the data of a non-unital
derived divided power $A$-algebra on $I=\operatorname*{fib}(A\rightarrow B)$.
To explain it more precisely, let us digress for a moment and delve into the
notion of ideal in the derived world. Let
$\mathrm{DAlg}_{\mathbb{Z}}^{\Delta^{1}}$ be the $\infty$-category of arrows
$A\rightarrow B$ in the $\infty$-category of derived rings.
$\mathrm{DAlg}_{\mathbb{Z}}^{\Delta^{1}}$ is monadic over the arrow category
$\mathrm{Mod}^{\Delta^{1}}_{\mathbb{Z}}$. Consider the functor
$\operatorname*{fib}:\mathrm{Mod}_{\mathbb{Z}}^{\Delta^{1}}\rightarrow\mathrm{Mod}_{\mathbb{Z}}^{\Delta^{1}}$
sending an arrow $(X\rightarrow Y)$ to the fiber
$(\operatorname*{fib}(X\rightarrow Y)\rightarrow X$. This functor is evidently
an equivalence. Therefore, the data of a derived ring map $(A\rightarrow B)$
can be equivalently characterized as the data of a derived ring $A$, together
with a certain data on the map $I=\operatorname*{fib}(A\rightarrow
B)\rightarrow A$. We refer to this data as the data of a derived Smith ideal,
and explain in more details what it amounts to in the text. A part of the data
of a Smith ideal $(I\rightarrow A)$ is the structure of a non-unital derived
$A$-algebra on $I$. We define a derived divided power map $(A\rightarrow B)$
as a map of derived rings $(A\rightarrow B)$, together with a lift of $I$ to a
non-unital derived divided power $A$-algebra. We call the corresponding
derived Smith ideal $(I\rightarrow A)$ a derived divided power Smith ideal,
and sometimes simply derived divided power algebra for short. The notion can
be extended to any ground derived ring $R$. We denote
$\mathrm{DAlg}^{\Delta^{1},\mathrm{pd}}_{R}$ the $\infty$-category of derived
divided power maps over $R$, and reserve a separate notation
$\mathrm{DAlg}^{\Delta^{1}_{\vee},\mathrm{pd}}_{R}$ for the $\infty$-category
of derived divided power Smith ideals over $R$. We freely interchange between
the two notions, as they are equivalent, but use separate notations depending
on whether we want to emphasize the data of a derived Smith ideal or not.

Furthermore, the theory of derived divided power Smith ideals admits a
filtered analogue. Let $R$ be a ground derived ring, and let
$\operatorname{Fil}^{\geq 0}\mathrm{Mod}_{R}$ be the $\infty$-category of non-
negatively filtered objects in $\mathrm{Mod}_{R}$, and
$\operatorname{Fil}^{\geq 1}\mathrm{Mod}_{R}\subset\operatorname{Fil}^{\geq
0}\mathrm{Mod}_{R}$ the full subcategory of positive filtrations. We define an
$\infty$-category of non-negatively filtered derived $R$-algebras
$\operatorname{Fil}^{\geq 0}\mathrm{DAlg}_{R}$ as the $\infty$-category of
algebras over a certain filtered version
$\mathrm{LSym}^{\geq\star}_{R}:\mathrm{Mod}_{R}\rightarrow\mathrm{Mod}_{R}$ of
the $\mathrm{LSym}$-monad. For any object
$A^{\geq\star}\in\operatorname{Fil}^{\geq 0}\mathrm{DAlg}_{R}$, the $0$-th
stage of the filtration $A^{\geq 0}$ is a derived $R$-algebra, and the
“augmentation fiber” $A^{\geq 1}$ is a non-unital filtered derived $A^{\geq
0}$-algebra. Now the monad
$\mathrm{L\Gamma}_{R}^{+}:\mathrm{Mod}_{R}\rightarrow\mathrm{Mod}_{R}$ admits
a filtered enhancement
$\mathrm{L\Gamma}_{R}^{+,\geq\star}:\operatorname{Fil}^{\geq
1}\mathrm{Mod}_{R}\rightarrow\operatorname{Fil}^{\geq 1}\mathrm{Mod}_{R}$,
thereby giving as the notion of a non-unital filtered derived divided power
$R$-algebra. We define an $\infty$-category $\operatorname{Fil}^{\geq
0}\mathrm{DAlg}^{\mathrm{pd}}_{R}$ of (non-negatively) filtered derived
divided power $R$-algebras as an $\infty$-category monadic over
$\operatorname{Fil}^{\geq 0}\mathrm{Mod}_{R}$, whose object consist of objects
$A^{\geq\star}$ in $\operatorname{Fil}^{\geq 0}\mathrm{DAlg}_{R}$ together
with a $A^{\geq 0}$-linear non-unital filtered derived divided power structure
on the augmentation fiber $A^{\geq 1}$. This monad is trivial in
characteristic $0$, i.e. when $R$ has characteristic $0$, the forgetful
functor induces an equivalence $\operatorname{Fil}^{\geq
0}\mathrm{DAlg}_{R}^{\mathrm{pd}}\simeq\operatorname{Fil}^{\geq
0}\mathrm{DAlg}_{R}$. In non-zero characteristic, the data of divided powers
on a discrete object $A^{\geq\star}\in\operatorname{Fil}^{\geq
0}\operatorname{CAlg}^{\mathrm{pd}}_{R,\heartsuit}$ roughly amounts to
specifying for any element $x\in A^{\geq i}$ in filtered stage $i\geq 1$ a
sequence of elements $\gamma_{n}(x)\in A^{\geq ni}$ for all $n\geq 1$, which
satisfy the usual divided power relations.

There is a functor of taking $0$-th graded piece
$\operatorname{gr}^{0}:\operatorname{Fil}^{\geq
0}\mathrm{DAlg}^{\mathrm{pd}}_{R}\rightarrow\mathrm{DAlg}_{R}$. It is not easy
to see that it preserves all limits and hence admits a left adjoint.

######  Definition 1.1.1.

The left adjoint of the functor $\operatorname{gr}^{0}$

$\mathrm{L\Omega}_{-/R}^{\geq\star}:\mathrm{DAlg}_{R}\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern
3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}\ignorespaces}}}}\ignorespaces\operatorname{Fil}^{\geq
0}\mathrm{DAlg}^{\mathrm{pd}}_{R}$

sends a derived $R$-algebra $A$ to the Hodge-filtered derived De Rham
cohomology $\mathrm{L\Omega}^{\geq\star}_{A/R}$ of $A$ relative to $R$.

This definition requires justification. Recall that there already exists a
functor
$\mathbb{L}\Omega_{-/R}:\operatorname{CAlg}_{R,\heartsuit}\rightarrow\operatorname{Fil}^{\geq
0}\operatorname{CAlg}_{R}$ with values in non-negatively filtered
$\mathbb{E}_{\infty}$-algebras obtained by resolving any $R$-algebra by
polynomial algebras, and applying the functor of Hodge-filtered De Rham
complex termwise. Extending this to all connective derived algebras by
commuting with arbitrary geometric realizations, we obtain a functor
$\mathbb{L}\Omega^{\geq\star}_{-/R}:\mathrm{DAlg}_{R,\geq
0}\rightarrow\operatorname{Fil}^{\geq 0}\operatorname{CAlg}_{R}$. To justify
the Definition 1.1.1, we need to show that the following triangle commutes

$\textstyle{\mathrm{DAlg}_{R,\geq
0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathbb{L}\Omega^{\geq\star}_{-/R}}$$\scriptstyle{\mathrm{L\Omega}^{\geq\star}_{-/R}}$$\textstyle{\operatorname{Fil}^{\geq
0}\mathrm{DAlg}^{\mathrm{pd}}_{R}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Fil}^{\geq
0}\operatorname{CAlg}_{R},}$ (1.5)

where the right vertical functor is the forgetful functor. To prove this
statement, we need a little more preliminary formalism. But already at this
point, we notice that there is at least one immediate corollary of the
definition.

######  Proposition 1.1.2.

Let $R$ be a discrete commutative ring, and $R\rightarrow A$ be a surjective
map such that the ideal $I=\ker(R\rightarrow A)$ is generated by a regular
sequence, and let $\operatorname{Env}^{\mathrm{pd},\geq\star}_{R}(I)$ be the
divided power envelope of $R$ at $I$ endowed with the divided power
filtration. Then there is an equivalence of filtered $R$-algebras

$\mathrm{L\Omega}^{\geq\star}_{A/R}\simeq\operatorname{Env}^{\mathrm{pd},\geq\star}_{R}(I).$

To prove that the diagram 1.5 commutes, by commutation with sifted colimits it
suffices to show that there is a natural equivalence
$\mathrm{L\Omega}_{A/R}^{\geq\star}\simeq\Omega^{\geq\star}_{A/R}$ for any
polynomial $R$-algebra $A$. To formulate this comparison statement, we need to
connect our definition of derived De Rham cohomology in terms of filtered
divided power algebras with the more familiar definition of the De Rham
cohomology as a cochain complex. Passing to the language of cochain complexes
can be done as in [R] using the equivalence between the complete filtered
derived category and the $\infty$-category of dg modules, or coherent cochain
complexes. Namely, let
$\widehat{\operatorname{Fil}\mathrm{Mod}}_{R}\subset\operatorname{Fil}\mathrm{Mod}_{R}$
be the full subcategory of complete objects. There is a functor of taking
associated graded object

$\operatorname{gr}:\widehat{\operatorname{Fil}\mathrm{Mod}}_{R}\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern
3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}\ignorespaces}}}}\ignorespaces\mathrm{Gr}\mathrm{Mod}_{R},$

and it turns out that it is monadic. We let $\mathrm{DG}_{-}\mathrm{Mod}_{R}$
be the resulting monadic $\infty$-category over $\mathrm{Gr}\mathrm{Mod}_{R}$,
so that the functor $\operatorname{gr}$ lifts to an equivalence
$\operatorname{gr}:\widehat{\operatorname{Fil}\mathrm{Mod}}_{R}\simeq\mathrm{DG}_{-}\mathrm{Mod}_{R}.$
We define an $\infty$-category
$\mathrm{DG}_{-}\mathrm{DAlg}^{\mathrm{pd}}_{R}$ of derived differential
graded divided power algebras by requiring that the functor
$\operatorname{gr}$ lifts to an equivalence
$\operatorname{gr}:\widehat{\operatorname{Fil}\mathrm{DAlg}}^{\mathrm{pd}}_{R}\simeq\mathrm{DG}_{-}\mathrm{DAlg}^{\mathrm{pd}}_{R}.$

######  Definition 1.1.3.

The functor of derived De Rham complex

$\mathrm{L\Omega}^{\bullet}_{A/R}:\mathrm{DAlg}_{R}\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern
3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}\ignorespaces}}}}\ignorespaces\mathrm{DG}_{-}\mathrm{DAlg}^{\mathrm{pd}}_{R}$
(1.6)

is the left adjoint of the limit preserving functor
$\operatorname{ev}^{0}:\mathrm{DG}_{-}\mathrm{DAlg}^{\mathrm{pd}}_{R}\rightarrow\mathrm{DAlg}_{R}$
extracting the $0$-th degree part of a differential graded derived divided
power algebra. Given a derived $R$-algebra $A$, the derived De Rham complex
$\mathrm{L\Omega}^{\bullet}_{A/R}$ can be computed as the composition
$\mathrm{L\Omega}^{\bullet}_{A/R}:=\operatorname{gr}\;\mathrm{L\Omega}^{\geq\star}_{A/R}$.

The $\infty$-category $\mathrm{DG}_{-}\mathrm{Mod}_{R}$ has a so-called
negative $t$-structure, whose heart is the abelian category of cochain
complexes. What we precisely show is that for a smooth map $R\rightarrow A$,
the derived De Rham complex $\mathrm{L\Omega}^{\bullet}_{A/R}$ is discrete in
this $t$-structure, and the corresponding cochain complex is the classical De
Rham complex (with the structure of a commutative differential graded
algebra). A precise formulation of this statement uses the décalage
isomorphism of Illusie, [I2, Proposition 4.3.2.1].

######  Proposition 1.1.4.

Let $R$ be a discrete commutative ring, and $A$ be a smooth $R$-algebra, and
$\Omega^{\bullet}_{A/R}$ be the De Rham complex of $A$ over $R$. Then the
derived De Rham complex $\mathrm{L\Omega}^{\bullet}_{A/R}$ is discrete in the
negative $t$-structure, and there is an equivalence of commutative
differential graded $R$-algebras

$\mathrm{L\Omega}^{\bullet}_{A/R}\simeq\Omega^{\bullet}_{A/R}.$

######  Remark 1.1.5.

At this point, let us clarify how the universal property given above is
different from the Raksit’s. The dichotomy lies in what kind of algebras one
considers as a target for the derived De Rham complex functor, and there are
essentially two main points of difference.

\- We have introduced an $\infty$-category $\mathrm{DG}_{-}\mathrm{Mod}_{R}$
such that the functor
$\operatorname{gr}:\operatorname{Fil}\mathrm{Mod}_{R}\rightarrow\mathrm{Gr}\mathrm{Mod}_{R}$
lifts to an equivalence
$\operatorname{gr}:\widehat{\operatorname{Fil}\mathrm{Mod}}_{R}\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern
3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
9.55pt\raise 4.15565pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.15565pt\hbox{$\scriptstyle{\sim}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}\ignorespaces}}}}\ignorespaces\mathrm{DG}_{-}\mathrm{Mod}_{R}$.
In Raksit’s terminology, objects of $\mathrm{DG}_{-}\mathrm{Mod}_{R}$ are
called $h_{-}$-differential graded objects. There also exists a notion of
$h_{+}$-differential graded objects. Namely, following [R], one defines a
graded algebra $\mathbb{D}_{+}:=R\oplus R[1](1)$ and the $\infty$-category of
$h_{+}$-differential graded objects as
$\mathrm{DG}_{+}\mathrm{Mod}_{R}:=\mathrm{Mod}_{\mathbb{D}_{+}}(\mathrm{Gr}\mathrm{Mod}_{R})$.
Raksit gives a simple universal property of the derived De Rham complex which
is analogous to the one given in Definition 1.1.3, but in the setting of
$h_{+}$-differential graded algebras, rather than in the setting of filtered
derived algebras or $h_{-}$-differential graded algebras.

\- The theory of the derived De Rham complex in the setting of
$h_{+}$-differential graded algebras turns out to be very different from the
setting of filtered derived algebras or $h_{-}$-differential graded algebras.
In the $h_{+}$-setting, there are no divided powers. In contrast, in the
filtered/$h_{-}$-setting, divided powers become crucial. The main reason why
one needs divided powers, is that the left adjoint of the functor
$\operatorname{ev}^{0}:\mathrm{DG}_{-}\mathrm{DAlg}_{R}\rightarrow\mathrm{DAlg}_{R}$
is _not_ the derived De Rham complex, but is rather related to the
_infinitesimal cohomology_.

One important property of (derived) De Rham cohomology is its relation to the
notions of the universal derivation and the universal square-zero extension.
This can be explained by means of the following two examples. Let $R$ be a
discrete commutative ring, and $A$ a smooth $R$-algebra. Then the naive
truncation $\Omega_{A/R}^{\\{0,1\\}}$ of the De Rham complex
$\Omega^{\bullet}_{A/R}$ in degrees $\leq 2$ coincides with the two-step
complex $d:A\rightarrow\Omega_{A/R}^{1}$, where the map $d$ is the universal
(De Rham) derivation of the map $R\rightarrow A$. its universality is that for
any other $R$-linear derivation $\delta:A\rightarrow M$ with values in an
$A$-module $M$, there is a unique map $f:\Omega^{1}_{A/R}\rightarrow M$ such
that the derivation $\delta$ factors as $\delta=f\circ d$. Another example is
the case of a surjective map $R\rightarrow A$ whose ideal $I$ is generated by
a regular sequence. As we mentioned above, there is an equivalence
$\mathrm{L\Omega}^{\geq\star}_{A/R}\simeq\operatorname{Env}_{R}^{\geq\star}(I)$.
Truncating the latter filtered object in degrees $\leq 2$, we obtain a square-
zero extension

$I/I^{2}\rightarrow R/I^{2}\rightarrow R/I.$

This square-zero extension also has a universal property. Suppose we have a
map $(I\rightarrow R)\rightarrow(J\rightarrow S)$ of algebras with an ideal,
and $J^{2}=0$. Then this map factors uniquely through a map
$(I/I^{2}\rightarrow R/I^{2})\rightarrow(J\rightarrow S)$. For this reason, we
might call the object $I/I^{2}\rightarrow R/I^{2}$ the universal square-zero
extension associated to the ideal $I\subset R$. The two examples we just
explained suggest two ideas:

1) truncation of the derived De Rham cohomology (or derived De Rham complex)
in degrees $\leq 2$ must have a universal property as a universal square-zero
extension (respectively, the universal derivation),

2) there should exist a canonical way of transforming the universal square-
zero extension into the universal derivation, and vice versa.

The second idea is in fact an instance of one of the foundational ideas of
derived algebraic geometry: square-zero extensions are controlled by
derivations. The passage from square-zero extensions to derivations is done by
taking associated gradeds for two-step filtration existing on any square-zero
extension, and interpreting the extension data as a derivation. A reasonable
expectation then is that the equivalence of square-zero extensions and
derivations is a specialization of the equivalence
$\widehat{\operatorname{Fil}\mathrm{Mod}}_{R}\simeq\mathrm{DG}_{-}\mathrm{Mod}_{R}$
to objects concentrated in degrees $0,1$ only. Indeed, consider subcategories
$\operatorname{Fil}^{[0,1]}\mathrm{Mod}_{R}\subset\operatorname{Fil}^{\geq
0}\mathrm{Mod}_{R}$ of two-step filtrations and
$\mathrm{DG}_{-}^{\\{0,1\\}}\mathrm{Mod}_{R}\subset\mathrm{DG}^{\geq
0}\mathrm{Mod}_{R}$ of differential two-step graded modules. We show that
$\operatorname{Fil}^{[0,1]}\mathrm{Mod}_{R}$ and
$\mathrm{DG}_{-}^{\\{0,1\\}}\mathrm{Mod}_{R}$ are symmetric monoidal
$\infty$-categories and there is a symmetric monoidal equivalence

$\operatorname{Fil}^{[0,1]}\mathrm{Mod}_{R}\simeq\mathrm{DG}_{-}^{\\{0,1\\}}\mathrm{Mod}_{R}.$

Moreover, we interpret the $\infty$-category
$\operatorname{Fil}^{[0,1]}\mathrm{DAlg}_{R}:=\mathrm{DAlg}(\operatorname{Fil}^{[0,1]}\mathrm{Mod}_{R})$
as the $\infty$-category of derived square-zero Smith ideals and
$\mathrm{DG}_{-}^{\\{0,1\\}}\mathrm{DAlg}_{R}:=\mathrm{DAlg}(\mathrm{DG}_{-}^{\\{0,1\\}}\mathrm{Mod}_{R})$
as the $\infty$-category of derived derivations. For instance, objects of the
latter can be thought as triples $(A,M,\delta)$ where $A\in\mathrm{DAlg}_{R}$,
$M\in\mathrm{Mod}_{A}$, and $\delta:A\rightarrow A\oplus M[1]$ is a
derivation. There exists a symmetric monoidal functor

$\textstyle{\operatorname{Fil}^{\geq
0}\mathrm{Mod}_{R}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{gr}}$$\textstyle{\mathrm{DG}_{-}^{\geq
0}\mathrm{Mod}_{R}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\\{0,1\\}}$$\textstyle{\mathrm{DG}_{-}^{\\{0,1\\}}\mathrm{Mod}_{R}}$

which induces a functor $\operatorname{Fil}^{\geq
0}\mathrm{DAlg}_{R}\rightarrow\mathrm{DG}^{\\{0,1\\}}_{-}\mathrm{DAlg}_{R}$,
which produces a universal derivation for any non-negatively filtered derived
algebra. We show the following:

######  Theorem 1.1.6.

Let $R\rightarrow A$ be a map of derived rings, and
$\mathrm{L\Omega}^{\geq\star}_{A/R}$ be the Hodge filtered derived De Rham
cohomology of $A$ over $R$. Then the differential two-step graded object
$\mathrm{L\Omega}^{\\{0,1\\}}_{A/R}$ is canonically equivalent to the
universal derivation $d:A\rightarrow A\oplus\mathbb{L}_{A/R}$.

This statement is to be expected, and its proof rather tautologically follows
by checking universal properties. The essential content of the discussion
above is Theorem 3.3.26 which interprets the equivalence
$\operatorname{Fil}^{[0,1]}\mathrm{DAlg}_{R}\simeq\mathrm{DG}_{-}^{\\{0,1\\}}\mathrm{DAlg}_{R}$
as an equivalence between square-zero Smith ideals and derivations.

In the end of the text, we explain how to set-up the theory of derived
crystalline cohomology relative to a general (derived) divided power base. We
consider the $\infty$-category $\mathbb{Z}^{{\mathrm{crys}}}$ consisting of
pairs $(A\rightarrow\overline{A},R)$ where $(A\rightarrow\overline{A})$ is a
derived divided power map (to which we will refer to as divided power base),
and $\overline{R}$ a derived $\overline{A}$-algebra, and define a functor
$\mathrm{R}\Gamma_{{\mathrm{crys}}}:\mathbb{Z}^{{\mathrm{crys}}}\rightarrow\mathrm{DAlg}_{\mathbb{Z}}^{\Delta^{1},\mathrm{pd}}$
with values in the $\infty$-category of derived divided power rings, whose
value on a pair $(A\rightarrow\overline{A},R)$ is the derived crystalline
cohomology of $R$ relative to $(A\rightarrow\overline{A})$, denoted
$\mathrm{R}\Gamma_{{\mathrm{crys}}}(R/(A\rightarrow\overline{R})$. We show
that it follows from our definition that derived crystalline cohomology
satisfies a base change property with respect to maps
$(A\rightarrow\overline{A})\rightarrow(B\rightarrow\overline{B})$ of divided
power bases. This property implies the following theorem, which summarizes the
main properties of derived crystalline cohomology. To formulate it, we notice
that for a given divided power base $(A\rightarrow\overline{A})$, the derived
divided power ring
$\mathrm{R}\Gamma_{{\mathrm{crys}}}(R/(A\rightarrow\overline{A}))$ is a
derived divided power algebra under $(A\rightarrow\overline{A})$, and hence, n
particular, the underlying derived ring of
$\mathrm{R}\Gamma_{{\mathrm{crys}}}(R/(A\rightarrow\overline{A}))$ is a
derived $A$-algebra.

######  Theorem 1.1.7.

Let $(A\rightarrow\overline{A})$ be a derived divided power base.

  1. 1.

Let $R$ be a derived $\overline{A}$-algebra. There is a natural equivalence

$\mathrm{R}\Gamma_{{\mathrm{crys}}}(R/(A\rightarrow\overline{A}))\underset{A}{\otimes}\overline{A}\simeq\mathrm{L\Omega}_{R/\overline{A}}.$

  2. 2.

Let $\overline{R}$ be a derive $\overline{A}$-algebra, and $R$ a derived
$A$-algebra endowed with an equivalence
$R\otimes_{A}\overline{A}\simeq\overline{R}$. Then there is a natural
equivalence

$\mathrm{L\Omega}_{R/A}\simeq\mathrm{R}\Gamma_{{\mathrm{crys}}}(\overline{R}/(A\rightarrow\overline{A})).$

  3. 3.

Let $(\mathcal{A}\rightarrow\overline{A})$ be the derived divided power
envelope of $(A\rightarrow\overline{A})$. Recall that
$\mathcal{A}\simeq\mathrm{L\Omega}_{\overline{A}/A}$. The divided power
structure on $(A\rightarrow\overline{A})$ gives a natural map
$(\mathcal{A}\rightarrow\overline{A})\rightarrow(A\rightarrow\overline{A})$.
Let $R$ be a derived $\overline{A}$-algebra, and consider its derived De Rham
cohomology $\mathrm{L\Omega}_{R/A}$ with the natural $\mathcal{A}$-linear
structure. There is a natural equivalence

$\mathrm{L\Omega}_{R/A}\underset{\mathcal{A}}{\otimes}A\simeq\mathrm{R}\Gamma_{{\mathrm{crys}}}(R/(A\rightarrow\overline{A})).$

The equivalences listed above are formulated as equivalences of derived
algebras. In the relevant section, we explain how to promote it to an
equivalence of derived divided power algebras.

The first two properties are the main properties expected of crystalline
cohomology. The first merely states, that derived crystalline cohomogy is a
natural lift of derived De Rham cohomology. The second says that we can
compute derived crystalline cohomology as derived De Rham cohomology of a
lift. The third property is a generalization of the idea that one can compute
the crystalline cohomology of an $\mathbb{F}_{p}$-algebra $R$ as the derived
De Rham cohomology of $R$ relative to $\mathbb{Z}$, based changed to
$\mathbb{Z}_{p}$ along the natural map
$\mathrm{L\Omega}_{\mathbb{F}_{p}/\mathbb{Z}}\rightarrow\mathbb{Z}_{p}$. This
observation is to the best of our knowledge, due to B.Bhatt, and appears in
the text [Bh12(2)].

Let us finish this introduction by mentioning the comparison with the stacky
and site-theoretic approach, which will be proven in a subsequent paper
[Magidson2]. In the work of [Dr20] of Drinfeld, and the works [BL22(2)] and
[Bh23] of B.Bhargav and J.Lurie there was developed a stacky approach to
derived De Rham cohomology. Let $X$ be a derived stack over a base ring $R$.
We can define a new prestack $X_{{\mathrm{dR}}}$, called De Rham stack of $X$,
understood merely as a fuctor $\mathrm{DAlg}_{R,\geq
0}\rightarrow\mathrm{Spc}$ which is defined by the formula

$X_{{\mathrm{dR}}}(R):=X(R/R^{\sharp}),$

where $R/R^{\sharp}$ is the quotient of $R$ by a certain generalized ideal
$R^{\sharp}\rightarrow R$ playing the role of nil-radical in the divided power
setting. One can show (see [Bh23], Theorem 2.5.6) that for a smooth affine
scheme $X$, there is an equivalence of $\mathbb{E}_{\infty}$ $R$-algebras:

$\Gamma(X_{{\mathrm{dR}}},\mathcal{O}_{X_{{\mathrm{dR}}}})\simeq|\Omega^{\bullet}_{A/R}|.$

More generally, it follows from [BL22(2), Theorem 7.20] and the crystalline
comparison for prismatic cohomology that the statement holds true for the
derived crystalline cohomology either when the algebra satisfies some
finiteness assumption, or is quasi-lci. It is expected that the result should
be true for derived De Rham cohomology for any $A$, if we replace the right-
hand side by Hodge-completed derived De Rham cohomology. At present we are not
aware of a reference for this statement proven in this generality. In the
paper [Magidson2], we characterize the construction
$X\longmapsto\Gamma(X_{{\mathrm{dR}}},\mathcal{O}_{X_{{\mathrm{dR}}}})$ in
terms of a universal property which makes manifest the fact that for any
affine scheme $X$, the $\mathbb{E}_{\infty}$-algebra
$\Gamma(X_{{\mathrm{dR}}},\mathcal{O}_{X_{{\mathrm{dR}}}})$ lifts to a
filtered derived divided power algebra in $\mathrm{Mod}_{R}$. Let
$\operatorname{Fil}^{\geq\star}\Gamma(X_{{\mathrm{dR}}},\mathcal{O}_{X_{{\mathrm{dR}}}})$
be the object of $\operatorname{Fil}^{\geq 0}\mathrm{DAlg}^{\mathrm{pd}}_{R}$
arising from this construction. In [Magidson2] we will prove the following:

######  Theorem 1.1.8.

Let $A$ be a connective derived commutative $R$-algebra, and
$X=\operatorname{Spec}A$. There is a natural filtered map

$\textstyle{\mathrm{L\Omega}_{A/R}^{\geq\star}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Fil}^{\geq\star}\Gamma(X_{{\mathrm{dR}}},\mathcal{O}_{X_{{\mathrm{dR}}}})}$

which induces an equivalence on Hodge completions.

As we mentioned above, the isomorphism above holds for derived De Rham
cohomology without Hodge completion for any map $R\rightarrow A$ satisfying
either a simple finiteness condition, or a certain quasi-lci condition. We do
not know whether the statement is true without Hodge completion for any map
$R\rightarrow A$, but if a counter-example exists, it has to be some
infinitely-generated and not quasi-lci algebra.

###  1.2 Notations and terminology.

We use the language of $\infty$-categories and higher algebra developed by
J.Lurie in [HTT] and [HA]. We freely use the notions of an _$\infty$
-category_, a _functor between $\infty$-categories_, an _$\infty$ -categorical
adjunction_ etc. The amount of higher categorical machinery used in the text,
is limited to a very basic level. Essentially the only technical statements we
use are the _adjoint functor theorem_ [HTT, Corollary 5.5.2.9], and the _Barr-
Beck-Lurie monadicity theorem_ formulated in [HA, Theorem 4.7.3.5]. In
general, most of our constructions are formulated in a homotopy invariant way.
In particular, all limits and colimits are in general assumed to be in the
homotopy sense. For better exposition, we often restrict our attention to some
discrete computations, in which case we specify that we are working with
discrete cochain complexes, algebras etc.

We let $\mathrm{Spc}$ be the $\infty$-category of _spaces_ , or _$\infty$
-groupoids_, and $\mathrm{Spt}$ the $\infty$-category of _spectra_. For a ring
spectrum $A$, we denote $\mathrm{Mod}_{A}$ the $\infty$-category of _$A$
-module spectra_. For example, our notation for the derived $\infty$-category
of abelian groups is $\mathrm{Mod}_{\mathbb{Z}}$. In a stable
$\infty$-category $\mathrm{C}$, we denote $[1]$ and $[-1]$ suspension and loop
functors respectively.

We often encounter pull-back diagrams of $\infty$-categories. Unless specified
otherwise, these are taken in the $\infty$-category of large
$\infty$-categories.

For a pair of $\infty$-categories $\mathrm{C}$ and $\mathrm{D}$, we let
$\operatorname{Fun}(\mathrm{C},\mathrm{D})$ be the $\infty$-category of
functors from $\mathrm{C}$ to $\mathrm{D}$. The $\infty$-category
$\operatorname{Fun}(\mathrm{C},\mathrm{C})$ has a monoidal structure given by
composition of functors. A monad on $\mathrm{C}$ is an associative algebra
object in this monoidal $\infty$-category.

Most $\infty$-categories we will encounter in the text will be presentable. We
let $\Pr^{L}$ to be the $\infty$-category of presentable $\infty$-categories
and continuous functors. This $\infty$-category has a symmetric monoidal
structure constructed by J.Lurie.

A presentable symmetric monoidal $\infty$-category is a commutative algebra
object in $\Pr^{L}$.

We use the $\bullet$-symbol for the grading on a graded object $X^{\bullet}$,
and the $\star$-symbol to indicate the filtration on a filtered object
$X^{\geq\star}$.

We use homological notation in the context of $t$-structures on a stable
$\infty$-category. For instance, for a stable $\infty$-category $\mathrm{C}$
with a $t$-structure, $\mathrm{C}_{\geq 0}$ is the full subcategory of
connective objects. For the full subcategory of discrete objects, we use the
symbol $\mathrm{C}_{\heartsuit}$.

Given a presentable $\infty$-category $\mathrm{C}$, we denote
$\mathrm{C}^{\omega}\subset\mathrm{C}$ the full $\infty$-subcategory
consisting of compact objects in $\mathrm{C}$.

Given a symmetric monoidal $\infty$-category $\mathrm{C}$, one can talk about
$\mathbb{E}_{\infty}$-algebra objects in $\mathrm{C}$. For us, $\mathrm{C}$
will typically be a stable symmetric monoidal $\infty$-category of the form
$\mathrm{Mod}_{R}$ for $R$ a derived commutative ring. In this setting, we can
also talk about derived commutative $R$-algebras.

In the context of derived algebraic geometry, we follow some basic terminology
of the texts [GaitsRozI], [GR2]. For instance, we use the notions of prestack
and quasicoherent sheaves on prestacks. A prestack over a derived ring $R$ is
a functor $X:\mathrm{DAlg}_{R,\geq 0}\rightarrow\mathrm{Spc}$, where
$\mathrm{DAlg}_{R,\geq 0}$ is the $\infty$-category of connective commutative
$R$-algebras over the base ring $R$. For an $A\in\mathrm{DAlg}_{R,\geq 0}$,
there is a representable prestack $\operatorname{Spec}(A)$, i.e. a derived
affine scheme. Any prestack $X$ can be written as a colimit of derived affine
schemes

$X=\underset{\operatorname{Spec}(A)\rightarrow
X}{\operatorname*{colim}}\;\;\operatorname{Spec}(A)$

The $\infty$-category of quasi-coherent sheaves on $X$ is the limit

$\operatorname{QCoh}(X)\simeq\underset{\operatorname{Spec}(A)\rightarrow
X}{\lim}\;\;\mathrm{Mod}_{A}$

taken along the diagram of all derived affine schemes mapping to $X$ using
base change functors
$-\underset{A}{\otimes}B:\mathrm{Mod}_{A}\rightarrow\mathrm{Mod}_{B}$ for any
map $\operatorname{Spec}(B)\rightarrow\operatorname{Spec}(A)$ over $X$.

## Acknowledgements.

I am grateful to Ben Antieau, Lukas Brantner, Chris Brav, Rune Haugseng, Adam
Holeman, Grigory Kondyrev, Zhouhang Mao, Akhil Mathew, Joost Nuiten, Artem
Prikhodko, Noah Riggenbach and Nick Rozenblyum for discussions on the topics
of this text. Many ideas of this text were developed by A.Raksit in [R] and
Zhouhang Mao in [Mao21] in a slightly different language, and we acknowledge
their important contribution to the field. The author was supported by by NSF
grants DMS-2152235 and DMS-21002010, and the grant 21-7-2-30-1 of the “BASIS”
foundation.

##  2 Preliminaries.

###  2.1 Higher category theory.

Here we will review some higher categorical facts used in the text. We will
often work with various $\infty$-categories built as pull-backs and/or lax
equalizers of other symmetric monoidal $\infty$-categories. We recall the
following construction from Nikolaus-Scholze [NS, Definition II.1.4].

######  Definition 2.1.1.

Assume given two $\infty$-categoties $\mathrm{C}$ and $\mathrm{D}$ and two
functors $F,G:\mathrm{C}\rightarrow\mathrm{D}$. The Lax equalizer
$\infty$-category $\mathrm{LEq}(F,G)$ is defined as the pull-back

$\textstyle{\mathrm{LEq}(F,G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{D}^{\Delta^{1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\partial^{0},\partial^{1})}$$\textstyle{\mathrm{C}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(F,G)}$$\textstyle{\mathrm{D}\times\mathrm{D}}$
(2.1)

We will often study the case $\mathrm{D}=\mathrm{C}$, $F=\operatorname{Id}$,
and $G$ is some endofunctor of $\mathrm{C}$. Moreover, we will use the fact
that if $\mathrm{C}$ is a symmetric monoidal $\infty$-category and $G$ is a
lax symmetric monoidal functor, then $\mathrm{LEq}(\operatorname{Id},G)$ gets
a symmetric monoidal structure. Moreover, commutative algebra objects in
$\mathrm{LEq}(\operatorname{Id},G)$ are precisely pairs $(A,f)$ where $A$ is a
commutative algebra object in $\mathrm{C}$, and $f:A\rightarrow G(A)$ is a map
of commutative algebra objects. To explain this, let us recall some basic
terminology regarding symmetric monoidal $\infty$-categories. Let
$\mathrm{Fin}_{*}$ be the category of finite pointed sets. In the approach of
J.Lurie, an $\infty$-operad is a functor
$p:\mathcal{O}_{\infty}\rightarrow\mathrm{Fin}_{*}$ satisfying conditions
listed in [HA, Definition 2.1.1.10]. A symmetric monoidal $\infty$-category is
an $\infty$-category $\mathrm{C}^{\otimes}$ equipped with a coCartesian
fibtration $\mathrm{C}^{\otimes}\rightarrow\mathrm{Fin}_{*}$ of
$\infty$-operads. A lax symmetric monoidal functor
$F:\mathrm{C}^{\otimes}\rightarrow\mathrm{D}^{\otimes}$ is simply an
$\infty$-operad map. Let $\operatorname{CAlg}(\mathrm{Cat})_{\mathrm{lax}}$ be
the $\infty$-category of symmetric monoidal $\infty$-categories and lax
symmetric monoidal functors. There is a fully faithful embedding
$\operatorname{CAlg}(\mathrm{Cat})_{\mathrm{lax}}\subset\mathrm{Op}_{\infty}$,
where $\mathrm{Op}_{\infty}$ is the $\infty$-category of operads defined as in
[HA, Definition 2.1.4.1]. Given a symmetric monoidal $\infty$-category
$p:\mathrm{C}^{\otimes}\rightarrow\mathrm{Fin}_{*}$, we let
$\mathrm{C}=\mathrm{C}_{\langle 1\rangle}:=p^{-1}(\langle 1\rangle)$. The
functor
$\operatorname{CAlg}(\mathrm{Cat})_{\mathrm{lax}}\rightarrow\mathrm{Cat}$,
$(p:\mathrm{C}^{\otimes}\rightarrow\mathrm{Fin}_{*})\longmapsto\mathrm{C}$ is
conservative and preserves all limits222Caution: the inclusion
$\operatorname{CAlg}(\mathrm{Cat})_{\mathrm{lax}}\hookrightarrow\mathrm{Op}_{\infty}$
is, however, not closed under all limits..

In the future, we will often write the data of a symmetric monoidal
$\infty$-category $p:\mathrm{C}^{\otimes}\rightarrow\mathrm{Fin}_{*}$ simply
as $\mathrm{C}$ keeping symmetric monoidal structure in mind. The tautological
coCartesian fibration $\mathrm{Fin}_{*}\rightarrow\mathrm{Fin}_{*}$ is the
commutative operad. For a given symmetric monoidal $\infty$-category
$\mathrm{C}:=(p:\mathrm{C}^{\otimes}\rightarrow\mathrm{Fin}_{*})$, the
collection of all operad maps
$\mathrm{Fin}_{*}\rightarrow\mathrm{C}^{\otimes}$ organize into an
$\infty$-category of commutative algebra objects in $\mathrm{C}$, denoted
$\operatorname{CAlg}(\mathrm{C})$. The following proposition follows
immediately from definitions.

######  Proposition 2.1.2.

Assume we are given

$\textstyle{\mathrm{C}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{\underset{i}{\lim}\;\mathrm{C}_{i}}$
(2.2)

a limit diagram in $\operatorname{CAlg}(\mathrm{Cat})_{\mathrm{lax}}$. Then
the natural map in $\mathrm{Cat}$

$\textstyle{\operatorname{CAlg}(\mathrm{C})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{\underset{i}{\lim}\;\mathrm{C}_{i}.}$

is an equivalence.

######  Proposition 2.1.3.

Assume $\mathrm{C}$ and $\mathrm{D}$ are symmetric monoidal
$\infty$-categories, $F$ is a symmetric monoidal functor, and $G$ is lax
symmetric monoidal. The $\infty$-category $\mathrm{LEq}(F,G)$ has a symmetric
monoidal structure, and there is an equivalence

$\operatorname{CAlg}(\mathrm{LEq}(F,G))\simeq\mathrm{LEq}(F,G:\operatorname{CAlg}(\mathrm{C})\rightarrow\operatorname{CAlg}(\mathrm{D})).$

###### Proof.

Using [NS, Construction IV.2.1 (ii)], there is a symmetric monoidal
$\infty$-category $\mathrm{LEq}(F,G)^{\otimes}$ defined as a pull-back diagram
of symmetric monoidal $\infty$-categories and lax symmetric monoidal functors,
whose underlying $\infty$-category is $\mathrm{LEq}(F,G)$. Taking the
$\infty$-category of commutative algebra objects and using 2.1.2, we get a
pull-back square of $\infty$-categories

$\textstyle{\operatorname{CAlg}(\mathrm{LEq}(F,G))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{CAlg}(\mathrm{D}^{\Delta^{1}})\simeq\operatorname{CAlg}(\mathrm{D})^{\Delta^{1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(ev^{0},\operatorname{ev}^{1})}$$\textstyle{\operatorname{CAlg}(\mathrm{C})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(F,G)}$$\textstyle{\operatorname{CAlg}(\mathrm{D}\times\mathrm{D})\simeq\operatorname{CAlg}(\mathrm{D})\times\operatorname{CAlg}(\mathrm{D}),}$

and the second statement follows. ∎

######  Remark 2.1.4.

Note that if $F$ is only lax symmetric monoidal, but not strictly symmetric
monoidal, there is no natural symmetric monoidal structure on
$\mathrm{LEq}(F,G)$. This is an example demonstrating that in general the
$\infty$-category $\operatorname{CAlg}(\mathrm{Cat})_{\mathrm{lax}}$ is not
closed under all limits inside $\mathrm{Op}_{\infty}$. It is crucial that when
$F$ is strictly symmetric monoidal, one can construct a symmetric monoidal
structure on $\mathrm{LEq}(F,G)$ making the square 2.1 a square of symmetric
monoidal $\infty$-categories and lax symmetric monoidal functors. In other
words, the commutative square 2.2 needs to be _given_ , and in general, we
_can not define_ $\mathrm{C}$ as a pull-back of symmetric monoidal
$\infty$-categories and lax symmetric monoidal functors.

Over a ring of non-zero characteristic, in addition to
$\mathbb{E}_{\infty}$-algebras, there are also theories of derived commutative
algebras as well as derived divided power algebras. We will review some basics
of derived commutative algebras following [R] in the next subsection, and the
theory of derived divided power algebras in Section 2.3 via the approach of
[BCN21]. We will also formulate and prove a version of Proposition 2.1.2 in
the setting of derived algebras in Section 2.3. To this end, we will introduce
the abstract framework in which the theories of derived commutative algebras
and derived divided power algebras fit.

Recall that given an $\infty$-category $\mathrm{C}$, a monad on $\mathrm{C}$
is an algebra object in the monoidal $\infty$-category
$\operatorname{Fun}(\mathrm{C},\mathrm{C})$ with respect to the composition
product. To formulate and prove a version of Proposition 2.1.2 in the setting
of algebras over a monad, we ought to have, given a pair of
$\infty$-categories $(\mathrm{C},\mathrm{T}_{\mathrm{C}})$ and
$(\mathrm{D},\mathrm{T}_{\mathrm{D}})$ endowed with monads, the notion of a
lax equivariant functor
$F:(\mathrm{C},\mathrm{T}_{\mathrm{C}})\rightarrow(\mathrm{D},\mathrm{T}_{\mathrm{D}})$
with respect to these monads action. In order to speak about such lax
equivariant functors, we use some basic $(\infty,2)$-categorical formalism. We
generally follow the approach of [GaitsRozI] to $(\infty,2)$-categories, and
use some elements of the formalism developed by R.Haugseng in the text [H]. We
will give references for all definitions and statements, but will not present
a detailed overview of the formalism and restrict to simply explaining what
various constructions mean on the naive level of 2-categories.

######  Notation 2.1.5.

We denote $\mathrm{Cat}_{(\infty,2)}$ the $(\infty,1)$-category of
$(\infty,2)$-categories. There exist several approaches to
$(\infty,2)$-categories in the literature. For instance, one can think of
objects of $\mathrm{Cat}_{(\infty,2)}$ as the $\infty$-categories of
$\infty$-categories enriched in the symmetric monoidal $\infty$-category
$\mathrm{Cat}$. This means that an $(\infty,2)$-category C is characterized by
the data of a collection of objects $X\in\textbf{C}$ and for any pair
$X,Y\in\textbf{C}$, an $(\infty,1)$-category $\operatorname{Hom}(X,Y)$
together with composition functors

$\operatorname{Hom}(X,Y)\times\operatorname{Hom}(Y,Z)\rightarrow\operatorname{Hom}(X,Z)$

with a hierarchy of higher coherences333See [H, Remark 2.14] for discussion
and references to this approach. We do not have to choose a specific model,
because most our statements and constructions already exist in the literature,
and we only need to reference them and apply to concrete problems.. The
elements of $\operatorname{Hom}(X,Y)$ are usually called 1-morphisms in C, and
morphisms of $\operatorname{Hom}(X,Y)$ are usually called 2-morphisms in C. We
use bold letters like $\textbf{C},\textbf{D},...$ for objects of
$\mathrm{Cat}_{(\infty,2)}$. Given an $(\infty,2)$-category C, we denote
$\imath_{1}\textbf{C}$ the underlying $(\infty,1)$-category obtained by
discarding all non-invertible $2$-morphisms. We denote Cat the
$(\infty,2)$-category of $(\infty,1)$-categories444We usually call
$(\infty,1)$-categories simply “$\infty$-categories”, unless there is any
possibility of confusion, or unless we want to be extra pedantic.. The
construction of Cat uses the natural enhancement of the usual
$(\infty,1)$-category of $(\infty,1)$-categories by letting the
$\operatorname{Hom}$-category between two objects
$\mathrm{C},\mathrm{D}\in\mathrm{Cat}$ to be the $\infty$-category
$\operatorname{Fun}(\mathrm{C},\mathrm{D})$ of all functors and not
necessarily invertible natural transformations. Given two
$(\infty,1)$-categories $\mathrm{C},\mathrm{D}$, there is an
$(\infty,1)$-category $\operatorname{Fun}(\mathrm{C},\mathrm{D})$ whose
objects are functors $F:\mathrm{C}\rightarrow\mathrm{D}$, and morphisms are
not necessarily invertible natural transformations $F\Rightarrow G$.

######  Construction 2.1.6.

Given a pair of $(\infty,2)$-categoies $\textbf{C},\textbf{D}$, there is an
$\infty$-category $\operatorname{Fun}(\textbf{C},\textbf{D})_{\mathrm{lax}}$
of 2-functors $\textbf{C}\rightarrow\textbf{D}$ and lax natural
transformations. A lax natural transformation $\eta:F\Rightarrow G$ between
2-functors $F,G:\textbf{C}\rightarrow\textbf{D}$ is the data assigning to any
1-morphism $f:X\rightarrow Y$ in C a lax square

$\textstyle{F(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F(f)}$$\scriptstyle{\eta_{X}}$$\textstyle{G(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{G(f)}$$\textstyle{F(Y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta_{Y}}$$\textstyle{G(Y),}$

i.e. a not necessarily invertible map
$\eta_{f}:G(f)\circ\eta_{X}\rightarrow\eta_{Y}\circ F(f)$. Dually, there is
also a notion of an oplax natural transformation for two functors
$F,G:\textbf{C}\rightarrow\textbf{D}$.

See [H, Definition 2.9] for a precise definition of
$\operatorname{Fun}(\textbf{C},\textbf{D})_{(\mathrm{op})\mathrm{lax}}$ using
Gray tensor product. We will give a very short overview here. There exists a
bifunctor
$\otimes^{(\mathrm{op})\mathrm{lax}}:\mathrm{Cat}_{(\infty,2)}\times\mathrm{Cat}_{(\infty,2)}\rightarrow\mathrm{Cat}_{(\infty,2)}$
called (op)lax Gray tensor product which preserves colimits in each variable,
and such that for any triple of $(\infty,2)$-categories
$\textbf{C},\textbf{D},\textbf{E}$, we have

$\operatorname{Map}_{\mathrm{Cat}_{(\infty,2)}}(\textbf{C},\operatorname{Fun}(\textbf{D},\textbf{E})_{\mathrm{op}\mathrm{lax}})\simeq\operatorname{Map}_{\mathrm{Cat}_{(\infty,2)}}(\textbf{D}\otimes^{\mathrm{op}\mathrm{lax}}\textbf{C},\textbf{E})$

$\simeq\operatorname{Map}_{\mathrm{Cat}_{(\infty,2)}}(\textbf{C}\otimes^{\mathrm{lax}}\textbf{D},\textbf{E})\simeq\operatorname{Map}_{\mathrm{Cat}_{(\infty,2)}}(\textbf{D},\operatorname{Fun}(\textbf{C},\textbf{E})_{\mathrm{lax}}).$

Unwinding the definition, one obtains that an (op)lax natural transformation
between functors $F,G:\textbf{C}\rightarrow\textbf{D}$ is a functor of
$(\infty,2)$-categories
$\textbf{C}\otimes^{(\mathrm{op})\mathrm{lax}}\Delta^{1}\rightarrow\textbf{D}$
whose restriction to $0,1\in\Delta^{1}$ is $F$ and $G$ respectively.

######  Example 2.1.7.

Let $\mathbb{N}=\\{0,1,2,...\\}$ be the monoid of natural numbers by addition.
We will apply the previous construction to the case
$\textbf{C}:=\mathrm{B}\mathbb{N}$, which is an ordinary category with one
object and a unique non-identity morphism, and $\textbf{D}:=\textbf{Cat}$. The
data of a functor $\mathrm{B}\mathbb{N}\rightarrow\textbf{Cat}$ is the same as
the data of a pair $(\mathrm{C},\mathrm{T}_{C})$ where $\mathrm{C}$ is an
$\infty$-category, and $\mathrm{T}:\mathrm{C}\rightarrow\mathrm{C}$ an
endofunctor. Giving a lax natural transformation
$F:(\mathrm{C},\mathrm{T}_{\mathrm{C}})\rightarrow(\mathrm{D},\mathrm{T}_{\mathrm{D}})$
is equivalent to giving a functor $F:\mathrm{C}\rightarrow\mathrm{D}$ and for
any $X\in\mathrm{C}$ a lax equivariant structure map
$\eta_{X}:\mathrm{T}_{\mathrm{D}}\circ F(X)\rightarrow
F\circ\mathrm{T}_{\mathrm{C}}(X)$.

######  Construction 2.1.8.

This construction follows [H, Definition 4.2, Definition 5.2] and the
references therein.

  1. 1.

There exist a universal $2$-category mnd containing a monad. The universal
property of mnd is that giving a functor $\textbf{mnd}\rightarrow\textbf{Cat}$
is the same as giving an $\infty$-category $\mathrm{C}$ endowed with a monad
$\mathrm{T}:\mathrm{C}\rightarrow\mathrm{C}$. The $2$-category mnd contains a
single object with $\hom$-category being the augmented simplex category, i.e.
$\textbf{mnd}\simeq\mathrm{B}\Delta_{a}$. We define the $(\infty,2)$-category
of monads and (op)lax-equivariant 1-morphisms in C as

$\mathrm{Mnd}(\textbf{cat})_{(\mathrm{op})\mathrm{lax}}:=\operatorname{Fun}(\textbf{mnd},\textbf{Cat})_{(\mathrm{op})\mathrm{lax}}.$

To unravel the definition above, objects of
$\mathrm{Mnd}(\mathrm{Cat})_{(\mathrm{op})\mathrm{lax}}$ are pairs
$(\mathrm{C},\mathrm{T}_{\mathrm{C}})$ consisting of an $\infty$-category
$\mathrm{C}$ endowed with a monad $\mathrm{T}_{\mathrm{C}}$ and maps are
functors $F:\mathrm{C}\rightarrow\mathrm{D}$ which are (op)lax-equivariant
with respect to the monads $\mathrm{T}_{\mathrm{C}}$ and
$\mathrm{T}_{\mathrm{D}}$. We also denote
$\mathrm{Mnd}(\mathrm{Cat})_{(\mathrm{op})\mathrm{lax}}:=\imath_{1}\operatorname{Fun}(\textbf{mnd},\textbf{Cat})_{(\mathrm{op})\mathrm{lax}}$
for simplicity.

More generally, the notion of a monad makes sense internally to any
$(\infty,2)$-category. Given $\textbf{C}\in\mathrm{Cat}_{(\infty,2)}$, we
define the $(\infty,2)$-category of monads and (op)lax equivariant
$1$-morphisms in C as the $(\infty,2)$-category of functors
$\mathrm{Mnd}(\textbf{C})_{(\mathrm{op})\mathrm{lax}}:=\operatorname{Fun}(\textbf{mnd},\textbf{C})_{(\mathrm{op})\mathrm{lax}}.$

  2. 2.

There exits a universal $2$-category adj containing an adjunction with the
universal property that the data of a functor
$\textbf{adj}\rightarrow\textbf{Cat}$ is equivalent to the data of a pair of
$\infty$-categories $\mathrm{C},\mathrm{D}$ endowed with an adjunction
$\textstyle{\mathrm{D}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{G}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathrm{C}}$$\scriptstyle{F}$.
We define the $(\infty,2)$-category of adjunctions and (op)lax-equivariant
functors as

$\mathrm{Adj}(\textbf{Cat})_{(\mathrm{op})\mathrm{lax}}:=\operatorname{Fun}(\textbf{adj},\textbf{Cat})_{(\mathrm{op})\mathrm{lax}},$

and use the simplified notation
$\mathrm{Adj}(\mathrm{Cat})_{(\mathrm{op})\mathrm{lax}}:=\imath_{1}\operatorname{Fun}(\textbf{adj},\textbf{Cat})_{(\mathrm{op})\mathrm{lax}}$
for the underlying $(\infty,1)$-category. There exists a forgetful functor
$\mathrm{Adj}(\textbf{Cat})_{\mathrm{lax}}\rightarrow\mathrm{Mnd}(\textbf{Cat})_{\mathrm{lax}}$
sending any adjunction
$\textstyle{\mathrm{D}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{G}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathrm{C}}$$\scriptstyle{F}$
to the monad $(\mathrm{C},G\circ F)$.

The notion of an adjunction is well-defined internally to any
$(\infty,2)$-category. Given $\textbf{C}\in\mathrm{Cat}_{(\infty,2)}$, the
$\infty$-category of adjunctions and (op)lax equivariant $1$-morphisms in C is
by definition, the $(\infty,2)$-category of functors
$\mathrm{Adj}(\textbf{C}):=\operatorname{Fun}(\textbf{adj},\textbf{C})_{(\mathrm{op})\mathrm{lax}}.$

  3. 3.

According to [H, Corollary 5.8], the forgetful functor
$\mathrm{Adj}(\textbf{Cat})_{\mathrm{lax}}\rightarrow\mathrm{Mnd}(\textbf{Cat})_{\mathrm{lax}}$
admits a fully faithful right adjoint
$\mathrm{Mnd}(\textbf{Cat})_{\mathrm{lax}}\rightarrow\mathrm{Adj}(\textbf{Cat})_{\mathrm{lax}}$
sending an object
$(\mathrm{C},\mathrm{T})\in\mathrm{Mnd}(\textbf{Cat})_{\mathrm{lax}}$ to the
adjunction

$\textstyle{\operatorname{Alg}_{\mathrm{T}}(\mathrm{C})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{U_{\mathrm{T}}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathrm{C},}$$\scriptstyle{F_{\mathrm{T}}}$

where $\operatorname{Alg}_{\mathrm{T}}(\mathrm{C})$ is the $\infty$-category
of $\mathrm{T}$-algebras in $\mathrm{C}$, $F_{\mathrm{T}}$ is the free algebra
functor, and $U_{\mathrm{T}}$ is the forgetful functor. The claim that
$\operatorname{Alg}_{\mathrm{T}}(\mathrm{C})$ is the conventional
$\infty$-category of $\mathrm{T}$-algebras in $\mathrm{C}$ as defined in [HA]
is established in [H, Proposition 8.10].

######  Proposition 2.1.9.

Assume $(\mathrm{C},\mathrm{T}_{\mathrm{C}})\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern
3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
9.55pt\raise 4.15565pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.15565pt\hbox{$\scriptstyle{\sim}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}\ignorespaces}}}}\ignorespaces\underset{i}{\lim}(\mathrm{C}_{i},\mathrm{T}_{\mathrm{C}_{i}})$
is a limit diagram in $\mathrm{Mnd}(\mathrm{Cat})_{\mathrm{lax}}$. In this
situation, the natural functor induces an equivalence:

$\textstyle{\operatorname{Alg}_{\mathrm{T}_{\mathrm{C}}}(\mathrm{C})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{\underset{i}{\lim}\operatorname{Alg}_{\mathrm{T}_{\mathrm{C}_{i}}}(\mathrm{C}_{i})}$

###### Proof.

This follows immediately from part 3 of Construction 2.1.8 as right adjoints
commute with limits, and the functor
$\mathrm{Adj}(\mathrm{Cat})_{\mathrm{lax}}\rightarrow\mathrm{Cat}$ sending an
adjunction
$\textstyle{\mathrm{D}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{G}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathrm{C}}$$\scriptstyle{F}$
to the $\infty$-category $\mathrm{D}$ (the source of the right adjoint)
commutes with limits as well. ∎

######  Construction 2.1.10.

Let $\mathrm{A}$ be a monoidal $\infty$-category, BA the corresponding
$(\infty,2)$-category which has a single object with $\hom$-category being
$\mathrm{A}$. We let

$\mathrm{Mod}_{\mathrm{A}}(\textbf{Cat})_{(\mathrm{op})\mathrm{lax}}:=\operatorname{Fun}(\textbf{BA},\textbf{Cat})_{(\mathrm{op})\mathrm{lax}}$

be the $(\infty,2)$-category of $(\infty,1)$-categories with an
$\mathrm{A}$-action and (op)lax $\mathrm{A}$-equivariant functors between
them, and following our usual convention let
$\mathrm{Mod}_{\mathrm{A}}(\mathrm{Cat})_{(\mathrm{op})\mathrm{lax}}:=\imath_{1}\mathrm{Mod}_{\mathrm{A}}(\textbf{Cat})_{(\mathrm{op})\mathrm{lax}}$.
Giving an algebra object $\mathrm{T}\in\operatorname{Alg}(\mathrm{A})$ is
equivalent to giving a functor of $(\infty,2)$-categories
$\textbf{mnd}\rightarrow\textbf{BA}$. Consequently, given a monad $\mathrm{T}$
in $\operatorname{Alg}(\mathrm{A})$, we get a limit-preserving restriction
functor

$\mathrm{Mod}_{\mathrm{A}}(\mathrm{Cat})_{\mathrm{lax}}\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern
3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}\ignorespaces}}}}\ignorespaces\mathrm{Mnd}(\mathrm{Cat})_{\mathrm{lax}}$
(2.3)

which takes any $\infty$-category $\mathrm{C}$ with $\mathrm{A}$-action to the
monad $(\mathrm{C},\mathrm{T}_{\mathrm{C}})$, where $\mathrm{T}_{\mathrm{C}}$
is the image of $\mathrm{T}\in\operatorname{Alg}(\mathrm{A})$ under the
monoidal functor
$\mathrm{A}\rightarrow\operatorname{Fun}(\mathrm{C},\mathrm{C})$.

We shall now recall the construction of passing to adjoints in the
$(\infty,2)$-categorical setting, and how it interchanges oplax and lax
natural transformations. Given a pair of $(\infty,2)$-categories
$\textbf{C},\textbf{D}$, let
$\operatorname{Fun}(\textbf{C},\textbf{D})_{\mathrm{oplax},L}$ be the
subcategory of $\operatorname{Fun}(\textbf{C},\textbf{D})_{\mathrm{oplax}}$
with all maps being left adjoint functors, and similarly
$\operatorname{Fun}(\textbf{C},\textbf{D})_{\mathrm{lax},R}$ the subcategory
of $\operatorname{Fun}(\textbf{C},\textbf{D})_{\mathrm{lax}}$ with all maps
being right adjoint functors. [GR2, Ch.12, Corollary 3.1.9] states that taking
adjoints gives a natural equivalence

$\operatorname{Fun}(\textbf{C},\textbf{D})_{\mathrm{oplax},L}\simeq\operatorname{Fun}(\textbf{C}^{(1,2)-\mathrm{op}},\textbf{D})_{\mathrm{lax},R},$

where the superscript ${(1,2)-\mathrm{op}}$ stands for inverting all $1$ and
$2$-morphisms.

Applying this general fact to the construction 2.3, we get a commutative
diagram of $\infty$-categories

$\textstyle{(\mathrm{Mod}_{\mathrm{A}}(\mathrm{Cat})_{L})^{\mathrm{op}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(\mathrm{Mnd}(\mathrm{Cat})_{L})^{\mathrm{op}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{Mod}_{\mathrm{A}}(\mathrm{Cat})_{\mathrm{lax},R}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{Mnd}(\mathrm{Cat})_{\mathrm{lax},R},}$

where the vertical functors are taking right adjoints of $1$-morphisms.

We will use the following Proposition in several crucial places.

######  Proposition 2.1.11.

Assume $\mathrm{A}$ is a monoidal $\infty$-category with a fixed algebra
object $\mathrm{T}\in\operatorname{Alg}(\mathrm{C})$. Suppose that we are
given a diagram $\\{\mathrm{C}_{i}\\}_{i\in I}$ in
$\mathrm{Mod}_{\mathrm{A}}(\mathrm{Cat})_{L}$ such that the corresponding
diagram in $\mathrm{Mod}_{\mathrm{A}}(\mathrm{Cat})_{\mathrm{lax}}$ obtained
by passing to right adjoints, has a limit
$\mathrm{C}\simeq\underset{i}{\lim}\;\mathrm{C}_{i}$ in
$\mathrm{Mod}_{\mathrm{A}}(\mathrm{Cat})_{\mathrm{lax}}$. Then Construction
2.1.10 produces an equivalence

$\textstyle{\operatorname{Alg}_{\mathrm{T}_{\mathrm{C}}}(\mathrm{C})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{\underset{i}{\lim}\operatorname{Alg}_{\mathrm{T}_{\mathrm{C}_{i}}}(\mathrm{C}_{i}).}$

###  2.2 Non-abelian derived functors.

In this section we will review some basic ideas from the theory of derived
commutative rings. Many ideas of this subsection are due to L.Brantner and
A.Mathew and were first developed in the paper [BM19] in the case
$\mathrm{C}=\mathrm{Mod}_{k}$, $k$ a field. We will closely follow the
terminology of the paper of A.Raksit [R].

######  Definition 2.2.1.

Let $\operatorname{CAlg}^{\mathrm{poly}}_{\mathbb{Z}}$ be the category of all
finitely generated polynomial rings. The $\infty$-category of connective
derived commutative rings $\mathrm{DAlg}_{\mathbb{Z},\geq 0}$ is obtained by
freely adjoining all sifted colimits to
$\operatorname{CAlg}^{\mathrm{poly}}_{\mathbb{Z}}$,

$\mathrm{DAlg}_{\mathbb{Z},\geq
0}:=\mathcal{P}_{\Sigma}(\operatorname{CAlg}^{\mathrm{poly}}_{\mathbb{Z}}).$

Given an object $R$, we let $\mathrm{DAlg}_{R,\geq
0}:=(\mathrm{DAlg}_{\mathbb{Z},\geq 0})_{R/}$ be the under category of
connective derived commutative $R$-algebras.

######  Remark 2.2.2.

The strategy of defining $R$-algebras used above will be used very frequently
throughout the text. In many cases, we will define some $\infty$-category of
derived rings over $\mathbb{Z}$ first, and then define $R$-linear objects as
an under-category. Consequently, many constructions with derived rings of
different types can be done in the case $R$ is a discrete ring, or even
$R=\mathbb{Z}$, and then generalized to under-objects.

######  Remark 2.2.3.

Let $R$ be a discrete commutative ring. The $\infty$-category
$\mathrm{Mod}_{R}$ of left $R$-modules is a stable presentable symmetric
monoidal $\infty$-category. There is a $t$-structure $(\mathrm{Mod}_{R,\geq
0},\mathrm{Mod}_{R,\leq 0})$ characterized by the property that
$M\in\mathrm{Mod}_{R}$ is connective if the underlying spectrum is connective.
This $t$-structure is compatible with the symmetric monoidal structure and is
right and left complete. In addition, there is a full subcategory
$\mathrm{Mod}_{R,0}\subset\mathrm{Mod}_{R,\geq 0}$ of finitely generated free
modules which is closed under direct sums and tensor powers. This
$\infty$-category compactly and projectively generates $\mathrm{Mod}_{R,\geq
0}$, i.e. $\mathrm{Mod}_{R,\geq
0}\simeq\mathcal{P}_{\Sigma}(\mathrm{Mod}_{R,0})$.

We will assume that $R$ is discrete throughout this subsection, unless
explicitly stated.

The $\infty$-category $\mathrm{DAlg}_{R,\geq 0}$ is monadic over
$\mathrm{Mod}_{R,\geq 0}$. It turns out that it is possible to extend this
monad to a monad on $\mathrm{Mod}_{R}$, so that one gets the free derived
commutative algebra monad
$\mathrm{LSym}_{\mathrm{R}}:\mathrm{Mod}_{R}\rightarrow\mathrm{Mod}_{R}$. The
paper [R] performs this construction by using the natural filtration on the
free symmetric algebra. We will briefly review the Raksit’s construction
below.

######  Example 2.2.4.

Let $X\in\mathrm{Mod}_{R}$. The symmetric power of $X$ is defined by the
formula

$\operatorname{Sym}_{\mathrm{C}}(X):=(\otimes^{n}X)_{h\Sigma_{n}},$

where the subscript $(-)_{h\Sigma_{n}}$ denotes homotopy orbits with respect
to the symmetric group action. The functor $\operatorname{Sym}^{\leq
n}=\oplus_{i=1}^{n}\operatorname{Sym}^{i}_{\mathrm{R}}:\mathrm{Mod}_{R}\rightarrow\mathrm{Mod}_{R}$
is excisively polynomial of degree $n$. The (discrete) symmetric power of
$X\in\mathrm{Mod}_{R,\heartsuit}$ is defined as

$\operatorname{Sym}_{R,\heartsuit}(X):=(\otimes^{n}X)_{\Sigma_{n}},$

where the coinvariants are taken in the naive sense. The functors
$\operatorname{Sym}^{\leq
n}_{R,\heartsuit}:=\oplus_{i=1}^{n}\operatorname{Sym}^{i}_{R,\heartsuit}:\mathrm{Mod}_{R,\heartsuit}\rightarrow\mathrm{Mod}_{R,\heartsuit}$
are additively polynomial of degree $n$.

######  Construction 2.2.5.

Let $\mathrm{T}:\mathrm{Mod}_{R,0}\rightarrow\mathrm{Mod}_{R}$ a functor. We
define the right-left Kan extension of $\mathrm{T}$,
$\mathrm{T}^{RL}:\mathrm{Mod}_{R}\rightarrow\mathrm{Mod}_{R}$ as the right Kan
extension along the inclusion
$\mathrm{Mod}_{R,0}\subset\mathrm{Mod}^{\omega}_{R,\leq 0}$, followed by the
left Kan extension to all of $\mathrm{Mod}_{R}$.

######  Definition 2.2.6.

Let $\mathrm{D}$ be an $\infty$-category containing all totalizations and
geometric realizations. We say that a functor $G:\mathrm{Mod}_{R,\leq
0}^{\omega}\rightarrow\mathrm{D}$ preserves _finite coconnective geometric
realizations_ if for any simplicial object $A_{\ast}$ in $\mathrm{Mod}_{R,\leq
0}^{\omega}$ which is $n$-skeletal for some $n$ such that the geometric
realization $|A_{\ast}|$ belongs to $\mathrm{Mod}_{R,\leq 0}^{\omega}$, the
colimit of $G(A_{\ast})$ exists in $\mathrm{D}$ and the natural map
$|G(A_{\ast})|\rightarrow G(|A_{\ast}|)$ is an equivalence. A functor
$G:\mathrm{Mod}_{R}^{\mathrm{free},\mathrm{fg}}\rightarrow\mathrm{D}$ is
right-left extendable if the right Kan extension $RG:\mathrm{Mod}_{R,\leq
0}^{\omega}\rightarrow\mathrm{D}$ preserves finite coconnective geometric
realizations.

######  Definition 2.2.7.

This definition concerns the two notions of polynomiality for functors of
$\infty$-categories.

  * •

Let $\mathrm{C}$ be an additive $\infty$-category, and $\mathrm{D}$ an
$\infty$-category containing small colimits. An additively polynomial functor
$F:\mathrm{C}\rightarrow\mathrm{D}$ of degree $0$ is a constant functor.
Assume a polynomial functor of degree $n-1$ has been defined. A functor
$F:\mathrm{C}\rightarrow\mathrm{D}$ is additively polynomial of degree $n$ if
the derivative $DF_{X}(Y)=\operatorname*{fib}(F(X\oplus Y)\rightarrow F(X))$
is of degree $n-1$. A functor $F$ is additively polynomial if it is additively
polynomial of some degree.

  * •

Let $\mathrm{C}$ be an $\infty$-category containing finite colimits, and
$\mathrm{D}$ an $\infty$-category containing all limits. A functor
$F:\mathrm{C}\rightarrow\mathrm{D}$ is excisively polynomial if it is
$n$-excisive of some degree $n$ in the sense of Goodwillie calculus (see [R,
Definition 4.2.7]).

We let
$\operatorname{Fun}_{\mathrm{addpoly}}(\mathrm{Mod}_{R,0},\mathrm{Mod}_{R})$
be the $\infty$-category of additively polynomial functors
$\mathrm{Mod}_{R,0}\rightarrow\mathrm{Mod}_{R}$ and
$\operatorname{Fun}_{\mathrm{excpoly}}^{\Sigma}(\mathrm{Mod}_{R},\mathrm{Mod}_{R})$
be the $\infty$-category of excisively polynomial sifted colimit preserving
functors $\mathrm{Mod}_{R}\rightarrow\mathrm{Mod}_{R}$.

For future reference, we record the following result proven in [BM19,
Proposition 3.34, Theorem 3.35] in the case $R$ is a field, and generalized to
an arbitrary derived algebraic context in [R, Proposition 4.2.14, Proposition
4.2.15].

######  Proposition 2.2.8.

Let $R$ be a connective derived commutative ring. The restriction functor

$\textstyle{\operatorname{Fun}^{\Sigma}_{\mathrm{excpoly}}(\mathrm{Mod}_{R},\mathrm{Mod}_{R})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Fun}_{\mathrm{addpoly}}(\mathrm{Mod}_{R,0},\mathrm{Mod}_{R})}$

is an equivalence of $\infty$-categories with inverse given by right-left
extension.

######  Remark 2.2.9.

For any right-left extendable functor
$\mathrm{T}:\mathrm{Mod}_{R,0}\rightarrow\mathrm{D}$, the restriction
$\mathrm{T}^{RL}|_{\mathrm{Mod}_{R,\geq 0}}$ of its right-left extension to
the full subcategory of connective objects coincides with the left Kan
extension of $F$ along the inclusion
$\mathrm{Mod}_{R,0}\subset\mathrm{Mod}_{R,\geq 0}$. In this case, we will use
the notation $\mathrm{LT}:\mathrm{Mod}_{R}\rightarrow\mathrm{D}$ for the
derived functor defined on all $\mathrm{Mod}_{R}$.

######  Definition 2.2.10.

Let $\mathbb{Z}_{\geq 0}^{\times}$ be the category of non-negative integers
considered with monoidal structure given by multiplication. A filtered (sifted
colimit preserving) monad is a lax monoidal functor

$\textstyle{\mathrm{T}^{\leq\star}:\mathbb{Z}^{\times}_{\geq
0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{End}_{\Sigma}(\mathrm{Mod}_{R}).}$
(2.4)

A.Raksit shows in [R] that for a filtered monad $\mathrm{T}^{\leq\star}$, the
colimit $\mathrm{T}:=\underset{\mathbb{Z}_{\geq
0}}{\operatorname*{colim}}\;\mathrm{T}^{\leq\star}$ is a monad on
$\mathrm{Mod}_{R}$. The advantage of using filtered monads is that many monads
are not excisively polynomial on the nose, but have filtrations whose stages
are excisively polynomial functors.

######  Example 2.2.11.

The functor

$\textstyle{\operatorname{Sym}_{R}^{\leq\star}:\mathbb{Z}^{\times}_{\geq
0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{End}_{\Sigma}(\mathrm{Mod}_{R})}$
(2.5)
$\textstyle{n\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(X\longmapsto\oplus_{i=1}^{n}\operatorname{Sym}^{i}_{R}(X))}$

is a filtered monad such that $\underset{\mathbb{Z}_{\geq
0}}{\operatorname*{colim}}\operatorname{Sym}_{R}^{\leq\star}\simeq\operatorname{Sym}_{R}$.
Moreover, the stages of the filtration are sifted colimit preserving
excisively polynomial functors. It follows that the functor
$\operatorname{Sym}_{R}^{\leq\star}$ lands in the full subcategory
$\operatorname{End}_{\Sigma}^{\mathrm{excpoly}}(\mathrm{Mod}_{R})\subset\operatorname{End}(\mathrm{Mod}_{R}),$
which is equivalent to
$\operatorname{Fun}^{\mathrm{addpoly}}(\mathrm{Mod}_{R,0},\mathrm{Mod}_{R})$
by Proposition 2.2.8. In other words, the filtered monad
$\operatorname{Sym}_{\mathrm{R}}^{\leq\star}$ is right-left extended from
$\mathrm{Mod}_{R,0}\subset\mathrm{Mod}_{R}$. As it preserves the connective
subcategory $\mathrm{Mod}_{R,\geq 0}\subset\mathrm{Mod}_{R}$, the monad
$\operatorname{Sym}_{R}^{\leq\star}$ is in particular, uniquely determined by
the filtered monad $\operatorname{Sym}^{\leq\star}_{\mathrm{Mod}_{R,\geq 0}}$
on $\mathrm{Mod}_{R,\geq 0}$.

######  Construction 2.2.12.

Let $R$ be a _discrete_ commutative ring. Let
$\operatorname{Sym}:\mathrm{Mod}_{R,0}\rightarrow\mathrm{Mod}_{R}$ be the free
commutative algebra functor, $\operatorname{Sym}_{R}(M):=\bigoplus_{n\geq
0}(M^{\otimes^{n}})_{\Sigma_{n}}$. Construction 2.2.5 supplies a derived
functor $\mathrm{LSym}_{R}:\mathrm{Mod}_{R}\rightarrow\mathrm{Mod}_{R}$ which
we will refer to as the free derived commutative algebra in
$\mathrm{Mod}_{R}$. Then [R, Construction 4.2.19] shows that the functor
$\mathrm{LSym}_{R}\in\operatorname{End}^{\Sigma}(\mathrm{Mod}_{R})$ has a
monad structure. This is achieved via a version of Example 2.2.11 in the
derived setting. Let $\operatorname{End}^{\Sigma}_{0}(\mathrm{Mod}_{R,\geq
0})$ be the full subcategory of
$\operatorname{End}^{\Sigma}(\mathrm{Mod}_{R})$ consisting of functors which
preserve the full subcategory $\mathrm{Mod}_{R,0}\subset\mathrm{Mod}_{R,\geq
0}$, and $\operatorname{End}^{\Sigma}_{1}(\mathrm{Mod}_{R,\geq 0})$ the full
subcategory of functors satisfying $\pi_{0}F(X)\in\mathrm{Mod}_{R,0}$ for any
$X\in\mathrm{Mod}_{R,0}$. Raksit shows in [R, Remark 4.2.18] that the
inclusion $\operatorname{End}_{0}^{\Sigma}(\mathrm{Mod}_{R,\geq
0})\subset\operatorname{End}_{1}^{\Sigma}(\mathrm{Mod}_{R,\geq 0})$ admits a
_monoidal_ left adjoint
$\tau:\operatorname{End}^{\Sigma}_{1}(\mathrm{Mod}_{R,\geq
0})\rightarrow\operatorname{End}^{\Sigma}_{0}(\mathrm{Mod}_{R,\geq 0})$. As we
observed in Example 2.2.11, the filtered monad
$\operatorname{Sym}^{\leq\star}_{R}$ is uniquely determined by its restriction
to the connective subcategory $\mathrm{Mod}_{R,\geq
0}\subset\mathrm{Mod}_{R}$. Moreover, for any $n$, the functor
$\operatorname{Sym}^{\leq n}_{R}$ lies in
$\operatorname{End}_{1}^{\Sigma}(\mathrm{Mod}_{R})$. We can define a new
filtered monad $\mathrm{LSym}^{\leq\star}_{R}$ by the formula
$\mathrm{LSym}^{\leq\star}_{R}:=\tau(\operatorname{Sym}_{R}^{\leq\star})$.
Moreover, it is an excisively polynomial filtered monad, therefore it extends
to a filtered monad on all of $\mathrm{Mod}_{R}$. Defining

$\mathrm{LSym}_{R}:=\underset{\mathbb{Z}_{\geq
0}}{\operatorname*{colim}}\;\mathrm{LSym}_{R}^{\leq\star},$

we get a monad on $\mathrm{Mod}_{R}$.

######  Definition 2.2.13.

Let us apply the Construction 2.2.12 to the initial ring $R=\mathbb{Z}$. A
derived commutative ring is an algebra over the monad
$\mathrm{LSym}_{\mathbb{Z}}:\mathrm{Mod}_{\mathbb{Z}}\rightarrow\mathrm{Mod}_{\mathbb{Z}}$
from Construction 2.2.12. We let
$\mathrm{DAlg}_{\mathbb{Z}}:=\operatorname{Alg}_{\mathrm{LSym}_{\mathbb{Z}}}(\mathrm{Mod}_{\mathbb{Z}})$
to be the $\infty$-category of derived commutative rings. The forgetful
functor
$\mathrm{DAlg}_{\mathbb{Z}}\rightarrow\operatorname{CAlg}_{\mathbb{Z}}$
commutes with all limits and colimits. Given any other (at this point, not
necessarily discrete) object $R\in\mathrm{DAlg}_{\mathbb{Z}}$, we define the
$\infty$-category of derived commutative $R$-algebras as the under category

$\mathrm{DAlg}_{R}:=(\mathrm{DAlg}_{\mathbb{Z}})_{R/}.$

The $\infty$-category $\mathrm{DAlg}_{R}$ is monadic over $\mathrm{Mod}_{R}$
with the monad $\mathrm{LSym}_{R}:\mathrm{Mod}_{R}\rightarrow\mathrm{Mod}_{R}$
which satisfies the formula

$\mathrm{LSym}_{R}(V\otimes R)\simeq\mathrm{LSym}_{\mathbb{Z}}(V)\otimes R$

for any induced $R$-module $V\otimes R$.

###  2.3 Symmetric sequences and derived pd operads.

Construction 2.2.12 produced the monad $\mathrm{LSym}_{R}$ from the monad of
free $\mathbb{E}_{\infty}$-algebra on $\mathrm{Mod}_{R}$ for $R$ a discrete
commutative ring. There exists another monad on the abelian symmetric monoidal
category $\mathrm{Mod}_{R,\heartsuit}$ called the free non-unital divided
power algebra monad $\Gamma^{+}_{R,\heartsuit}$ whose underlying functor is

$\Gamma^{+}_{R,\heartsuit}(X)=\bigoplus_{n=1}^{\infty}(X^{\otimes^{n}})^{\Sigma_{n}}.$
(2.6)

This monad can be defined on the $\infty$-category $\mathrm{Mod}_{R}$ for any
connective derived commutative ring by means of the theory of derived
symmetric sequences. Let us recall the following classical definition.

######  Definition 2.3.1.

Let $\mathrm{Spt}^{\mathrm{B}\Sigma_{n}}$ be the $\infty$-category of
representations of the symmetric group $\Sigma_{n}$ in spectra. We let the
$\infty$-category of symmetric sequences to be the product:

$\mathrm{SSeq}:=\underset{n\geq 0}{\prod}\mathrm{Spt}^{\mathrm{B}\Sigma_{n}}.$

Below we will recall the universal property of the $\infty$-category of
symmetric sequences which we learnt from [GaitsRozI, Section 1.1].

######  Remark 2.3.2.

The $\infty$-category $\mathrm{SSeq}$ is the free stable presentable symmetric
monoidal $\infty$-category on the unit object $\mathrm{Spt}\in\Pr^{L}$. The
tensor product of two symmetric sequences $\\{P(n)\\}$ and $\\{Q(n)\\}$ is
given by the formula

$(P\otimes
Q)(n)=\bigoplus_{i+j=n}\operatorname{Ind}_{\Sigma_{i}\times\Sigma_{j}}^{\Sigma_{n}}P(i)\otimes
Q(j).$

The universal property of being free on one generator implies that there is an
equivalence

$\mathrm{SSeq}\simeq\operatorname{Fun}_{\mathrm{Cat}}(*,\mathrm{SSeq})\simeq\operatorname{Fun}_{\Pr^{L}}(\mathrm{Spt},\mathrm{SSeq})\simeq\operatorname{End}_{\operatorname{CAlg}(\Pr^{L})}(\mathrm{SSeq}).$
(2.7)

The equivalence 2.7 gives a monoidal structure on the $\infty$-category
$\mathrm{SSeq}$, called the composition product. For two objects
$\\{P(n)\\},\\{Q(n)\\}\in\mathrm{SSeq}$, there is a formula

$P\circ Q\simeq\bigoplus_{n=0}^{\infty}(P(n)\otimes
Q^{\otimes^{n}})_{h\Sigma_{n}},$

where the tensor product is taken with respect to the symmetric monoidal
structure in $\mathrm{SSeq}$. An operad (in spectra) is an algebra object in
$\mathrm{SSeq}$ with respect to the composition product symmetric monoidal
structure.

######  Construction 2.3.3.

The forgetful functor $\operatorname{CAlg}(\Pr^{L})\rightarrow\mathrm{Cat}$ is
represented by $\mathrm{SSeq}$. Consequently, the monoidal $\infty$-category
$\mathrm{SSeq}$ acts on the underlying $\infty$-category of any symmetric
monoidal stable presentable $\infty$-category $\mathrm{C}$. The action functor
$-\circ:\mathrm{SSeq}\rightarrow\operatorname{End}(\mathrm{C})$ is given by
the formula

$P\circ X=\bigoplus_{n=0}^{\infty}(P(n)\otimes
X^{\otimes^{n}})_{h\Sigma_{n}}.$

Notice that this action preserves sifted colimits. Given an operad
$\mathcal{O}\in\operatorname{Alg}(\mathrm{SSeq})$, we obtain a sifted colimit
preserving monad
$\mathrm{T}_{\mathcal{O}}:=\mathcal{O}\circ-:\mathrm{C}\rightarrow\mathrm{C}$
acting on $\mathrm{C}$.

It was observed by B.Fresse [Fr] and later in the $\infty$-categorical context
by Francis and Gaitsgory [FrG] that there is another monoidal structure on
$\mathrm{SSeq}$ which acts naturally on any stable presentable symmetric
monoidal $\infty$-category. The product in this monoidal structure is given by
the formula

$P\overline{\circ}Q=\bigoplus_{n=0}^{\infty}(P(n)\otimes
Q^{\otimes^{n}})^{h\Sigma_{n}},$

and the action on any symmetric monoidal stable presentable $\infty$-category
$\mathrm{C}$ is given by

$P\overline{\circ}X=(\bigoplus_{n=0}^{\infty}P\otimes
X^{\otimes^{n}})^{h\Sigma_{n}}.$

The rigorous construction of this monoidal structure and the action on any
symmetric monoidal stable presentable $\infty$-category are not essential to
the current text. We would only like to remark that in the particular case of
the object $(0,\mathbb{S},\mathbb{S},...)$, the monad $\bigoplus_{n\geq
1}(X^{\otimes^{n}})^{h\Sigma_{n}}$ can be called the monad of “free non-unital
$\mathbb{E}_{\infty}$-divided power algebra”. One crucial difference with the
free $\mathbb{E}_{\infty}$-algebra monad is that the free
$\mathbb{E}_{\infty}$-divided power algebra monad does not preserve connective
objects, and therefore can not be treated in the same way as in Construction
2.2.12. Instead, we will construct a natural action of the monoidal category
of discrete symmetric sequences on $\mathrm{Mod}_{P,\heartsuit}$ for any
polynomial ring $P$, and produce an action of derived symmetric sequences on
$\mathrm{Mod}_{R}$ for any connective derived commutative ring $R$ using the
formalism of derived functors. We should note that such construction is
essentially contained in [BCN21], and our goal here is to merely review it and
fix the relevant notation.

######  Construction 2.3.4.

Here we recall the construction of derived symmetric sequences given in
[BCN21, Notation 3.52]. Let
$\mathbb{Z}[\mathcal{O}_{\Sigma_{n}}]\subset\mathrm{Mod}_{\mathbb{Z}[\Sigma_{n}],\heartsuit}$
be the full additive subcategory in the abelian category of
$\Sigma_{n}$-representations spanned by $\mathbb{Z}$-linearizations of finite
$\Sigma_{n}$-sets. Let $\mathbb{Z}[\mathcal{O}_{\Sigma}]=\bigoplus_{n\geq
0}\mathbb{Z}[\mathcal{O}_{\Sigma_{n}}]$, understood as direct sum of additive
categories. An object of $\mathbb{Z}[\mathcal{O}_{\Sigma}]$ consists of a
_finite_ collection $\\{P(i)\\}$ of $\Sigma_{i}$-representations in
$\mathbb{Z}[\mathcal{O}_{\Sigma}]$, i.e. $P(n)=0$ for any large enough $n$. We
define the $\infty$-category of derived symmetric sequences as the
$\infty$-category of left modules over the additive category
$\mathbb{Z}[\mathcal{O}_{\Sigma}]$, i.e. the $\infty$-category of additive
functors

$\mathrm{SSeq}^{\mathrm{gen}}:=\mathrm{Mod}_{\mathbb{Z}[\mathcal{O}_{\Sigma}]}=\operatorname{Fun}^{\oplus}(\mathbb{Z}[\mathcal{O}]^{\mathrm{op}},\mathrm{Spt}).$

The additive category $\mathbb{Z}[\mathcal{O}_{\Sigma}]$, understood as a
subcategory $\mathrm{SSeq}_{\heartsuit}\subset\mathrm{SSeq}$ of the
$\infty$-category of symmetric sequences in spectra, is closed under tensor
product and composition defined in Construction 2.7 (for explanation, see
[BCN21, Construction 3.58]). Moreover, the duality equivalence
$(-)^{\vee}:\mathbb{Z}[\mathcal{O}_{\Sigma}]^{\mathrm{op}}\simeq\mathbb{Z}[\mathcal{O}_{\Sigma}]$
allows us to define a new monoidal structure on
$\mathbb{Z}[\mathcal{O}_{\Sigma}]$ by the formula:

$P\overline{\circ}Q:=(P^{\vee}\circ
Q^{\vee})^{\vee}=\bigoplus_{n=0}^{\infty}(P(n)\otimes
Q^{\otimes^{n}})^{\Sigma_{n}}.$

For any pair of objects $P,Q\in\mathbb{Z}[\Sigma]$, there is a Norm map

$\operatorname{Nm}:(P(n)\otimes
Q^{\otimes^{n}})_{\Sigma_{n}}\rightarrow(P(n)\otimes
Q^{\otimes^{n}})^{\Sigma_{n}},$

and this induces a lax monoidal functor
$(\mathbb{Z}[\mathcal{O}_{\Sigma}],\circ)\rightarrow(\mathbb{Z}[\mathcal{O}_{\Sigma}],\overline{\circ})$.

[BCN21, Proposition 3.72] shows that the composition product
$\overline{\circ}$ right-left extends to a monoidal structure on
$\mathrm{SSeq}^{\mathrm{gen}}$, and restriction by the Norm
$(\mathbb{Z}[\mathcal{O}_{\Sigma}],\circ)\rightarrow(\mathbb{Z}[\mathcal{O}_{\Sigma}],\overline{\circ})$
extends to a right lax monoidal functor
$\operatorname{Nm}^{*}:(\mathrm{SSeq}^{\mathrm{gen}},\circ)\rightarrow(\mathrm{SSeq}^{\mathrm{gen}},\overline{\circ})$
which is identical on objects.

######  Notation 2.3.5.

Let us fix some notation related to Construction 2.3.4.

  1. 1.

$\mathrm{SSeq}^{\mathrm{gen}}_{0}$ is the monoidal category
$(\mathbb{Z}[\mathcal{O}_{\Sigma}],\circ)$.

  2. 2.

$\mathrm{SSeq}^{\mathrm{gen},\mathrm{pd}}_{0}$ is the monoidal category
$(\mathbb{Z}[\mathcal{O}_{\Sigma}],\overline{\circ})$.

  3. 3.

$\mathrm{SSeq}^{\mathrm{gen}}$ is the presentable monoidal $\infty$-category
$(\mathrm{SSeq}^{\mathrm{gen}},\circ)$.

  4. 4.

$\mathrm{SSeq}^{\mathrm{gen},\mathrm{pd}}$ is the presentable monoidal
$\infty$-category $(\mathrm{SSeq}^{\mathrm{gen}},\overline{\circ})$.

######  Remark 2.3.6.

To construct a compatible action of $\mathrm{SSeq}^{\mathrm{gen}}$ and
$\mathrm{SSeq}^{\mathrm{gen},\mathrm{pd}}$ on some stable $\infty$-category,
it is sufficient to construct an action of the latter
$\mathrm{SSeq}^{\mathrm{gen},\mathrm{pd}}$ as the action of the former
$\mathrm{SSeq}^{\mathrm{gen}}$ can then be obtained by restricting along the
lax monoidal functor
$\operatorname{Nm}^{*}:\mathrm{SSeq}^{\mathrm{gen}}\rightarrow\mathrm{SSeq}^{\mathrm{gen},\mathrm{pd}}$.

Following observation of [BCN21, Proposition 3.72 (4)], we record the
following proposition. We say that a derived symmetric sequence
$P\in\mathrm{SSeq}^{\mathrm{gen}}$ is concentrated in arities $\geq 1$ if
$P(0)\simeq 0$.

######  Proposition 2.3.7.

The functor $\operatorname{Nm}^{*}$ is a _monoidal_ equivalence on the full
subcategories of derived symmetric sequences concentrated in arities $\geq 1$.

###### Proof.

Take two objects $P,Q\in R[\Sigma]$ which are trivial in arity $0$555In fact,
in the subsequent argument it suffices to assume that only $Q$ is trivial in
arity $0$., and consider the symmetric sequence $P(n)\otimes Q^{\otimes^{n}}$.
We wish to prove that the map $\operatorname{Nm}_{\Sigma_{n}}:(P(n)\otimes
Q^{\otimes^{n}})_{\Sigma_{n}}\rightarrow(P(n)\otimes
Q^{\otimes^{n}})^{\Sigma_{n}}$ is an equivalence. To see this, we look at the
formula for the arity $r$ module of the symmetric sequence $Q^{\otimes^{n}}$:

$Q^{\otimes^{n}}(r)=\bigoplus_{i_{1}+...+i_{n}=r}\operatorname{Ind}_{\Sigma_{i_{1}}\times...\times\Sigma_{i_{n}}}^{\Sigma_{r}}Q(i_{1})\otimes...\otimes
Q(i_{n}).$ (2.8)

By assumption, $Q(i)=0$ unless $i\geq 1$. It implies that the right hand side
in Equation 2.8 is $0$ unless $n\leq r$, because if $n>r$, then in order for
the sum $i_{1}+...+i_{n}$ to be equal $r$, at least one of the indices must be
$0$, which forces the tensor product to vanish. In the case $n\geq r$, we
claim that the right hand side of Equation 2.8 is an induced
$\Sigma_{n}$-representation. Indeed, consider a set of indices
$i_{1},...,i_{n}\geq 1$ such that $i_{1}+...+i_{n}=r$. The sum in the formula
2.8 contains as a summand the $\Sigma_{n}$-submodule

$\bigoplus_{\sigma\in\Sigma_{n}}\operatorname{Ind}_{\Sigma_{i_{\sigma(1)}}\times...\times\Sigma_{i_{\sigma(n)}}}^{\Sigma_{r}}(Q(i_{\sigma(1)})\otimes...\otimes
Q(i_{\sigma(n)})),$ (2.9)

where the sum is taken with respect to all permuations of the indices. The
$\Sigma_{n}$-submodule in Equation 2.9 is $\Sigma_{n}$-induced. As the sum in
Equation 2.8 splits into the sum of $\Sigma_{n}$\- submodules of the form 2.9
for various collections $i_{1},...,i_{n}$, it follows that the right hand side
of 2.8 is an induced $\Sigma_{n}$-module, and the Norm map induces an
isomorphism between its $\Sigma_{n}$-orbits and $\Sigma_{n}$-fixed points, or
equivalently, its Tate fixed points module vanishes. But the full subcategory
of $\Sigma_{n}$-representations whose Tate fixed points vanishes, is a
$\otimes$-ideal in the category of all $\Sigma_{n}$-representations.
Therefore, the Tate fixed points module of $X(n)\otimes Q^{\otimes^{n}}(r)$
also vanishes for $n\geq r$, and the Norm map $(P(n)\otimes
Q^{\otimes^{n}}(r))_{\Sigma_{n}}\rightarrow(P(n)\otimes
Q^{\otimes^{n}}(r))^{\Sigma_{n}}$ is an equivalence, as desired. ∎

######  Definition 2.3.8.

A derived (pd) operad is an algebra object in the monoidal $\infty$-category
$\mathrm{SSeq}^{\mathrm{gen}}$ (respectively, in the monoidal
$\infty$-category $\mathrm{SSeq}^{\mathrm{gen},\mathrm{pd}}$).

######  Remark 2.3.9.

Proposition 2.3.7 implies that given a derived operad $O$ concentrated in
arities $\geq 1$ gives rise to a pd operad. As an example, consider derived
symmetric sequence $\mathcal{O}:=(0,\mathbb{Z},\mathbb{Z},...)$. We compute
that

$\mathcal{O}\circ\mathcal{O}(k)\simeq\bigoplus_{n\geq
1}\bigl{(}\bigoplus_{i_{1}+...+i_{k}=n}\mathbb{Z}\bigr{)}.$

The map $\mathcal{O}\circ\mathcal{O}\rightarrow\mathcal{O}$ given by the
natural map $\bigoplus_{n\geq
1}\bigl{(}\bigoplus_{i_{1}+...+i_{k}=n}\mathbb{Z}\bigr{)}\rightarrow\mathbb{Z}$
in every arity $k$, endows $\mathcal{O}$ with the structure of a derived
operad. This derived operad is the non-unital derived commutative operad. As
it vanishes in arity $0$, there is a corresponding deried pd operad which is
called the non-unital pd commutative operad.

Below we will construct an action of the monoidal $\infty$-category
$\mathrm{SSeq}^{\mathrm{gen},\mathrm{pd}}$ on $\mathrm{Mod}_{R}$ for a
discrete commutative ring $R$.

######  Construction 2.3.10.

Let $R$ be a discrete commutative ring, and consider the $\infty$-category
$\mathrm{Mod}_{R}$ and the full subcategory of finitely generated free modules
inside all connective $R$-modules
$\mathrm{Mod}_{R,0}\subset\mathrm{Mod}_{R,\geq 0}$.

The subcategory $\mathrm{Mod}_{R,0}$ has the following properties.

\- For any object $X\in\mathrm{Mod}_{R,0}$, the dual $X^{\vee}$ is in
$\mathrm{Mod}_{R,0}$ again;

\- For any $X\in\mathscr{\mathrm{Mod}}_{R,0}$ and any finitely generated free
$\Sigma_{n}$-module $V$, the object $(V\otimes
X^{\otimes{n}})_{\Sigma_{n}}\in\mathrm{Mod}_{R,\heartsuit}$ is in
$\mathrm{Mod}_{R,0}$ again.

Together, these two conditions imply that for any finitely generated free
$\Sigma_{n}$-module $V$, the functor $X\longmapsto(V\otimes
X^{\otimes^{n}})^{\Sigma_{n}}$ on $\mathrm{Mod}_{R,\heartsuit}$ preserves the
full subcategory $\mathrm{Mod}_{R,0}\subset\mathrm{Mod}_{R,\heartsuit}$.

Therefore, there is a functor

$\mathbb{Z}[\mathcal{O}_{\Sigma_{n}}]\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}\ignorespaces}}}}\ignorespaces\operatorname{End}(\mathrm{Mod}_{R,0})$

sending

$V\longmapsto\bigl{(}X\longmapsto(V(n)\otimes
X^{\otimes^{n}})^{\Sigma_{n}}\bigr{)}.$

Note that we switched invariants and coinvariants in the formula above by
using the isomorphism

$(V(n)\otimes
X^{\otimes^{n}})^{\Sigma_{n}}\simeq\bigl{(}(V^{\vee}(n)\otimes(X^{\vee})^{\otimes^{n}})_{\Sigma_{n}}\bigr{)}^{\vee}$

together with two conditions listed above. Moreover, for each $n$, the functor
$X\longmapsto(V\otimes X^{\otimes^{n}})_{\Sigma_{n}}$ is additively polynomial
of degree $n$, and it follows by duality that the functor
$X\longmapsto(V\otimes X^{\otimes^{n}})^{\Sigma_{n}}$ is additively polynomial
of degree $n$ again. Since each object $\\{P(i)\\}$ in
$\mathbb{Z}[\mathcal{O}_{\Sigma}]$ is non-zero for only finitely many $i$, it
follows $\mathrm{SSeq}^{\mathrm{gen},\mathrm{pd}}$ acts on
$\mathrm{Mod}_{R,0}$ by additively polynomial functors. To show that this
action extends to an action of $\mathrm{SSeq}^{\mathrm{gen},\mathrm{pd}}$ on
$\mathrm{Mod}_{R}$, we shall use [BCN21, Proposition 3.48] which states that
the functor
$\mathrm{Mod}:\mathrm{Cat}^{\mathrm{add}}\rightarrow\mathrm{Cat}^{\operatorname{St},\Sigma}$
from the $\infty$-category of small additive $\infty$-categories and additive
functors $\mathrm{Cat}^{\mathrm{add}}$ to the $\infty$-category
$\mathrm{Cat}^{\operatorname{St},\Sigma}$ of presentable stable
$\infty$-categories and sifted colimit preserving functors, refines to a
symmetric monoidal functor
$\mathrm{Mod}:\mathrm{Cat}^{\mathrm{add},\mathrm{poly}}\rightarrow\mathrm{Cat}^{\operatorname{St},\Sigma}$,
where $\mathrm{Cat}^{\mathrm{add},\mathrm{poly}}$ is the $\infty$-category of
small additive $\infty$-categories and additively polynomial functors between
them.

######  Construction 2.3.11.

The preceeding discussion of the action of
$\mathrm{SSeq}^{\mathrm{gen},\mathrm{pd}}$ on $\mathrm{Mod}_{R}$ for any
discrete commutative ring $R$ generalizes to any connective derived
commutative ring. To this end, we have constructed a functor

$\textstyle{\mathrm{SSeq}^{\mathrm{gen},\mathrm{pd}}\curvearrowright\mathrm{Mod}_{(-)}:\operatorname{CAlg}_{\mathbb{Z}}^{\mathrm{poly}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{Mod}_{\mathrm{SSeq}^{\mathrm{gen},\mathrm{pd}}}(\mathrm{Cat}^{\operatorname{St},\Sigma})}$

which sends a polynomial ring $R$ to the $\infty$-category $\mathrm{Mod}_{R}$
endowed with the sifted colimit preserving action of
$\mathrm{SSeq}^{\mathrm{gen},\mathrm{pd}}$ on $\mathrm{Mod}_{R}$ constructed
in Construction 2.3.10. Left Kan extending along the inclusion
$\operatorname{CAlg}^{\mathrm{poly}}_{\mathbb{Z}}\subset\mathrm{DAlg}_{\mathbb{Z},\geq
0}$, we obtain a sifted colimit preserving functor

$\textstyle{\mathrm{SSeq}^{\mathrm{gen},\mathrm{pd}}\curvearrowright\mathrm{Mod}_{(-)}:\mathrm{DAlg}_{\mathbb{Z},\geq
0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{Mod}_{\mathrm{SSeq}^{\mathrm{gen},\mathrm{pd}}}(\mathrm{Cat}^{\operatorname{St},\Sigma})}$

such that the composition with the forgetful functor
$\mathrm{Mod}_{\mathrm{SSeq}^{\mathrm{gen},\mathrm{pd}}}(\mathrm{Cat}^{\operatorname{St},\Sigma})\rightarrow\mathrm{Cat}^{\operatorname{St},\Sigma}$
is equivalent to the functor of left modules $R\longmapsto\mathrm{Mod}_{R}$.
The latter follows from the fact that the forgetful functor preserves sifted
colimits and the functor $\mathrm{Mod}_{(-)}:\mathrm{DAlg}_{\mathbb{Z},\geq
0}\rightarrow\mathrm{Cat}^{\operatorname{St},\Sigma}$ is itself left Kan
extended from $\operatorname{CAlg}^{\mathrm{poly}}_{\mathbb{Z}}$.

######  Definition 2.3.12.

Let $R$ be a connective derived ring. Using Remark 2.3.9, Construction 2.3.11
gives rise to a monad
$\mathrm{L\Gamma}^{+}_{\mathrm{R}}\in\operatorname{End}^{\Sigma}(\mathrm{Mod}_{R})$
on $\mathrm{Mod}_{R}$ whose underlying functor satisfies
$\mathrm{L\Gamma}^{+}_{R}=\bigoplus_{n\geq 1}(X^{\otimes^{n}})^{\Sigma_{n}}$
for any $X\in\mathrm{Mod}_{R,0}$. We refer to this as the free non-unital
divided power (pd) algebra monad, and algebras over this monad are called non-
unital derived divided power (pd) algebras. We denote

$\mathrm{DAlg}_{R}^{\mathrm{nu},\mathrm{pd}}:=\operatorname{Alg}_{\mathrm{L\Gamma}_{R}^{+}}(\mathrm{Mod}_{R}).$

A map $R\rightarrow S$ of connective derived commutative rings gives a base
change functor
$-\otimes_{R}S:\mathrm{DAlg}^{\mathrm{nu},\mathrm{pd}}_{R}\rightarrow\mathrm{DAlg}^{\mathrm{nu},\mathrm{pd}}_{S}$,
as well as restriction
$\mathrm{DAlg}_{S}^{\mathrm{nu},\mathrm{pd}}\rightarrow\mathrm{DAlg}^{\mathrm{nu},\mathrm{pd}}_{R}.$

######  Construction 2.3.13.

Let $X$ be a derived prestack, i.e. a functor
$X:\mathrm{DAlg}_{\mathbb{Z},\geq 0}\rightarrow\mathrm{Spc}$. Construction
2.3.11 implies that the $\infty$-category $\operatorname{QCoh}(X)$ of quasi-
coherent sheaves on $X$ has an action of
$\mathrm{SSeq}^{\mathrm{gen},\mathrm{pd}}$. Indeed, this follows by writing
the $\infty$-category $\operatorname{QCoh}(X)$ as the limit

$\operatorname{QCoh}(X)=\underset{\operatorname{Spec}R\rightarrow
X}{\lim}\mathrm{Mod}_{R}$

along pull-back functors for all affines $\operatorname{Spec}A\rightarrow X$
mapping to $X$, and using the fact that the forgetful functor
$\mathrm{Mod}_{\mathrm{SSeq}^{\mathrm{gen},\mathrm{pd}}}(\mathrm{Cat}^{\operatorname{St},\Sigma})\rightarrow\mathrm{Cat}^{\operatorname{St},\Sigma}$
preserves limits. Consequently, we can simply _define_ the $\infty$-category
of derived commutative (respectively, derived non-unital divided power)
algebra objects in $\operatorname{QCoh}(X)$ as the limit

$\mathrm{DAlg}^{(\mathrm{nu},\mathrm{pd})}(\operatorname{QCoh}(X))\simeq\underset{\operatorname{Spec}R\rightarrow
X}{\lim}\mathrm{DAlg}^{(\mathrm{nu},\mathrm{pd})}_{R}.$

In some cases we have to work with a symmetric monoidal $\infty$-category
$\mathrm{C}$ which is not (or at least, not known to be) of the form
$\operatorname{QCoh}(X)$ for a derived prestack $X$. In such a case, one
approach to constructing an action of
$\mathrm{SSeq}^{\mathrm{gen},\mathrm{pd}}$ is to specify a $t$-structure
$(\mathrm{C}_{\geq 0},\mathrm{C}_{\leq 0})$ and a choice of a subcategory
$\mathrm{C}^{0}$ of compact generators in $\mathrm{C}$ satisfying certain
properties which will be listed below (a derived algebraic context in the
sense of [R]). For such a symmetric monoidal $\infty$-category $\mathrm{C}$,
we will be able to construct a sifted colimit preserving action of
$\mathrm{SSeq}^{\mathrm{gen},\mathrm{pd}}$ on $\mathrm{C}$. This should be
compared with the construction of derived algebras in a derived algebraic
context in [R], albeit the types of $\infty$-categories we consider below are
slightly different from derived algebraic contexts of Raksit.

######  Construction 2.3.14.

Let $\mathrm{C}$ be a symmetric monoidal $\infty$-category equipped with a
compatible $t$-structure $(\mathrm{C}_{\geq 0},\mathrm{C}_{\leq 0})$ and a
full subcategory
$\mathrm{C}_{0}\subset\mathrm{C}_{\heartsuit}\subset\mathrm{C}_{\geq 0}$
consisting of compact projective generators of $\mathrm{C}_{\geq 0}$. We
require that the $t$-structure is left complete, and the full subcategory
$\mathrm{C}_{0}$ is closed in $\mathrm{C}_{\heartsuit}$ under the action of
$\mathrm{SSeq}^{\mathrm{gen},\mathrm{pd}}_{0}$, _and_ the action of
$\mathrm{SSeq}^{\mathrm{gen}}$. Using the strategy of Construction 2.3.10,
there is a sifted colimit preserving action of the monoidal $\infty$-category
$\mathrm{SSeq}^{\mathrm{gen},\mathrm{pd}}$ on $\mathrm{C}$. Moreover, this
construction is functorial in $\mathrm{C}$ in the following sense. Notice that
for $\mathrm{C}$ as above, there is an equivalence
$\mathrm{C}\simeq\mathrm{Mod}(\mathrm{C}_{0})$. Consequently, using [BCN21,
Proposition 3.48] again as in Construction 2.3.10, we obtain a functor

$\mathrm{Mod}_{\mathrm{SSeq}^{\mathrm{gen},\mathrm{pd}}_{0}}(\mathrm{Cat}^{\mathrm{add},\mathrm{poly}})\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern
3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}\ignorespaces}}}}\ignorespaces\mathrm{Mod}_{\mathrm{SSeq}^{\mathrm{gen},\mathrm{pd}}}(\mathrm{Cat}^{\operatorname{St},\Sigma}).$

Therefore, the action of $\mathrm{SSeq}^{\mathrm{gen},\mathrm{pd}}$ on
$\mathrm{C}$ is functorial in symmetric monoidal right $t$-exact functors
$F:\mathrm{C}\rightarrow\mathrm{D}$ such that
$F(\mathrm{C}_{0})\subset\mathrm{D}_{0}$. The same construction applies to the
action of $\mathrm{SSeq}^{\mathrm{gen}}$. Note that if we drop the condition
that $\mathrm{C}_{0}$ is closed under the action of
$\mathrm{SSeq}^{\mathrm{gen}}_{0}$, we still get a
$\mathrm{SSeq}^{\mathrm{gen}}$-action on $\mathrm{C}$ by restricting along
$\operatorname{Nm}^{*}:\mathrm{SSeq}^{\mathrm{gen}}\rightarrow\mathrm{SSeq}^{\mathrm{gen},\mathrm{pd}}$,
however, the naturality of this action could be more subtle without the
aforementioned condition.

######  Observation 2.3.15.

In studying the $\mathrm{L\Gamma}^{+}$-monad defined in Definition 2.3.12 in
such a context as in Construction 2.3.14, restriction
$\mathrm{L\Gamma}^{+}|_{\mathrm{C}_{0}}$ to $\mathrm{C}_{0}\subset\mathrm{C}$
does not yield a monad. However, as each $\mathrm{L\Gamma}^{n}$ preserves
$\mathrm{C}_{0}$, it follows that $\mathrm{L\Gamma}^{+}$ preserves the full
subcategory $\operatorname{Ind}(\mathrm{C}_{0})\subset\mathrm{C}$, and hence
$\mathrm{L\Gamma}^{+}|_{\operatorname{Ind}(\mathrm{C}_{0})}:\operatorname{Ind}(\mathrm{C}_{0})\rightarrow\operatorname{Ind}(\mathrm{C}_{0})$
yields a monad. Moreover, the original monad $\mathrm{L\Gamma}^{+}$ on
$\mathrm{C}$ is uniquely determined by
$\mathrm{L\Gamma}^{+}|_{\operatorname{Ind}(\mathrm{C}_{0})}$ in the following
sense. Let
$\operatorname{End}_{\Sigma,RL}^{\operatorname{Ind}(\mathrm{C}_{0})}(\mathrm{C})\subset\operatorname{End}_{\Sigma}(\mathrm{C})$
be the subcategory of endofunctors preserving the full subcategory
$\operatorname{Ind}(\mathrm{C}_{0})\subset\mathrm{C}$ which are right-left
extended from $\mathrm{C}_{0}\subset\mathrm{C}$. Then the restriction functor

$\textstyle{\operatorname{End}^{\operatorname{Ind}(\mathrm{C}_{0})}_{\Sigma,RL}(\mathrm{C})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{\operatorname{End}_{\omega}(\operatorname{Ind}(\mathrm{C}_{0}))}$

is an equivalence. We will use this observation for constructing maps of
monads right-left extended from $\mathrm{C}_{0}$.

The same applies to the monad $\mathrm{LSym}:\mathrm{C}\rightarrow\mathrm{C}$,
as each $\mathrm{LSym}^{n}:\mathrm{C}\rightarrow\mathrm{C}$ preserves the full
subcategory $\mathrm{C}_{0}$.

Let us finish this section with some observations on how lax symmetric
monoidal functors are incorporated in this setting.

######  Construction 2.3.16.

Let $\mathrm{A}_{0}$ be a small additive $\infty$-category, consider the
$\infty$-category

$\mathrm{Mod}_{\mathrm{A}^{0}}(\mathrm{Cat}^{\mathrm{add},\mathrm{poly}})_{\mathrm{lax}}:=\operatorname{Fun}(\mathrm{B}\mathrm{A}_{0},\mathrm{Cat}^{\mathrm{add},\mathrm{poly}})_{\mathrm{lax}}.$

Composing with the monoidal functor
$\mathrm{Mod}:\mathrm{Cat}^{\mathrm{add},\mathrm{poly}}\rightarrow\mathrm{Cat}^{\operatorname{St},\Sigma}$,
we obtain a functor

$\operatorname{Fun}(\mathrm{B}\mathrm{A}_{0},\mathrm{Cat}^{\mathrm{add},\mathrm{poly}})_{\mathrm{lax}}\rightarrow\operatorname{Fun}(\mathrm{B}\mathrm{A}_{0},\mathrm{Cat}^{\operatorname{St},\Sigma})_{\mathrm{lax}}.$

Notice that for any $\mathrm{C}\in\mathrm{Cat}^{\operatorname{St},\Sigma}$,
the endomorphism object $\operatorname{Fun}_{\Sigma}(\mathrm{C},\mathrm{C})$
is a presentable stable monoidal $\infty$-category. Therefore, any monoidal
functor
$\mathrm{A}_{0}\rightarrow\operatorname{Fun}_{\Sigma}(\mathrm{C},\mathrm{C})$
extends to a continuous monoidal functor
$\mathrm{A}:=\mathrm{Mod}(\mathrm{A}_{0})\rightarrow\operatorname{Fun}_{\Sigma}(\mathrm{C},\mathrm{C})$,
and we obtain a functor

$\mathrm{Mod}_{\mathrm{A}_{0}}(\mathrm{Cat}^{\mathrm{add},\mathrm{poly}})_{\mathrm{lax}}\rightarrow\mathrm{Mod}_{\mathrm{A}}(\mathrm{Cat}^{\operatorname{St},\Sigma})_{\mathrm{lax}},$

where
$\mathrm{Mod}_{\mathrm{A}_{0}}(\mathrm{Cat}^{\mathrm{add},\mathrm{poly}})_{\mathrm{lax}}:=\operatorname{Fun}(\textbf{BA},\mathrm{Cat}^{\operatorname{St},\Sigma})_{\mathrm{lax}}$.

A consequence of this construction is the following example.

######  Example 2.3.17.

Assume

$I\rightarrow\mathrm{Mod}_{\mathrm{A}_{0}}(\mathrm{Cat}^{\mathrm{add},\mathrm{poly}})_{\mathrm{lax}}$

is a diagram. Then applying
$\mathrm{Mod}:\mathrm{Cat}^{\mathrm{add},\mathrm{poly}}\rightarrow\mathrm{Cat}^{\operatorname{St},\Sigma}$,
we obtain a diagram

$I\rightarrow\mathrm{Mod}_{\mathrm{A}}(\mathrm{Cat}^{\operatorname{St},\Sigma})_{\mathrm{lax}}.$

We will sometimes apply this example in the setting
$\mathrm{A}_{0}:=\mathrm{SSeq}^{\mathrm{gen},\mathrm{pd}}_{0}$ and
$\mathrm{A}:=\mathrm{SSeq}^{\mathrm{gen},\mathrm{pd}}$.

###  2.4 Recollection on gradings and filtrations.

######  Definition 2.4.1.

We let $\mathbb{Z}^{\mathrm{ds}}$ be the discrete category whose objects are
integers $n\in\mathbb{Z}$ and all morphisms are identities, and
$\mathbb{Z}_{\leq*}$ be the nerve of the partially ordered set of integers.
Note that both $\mathbb{Z}^{\mathrm{ds}}$ and $\mathbb{Z}_{\leq*}$ are
symmetric monoidal categories with respect to the operation of addition. In
what follows, we will work over a connective derived commutative ring $R$.

  1. 1.

We define the $\infty$-category of graded objects in $\mathrm{Mod}_{R}$,
denoted $\mathrm{Gr}\mathrm{Mod}_{R}$ to be the $\infty$-category of functors

$\mathrm{Gr}\mathrm{Mod}_{R}:=\operatorname{Fun}((\mathbb{Z}^{\mathrm{ds}})^{\mathrm{op}},\mathrm{Mod}_{R}).$

An object $X\in\mathrm{Gr}\mathrm{Mod}_{R}$ is the same data as a collection
of objects $\\{X^{n}\\}_{n\in\mathbb{Z}}$ in $\mathrm{Mod}_{R}$ parametrized
by all integers $n$. We will often denote a graded object $X$ as $X^{\bullet}$
to emphasize the grading. The $\infty$-category $\mathrm{Gr}\mathrm{Mod}_{R}$
carries a _Day convolution symmetric monoidal structure_ with tensor product
given by the formula

$(X^{\bullet}\otimes Y^{\bullet})^{n}:=\bigoplus_{p+q=n}X^{p}\otimes Y^{q}.$
(2.10)

  2. 2.

W let the $\infty$-category of filtered objects666To be more precise, we are
dealing with _decreasing filtrations_ here.
$\operatorname{Fil}\mathrm{Mod}_{R}$ to be the $\infty$-category of functors

$\operatorname{Fil}\mathrm{Mod}_{R}:=\operatorname{Fun}((\mathbb{Z}_{\leq*})^{\mathrm{op}},\mathrm{Mod}_{R}).$

An object $X\in\operatorname{Fil}\mathrm{Mod}_{R}$ is the data of a collection
of objects $\\{X^{\geq n}\\}_{n\in\mathbb{Z}}$ in $\mathrm{Mod}_{R}$, and maps
$X^{\geq m}\rightarrow X^{\geq n}$ for any pair of integers $m$ and $n$ such
that $m\geq n$. We can depict an object
$X\in\operatorname{Fil}\mathrm{Mod}_{R}$ as a diagram

$\\{...\rightarrow X^{\geq 2}\rightarrow X^{\geq 1}\rightarrow X^{\geq
0}\rightarrow X^{\geq-1}\rightarrow X^{\geq-2}\rightarrow...\\}$

in $\mathrm{Mod}_{R}$. We will often denote the data of a filtered object by
the symbol $X^{\geq\star}$ to emphasize the filtration. The $\infty$-category
$\operatorname{Fil}\mathrm{Mod}_{R}$ has a _Day convolution symmetric monoidal
structure_ with tensor product given by the formula

$(X^{\geq\star}\otimes Y^{\geq\star})^{n}:=\operatorname*{colim}_{p+q\geq
n}X^{\geq p}\otimes Y^{\geq q}.$ (2.11)

  3. 3.

The associated graded object functor
$\operatorname{gr}:\operatorname{Fil}\mathrm{Mod}_{R}\rightarrow\mathrm{Gr}\mathrm{Mod}_{R}$
is the functor sending a filtered object
$X^{\geq\star}\in\operatorname{Fil}\mathrm{Mod}_{R}$ to a graded object
$\operatorname{gr}^{\bullet}(X)$ with the $n$-th graded piece defined by the
formula

$\operatorname{gr}(X)^{n}:=X^{n}/X^{n+1}=\operatorname*{cofib}(X^{n}\rightarrow
X^{n+1}).$

By [L, Proposition 3.2.1], this functor is symmetric monoidal (alternatively,
see Remark 2.4.3).

  4. 4.

We define a forgetful functor
$(\operatorname{Fil}\rightarrow\mathrm{Gr}):\operatorname{Fil}\mathrm{Mod}_{R}\rightarrow\mathrm{Mod}_{R}$
which sends a filtered object $X:=X^{\geq\star}$ to the underlying graded
object whose weighted pieces are
$(\operatorname{Fil}\rightarrow\mathrm{Gr})(X)^{n}:=X^{\geq n}.$ This functor
is right lax symmetric monoidal (see Remark 2.4.3 for explanation)

  5. 5.

We let
$\mathrm{ins}^{n}:\mathrm{Mod}_{R}\rightarrow\mathrm{Gr}\mathrm{Mod}_{R}$ be
the insertion functor sending an object $X\in\mathrm{Mod}_{R}$ to the graded
object $\mathrm{ins}^{n}(X)^{\bullet}$ which has $\mathrm{ins}^{n}(X)^{i}=X$
when $i=n$, and $\mathrm{ins}^{n}(X)^{i}=0$ if $i\neq 0$. It has a right
adjoint evaluation functor
$\operatorname{ev}^{n}:\mathrm{Gr}\mathrm{Mod}_{R}\rightarrow\mathrm{Mod}_{R}$
sending a graded object $X^{\bullet}$ to its $n$-th graded piece
$X^{n}\in\mathrm{Mod}_{R}$. For $n=0$, the functor $\mathrm{ins}^{0}$ is
symmatric monoidal, and the functor $\operatorname{ev}^{0}$ is _right-lax
symmetric monoidal_.

  6. 6.

Similarly, we let
$\mathrm{ins}^{n}:\mathrm{Mod}_{R}\rightarrow\operatorname{Fil}\mathrm{Mod}_{R}$
be the insertion functor sending an $R$-module $M$ to the filtered object
which has $M$ in degree $n$, and is $0$ otherwise. It has a left adjoint
$\operatorname{gr}^{n}:\operatorname{Fil}\mathrm{Mod}_{R}\rightarrow\mathrm{Mod}_{R}$
and a right adjoint $\operatorname{ev}^{n}$. The functor
$\mathrm{ins}^{0}:\mathrm{Mod}_{R}\rightarrow\operatorname{Fil}\mathrm{Mod}_{R}$
is symmetric monoidal, and its right adjoint
$\operatorname{ev}^{0}:\operatorname{Fil}\mathrm{Mod}_{R}\rightarrow\mathrm{Mod}_{R}$
is lax symmetric monoidal.

  7. 7.

Let $X=X^{\bullet}\in\mathrm{Gr}\mathrm{Mod}_{R}$. We define a new graded
object $X(i)$, called the $i$-th twist of $X$, by the formula
$X(i)^{n}=X^{n-i}$. For $X\in\mathrm{Mod}_{R}$, we will also often denote
$X(i):=\mathrm{ins}^{i}(X)$.

  8. 8.

There exists a constant filtration functor
$\operatorname{const}:\mathrm{Mod}_{R}\rightarrow\operatorname{Fil}\mathrm{Mod}_{R}$
which sends any $R$-module $M$ to the filtered object

$\textstyle{\operatorname{const}(M):=\\{...\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{Id}}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{Id}}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{Id}}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{Id}}$$\textstyle{...\\}.}$

It has a left adjoint functor sending a filtered object $X^{\geq\star}$ to the
colimit
$\underset{\mathbb{Z}^{\mathrm{op}}_{\leq*}}{\operatorname*{colim}}\>X^{\geq\star}\in\mathrm{Mod}_{R}$.
We will often use the notation
$|X|=\underset{\mathbb{Z}^{\mathrm{op}}_{\leq*}}{\operatorname*{colim}}\>X^{\geq\star}$.

  9. 9.

The $\infty$-category $\mathrm{Gr}\mathrm{Mod}_{R}$ contains a full symmetric
monoidal subcategory $\mathrm{Gr}^{\geq
0}\mathrm{Mod}_{R}\subset\mathrm{Gr}\mathrm{Mod}_{R}$ consisting of non-
negatively graded objects, i.e. graded objects $X^{\bullet}$ with $X^{n}=0$
for $n<0$.

  10. 10.

A filtered object $X^{\geq\star}$ is non-negatively filtered if all maps
$X^{n}\rightarrow X^{n-1}$ are equivalences for $n\leq 0$. There is a
symmetric monoidal subcategory $\operatorname{Fil}^{\geq
0}\mathrm{Mod}_{R}\subset\operatorname{Fil}\mathrm{Mod}_{R}$ spanned by non-
negatively filtered objects in all filtered objects.

In the next sections, we will mainly work with non-negative
filtrations/gradings. Therefore, by a slight abuse of notation, we will often
speak of non-negatively filtered/graded objects as simply filtered/graded
objects.

######  Construction 2.4.2.

Let us work in the case $R=\mathbb{Z}$. Let $\mathbb{Z}[t]$ be the free
commutative ring on one generator, and assign the element $t$ degree $-1$. By
[L, Proposition 3.1.6], there is a symmetric monoidal Rees equivalence (or
Rees construction),

$\mathrm{Rees}:\operatorname{Fil}\mathrm{Mod}_{\mathbb{Z}}\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern
3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
9.55pt\raise 4.15565pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.15565pt\hbox{$\scriptstyle{\sim}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}\ignorespaces}}}}\ignorespaces\mathrm{Mod}_{\mathbb{Z}[t]}(\mathrm{Gr}\mathrm{Mod}_{\mathbb{Z}}).$
(2.12)

This equivalence sends a filtered object $X^{\geq\star}$ to the underlying
graded object with a graded action of the graded algebra $\mathbb{Z}[t]$
induced by the filtration.

The perspective of the Rees construction is closely related with the stacky
approach to filtrations. Let $\mathbb{G}_{m}$ be the multiplicative group
scheme, and $\mathrm{B}\mathbb{G}_{m}$ the classifying stack of
$\mathbb{G}_{m}$. Then there is an equivalence of symmetric monoidal
$\infty$-categories (see [Maul21, Theorem 4.1])

$\mathrm{Gr}\mathrm{Mod}_{\mathbb{Z}}\simeq\operatorname{QCoh}(\mathrm{B}\mathbb{G}_{m}).$

Also, let $\mathbb{A}^{1}/\mathbb{G}_{m}$ be the quotient of affine line by
the standard $\mathbb{G}_{m}$-action. There is an equivalence of symmetric
monoidal categories:

$\operatorname{QCoh}(\mathbb{A}^{1}/\mathbb{G}_{m})\simeq\mathrm{Mod}_{\mathbb{Z}[t]}(\mathrm{Gr}\mathrm{Mod}_{\mathbb{Z}}).$
(2.13)

Combining equivalences 2.12 and 2.13, we obtain an equivalence

$\textstyle{\operatorname{Fil}\mathrm{Mod}_{\mathbb{Z}}\simeq\operatorname{QCoh}(\mathbb{A}^{1}/\mathbb{G}_{m})}$

giving a description of the $\infty$-category of filtered objects as quasi-
coherent sheaves on a stack.

For a connective derived commutative algebra $R$, we have

$\operatorname{Fil}\mathrm{Mod}_{R}\simeq\operatorname{Fil}\mathrm{Mod}_{\mathbb{Z}}\underset{\mathrm{Mod}_{\mathbb{Z}}}{\otimes}\mathrm{Mod}_{R}\simeq\operatorname{QCoh}(\mathbb{A}^{1}/\mathbb{G}_{m}\times\operatorname{Spec}R).$

It follows from Construction 2.3.13 that $\operatorname{Fil}\mathrm{Mod}_{R}$
supports the notion of derived commutative (pd) algebras therein.

######  Remark 2.4.3.

From the point of view of Construction 2.4.2, the functor
$\operatorname{gr}:\operatorname{Fil}\mathrm{Mod}_{R}\rightarrow\mathrm{Gr}\mathrm{Mod}_{R}$
is the pullback of quasi-coherent sheaves along the inclusion of the
$0$-section
$\mathrm{B}\mathbb{G}_{m}\rightarrow\mathbb{A}^{1}/\mathbb{G}_{m}$. The
functor
$(\operatorname{Fil}\rightarrow\mathrm{Gr}):\operatorname{Fil}\mathrm{Mod}_{R}\rightarrow\mathrm{Gr}\mathrm{Mod}_{R}$
can be interpreted as the push-forward of quasi-coherent sheaves along the map
$\mathbb{A}^{1}/\mathbb{G}_{m}\rightarrow\mathrm{B}\mathbb{G}_{m}.$ It follows
from this interpretation that $\operatorname{gr}$ is symmetric monoidal, and
$(\operatorname{Fil}\rightarrow\mathrm{Gr})$ is right lax symmetric monoidal.

######  Construction 2.4.4.

For a general connective derived commutative ring $R$, we can construct
several different $t$-structures on $\mathrm{Gr}\mathrm{Mod}_{R}$. We will
record the construction of two of them.

  1. 1.

The neutral $t$-structure on $\mathrm{Gr}\mathrm{Mod}_{R}$ has connective part
$\mathrm{Gr}\mathrm{Mod}_{R,\geq 0}$ spanned by graded objects $X^{\bullet}$
for which $X^{n}\in\mathrm{Mod}_{R,\geq 0}$ for all $n$. The coconnective part
of the neutral $t$-structure consists of objects $X^{\bullet}$ for which
$X^{n}\in\mathrm{Mod}_{R,\leq 0}$ for all $n$. The neutral $t$-structure on
$\mathrm{Gr}\mathrm{Mod}_{R}$ has the heart
$\mathrm{Gr}\mathrm{Mod}_{R,\heartsuit}$ consisting of graded objects in the
abelian category of $t$-discrete objects in $\mathrm{Mod}_{R}$.

  2. 2.

Another important $t$-structure on $\mathrm{Gr}\mathrm{Mod}_{R}$ is the
negative $t$-structure. It has the connective part
$\mathrm{Gr}\mathrm{Mod}_{R,\geq 0^{-}}$ spanned by graded objects
$X^{\bullet}$ which have $X^{n}\in\mathrm{Mod}_{R,\geq-n}$ for any $n$, and
the coconnective part spanned by objects with
$X^{n}\in\mathrm{Mod}_{R,\leq-n}$ for any $n$. The heart of this $t$-structure
is equivalent to the $\infty$-category of graded objects in
$\mathrm{Mod}_{R,\heartsuit}$ again, albeit in a non-tautological way: a
typical $t$-discrete object $X^{\bullet}$ in the negative $t$-structure has
each $X^{n}\in\mathrm{Mod}_{R},[-n,-n],$ which is a discrete $R$-module
concentrated in degree $-n$.

######  Remark 2.4.5.

The negative $t$-structure is compatible with the symmetric monoidal
structure. Indeed, given two objects
$X^{\bullet},Y^{\bullet}\in\mathscr{\mathrm{Gr}}\mathrm{Mod}_{R,\geq 0^{-}}$,
we have $X^{p}\otimes Y^{q}\in\mathrm{Mod}_{R,\leq-p-q}$ since
$X^{p}\in\mathrm{Mod}_{R,\leq-p}$ and $Y^{q}\in\mathrm{Mod}_{R,\leq-q}$, and
the formula 2.10 implies that $X^{\bullet}\otimes Y^{\bullet}$ is connective
in the negative $t$-structure. In particular, the heart
$\mathrm{Gr}\mathrm{Mod}_{R,\heartsuit^{-}}$ becomes a symmetric monoidal
$\infty$-category.

Recall that to define a discrete additive symmetric monoidal category
$\mathrm{C}$, we need to give a bilinear functor
$-\otimes-:\mathrm{C}\times\mathrm{C}\rightarrow\mathrm{C}$, and a natural
$\Sigma_{2}$-action on the tensor product $X\otimes Y$ of any pair of elements
(a “braiding”), which is subject to the constraints listed, for instance, in
[ML, Chapter XI, 1]. The derived $\infty$-category
$\mathrm{D}(\mathrm{C}):=\operatorname{Fun}^{\oplus}(\mathrm{C}^{\mathrm{op}},\mathrm{Spt})$
then acquires a symmetric monoidal structure via right-left extension.

######  Construction 2.4.6.

In this construction, we will define the so-called Koszul sign rule symmetric
monoidal structure on $\mathrm{Gr}\mathrm{Mod}_{\mathbb{Z}}$. First, we define
a tensor product on the abelian category
$\mathrm{Gr}\mathrm{Mod}_{\mathbb{Z},\heartsuit}$ such that for two objects
$X,Y\in\mathrm{Gr}\mathrm{Mod}_{\mathbb{Z}}$, the tensor product of $X$ and
$Y$ coincides with the usual tensor product $X\otimes Y$ of graded modules,
but the action of $\Sigma_{2}$ is twisted by the sign permutation, i.e. for an
element $x\otimes y\in X\otimes Y$, the generator $\sigma\in\Sigma_{2}$ acts
by $\sigma(x\otimes y)=-y\otimes x$. This defines a symmetric monoidal
structure on $\mathrm{Gr}\mathrm{Mod}_{\mathbb{Z},\heartsuit}$ which restricts
to a symmetric monoidal structure on the additive category