$\operatorname{Proj}^{\omega}(\mathrm{Gr}\mathrm{Mod}_{\mathbb{Z},\heartsuit})$
of compact projective objects in
$\mathrm{Gr}\mathrm{Mod}_{\mathbb{Z},\heartsuit}$, and right-left extends to a
symmetric monoidal structure on $\mathrm{Gr}\mathrm{Mod}_{\mathbb{Z}}$.

It turns out that the heart of the negative $t$-structure is equivalent to
$\mathrm{Gr}\mathrm{Mod}_{R,\heartsuit}$ with Koszul sign rule symmetric
monoidal structure via the following construction.

######  Construction 2.4.7.

The $\infty$-category $\mathrm{Gr}\mathrm{Mod}_{R}$ carries a shear auto-
equivalence
$\textstyle{[\bullet]:\mathrm{Gr}\mathrm{Mod}_{R}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{\mathrm{Gr}\mathrm{Mod}_{R}}$
defined by the formula $(X[\bullet])^{i}=X^{i}[i]$ for any graded object
$X:=X^{\bullet}$.

Below will establish the main properties of the shear equivalence. These
statements are well-known. For instance, Proposition 2.4.9 is essentially an
equivalent way of formulating property 2) of Proposition 2.4.8 below, and it
was already established by L.Illusie in the text [I1].

######  Proposition 2.4.8.

The shear construction has the following properties:

  1. 1.

It is left and right $t$-exact with respect to the standard $t$-structure on
the target, and the _negative_ $t$-structure on the source.

  2. 2.

It intertwines the _Koszul sign rule_ symmetric monoidal structure on
$\mathrm{Gr}\mathrm{Mod}_{\mathbb{Z}}$ with the standard symmetric monoidal
structure. In particular, it induces an equivalence of abelian symmetric
monoidal categories
$\mathrm{Gr}\mathrm{Mod}_{\mathbb{Z},\heartsuit}\simeq\mathrm{Gr}\mathrm{Mod}_{\mathbb{Z},\heartsuit^{-}}$,
where the left hand side is endowed with the Koszul sign rule tensor product.

###### Proof.

Identification of $t$-structures follows immediately from definition of the
negative $t$-structure and the formula for the shear equivalence.

Since both symmetric monoidal structures are right-left extended the symmetric
monoidal structures on the hearts, it suffices to show the statement in the
following setting. Given two objects
$X,Y\in\operatorname{Proj}^{\omega}(\mathrm{Gr}\mathrm{Mod}_{\mathbb{Z},\heartsuit})$,
the Koszul sign rule tensor product $X\otimes Y$ carries a $\Sigma_{2}$-action
twisted by sign permutation; we need to prove that the $\Sigma_{2}$-action on
the graded object $(X\otimes Y)[\bullet]$ induced by functoriality, is
equivalent to the usual $\Sigma_{2}$-action. In this situation, the
$\Sigma_{2}$-action on $(X\otimes Y)[\bullet]$ is completely determined by the
$\Sigma_{2}$-action on its homotopy groups. The statement then follows from
the fact that the induced braiding on the homotopy groups of the tensor
product of two objects, satisfies the Koszul sign rule. ∎

######  Proposition 2.4.9.

Let $R$ be a connective derived commutative ring. There are equivalences

$\mathrm{LSym}^{n}_{R}(X[1])\simeq\mathrm{L\Lambda}^{n}_{R}(X)[n],$

$\mathrm{L\Gamma}^{n}_{R}(X[-1])\simeq\mathrm{L\Lambda}^{n}_{R}(X)[-n].$

for any $n\geq 1$ and $X\in\mathrm{Mod}_{R}$.

###### Proof.

It is sufficient to prove either statement in the case $R$ is a polynomial
ring. In this case, since all functors involved are derived from the full
subcategory of finitely generated free modules, using base change from
$\mathbb{Z}$, it suffices to prove that
$\mathrm{LSym}_{\mathbb{Z}}^{n}(\mathbb{Z}^{d}[1])\simeq\Lambda_{\mathbb{Z}}^{n}(\mathbb{Z}^{d})[n]$
for any $d$. This follows from property 2 of Proposition 2.4.8. The statement
for derived divided power algebra follows by duality, as exterior powers
commute with taking duals. ∎

Given an $R$-module $X$, let
$\mathrm{L\Gamma}_{R}(X):=R\oplus\mathrm{L\Gamma}^{+}_{R}(X)$. This is the
unital augmented algebra corresponding to the non-unital algebra
$\mathrm{L\Gamma}_{R}^{+}(X)$. Also, let $\mathrm{L\Lambda}_{R}(X)$ be the
free derived exterior algebra on $X$.

######  Corollary 2.4.10.

Let $X\in\mathrm{Mod}_{R}$. There is an equivalence of derived graded
$R$-algebras

$\mathrm{L\Gamma}_{R}(X[-1])[\bullet]\simeq\mathrm{L\Lambda}_{R}(X),$

where the right hand side is understood as a derived graded algebra with
respect to the Koszul sign rule symmetric monoidal structure.

Assume $X^{\geq\star}$ is a filtered object in $\mathrm{Mod}_{R}$. The the
graded object $\operatorname{gr}^{\bullet}(X)$ can be endowed with an
additional structure. Namely, for any $n$, there is a cofiber sequence

$X^{\geq n+1}\rightarrow X^{\geq n}\rightarrow\operatorname{gr}^{n}(X)$

which produces a boundary map $\operatorname{gr}^{n}(X)\rightarrow X^{\geq
n+1}[1]$. Composing with the quotient map $X^{\geq
n+1}[1]\rightarrow\operatorname{gr}^{n+1}(X)[1]$, we obtain a map
$\operatorname{gr}^{n}(X)\rightarrow\operatorname{gr}^{n+1}(X)[1]$ or
equivalently, a map
$\operatorname{gr}^{n}(X)[-1]\rightarrow\operatorname{gr}^{n+1}(X)$. In other
words, the associated graded object of a filtered object inherits a graded map
$\operatorname{gr}(X)(1)[-1]\rightarrow\operatorname{gr}(X)$ encoding
successive extensions in the filtration. This structure can be understood in
terms of an action of a certain algebra, whose definition we will give below.

######  Definition 2.4.11.

Let $R$ be a connective derived commutative ring. Following [R], we define a
square-zero extension algebra

$\mathbb{D}_{-}:=R\oplus R[-1](1).$

######  Remark 2.4.12.

Vanishing of higher exterior powers of a free module of rank $1$ together with
Proposition 2.4.10 imply that there is an equivalence

$\mathrm{L\Gamma}_{R}\bigl{(}R[-1](1)\bigr{)}\simeq\bigl{(}\mathrm{L\Lambda}_{R}(R)\bigr{)}[-\bullet]\simeq\bigl{(}R\oplus
R(1)\bigr{)}[-\bullet]\simeq R\oplus R[-1](1).$

The diagonal map $R\rightarrow R\oplus R$ induces a map

$\mathbb{D}_{-}=\mathrm{L\Gamma}_{R}\bigl{(}R[-1](1)\bigr{)}\rightarrow\mathrm{L\Gamma}_{R}\bigl{(}R[-1](1)\bigr{)}\otimes\mathrm{L\Gamma}_{R}\bigl{(}R[-1](1)\bigr{)}\simeq\mathbb{D}_{-}\otimes\mathbb{D}_{-},$

which endows $\mathbb{D}_{-}$ with the structure of a _cocommutative Hopf
algebra_ in $\mathrm{Gr}\mathrm{Mod}_{R}$.

######  Definition 2.4.13.

For a connective derived commutative ring $R$, we define the $\infty$-category
$\mathrm{DG}_{-}\mathrm{Mod}_{R}$ of dg objects in $\mathrm{Mod}_{R}$, or dg
$R$-modules, as the $\infty$-category of graded modules over the graded
algebra $\mathbb{D}_{-}$:

$\mathrm{DG}_{-}\mathrm{Mod}_{R}:=\mathrm{Mod}_{\mathbb{D}_{-}}(\mathrm{Gr}\mathrm{Mod}_{R}).$

Using the cocommutative Hopf algebra structure on $\mathbb{D}_{-}$ observed in
Remark 2.4.12, the $\infty$-category gets a symmetric monoidal structure
making the forgetful functor
$\mathrm{DG}_{-}\mathrm{Mod}_{R}\rightarrow\mathrm{Gr}\mathrm{Mod}_{R}$
symmetric monoidal.

######  Remark 2.4.14.

The functor
$\operatorname{gr}:\operatorname{Fil}\mathrm{Mod}_{R}\rightarrow\mathrm{Gr}\mathrm{Mod}_{R}$
lifts to a symmetric monoidal functor
$\operatorname{gr}:\operatorname{Fil}\mathrm{Mod}_{R}\rightarrow\mathrm{DG}_{-}\mathrm{Mod}_{R}$.
To explain this, recall that in terms of the Rees equivalence, the functor
$\operatorname{gr}:\operatorname{Fil}\mathrm{Mod}_{R}\rightarrow\mathrm{Gr}\mathrm{Mod}_{R}$
is equivalent to the base change
$-\underset{R[t]}{\otimes}R:\mathrm{Gr}\mathrm{Mod}_{R[t]}\rightarrow\mathrm{Gr}\mathrm{Mod}_{R}$.
Consequently, the associated graded object $\operatorname{gr}(X)$ of any
filtered object $X$ carries a natural action of the algebra
$\underline{\operatorname{Hom}}_{R[t]}(R,R)$, where the notation
$\underline{\operatorname{Hom}}$ means the internal graded hom-object. The
existence of a natural $\mathbb{D}_{-}$-action then follows from the
equivalence of graded $R$-algebras

$\underline{\operatorname{Hom}}_{R[t]}(R,R)\simeq\underline{\operatorname{Hom}}_{R}(R\underset{R[t]}{\otimes}R,R)\simeq\underline{\operatorname{Hom}}_{R}\bigl{(}\mathrm{LSym}_{R}(R(-1)[1]),R\bigr{)}\simeq\mathrm{L\Gamma}_{R}\bigl{(}R(1)[-1]\bigr{)}\simeq\mathbb{D}_{-},$

We can think of dg modules as representations of a derived algebraic group
scheme whose algebra of distributions is $\mathbb{D}_{-}$. In representation
theory, there are functors of _homology_ and _cohomology_ , or speaking in a
more higher algebraic language, _homotopy orbits_ and _homotopy fixed points_
of a representation.

######  Construction 2.4.15.

Let
$\operatorname{triv}:\mathrm{Gr}\mathrm{Mod}_{R}\rightarrow\mathrm{DG}_{-}\mathrm{Mod}_{R}$
be the functor endowing a graded module with a trivial
$\mathbb{D}_{-}$-action. In other words, it is restricting along the
augmentation map $\mathbb{D}_{-}\rightarrow R$. There is an adjunction

$\textstyle{\mathrm{Gr}\mathrm{Mod}_{R}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{triv}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathrm{DG}_{-}\mathrm{Mod}_{R}.}$$\scriptstyle{(-)_{h\mathbb{D}_{-}}}$$\scriptstyle{(-)^{h\mathbb{D}_{-}}}$

For $X\in\mathrm{DG}_{-}\mathrm{Mod}_{R}$, we call $X_{h\mathbb{D}_{-}}$ the
$\mathbb{D}_{-}$-homotopy orbits object of $X$, and $X^{h\mathbb{D}_{-}}$
homotopy fixed points object.

######  Remark 2.4.16.

Note that as the functor $\operatorname{triv}$ is symmetric monoidal, the
right adjoint $(-)^{h\mathbb{D}_{-}}$ is right-lax symmetric monoidal. It is
given explicitly by the formula

$X^{h\mathbb{D}_{-}}:=\underline{\operatorname{Hom}}_{\mathbb{D}_{-}}(R,X),$

where the $\underline{\operatorname{Hom}}$ is the graded internal hom-object.
Similarly as in Remark 2.4.14, we can compute that there is an equivalence

$R^{h\mathbb{D}_{-}}=\underline{\operatorname{Hom}}_{\mathbb{D}_{-}}(R,R)=R[t]$

of graded algebras, where $t$ has degree $-1$. In particular, the graded
object $X^{h\mathbb{D}_{-}}$ carries an operation of degree $-1$, which is the
same data as a filtration. We denote

$(-)^{\geq\star}:\mathrm{DG}_{-}\mathrm{Mod}_{R}\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern
3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\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{}$}}}}}}}\ignorespaces}}}}\ignorespaces\operatorname{Fil}\mathrm{Mod}_{R}$

the resulting functor lifting $(-)^{h\mathbb{D}_{-}}$.

We record the following proposition without proof, which can be found in [R,
Theorem 3.2.14]. Let
$\widehat{\operatorname{Fil}\mathrm{Mod}}_{R}\subset\operatorname{Fil}\mathrm{Mod}_{R}$
be the $\infty$-category of complete filtered objects in $\mathrm{Mod}_{R}$,
i.e. the full subcategory of $\operatorname{Fil}\mathrm{Mod}_{R}$ spanned by
the objects $X^{\geq\star}$ which satisfy $\underset{n}{\lim}\>X^{\geq
n}\simeq 0$.

######  Theorem 2.4.17.

Let $R$ be a derived commutative ring. The functors $\operatorname{gr}$ and
$(-)^{h\mathbb{D}_{-}}$ form a mutually inverse equivalence of symmetric
monoidal $\infty$-categories

$\textstyle{\widehat{\operatorname{Fil}\mathrm{Mod}}_{R}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{gr}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathrm{DG}_{-}\mathrm{Mod}_{R}.}$$\scriptstyle{(-)^{\geq\star}}$

######  Remark 2.4.18.

If $R$ is connective, then the algebra $\mathbb{D}_{-}$ is connective in the
negative $t$-structure, therefore the negative $t$-structure on
$\mathrm{Gr}\mathrm{Mod}_{R}$ extends to a $t$-structure on
$\mathrm{DG}_{-}\mathrm{Mod}_{R}$ which we call negative $t$-structure as
well. The heart of this $t$-structure is equivalent to the _abelian category
of cochain complexes_ of $\pi_{0}(R)$-modules. Note that when $R$ is discrete,
the algebra $\mathbb{D}_{-}$ is discrete in the negative $t$-structure too.

######  Remark 2.4.19.

Let us work over $R=\mathbb{Z}$. The structure of a $\mathbb{D}_{-}$-action
can be understood in stacky terms as follows. Following A.Preygel [Pr], let
$\Omega\mathbb{A}^{1}$ be the based loop space of $\mathbb{A}^{1}$ defined as
a pull-back square in derived affine schemes

$\Omega\mathbb{A}^{1}:=\\{0\\}\underset{\mathbb{A}^{1}}{\times}\\{0\\},$

where both maps
$\\{0\\}=\operatorname{Spec}(\mathbb{Z})\rightarrow\mathbb{A}^{1}$ are
inclusions of the zero section. $\Omega\mathbb{A}^{1}$ becomes a group object
in derived schemes with respect to composition of loops. The zero section in
$\mathbb{A}^{1}$ is fixed by the action of $\mathbb{G}_{m}$, and induces an
action on $\Omega\mathbb{A}^{1}$. Therefore, we can consider the quotient
stack $\Omega\mathbb{A}^{1}/\mathbb{G}_{m}$ which is a derived group stack
over $\mathbb{G}_{m}$. Let
$\operatorname{Rep}(\Omega\mathbb{A}^{1}/\mathbb{G}_{m})$ be the category of
graded $\Omega\mathbb{A}^{1}$-representations (understood as, say, comodules
over the algebra of functions on $\Omega\mathbb{A}^{1}$). In these terms, the
functor
$\textstyle{\operatorname{gr}:\operatorname{Fil}\mathrm{Mod}_{R}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{Gr}\mathrm{Mod}_{\mathbb{D}_{-}}}$
is given by the pull-back

$\textstyle{-\underset{\mathbb{A}^{1}/\mathbb{G}_{m}}{\times}\mathrm{B}\mathbb{G}_{m}:\operatorname{QCoh}(\mathbb{A}^{1}/\mathbb{G}_{m})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Rep}(\Omega\mathbb{A}^{1}/\mathbb{G}_{m})}$

along the inclusion of zero section
$i:\mathrm{B}\mathbb{G}_{m}\rightarrow\mathbb{A}^{1}/\mathbb{G}_{m}.$

######  Definition 2.4.20.

Let $R$ be a connective derived ring. Using Remark 2.3.13 and Construction
2.4.2, there is a notion of derived commutative algebras in
$\operatorname{Fil}\mathrm{Mod}_{R}$, $\mathrm{Gr}\mathrm{Mod}_{R}$,
$\operatorname{Fil}^{\geq 0}\mathrm{Mod}_{R}$ and $\mathrm{Gr}^{\geq
0}\mathrm{Mod}_{R}$.

We let

$\mathrm{DAlg}(\mathrm{Gr}\mathrm{Mod}_{R}),\mathrm{DAlg}(\operatorname{Fil}\mathrm{Mod}_{R}),\mathrm{DAlg}(\mathrm{Gr}^{\geq
0}\mathrm{Mod}_{R}),\mathrm{DAlg}(\operatorname{Fil}^{\geq
0}\mathrm{Mod}_{R})$

to be the respective $\infty$-categories of derived commutative algebras. We
will often refer to them as graded, filtered, non-negatively graded and non-
negatively filtered777Note that there is a potential abuse of terminology here
as according to standard conventions, a “graded/filtered derived commutative
algebra” could mean a graded/filtered object in the $\infty$-category of
derived commutative algebras. However, in the case of algebras, this seems to
be a standard terminology, so there is no risk of confusion. If we did need to
speak about filtered/graded objects in derived commutative rings, we would
call them by their longer name.  derived commutative algebras in
$\mathrm{Mod}_{R}$ respectively, and we use respective notations

$\mathrm{Gr}\mathrm{DAlg}_{R},\operatorname{Fil}\mathrm{DAlg}_{R},\mathrm{Gr}^{\geq
0}\mathrm{DAlg}_{R},\operatorname{Fil}^{\geq 0}\mathrm{DAlg}_{R}.$

Below we will give an equivalent, more explicit, description of the
$\infty$-categories of derived algebras given in Definition 2.4.20 in the case
when $R$ is discrete.

######  Construction 2.4.21.

Assume $R$ is a discrete ring. Let us unwind the natural $t$-structure on
$\operatorname{Fil}\mathrm{Mod}_{R}$ obtained from the equivalence
$\operatorname{Fil}\mathrm{Mod}_{R}\simeq\operatorname{QCoh}(\mathbb{A}^{1}/\mathbb{G}_{m})$.
Using the affine map
$p:\mathbb{A}^{1}/\mathbb{G}_{m}\rightarrow\mathrm{B}\mathbb{G}_{m},$ an
object $F\in\operatorname{QCoh}(\mathbb{A}^{1}/\mathbb{G}_{m})$ is connective
iff its push-forward $p_{*}F$ is connective as a quasi-coherent sheaf on
$\mathrm{B}\mathbb{G}_{m}$. Denote the graded components of $p_{*}F$ as
$\\{F^{i}\\}_{i\in\mathbb{Z}}$. Pulling-back along the map
$i:\operatorname{Spec}(R)\rightarrow\mathrm{B}\mathbb{G}_{m}$, we obtain that
$i^{*}p_{*}F$ is connective as an $R$-module. But we can identify
$i^{*}p_{*}F\simeq F^{0}$, and $F^{0}$ is a connective $R$-module. Similarly,
by shifting the grading, we obtain that all $F^{i}$’s must be connective
$R$-modules. Therefore, the natural $t$-structure on
$\operatorname{Fil}\mathrm{Mod}_{R}\simeq\operatorname{QCoh}(\mathbb{A}^{1}/\mathbb{G}_{m})$
has the connective part consisting of filtered objects $F^{\geq\star}$ which
are connective in all degrees. This $t$-structure is compatible with the
symmetric monoidal structure, and is left and right complete.

Let
$\operatorname{Fil}\mathrm{Mod}_{R,0}\subset\operatorname{Fil}\mathrm{Mod}_{R}$
be the subcategory spanned by direct sums of objects of the form $R(i)$ for
all $i\in\mathbb{Z}$. This is an additive subcategory of
$\operatorname{Fil}\mathrm{Mod}_{R,\heartsuit}$ which is closed under the
action of $\mathrm{SSeq}^{\mathrm{gen}}_{0}$. Consequently, using Construction
2.3.14, we obtain an action of $\mathrm{SSeq}^{\mathrm{gen}}$ on
$\operatorname{Fil}\mathrm{Mod}_{R}$. This action has the property that for
any map $f:\operatorname{Spec}A\rightarrow\mathbb{A}^{1}/\mathbb{G}_{m}$, the
pull-back
$f^{*}:\operatorname{Fil}\mathrm{Mod}_{R}\simeq\operatorname{QCoh}(\mathbb{A}^{1}/\mathbb{G}_{m})\rightarrow\mathrm{Mod}_{A}$
is $\mathrm{SSeq}^{\mathrm{gen}}$-linear. In the same way, we construct an
action of $\mathrm{SSeq}^{\mathrm{gen},\mathrm{pd}}$ on
$\operatorname{Fil}\mathrm{Mod}_{R}$. Therefore, we obtain two monads on
$\operatorname{Fil}\mathrm{Mod}_{R}$:

  1. 1.

The free filtered derived algebra monad
$\mathrm{LSym}^{\geq\star}_{R}:\operatorname{Fil}\mathrm{Mod}_{R}\rightarrow\operatorname{Fil}\mathrm{Mod}_{R}$;

  2. 2.

The free non-unital derived divided power algebra monad
$\mathrm{L\Gamma}^{+,\geq\star}_{R}:\operatorname{Fil}\mathrm{Mod}_{R}\rightarrow\operatorname{Fil}\mathrm{Mod}_{R}$.

######  Remark 2.4.22.

The monad
$\mathrm{LSym}^{\geq\star}_{R}:\operatorname{Fil}\mathrm{Mod}_{R}\rightarrow\operatorname{Fil}\mathrm{Mod}_{R}$
restricts to a filtered-colimit preserving endofunctor
$\operatorname{Sym}^{\geq\star}_{R,\heartsuit}:\operatorname{Ind}(\operatorname{Fil}\mathrm{Mod}_{R,0})\rightarrow\operatorname{Ind}(\operatorname{Fil}\mathrm{Mod}_{R,0})$
and is uniquely determined by it according to Observation 2.3.15. The same
applies to the monad $\mathrm{L\Gamma}^{+,\geq\star}_{R}$.

Let us mention a concrete description of the category
$\operatorname{Ind}(\operatorname{Fil}^{\geq 0}\mathrm{Mod}_{R,0})$.

######  Proposition 2.4.23.

The category $\operatorname{Ind}(\operatorname{Fil}^{\geq
0}\mathrm{Mod}_{R,0})$ is spanned by objects
$F^{\geq\star}\in\operatorname{Fil}^{\geq 0}\mathrm{Mod}_{R}$ where $F^{0}$ is
a flat $R$-module, and each map $F^{\geq i+1}\hookrightarrow F^{i}$ is
injective with flat cokernel.

The notion of filtered/graded derived $R$-algebra can be extended to the non-
connective case as follows.

######  Remark 2.4.24.

Let $R$ be a derived ring, and consider $R$ as a filtered (graded) derived
ring inserted in degree $0$, extended by $0$ in positive degrees, and in a
constant way in negative degrees. Then we can define

$\operatorname{Fil}\mathrm{DAlg}_{R}:=(\operatorname{Fil}\mathrm{DAlg}_{\mathbb{Z}})_{R/},\;\;\mathrm{Gr}\mathrm{DAlg}_{R}:=(\mathrm{Gr}\mathrm{DAlg}_{\mathbb{Z}})_{R/}.$

This recovers the previous definition when $R$ is connective, and extends it
immediately to the not necessarily connective case.

##  3 Filtrations and divided power algebras.

###  3.1 Smith ideals.

######  Construction 3.1.1.

Below we will introduce two constructions which feature prominently in the
text.

  1. 1.

Let $\Delta^{1}$ be the $1$-simplex considered as a category with two objects
$0,1$ and a single non-trivial arrow $0\rightarrow 1$. There is a symmetric
monoidal structure $\ast:\Delta^{1}\times\Delta^{1}\rightarrow\Delta^{1}$ on
$\Delta^{1}$ defined as:

$0\ast 0=0,0\ast 1=1,1\ast 0=1,1\ast 1=1$ (3.1)

Fix a (connective, derived) commutative ring $R$. Then the $\infty$-category
$\operatorname{Fun}((\Delta^{1})^{\mathrm{op}},\mathrm{Mod}_{R})$ is a
symmetric monoidal $\infty$-category with respect to the _Day convolution
symmetric monoidal structure_. The formula for the tensor product of two
objects $X^{1}\rightarrow X^{0}$ and $Y^{1}\rightarrow Y^{0}$ is as follows:

$(X^{1}\rightarrow X^{0})\otimes(Y^{1}\rightarrow Y^{0})\simeq X^{0}\otimes
Y^{1}\bigsqcup_{X^{1}\otimes X^{1}}X^{1}\otimes Y^{0}\rightarrow X^{0}\otimes
Y^{0}.$

We use the notation

$\mathrm{Mod}^{\Delta^{1}_{\vee}}_{R}:=\operatorname{Fun}((\Delta^{1})^{\mathrm{op}},\mathrm{Mod}_{R})$

considered with Day convolution symmetric monoidal structure satisfying the
formula above.

  2. 2.

Let $\\{0,1\\}$ be the discrete 2 point category endowed with the symmetric
monoidal structure defined on objects by the same formulas as 3.1. The
$\infty$-category
$\operatorname{Fun}(\\{0,1\\}^{\mathrm{op}},\mathrm{Mod}_{R})$ gets a Day
convolution symmetric monoidal structure with tensor product

$(X^{1},X^{0})\otimes(Y^{1},Y^{0})\simeq(X^{0}\otimes Y^{1}\oplus X^{1}\otimes
Y^{1}\oplus X^{1}\otimes Y^{0},X^{0}\otimes Y^{0}).$

We denote

$\mathrm{Mod}^{\\{0,1\\},\vee}_{R}:=\operatorname{Fun}(\\{0,1\\}^{\mathrm{op}},\mathrm{Mod}_{R})$

considered with Day convolution symmetric monoidal structure.

We will refer to both symmetric monoidal structures defined in the
construction above as “push-out symmetric monoidal structures”.

######  Definition 3.1.2.

Let $R$ be a (connective, derived) commutative ring.

  1. 1.

Let

$\operatorname{CAlg}_{R}^{\Delta^{1}_{\vee}}:=\operatorname{CAlg}(\mathrm{Mod}^{\Delta^{1}_{\vee}}_{R})$

be the $\infty$-category of $\mathbb{E}_{\infty}$-algebras in
$\mathrm{Mod}^{\Delta^{1}_{\vee}}_{R}$.

  2. 2.

Similarly, let

$\operatorname{CAlg}_{R}^{\\{0,1\\},\vee}:=\operatorname{CAlg}(\mathrm{Mod}_{R}^{\\{0,1\\},\vee})$

be the $\infty$-category of commutative algebra objects in
$\mathrm{Mod}_{R}^{\\{0,1\\},\vee}$.

In what follows we will clarify the meanings of these two $\infty$-categories.
As a warm-up, let us give a description of discrete objects in
$\operatorname{CAlg}_{R}^{\Delta^{1}_{\vee}}$ and
$\operatorname{CAlg}_{R}^{\\{0,1\\},\vee}$ respectively.

######  Definition 3.1.3.

Recall the classical notion of a Smith ideal888This notion was introduced by
Jess Smith and was reviewed and used recently in the texts [BI22], [Mao21],
[Dr20] and others.. Let $A$ be a discrete commutative ring. A Smith ideal in
$\mathrm{Mod}_{A}$ is an $A$-module $M$ endowed with an $A$-linear map
$i:I\rightarrow A$ such that $i(x)y=xi(y)$. The map $I\otimes_{A}I\rightarrow
I$ given by sending $(x,y)\longmapsto i(x)y$ defines a non-unital $A$-algebra
structure on $I$, and the quotient $A/I:=\operatorname*{cofib}(I\rightarrow
A)$ has a structure of a simplicial commutative ring, which is in general non-
discrete unless $I\rightarrow A$ is injective.

######  Proposition 3.1.4.

Let $R$ be a discrete commutative ring. Let
$\operatorname{CAlg}_{R,\heartsuit}^{\Delta^{1}_{\vee}}$ and
$\operatorname{CAlg}_{R,\heartsuit}^{\\{0,1\\},\vee}$ be the subcategories of
$\operatorname{CAlg}_{R}^{\Delta^{1}_{\vee}}$ and
$\operatorname{CAlg}_{R}^{\\{0,1\\},\vee}$ respectively, consisting of
discrete objects with respect to the canonical $t$-structure on
$\mathrm{Mod}_{R}$.

  1. 1.

The category $\operatorname{CAlg}_{R,\heartsuit}^{\Delta^{1}_{\vee}}$ is
equivalent to the category of pairs $(I\rightarrow A)$ where $A$ is a
commutative $R$-algebra, and $i:I\rightarrow A$ is a Smith ideal in $A$.

  2. 2.

The category $\operatorname{CAlg}_{R,\heartsuit}^{\\{0,1\\},\vee}$ is
equivalent to the category of pairs $(A,B)$ where $A$ is a commutative
$R$-algebra, and $B$ is a non-unital $A$-algebra.

###### Proof.

  1. 1.

A unital commutative algebra structure on an object $i:A^{1}\rightarrow A^{0}$
amounts to giving a unit map $\varepsilon:(0\rightarrow
R)\rightarrow(A^{1}\rightarrow A^{0})$, and a multiplication map

$\textstyle{A^{1}\otimes A^{0}\underset{A^{1}\otimes
A^{1}}{\bigsqcup}A^{0}\otimes
A^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{m^{1}}$$\textstyle{A^{0}\otimes
A^{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{m^{0}}$$\textstyle{A^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A^{0}}$

which satisfy associativity and commutativity conditions. The maps $m^{0}$ and
$\varepsilon^{0}$ give a unital, commutative and associative multiplication on
$A^{0}$. The map

$m^{1}:A^{1}\otimes A^{0}\bigsqcup_{A^{1}\otimes A^{1}}A^{0}\otimes
A^{1}\rightarrow A^{1}$

is equivalent to a pair of maps $A^{0}\otimes A^{1}\rightarrow A^{1}$ and
$A^{1}\otimes A^{0}\rightarrow A^{1}$ which agree on $A^{1}\otimes A^{1}$.
This defines a left and right actions of $A^{0}$ on $A^{1}$ satisfying the
condition $i(x)y=xi(y)$. Commutativity and associativity constraints say that
$A^{1}$ is symmetric $A^{0}$-bimodule, and $i:A^{1}\rightarrow A^{0}$ is an
$A^{0}$-module map. This is the same as a Smith ideal.

  2. 2.

The proof of this part goes the same way as in part 1 by unwinding the
definition of a commutative algebra object in tensor product defined in
Construction 3.1.1, 2.

∎

######  Construction 3.1.5.

Let $R$ be a derived commutative ring, and we will consider the
$\infty$-category of arrows $\mathrm{Mod}_{R}^{\Delta^{1}}$ witht the
_pointwise_ symmetric monoidal structure. The functor
$\operatorname*{cofib}:\mathrm{Mod}^{\Delta^{1}_{\vee}}_{R}\rightarrow\mathrm{Mod}_{R}^{\Delta^{1}}$
sending $(X\rightarrow
Y)\longmapsto(Y\rightarrow\operatorname*{cofib}(X\rightarrow Y))$ admits an
oplax symmetric monoidal structure. To see this, we first note that the
functor
$\operatorname{ev}^{0}:\mathrm{Mod}^{\Delta^{1}_{\vee}}_{R}\rightarrow\mathrm{Mod}_{R},(X\rightarrow
Y)\longmapsto Y$ is symmetric monoidal. Then it remains to construct an oplax
symmetric monoidal structure on the functor
$\mathrm{Mod}^{\Delta^{1}_{\vee}}_{R}\rightarrow\mathrm{Mod}_{R},(X\rightarrow
Y)\longmapsto\operatorname*{cofib}(X\rightarrow Y).$ But this functor is the
left adjoint of the symmetric monoidal inclusion
$\mathrm{Mod}_{R}\rightarrow\mathrm{Mod}^{\Delta^{1}_{\vee}}_{R}$ which sends
$X\in\mathrm{Mod}_{R}$ to $(0\rightarrow
X)\in\mathrm{Mod}^{\Delta^{1}_{\vee}}_{R}$, and therefore admits an oplax
symmetric monoidal structure.

######  Proposition 3.1.6.

The oplax symmetric monoidal structure on the functor
$\operatorname*{cofib}:\mathrm{Mod}^{\Delta^{1}_{\vee}}_{R}\rightarrow\mathrm{Mod}_{R}^{\Delta^{1}}$
is strict.

###### Proof.

We need to show the following. For two arrows $(X\rightarrow Y)$ and
$(X^{\prime}\rightarrow Y^{\prime})$ the natural map

$\textstyle{\operatorname*{cofib}(X\otimes Y^{\prime}\underset{X\otimes
X^{\prime}}{\bigsqcup}Y\otimes
X^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y\otimes
Y^{\prime})\rightarrow Y/X\otimes Y^{\prime}/X^{\prime}}$

is an equivalence. This is equivalent to showing that the diagram

$\textstyle{(Y\otimes Y^{\prime})/(X\otimes
X^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y/X\otimes
Y^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y\otimes
Y^{\prime}/X^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y/X\otimes
Y^{\prime}/X^{\prime}}$

is a push-out square. This is in turn equivalent to showing that the map
induced on horizontal cofibers is an equivalence. The cofiber of the lower
horizontal arrow is equivalent to $X\otimes Y^{\prime}/X^{\prime}[1]$. The
cofiber of the upper hozitontal arrow can be read of from the diagram

$\textstyle{X\otimes
X^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f\otimes
f^{\prime}}$$\scriptstyle{\operatorname{Id}\otimes
f^{\prime}}$$\textstyle{X\otimes
Y^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f\otimes\operatorname{Id}}$$\textstyle{X\otimes
Y^{\prime}/X^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y\otimes
Y^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{=}$$\textstyle{Y\otimes
Y^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(Y\otimes
Y^{\prime})/(X\otimes
X^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y/X\otimes
Y^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X\otimes
Y^{\prime}/X^{\prime}[1].}$

Therefore, the natural map $\operatorname*{cofib}((Y\otimes
Y^{\prime})/(X\otimes X^{\prime})\rightarrow Y/X\otimes
Y^{\prime})\rightarrow\operatorname*{cofib}(Y\otimes
Y^{\prime}/X^{\prime}\rightarrow Y/X\otimes Y^{\prime}/X^{\prime})$ is an
equivalence, and this finishes the proof. ∎

######  Proposition 3.1.7.

The functor
$\operatorname*{cofib}:\mathrm{Mod}^{\Delta^{1}_{\vee}}_{R}\rightarrow\mathrm{Mod}_{R}^{\Delta^{1}}$
is an equivalence of symmetric monoidal $\infty$-categories where the source
is endowed with the push-out symmetric monoidal structure, and the target with
the pointwise symmetric monoidal structure.

###### Proof.

The inverse functor sends a map $X\rightarrow Y$ to
$\operatorname*{fib}(X\rightarrow Y)\rightarrow X$. ∎

The following corollary is immediate from Proposition 3.1.7:

######  Corollary 3.1.8.

There is an equivalence of $\infty$-categories:

$\textstyle{\operatorname*{cofib}:\operatorname{CAlg}_{R}^{\Delta^{1}_{\vee}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{\operatorname{CAlg}_{R}^{\Delta^{1}}.}$

Let us now clarify the meaning of the $\infty$-category
$\operatorname{CAlg}_{R}^{\\{0,1\\},\vee}$. As a preparation for the next
construction, observe that given an object of
$(A^{1},A^{0})\in\operatorname{CAlg}^{\\{0,1\\},\vee}_{R,\heartsuit}$, we can
define a Smith ideal by the formula $(A^{1}\rightarrow A^{0}\oplus A^{1})$. On
the other hand, any Smith ideal $(I\rightarrow A)$ gives rise to a pair
$(A,I)\in\operatorname{CAlg}^{\\{0,1\\},\vee}_{R,\heartsuit}$ where $I$ is
considered merely as a non-unital $A$-algebra. Below we will construct an
adjunction of symmetric monoidal $\infty$-categories which gives rise to these
constructions on the $\infty$-categories of commutative algebra objects.

######  Construction 3.1.9.

The tautological inclusion functor $\\{0,1\\}\rightarrow\Delta^{1}$ is
strictly symmetric monoidal with respect to the symmetric monoidal structure
described in Construction 3.1.1, and consequently induces an adjunction

$\textstyle{\mathrm{Mod}^{\\{0,1\\},\vee}_{R}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{aug}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathrm{Mod}^{\Delta^{1}_{\vee}}_{R},}$$\scriptstyle{(\operatorname{Fil}\rightarrow\mathrm{Gr})}$

where the left adjoint $\mathrm{aug}$ is strictly symmetric monoidal, and the
right adjoint $(\operatorname{Fil}\rightarrow\mathrm{Gr})$ is lax symmetric
monoidal.

Unwinding the definition, the functor
$(\operatorname{Fil}\rightarrow\mathrm{Gr})$ sends an arrow $(Y\rightarrow X)$
to the pair $(Y,X)$. The left adjoint $\mathrm{aug}$ sends a pair
$(X^{1},X^{0})$ to the arrow $(X^{1}\rightarrow X^{1}\oplus X^{0})$.

######  Remark 3.1.10.

The functor $\mathrm{aug}$ preserves all colimits being a left adjoint. It
also preserves all limits, as follows from the formula. Moreover, it is
conservative: if a map $(X^{1},X^{0})\rightarrow(Y^{1},Y^{0})$ induces an
equivalence $(X^{1}\rightarrow X^{1}\oplus X^{0})\simeq(Y^{1}\rightarrow
Y^{1}\oplus Y^{0})$, then we immediately get that $X^{1}\simeq Y^{1}$ is an
equivalence, and taking cofibers that $X^{0}\simeq Y^{0}$ too. It follows that
$\mathrm{aug}$ is monadic and comonadic.

######  Remark 3.1.11.

It follows from proposition above that the adjunction of Construction 3.1.9
refines to an adjunction

$\textstyle{\operatorname{CAlg}^{\\{0,1\\},\vee}_{R}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{aug}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\operatorname{CAlg}^{\Delta^{1}_{\vee}}_{R}.}$$\scriptstyle{(\operatorname{Fil}\rightarrow\mathrm{Gr})}$

Based on Remark 3.1.10, the $\infty$-category
$\operatorname{CAlg}^{\\{0,1\\},\vee}_{R}$ should have a description as an
$\infty$-category of objects of $\operatorname{CAlg}^{\Delta^{1}_{\vee}}_{R}$
endowed with some additional data. Such a description will be given below.

######  Construction 3.1.12.

Let $\mathrm{Mod}_{R}^{\\{0,1\\}}$ be the $\infty$-category of functors
$\operatorname{Fun}(\\{0,1\\},\mathrm{Mod}_{R})$ endowed with the _pointwise_
symmetric monoidal structure, i.e. the tensor product of two objects
$(X^{0},X^{1})$ and $(Y^{0},Y^{1})$ is given by the formula

$(X^{0},X^{1})\otimes(Y^{0},Y^{1})\simeq(X^{0}\otimes Y^{0},X^{1}\otimes
Y^{1}),$

and the unit object is the pair $(R,R)\in\mathrm{Mod}_{R}^{\\{0,1\\}}$.

The symmetric monoidal $\infty$-category $\mathrm{Mod}_{R}^{\\{0,1\\}}$
carries a symmetric monoidal equivalence
$\varepsilon:\mathrm{Mod}_{R}^{\\{0,1\\}}\simeq\mathrm{Mod}_{R}^{\\{0,1\\}}$
defined by the formula $\varepsilon(X^{1},X^{0}):=(X^{0},X^{1})$. We define a
stable $\infty$-category of augmented arrows as the pull-back

$\textstyle{\mathrm{Mod}_{R}^{\Delta^{1},\mathrm{aug}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{Mod}_{R}^{\Delta^{1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\operatorname{Fil}\rightarrow\mathrm{Gr})}$$\textstyle{\mathrm{Mod}_{R}^{\Delta^{1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varepsilon\circ(\operatorname{Fil}\rightarrow\mathrm{Gr})}$$\textstyle{\mathrm{Mod}_{R}^{\\{0,1\\}}.}$
(3.2)

Objects of $\mathrm{Mod}_{R}^{\Delta^{1},\mathrm{aug}}$ can be thought as
arrows $p:Y\rightarrow X$ endowed with a section $e:X\rightarrow Y$ such that
$p\circ e\simeq\operatorname{Id}_{Y}$. We can endow
$\mathrm{Mod}_{R}^{\Delta^{1},\mathrm{aug}}$ with _pointwise_ symmetric
monoidal structure, so that the diagram 3.2 is a pull-back diagram in
$\operatorname{CAlg}(\mathrm{Cat})_{\mathrm{lax}}$. There exists a symmetric
monoidal functor

$\textstyle{\mathrm{Mod}_{R}^{\\{0,1\\},\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{Mod}_{R}^{\Delta^{1},\mathrm{aug}},}$$\textstyle{\;\;\;\;\;(X^{1},X^{0})\longmapsto(X^{0}\oplus
X^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
X^{0})}$$\scriptstyle{e}$ (3.3)

where the arrow $p:(X^{1}\oplus X^{0}\rightarrow X^{0})$ is the natural
projection, endowed with the $0$-section $e:X^{0}\hookrightarrow X^{1}\oplus
X^{0}$.

######  Proposition 3.1.13.

The functor 3.3 induces a symmetric monoidal equivalence

$\textstyle{\mathrm{Mod}_{R}^{\\{0,1\\},\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{\mathrm{Mod}_{R}^{\Delta^{1},\mathrm{aug}}.}$

###### Proof.

The inverse functor sends a split arrow $Z\rightarrow X$ to the pair $(Y,X)$,
where $Y:=\operatorname*{fib}(Z\rightarrow X)$. ∎

######  Corollary 3.1.14.

There is an equivalence of $\infty$-categories

$\operatorname{CAlg}_{R}^{\\{0,1\\},\vee}\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern
3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\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{}$}}}}}}}\ignorespaces}}}}\ignorespaces\operatorname{CAlg}^{\Delta^{1}_{\vee},\mathrm{aug}}_{R}:=\operatorname{CAlg}(\mathrm{Mod}^{\Delta^{1}_{\vee},\mathrm{aug}}_{R})$

######  Remark 3.1.15.

Given the description of $\mathrm{Mod}_{R}^{\Delta^{1},\mathrm{aug}}$ as a
pull-back diagram 3.2 in $\operatorname{CAlg}(\mathrm{Cat})_{\mathrm{lax}}$,
Corollary 3.1.14 together with Proposition 2.1.2 imply that the
$\infty$-category
$\operatorname{CAlg}^{\\{0,1\\},\vee}_{R}:=\operatorname{CAlg}(\mathrm{Mod}_{R}^{\\{0,1\\},\vee})$
is equivalent to the $\infty$-category of arrows $p:B\rightarrow A$ in
$\operatorname{CAlg}_{R}$ endowed with a section $i:A\rightarrow B$, or
equivalently, pairs $(A,A^{+})$ where $A\in\operatorname{CAlg}_{R}$ and
$A^{+}\in\operatorname{CAlg}^{\mathrm{nu}}_{A}$.

We will now move to the study of Smith ideals in the derived world.

######  Construction 3.1.16.

We define a derived algebra object $\mathrm{Mod}_{R}^{\Delta^{1}}$ as an arrow
$A\rightarrow B$ in the $\infty$-category $\mathrm{DAlg}_{R}$. The resulting
$\infty$-category $\operatorname{Fun}(\Delta^{1},\mathrm{DAlg}_{R})$ is
monadic over $\mathrm{Mod}_{R}^{\Delta^{1}}$, and when $R$ is discrete, we can
describe the monad as the right-left extension of the monad of free
commutative algebra defined on the subcategory
$(\mathrm{Mod}_{R,0})^{\Delta^{1}}\subset\mathrm{Mod}_{R}^{\Delta^{1}}$ using
the standard (pointwise) $t$-structure on $\mathrm{Mod}_{R}^{\Delta^{1}}$. We
denote this monad as
$(\mathrm{LSym}_{R})^{\Delta^{1}}:\mathrm{Mod}_{R}^{\Delta^{1}}\rightarrow\mathrm{Mod}_{R}^{\Delta^{1}}$.
It is uniquely determined by its restriction to the full subcategory
$\operatorname{Ind}(\mathrm{Mod}_{R,0}^{\Delta^{1}})\subset\mathrm{Mod}_{R}^{\Delta^{1}}$
consisting of pairs $(f:X\rightarrow Z)$ with both $X$ and $Z$ being flat
$R$-modules.

Below we will give another construction of the $\infty$-category of derived
algebras in $\mathrm{Mod}_{R}^{\Delta^{1}}$, which is based on a different
$t$-structure on $\mathrm{Mod}_{R}^{\Delta^{1}}$ and another choice of a
subcategory
$\mathrm{Mod}_{R,0}^{\Delta^{1}}\subset\mathrm{Mod}_{R}^{\Delta^{1}}$.

######  Construction 3.1.17.

Let $R$ be a discrete ring. The $\infty$-category
$\mathrm{Mod}_{R}^{\Delta^{1}}$ has a geometric $t$-structure, whose
connective part $\mathrm{Mod}_{R,\geq 0}^{\Delta^{1}}$ consists of arrows
$(f:X\rightarrow Z)$ with both $X$ and $Z$ being connective, and the map $f$
being surjective on $\pi_{0}$. Equivalently, an arrow $(f:X\rightarrow Z)$ is
in $\mathrm{Mod}_{R,\geq 0}^{\Delta^{1}}$ if $X$ and $\operatorname*{fib}(f)$
are connective. Consequently, an arrow $(f:X\rightarrow Z)$ is coconnective in
geometric $t$-structure if $X$ and $\operatorname*{fib}(f)$ are coconnective.
Below we will list the main observations regarding this $t$-structure.

  * •

The geometric $t$-structure is compatible with the pointwise symmetric
monoidal structure, and is left and right complete.

  * •

The geometric $t$-structure is obtained by transferring the standard
(pointwise) $t$-structure on $\mathrm{Mod}_{R}^{\Delta^{1}}$ by means of the
self-equivalence
$\operatorname*{cofib}:\mathrm{Mod}_{R}^{\Delta^{1}}\simeq\mathrm{Mod}_{R}^{\Delta^{1}}$.

  * •

The heart $\mathrm{Mod}_{R,\heartsuit}^{\Delta^{1}}$ consists of arrows
$(f:X\twoheadrightarrow Z)$ for which $X$ and $\operatorname*{fib}(f)$ are
discrete $R$-modules. Note that $Z$ itself might not be discrete, as long as
$f:X\rightarrow Z$ is surjective on $\pi_{0}$.

  * •

There is an additive symmetric monoidal subcategory
$\mathrm{Mod}_{R,\heartsuit}^{\Delta^{1},\mathrm{surj}}\subset\mathrm{Mod}_{R,\heartsuit}^{\Delta^{1}}$
consisting of surjective maps $(f:X\twoheadrightarrow Z)$ where both $X$ and
$Z$ are discrete.

  * •

Let $\mathrm{Mod}_{R,0}^{\Delta^{1}}\subset\mathrm{Mod}_{R,\geq
0}^{\Delta^{1}}$ be the subcategory spanned by direct sums of objects
$R\rightarrow 0$ and $\operatorname{Id}:R\rightarrow R$. Then
$\mathrm{Mod}_{R,0}^{\Delta^{1}}$ is an additive subcategory of
$\mathrm{Mod}_{R,\heartsuit}^{\Delta^{1}}$, and it is closed under the action
of $\mathrm{SSeq}^{\mathrm{gen}}_{0}$ and
$\mathrm{SSeq}^{\mathrm{gen},\mathrm{pd}}_{0}$. Moreover, the objects
$R\rightarrow 0$ and $\operatorname{Id}:R\rightarrow R$ are compact and
projective generators of $\mathrm{Mod}_{R,\geq 0}$. Consequently, using
Construction 2.3.14, we obtain an action of $\mathrm{SSeq}^{\mathrm{gen}}$ on
$\mathrm{Mod}_{R}^{\Delta^{1}}$. In particular, we get an $\mathrm{LSym}$
monad
$\mathrm{LSym}_{R}^{\Delta^{1}}:\mathrm{Mod}^{\Delta^{1}}_{R}\rightarrow\mathrm{Mod}_{R}^{\Delta^{1}}$.
We denote the $\infty$-category of $\mathrm{LSym}^{\Delta^{1}}_{R}$-algebras
as $\mathrm{DAlg}_{R}^{\Delta^{1}}$ and refer to its objects derived
($R$-linear) arrows. A justification for this terminology will be given below
when we show that this is indeed the same $\infty$-category as the one defined
in Construction 3.1.16.

  * •

The additive subcategory
$\operatorname{Ind}(\mathrm{Mod}_{R,0}^{\Delta^{1}})\subset\mathrm{Mod}_{R}^{\Delta^{1}}$
is spanned by objects of the form $(f:X\twoheadrightarrow Z)$ where $X$ and
$Z$ are flat $R$-modules. The monad $\mathrm{LSym}^{\Delta^{1}}_{R}$ is
uniquely determined by its restriction to
$\operatorname{Ind}(\mathrm{Mod}_{R,0}^{\Delta^{1}})$.

  * •

There exists a tautological forgetful functor
$\mathrm{DAlg}_{R}^{\Delta^{1}}\rightarrow\operatorname{Fun}(\Delta^{1},\mathrm{DAlg}_{R})$.
To construct it, recall that in Construction 3.1.16 we let
$(\mathrm{Mod}_{R,0})^{\Delta^{1}}$ to be the category of all arrows of
finitely generated free $R$-modules; there is an inclusion
$\mathrm{Mod}_{R,0}^{\Delta^{1}}\subset(\mathrm{Mod}_{R,0})^{\Delta^{1}}$,
which induces a colimit-preserving fully faithful functor
$\operatorname{Ind}(\mathrm{Mod}_{R,0}^{\Delta^{1}})\hookrightarrow\operatorname{Ind}((\mathrm{Mod}_{R,0})^{\Delta^{1}})$
making commutative the following diagram:

$\textstyle{\operatorname{Ind}(\mathrm{Mod}_{R,0}^{\Delta^{1}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{LSym}_{R}^{\Delta^{1}}}$$\textstyle{\operatorname{Ind}((\mathrm{Mod}_{R,0})^{\Delta^{1}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\mathrm{LSym}_{R})^{\Delta^{1}}}$$\textstyle{\operatorname{Ind}(\mathrm{Mod}_{R,0}^{\Delta^{1}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Ind}((\mathrm{Mod}_{R,0})^{\Delta^{1}}).}$

Consequently, by right-left extension we obtain a map of monads
$\mathrm{LSym}^{\Delta^{1}}_{R}\rightarrow(\mathrm{LSym}_{R})^{\Delta^{1}}$,
and a functor
$\mathrm{DAlg}^{\Delta^{1}}_{R}\rightarrow\operatorname{Fun}(\Delta^{1},\mathrm{DAlg}_{R})$.

######  Proposition 3.1.18.

The functor
$\mathrm{DAlg}_{R}^{\Delta^{1}}\rightarrow(\mathrm{DAlg}_{R})^{\Delta^{1}}$ is
an equivalence.

###### Proof.

The forgetful functor is a functor of monadic $\infty$-categories over
$\mathrm{Mod}_{R}^{\Delta^{1}}$. Both monads preserve sifted colimits and are
filtered by subfunctors which preserve finite totalizations, the monad
$(\mathrm{LSym}_{R})^{\Delta^{1}}$ is determined by its values on the objects
$0\rightarrow R$ and $R\rightarrow 0$, and we have an equivalence
$\mathrm{LSym}^{\Delta^{1}}_{R}(R\rightarrow
0)\simeq(\mathrm{LSym}_{R})^{\Delta^{1}}(R\rightarrow 0)$ as $R\rightarrow 0$
belongs to the connective part of the geometric $t$-structure. Therefore, it
only remains to show that the natural map

$\mathrm{LSym}^{\Delta^{1}}(0\rightarrow R)\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern
3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\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{}$}}}}}}}\ignorespaces}}}}\ignorespaces(\mathrm{LSym}_{R})^{\Delta^{1}}(0\rightarrow
R)$ (3.4)

is an equivalence.

The object $(0\rightarrow R)\in\mathrm{Mod}_{R}^{\Delta^{1}}$ is connective in
the standard $t$-structure, and the free
$\mathrm{LSym}^{\Delta^{1}}_{R}$-algebra on it is the unit map $(R\rightarrow
R[x]).$ On the other hand, $(0\rightarrow R)$ is not connective in the
geometric $t$-structure. To compute
$\mathrm{LSym}^{\Delta^{1}}_{R}(0\rightarrow R)$, we can present
$(0\rightarrow R)$ as a totalization

$(0\rightarrow R)\simeq(0\rightarrow
R[1])[-1]\simeq\underset{n\in\Delta}{\operatorname{Tot}}(0\rightarrow
R[1]^{\oplus^{n}}),$ (3.5)

and use the colimit formula of Construction 2.2.12

$\mathrm{LSym}^{\Delta^{1}}_{R}:=\underset{i}{\operatorname*{colim}}\;\mathrm{LSym}^{\Delta^{1},\leq
i}_{R},$ (3.6)

where each filtered layer $\mathrm{LSym}^{\Delta^{1},\leq i}_{R}$ commutes
with finite totalizations being an $i$-excisive functor.

Since $(0\rightarrow R[1]^{\oplus^{n}})$ is connective in geometric
$t$-structure, we have by definition of the $\mathrm{LSym}_{R}^{\Delta^{1}}$
functor:

$\mathrm{LSym}_{R}^{\Delta^{1}}(0\rightarrow
R[1]^{\oplus^{n}})\simeq(R\rightarrow\mathrm{LSym}_{R}(R[1]^{\oplus^{n}}).$

Consequently, combining formulas 3.5 and 3.6, we obtain

$\mathrm{LSym}^{\Delta^{1}}_{R}(0\rightarrow
R)\simeq\underset{i}{\operatorname*{colim}}\;\underset{n\in\Delta}{\operatorname{Tot}}\Bigl{(}R\rightarrow\mathrm{LSym}^{\leq
i}_{R}(R^{\oplus^{n}}[1])\Bigr{)}.$

As the map 3.4 is compatible with filtrations, and the same totalization
formula works for the standard $(\mathrm{LSym}_{R})^{\Delta^{1},\leq
n}$-functors, we conclude that it is an equivalence. ∎

######  Example 3.1.19.

Let $R=\mathbb{Z}$. We compute in $\mathrm{DAlg}_{\mathbb{Z}}^{\Delta^{1}}$:

$\mathrm{LSym}^{\Delta^{1}}_{\mathbb{Z}}(\operatorname{Id}:\mathbb{Z}\rightarrow\mathbb{Z})\simeq(\operatorname{Id}:\mathbb{Z}[x]\rightarrow\mathbb{Z}[x]);$
$\mathrm{LSym}^{\Delta^{1}}_{\mathbb{Z}}(\mathbb{Z}\rightarrow
0)\simeq(\mathbb{Z}[x]\rightarrow\mathbb{Z},x\longmapsto 0).$

The corresponding Smith ideals in the polynomial ring on one generator are
$(0)\subset\mathbb{Z}[x]$ and $(x)\subset\mathbb{Z}[x]$ respectively.

######  Remark 3.1.20.

When $R$ is a discrete commutative ring, the initial object in the
$\infty$-category $\mathrm{DAlg}_{R}^{\Delta^{1}}$ is the arrow
$(\operatorname{Id}:R\rightarrow R)$. Therefore, when $R$ is a not necessarily
discrete derived algebra, we can define the $\infty$-category of geometric
$R$-linear arrows as the under-category

$\mathrm{DAlg}^{\Delta^{1}}_{R}:=(\mathrm{DAlg}_{\mathbb{Z}}^{\Delta^{1}})_{R/},$

where $R$ is considered as a tautological arrow
$(\operatorname{Id}:R\rightarrow R)$ in
$\mathrm{DAlg}_{\mathbb{Z}}^{\Delta^{1}}$.

######  Definition 3.1.21.

Let $R$ be a connective derived ring, and $\mathrm{DAlg}_{R}^{\Delta^{1}}$ be
the $\infty$-category of derived $R$-linear arrows, and
$\mathrm{DAlg}_{R}^{\Delta^{1},\mathrm{aug}}$ be the $\infty$-category of
objects of $(f:A\rightarrow B)\in\mathrm{DAlg}^{\Delta^{1}}_{R}$ endowed with
a splitting $(s:B\rightarrow A)$ in $\mathrm{DAlg}_{R}$.

  1. 1.

We define the $\infty$-category of derived Smith ideals as the pull-back of
$\infty$-categories along the bottom horizontal equivalence:

$\textstyle{\mathrm{DAlg}_{R}^{\Delta^{1}_{\vee}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname*{cofib}}$$\scriptstyle{\sim}$$\textstyle{\mathrm{DAlg}_{R}^{\Delta^{1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{Mod}^{\Delta^{1}_{\vee}}_{R}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname*{cofib}}$$\scriptstyle{\sim}$$\textstyle{\mathrm{Mod}_{R}^{\Delta^{1}}.}$

By construction, the forgetful functor
$\mathrm{DAlg}_{R}^{\Delta^{1}_{\vee}}\rightarrow\mathrm{Mod}^{\Delta^{1}_{\vee}}_{R}$
is monadic. We denote

$\mathrm{LSym}_{R}^{\Delta^{1}_{\vee}}:\mathrm{Mod}^{\Delta^{1}_{\vee}}_{R}\rightarrow\mathrm{DAlg}_{R}^{\Delta^{1}_{\vee}}$

the left adjoint of the forgetful functor.

  2. 2.

We also define an $\infty$-category $\mathrm{DAlg}_{R}^{\\{0,1\\},\vee}$,
monadic over $\mathrm{Mod}_{R}^{\\{0,1\\},\vee}$, by means of the pull-back
diagram

$\textstyle{\mathrm{DAlg}_{R}^{\\{0,1\\},\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{\mathrm{DAlg}_{R}^{\Delta^{1},\mathrm{aug}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{CAlg}^{\\{0,1\\},\vee}_{R}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{\operatorname{CAlg}^{\Delta^{1},\mathrm{aug}}_{R},}$

where the lower horizontal equivalence is the equivalence from Proposition
3.1.14, and let

$\mathrm{LSym}_{R}^{\\{0,1\\},\vee}:\mathrm{Mod}_{R}^{\\{0,1\\},\vee}\rightarrow\mathrm{DAlg}^{\\{0,1\\},\vee}_{R}$

be the free algebra functor.

######  Remark 3.1.22.

There is an adjunction

$\textstyle{\mathrm{DAlg}^{\\{0,1\\},\vee}_{R}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{aug}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathrm{DAlg}^{\Delta^{1}_{\vee}}_{R}}$$\scriptstyle{(\operatorname{Fil}\rightarrow\mathrm{Gr})}$

lifting the adjunction noted in Remark 3.1.11.

######  Remark 3.1.23.

Given a map $B\rightarrow A$ endowed with a section $A\rightarrow B$ is the
same as endowing $B$ with the structure of an augmented $A$-algebra. We have
$B\simeq A\oplus A^{+}$ where $A^{+}:=\operatorname*{fib}(B\rightarrow A)$,
and $A^{+}$ gets the structure of a non-unital $A$-algebra. Therefore, we can
think of the $\infty$-category $\mathrm{DAlg}^{\\{0,1\\},\vee}_{R}$ as the
$\infty$-category of pairs $(A,A^{+})$ where $A\in\mathrm{DAlg}_{R}$ and
$A^{+}$ is a non-unital derived $A$-algebra.

For a connective derived ring $R$, we will construct a $t$-exact lax symmetric
monoidal functor $\operatorname{Fil}^{\geq
0}\mathrm{Mod}_{R}\rightarrow\mathrm{Mod}_{R}^{\Delta^{1}_{\vee}}$ and extend
it to a functor $\operatorname{Fil}^{\geq
0}\mathrm{DAlg}_{R}\rightarrow\mathrm{DAlg}^{\Delta^{1}_{\vee}}_{R}.$

######  Construction 3.1.24.

Let $\Delta^{1}\hookrightarrow\mathbb{Z}_{\geq 0}$ be the inclusion of $0$ and
$1$. Notice that it is a lax symmetric monoidal functor with respect to the
symmetric monoidal structure on $\Delta^{1}$ described in Construction 3.1.1.
It induces a limit preserving lax symmetric monoidal functor on presheaf
categories $\operatorname{ev}^{[0,1]}:\operatorname{Fil}^{\geq
0}\mathrm{Mod}_{R}\rightarrow\mathrm{Mod}_{R}^{\Delta^{1}_{\vee}}$ which sends
an object $F^{\geq\star}$ to the arrow $F^{\geq 1}\rightarrow F^{\geq 0}$. In
fact, the lax symmetric monoidal structure is strict. Indeed, composing the
functor $\operatorname{ev}^{[0,1]}$ with the symmetric monoidal equivalence
$\operatorname*{cofib}:\mathrm{Mod}_{R}^{\Delta^{1}_{\vee}}\simeq\mathrm{Mod}_{R}^{\Delta^{1}}$,
we get the functor sending a filtered object $F^{\geq\star}$ to the arrow
$F^{\geq 0}\rightarrow\operatorname{gr}_{F}^{0}$, which is manifestly
symmetric monoidal. Therefore, so is the functor $\operatorname{ev}^{[0,1]}$.

  * •

Assume $R$ is a discrete ring. The functor $\operatorname{ev}^{[0,1]}$ is
continuous and sends $\operatorname{Fil}^{\geq
0}\mathrm{Mod}_{R,0}\subset\operatorname{Fil}^{\geq 0}\mathrm{Mod}_{R}$ to
$\mathrm{Mod}^{\Delta^{1}_{\vee}}_{R,0}\subset\mathrm{Mod}_{R}^{\Delta^{1}_{\vee}}$.
Consequently, we get a commutative square

$\textstyle{\operatorname{Ind}(\operatorname{Fil}\mathrm{Mod}_{R,0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{LSym}_{R}^{\geq\star}}$$\scriptstyle{\operatorname{ev}^{[0,1]}}$$\textstyle{\operatorname{Ind}(\mathrm{Mod}^{\Delta^{1}_{\vee}}_{R,0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{LSym}^{\Delta^{1}_{\vee}}_{R}}$$\textstyle{\operatorname{Ind}(\operatorname{Fil}\mathrm{Mod}_{R,0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{ev}^{[0,1]}}$$\textstyle{\operatorname{Ind}(\mathrm{Mod}^{\Delta^{1}_{\vee}}_{R,0}).}$

The functor $\operatorname{ev}^{[0,1]}$ is $t$-exact with respect to the
geometric $t$-structure $\mathrm{Mod}_{R}^{\Delta^{1}_{\vee}}$. Consequently,
using Observation 2.3.15 and the fact that both monads involved are right-left
extended, we obtain a functor

$\operatorname{ev}^{[0,1]}:\operatorname{Fil}^{\geq
0}\mathrm{DAlg}_{R}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\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{}$}}}}}}}\ignorespaces}}}}\ignorespaces\mathrm{DAlg}^{\Delta^{1}_{\vee}}_{R}.$

  * •

If $R$ is a not necessarily discrete connective derived ring, we consider $R$
as a filtered ring with $R$ inserted in degree $0$ and zero in all other
degrees, and notice that

$\operatorname{Fil}^{\geq 0}\mathrm{DAlg}_{R}\simeq(\operatorname{Fil}^{\geq
0}\mathrm{DAlg}_{\mathbb{Z}})_{R/},\;\;\;\;\mathrm{DAlg}_{R}^{\Delta^{1}_{\vee}}\simeq(\mathrm{DAlg}^{\Delta^{1}_{\vee}}_{\mathbb{Z}})_{R/}.$

In this case, the functor $\operatorname{ev}^{[0,1]}:\operatorname{Fil}^{\geq
0}\mathrm{DAlg}_{R}\rightarrow\mathrm{DAlg}^{\Delta^{1}_{\vee}}_{R}$ is
obtained from the previous part of the construction as an induced functor on
objects under $R$.

######  Remark 3.1.25.

Using Definition 3.1.21, let us describe limits and sifted colimits in the
$\infty$-category $\mathrm{DAlg}_{R}^{\Delta^{1}_{\vee}}$. In the
$\infty$-category $\mathrm{DAlg}_{R}^{\Delta^{1}}$ limits and colimits are
taken pointwise. Let
$S\rightarrow\mathrm{DAlg}_{R}^{\Delta^{1}_{\vee}},i\longmapsto(A_{i},I_{i})$
be a diagram in $\mathrm{DAlg}_{R}^{\Delta^{1}_{\vee}}$. Applying the functor
$\operatorname*{cofib}:\mathrm{DAlg}_{R}^{\Delta^{1}_{\vee}}\rightarrow\mathrm{DAlg}_{R}^{\Delta^{1}}$,
we get a diagram $\\{A_{i}\rightarrow A_{i}/I_{i}\\}$ in
$\mathrm{DAlg}_{R}^{\Delta^{1}}$. Let $(A\rightarrow
A/I):=\lim_{i}(A_{i}\rightarrow A_{i}/I_{i})$. Applying the functor
$\operatorname*{fib}$, we get that the colimit of the diagram
$\\{(A_{i},I_{i})\\}$ is given by $(I,A)$, where $A=\lim_{i}A_{i}$ and

$I=\operatorname*{fib}(\lim_{i}A\rightarrow\lim_{i}A_{i}/I_{i}))\simeq\lim_{i}\operatorname*{fib}(A_{i}\rightarrow
A_{i}/I_{i})\simeq\lim_{i}I_{i},$

where each term $I_{i}$ is endowed with the $A$-module structure obtained by
restricting via the natural map $A\rightarrow A_{i}$.

Similarly, let
$(A^{\prime},I^{\prime})=\operatorname*{colim}_{i}(A_{i},I_{i})$. Then
$A^{\prime}=\operatorname*{colim}_{i}A_{i}$, and $I^{\prime}$ is computed as

$I^{\prime}=\operatorname*{fib}(\operatorname*{colim}_{i}A_{i}\rightarrow\operatorname*{colim}_{i}A_{i}/I_{i}).$

Assume $S$ is a sifted diagram. Then as the forgetful functor
$\mathrm{DAlg}_{R}^{\Delta^{1}_{\vee}}\rightarrow\mathrm{Mod}_{R}^{\Delta^{1}}$
commutes with sifted colimits, and in the stable $\infty$-category
$\mathrm{Mod}_{R}$ finite limits commute with sifted colimits, we obtain that

$I^{\prime}=\operatorname*{colim}_{i}(I_{i}\otimes_{A_{i}}A^{\prime})$

as an $A^{\prime}$-module.

######  Remark 3.1.26.

Consider the category
$\operatorname{CAlg}_{\mathbb{Z},\heartsuit}^{\Delta^{1}_{\vee},\mathrm{inj}}$
of pairs $(I\rightarrow A)$ where $A$ is a discrete commutative ring, and
$I\hookrightarrow A$ is a usual ideal. One can show that
$\operatorname{CAlg}_{\mathbb{Z},\heartsuit}^{\Delta^{1}_{\vee},\mathrm{inj}}$
has all colimits, but the forgetful functor
$\operatorname{CAlg}_{\mathbb{Z},\heartsuit}^{\Delta^{1}_{\vee},\mathrm{inj}}\rightarrow\operatorname{CAlg}_{\mathbb{Z},\heartsuit},(I\rightarrow
A)\longmapsto A$ does not commute with colimits. An explicit example is worked
out in [Mao21, Example 3.33]. The computation of Zhouhang Mao is motivated by
[Stacks, Remark 23.3.5], where the same example is brought to demonstrate the
discrepancy between coproducts in the category of divided power algebras and
tensor products of underlying algebras. In the setting of derived Smith ideals
the same problem does not arise, even in the divided power case.

We will finish this subsection with the following construction which will be
useful in the sequel.

######  Construction 3.1.27.

Let
$\mathrm{ins}^{0}:\mathrm{DAlg}_{R}\rightarrow\mathrm{DAlg}_{R}^{\Delta^{1}_{\vee}}$
be the functor sending $A$ the trivial Smith ideal $0\rightarrow A$ of $A$.
There is an adjunction

$\textstyle{\mathrm{DAlg}_{R}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{ins}^{0}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathrm{DAlg}_{R}^{\Delta^{1}_{\vee}},}$$\scriptstyle{\operatorname{gr}^{0}}$$\scriptstyle{\operatorname{ev}^{0}}$

where $\operatorname{gr}^{0}(I\rightarrow A)=A/I$ and
$\operatorname{ev}^{0}(I\rightarrow A)=A$.

###  3.2 Divided power algebras.

Based on Definition 3.1.21, given a map $A\rightarrow B$, we can construct a
unique derived Smith ideal $I\rightarrow A$ in $A$, and formulate various
properties or additional structures on the map $A\rightarrow B$ in terms of
the Smith ideal $I$. In this text we will define divided power structure on a
map of derived commutative rings as the data of a non-unutal derived divided
power structure on the underlying non-unital derived commutative algebra of
$I$. Below we shall explain our reasons for introducing divided power
algebras.

There are two ways to think about De Rham cohomology theory, a more well-known
one is in terms of differential graded commutative algebras, and another one
is in terms of divided power algebras. Assume $A$ is a discrete commutative
algebra. Then there is a classical universal property of the De Rham complex
of $A$: for any map $A\rightarrow B^{0}$ into the $0$-th graded piece of a
differential graded commutative algebra $B^{\bullet}$, there is a unique map
of differential graded commutative algebras $\Omega^{\bullet}_{A}\rightarrow
B^{\bullet}$. Moreover, we have an equivalence of graded algebras
$\Omega^{\bullet}_{A}\simeq\Lambda^{\bullet}_{A}(\Omega^{1}_{A})$ with the
differential extended from the universal derivation
$d:A\rightarrow\Omega_{A}^{1}$ via the graded Leibnitz rule. A.Raksit [R]
generalized this universal property to the derived setting. In this text we
will develop another approach in terms of derived _divided power_ algebras,
rather than differential graded commutative algebras. It can be briefly
summarized as follows. One defines an $\infty$-category
$\operatorname{Fil}^{\geq 0}\mathrm{DAlg}^{\mathrm{pd}}_{R}$ of non-negatively
filtered derived divided power algebras, and considers the functor
$\operatorname{gr}^{0}:\operatorname{Fil}^{\geq
0}\mathrm{DAlg}^{\mathrm{pd}}_{R}\rightarrow\mathrm{DAlg}_{R}$. The derived De
Rham complex $\mathrm{L\Omega}^{\geq\star}_{-/R}$ is defined as the left
adjoint of this functor. The main reason why this recovers the classical De
Rham complex in the case of a smooth map $R\rightarrow A$, is essentially due
to the décalage isomorphism

$\mathrm{L\Gamma}^{n}(M[-1])\simeq\mathrm{L\Lambda}^{n}(M)[-n]$

demonstrated in Proposition 2.4.10. Namely, the associated graded object of
$\mathrm{L\Omega}^{\geq\star}_{A/R}$ is given by

$\mathrm{L\Omega}^{\bullet}_{A/R}\simeq\mathrm{L\Gamma}^{\bullet}_{A}(\mathbb{L}_{A}[-1])\simeq\mathrm{L\Lambda}^{\bullet}_{A}(\mathbb{L}_{A})[-n],$

where $\mathbb{L}_{A/R}$ is the cotangent complex of the map $R\rightarrow A$,
and in the case when $\mathbb{L}_{A/R}$ is a projective $A$-module, the
divided power structure on $\mathrm{L\Omega}_{A/R}$ is equivalent to the data
of a differential graded commutative algebra. The main reason why we prefer to
use think about De Rham cohomology in terms of derived divided power algebras,
is because various constructions are more natural and easier to define in this
setting. This particularly applies to stacky approaches to derived De Rham
cohomology, relation with crystalline and prismatic cohomology. We will show
that the relative derived De Rham cohomology $\mathrm{L\Omega}_{A/B}$ of a map
$A\rightarrow B$ can be understood as the derived pd filtration of the derived
pd envelope of $A$ with respect to the Smith ideal
$I=\operatorname*{fib}(A\rightarrow B)$. In the case when the map
$A\rightarrow B$ is a surjective map of discrete commutative algebras whose
ideal $I$ is generated by a regular sequence, this recovers the classical
divided power envelope. On the other hand, when the map $A\rightarrow B$ is a
smooth map of discrete commutative rings, the negative $t$-structure together
with décalage isomorphism enables us to think about the relative De Rham
complex $\Omega_{A/B}$ in terms of the exterior algebra on the module of
relative Kahler differentials $\Omega^{1}_{B/A}$.

We will finish this introduction with a historical remark. Divided power
algebras featured prominently in algebraic topology, i.e. in the context of
homology of loop spaces, and were brought to world of algebraic geometry by
A.Grothendieck [Gr] in the context of crystalline cohomology. The basic idea
of divided power algebras is that outside of characteristic $0$, we can not
form $\frac{x^{n}}{n!}$ of an element in an arbitrary algebra $A$, and a
divided power structure on $A$ is an additional structure which axiomatizes
the existence of such operations on some ideal. Initially, the definition of a
divided power algebra was given in terms of the data of operations
$\gamma_{n}(x)$ for all $n$ satisfying a number of conditions reflecting the
idea that $\gamma_{n}(x)$ should behave like $\frac{x^{n}}{n!}$. B.Fresse in
the text [Fr] explained an operadic approach to divided power algebras. The
manuscript of Brantner, Campos and Nuiten [BCN21] developed the notion of
derived pd operads in a very general setting, which we follow in this text.

####  3.2.1 Non-unital divided power algebras.

Recall the following classical definition.

######  Definition 3.2.1.

A (discrete) non-unital divided power ring is a non-unital commutative ring
$A$ endowed with operations $\gamma_{n}:A\rightarrow A$ satisfying the
following relations:

  1. 1.

$\gamma_{0}(x)=1,\gamma_{1}(x)=x$ for any $x\in A$;

  2. 2.

$\gamma_{n}(x+y)=\sum_{i+j=n}\gamma_{i}(x)\gamma_{j}(y)$ for any $x,y\in A$;

  3. 3.

$\gamma(ax)=a^{n}\gamma(x)$ for any $a,x\in A$;

  4. 4.

$\gamma_{m}(x)\gamma_{n}(x)=\frac{(m+n)!}{m!n!}\gamma_{m+n}(x)$;

  5. 5.

$\gamma_{n}(\gamma_{m}(x))=\frac{(mn)!}{(m!)^{n}n!}\gamma_{mn}(x)$.

We let $\operatorname{CAlg}^{\mathrm{pd}}_{\mathbb{Z},\heartsuit}$ to be the
category of discrete _non-unital_ 999There is no notion of a “unital” divided
power algebra, because the unit never supports divided powers unless we are in
characteristic $0$ divided power rings.

######  Remark 3.2.2.

It was observed by B.Fresse in [Fr] that this category is monadic over
$\mathrm{Mod}_{\mathbb{Z},\heartsuit}$ with the monad given by the formula

$\Gamma^{+}(M)=\bigoplus_{n\geq 1}\Gamma^{n}(M)=\bigoplus_{n\geq
1}(M^{\otimes^{n}})^{\Sigma_{n}}.$

There is a norm map $\operatorname{Sym}(M)\rightarrow\Gamma^{+}(M)$ given in
every degree by the map $\operatorname{Sym}^{n}(M)\rightarrow\Gamma^{n}(M)$
induced by the $\Sigma_{n}$-norm map

$\operatorname{Nm}_{\Sigma_{n}}:M^{\otimes^{n}}\rightarrow
M^{\otimes^{n}},x_{1}\otimes...\otimes
x_{n}\longmapsto\sum_{\sigma\in\Sigma_{n}}x_{\sigma(1)}\otimes...\otimes
x_{\sigma(n)}$

One can think of the structure of a non-unital divided power algerba $A$ as a
non-unital commutative algebra endowed with a factorization of the
multiplication maps $m_{n}:A^{\otimes^{n}}\rightarrow A$ through the norm as
follows

---
$\textstyle{A^{\otimes^{n}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{Nm}_{\Sigma_{n}}}$$\scriptstyle{m_{n}}$$\textstyle{A}$$\textstyle{A^{\otimes^{n}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces.}$$\scriptstyle{\gamma_{n}}$

for any $n$, and these factorizations satisfy some additional associativity
conditions.

We call $\Gamma^{+}(M)$ the free non-unital divided power algebra on $M$.
There is a subcategory
$\operatorname{CAlg}^{\mathrm{nu},\mathrm{pd}-\mathrm{poly}}_{\mathbb{Z}}\subset\operatorname{CAlg}^{\mathrm{nu},\mathrm{pd}}_{\mathbb{Z},\heartsuit}$
of compact projective generators consisting of free non-unital divided power
algebras on finitely many variables. One can then define an $\infty$-category

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

which is monadic over $\mathrm{Mod}_{\mathbb{Z},\geq 0}$, and one can show
that it is also comonadic over $\mathrm{DAlg}_{\mathbb{Z},\geq 0}$ (see 3.2.6
for precise argument).

######  Example 3.2.3.

The free non-untal divided power algebra on one generator
$\Gamma^{+}(\mathbb{Z})\simeq\mathbb{Z}\langle x\rangle^{+}$ can be described
as the smallest non-unital subalgebra of $\mathbb{Q}[x]$ containing all
elements of the form $\frac{x^{n}}{n!}.\in\mathbb{Q}[x]$ .

######  Remark 3.2.4.

Let $(-)^{\sharp}:\mathrm{DAlg}^{\mathrm{nu}}_{\mathbb{Z},\geq
0}\rightarrow\mathrm{DAlg}^{\mathrm{nu},\mathrm{pd}}_{\mathbb{Z},\geq 0}$ be
the right adjoint of the forgetful functor. For any connective
$A\in\mathrm{DAlg}^{\mathrm{nu}}_{\mathbb{Z},\geq 0}$, there is an equivalence
of spaces

$A^{\sharp}\simeq\operatorname{Map}_{\mathrm{DAlg}^{\mathrm{nu}}_{\mathbb{Z},\geq
0}}(\mathbb{Z}\langle x\rangle^{+},A).$

For a discrete $A$, the latter can be understood as the set of divided power
sequences in $A$, i.e. the subset of $\prod_{n=1}^{\infty}A$ consisting of
collections of elements $(x_{1},x_{2},...)$ satisfying

$x_{n}x_{m}=\frac{(m+n)!}{m!\>n!}x_{n+m}.$

######  Definition 3.2.5.

Let $R$ be a connective derived commutative ring. We define the
$\infty$-category of (non-unital) derived divided power algebras

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

as the $\infty$-category of algebras over the non-unital derived divided power
algebra monad $\mathrm{L\Gamma}^{+}_{R}$ provided by Construction 2.3.11.

######  Proposition 3.2.6.

Let $R$ be a connective derived commutative ring, and
$\mathrm{DAlg}_{R}^{\mathrm{nu}}$ the $\infty$-category of non-unital derived
commutative $R$-algebras. The forgetful functor
$\mathrm{DAlg}^{\mathrm{nu},\mathrm{pd}}_{R}\rightarrow\mathrm{DAlg}^{\mathrm{nu}}_{R}$
commutes with limits and colimits.

###### Proof.

We argue as in the proof in [R, Proposition 4.2.27]. Commutation with limits
follows from the fact that both sides are $\infty$-categories of algebras over
a monad, and the forgetful functor is restriction along a monad map.
Commutation with sifted colimits follows from commutation of the monad
$\mathrm{L\Gamma}^{+}_{R}:\mathrm{Mod}_{R}\rightarrow\mathrm{Mod}_{R}$ with
sifted colimits. For commutation with finite coproducts, one reduces to the
statement that for two objects
$X,Y\in\mathrm{Mod}^{\mathrm{free},\mathrm{fg}}_{R}$, the natural map
$\mathrm{L\Gamma}^{+}_{R}(X)\otimes\mathrm{L\Gamma}^{+}_{R}(Y)\rightarrow\mathrm{L\Gamma}^{+}_{R}(X\oplus
Y)$ is an equivalence, which follows from the case $R=\mathbb{Z}$ by base
change. ∎

######  Definition 3.2.7.

We let
$\operatorname{Env}^{\mathrm{pd}}:\mathrm{DAlg}^{\mathrm{nu}}_{R}\rightarrow\mathrm{DAlg}^{\mathrm{pd}}_{R}$
and
$\mathrm{coEnv}^{\mathrm{pd}}:\mathrm{DAlg}^{\mathrm{nu}}_{R}\rightarrow\mathrm{DAlg}^{\mathrm{pd}}_{R}$
be the left adjoint and the right adjoint of the forgetful functor,
respectively. We call them divided power (pd) envelope and divided power (pd)
coenvelope.

######  Construction 3.2.8.

Let $\mathrm{DAlg}^{\mathrm{pd}}_{R,\geq
0}\subset\mathrm{DAlg}^{\mathrm{pd}}_{R}$ be the full subcategory of objects
whose underlying $\mathrm{Mod}_{R}$-object is connective. As the free derived
divided power algebra monad
$\mathrm{L\Gamma}^{+}_{R}:\mathrm{Mod}_{R}\rightarrow\mathrm{Mod}_{R}$
preserves connective objects, it follows that the functor
$\mathrm{L\Gamma}^{+}_{R}:\mathrm{Mod}_{R,\geq
0}\rightarrow\mathrm{DAlg}^{\mathrm{pd}}_{R,\geq 0}$, left adjoint of the
forgetful functor $\mathrm{DAlg}^{\mathrm{pd}}_{R,\geq
0}\rightarrow\mathrm{Mod}_{R,\geq 0}$, is well-defined, and commutes with the
inclusion $\mathrm{DAlg}^{\mathrm{pd}}_{R,\geq
0}\hookrightarrow\mathrm{DAlg}^{\mathrm{pd}}_{R}$. Since the inclusion
commutes with all colimits, it follows that there exists a right adjoint
connective cover functor $\tau_{\geq
0}:\mathrm{DAlg}^{\mathrm{pd}}_{R}\rightarrow\mathrm{DAlg}^{\mathrm{pd}}_{R,\geq
0}$ which fits into a commutative diagram

$\textstyle{\mathrm{DAlg}^{\mathrm{pd}}_{R}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau_{\geq
0}}$$\textstyle{\mathrm{DAlg}^{\mathrm{pd}}_{R,\geq
0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{DAlg}^{\mathrm{nu}}_{R}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau_{\geq
0}}$$\textstyle{\mathrm{DAlg}^{\mathrm{nu}}_{R},}$

where the vertical functors are forgetful functors.

In what follows we will also need to work with divided power algebras over not
necessarily connective derived rings. To explain how to do it, consider how
this is done in the context of ordinary (without divided powers) derived
rings. First, we have the $\infty$-category $\mathrm{DAlg}_{\mathbb{Z}}$ of
all derived rings. Then given a not necessarily connective
$R\in\mathrm{DAlg}_{\mathbb{Z}}$, we can consider the $\infty$-category
$\mathrm{DAlg}_{R}:=(\mathrm{DAlg}_{\mathbb{Z}})_{R/}$ of under objects. For
connective $R$, this is equivalent to the $\infty$-category of algebras over
the monad $\mathrm{LSym}_{R}$. Over a not necessarily connective $R$, this
$\infty$-category is monadic over $\mathrm{Mod}_{R}$, and we can consider this
to be the definition of the monad $\mathrm{LSym}_{R}$ for not necessarily
connective $R$. We can work similarly in the setting of non-unital derived
divided power algebras.

######  Definition 3.2.9.

Recall that the $\infty$-category $\mathrm{DAlg}_{\mathbb{Z}}^{\mathrm{nu}}$
of non-unital derived rings is equivalent to the $\infty$-category of
augmented derived rings. Composing the forgetful functor
$\mathrm{DAlg}_{\mathbb{Z}}^{\mathrm{nu},\mathrm{pd}}\rightarrow\mathrm{DAlg}^{\mathrm{nu}}_{\mathbb{Z}}$
with the equivalence
$\mathrm{DAlg}^{\mathrm{nu}}_{\mathbb{Z}}\simeq\mathrm{DAlg}^{\mathrm{aug}}_{\mathbb{Z}}$,
we obtain a forgetful functor
$\mathrm{DAlg}_{\mathbb{Z}}^{\mathrm{pd},\mathrm{nu}}\rightarrow\mathrm{DAlg}_{\mathbb{Z}}^{\mathrm{aug}}$
sending $A^{+}\in\mathrm{DAlg}_{\mathbb{Z}}^{\mathrm{pd},\mathrm{nu}}$ to the
pair $(\mathbb{Z}\oplus A^{+},A^{+})$. Let $R$ be a derived ring, not
necessarily connective. Consider it as an object $(R,0)$ of
$\mathrm{DAlg}_{\mathbb{Z}}^{\mathrm{aug}}$ with $0$ augmentation ideal. Let
$(\mathrm{DAlg}^{\mathrm{aug}}_{\mathbb{Z}})_{R/}$ the $\infty$-category of
derived pairs receiving a map from $(R,0)$. Then the $\infty$-category of
derived divided power non-unital $R$-algebras can be defined as the
$\infty$-category of derived non-unital divided power $\mathbb{Z}$-algebras
whose underlying augmented derived ring $(\mathbb{Z}\oplus A^{+},A^{+})$
receives a map from $(R,0)$:

$\mathrm{DAlg}_{R}^{\mathrm{pd},\mathrm{nu}}:=\mathrm{DAlg}_{\mathbb{Z}}^{\mathrm{pd},\mathrm{nu}}\underset{\mathrm{DAlg}^{\mathrm{aug}}_{\mathbb{Z}}}{\times}(\mathrm{DAlg}^{\mathrm{aug}}_{\mathbb{Z}})_{R/}.$

There is a forgetful functor
$\mathrm{DAlg}_{R}^{\mathrm{pd},\mathrm{nu}}\rightarrow\mathrm{Mod}_{R}$ which
is monadic. We denote
$\mathrm{L\Gamma}_{R}^{+}:\mathrm{Mod}_{R}\rightarrow\mathrm{Mod}_{R}$ the
free algebra monad.

######  Remark 3.2.10.

If $R$ is connective, then Definition 3.2.9 is equivalent to Definition 3.2.5.
To see this, we can assume that $R$ is discrete, and then it is sufficient to
describe the category of discrete divided power $R$-algebras. Then any object
of the category defined as in Definition 3.2.9 gives rise to a divided power
$\mathbb{Z}$-algerba $A^{+}$ together with an action of $R$ on $A^{+}$ turning
$A^{+}$ commuting with divided power operations in the usual sense. The slogan
is that a divided power $R$-algebra is the same as a divided power ring
together with a $R$-linear structure on the underlying non-unital ring. This
will be often used throughout the text.

####  3.2.2 Divided power Smith ideals.

A global version of the notion of a non-unital divided power algebra is the
notion of a divided power pair, or a divided power Smith ideal. To define this
notion, recall that for any Smith ideal $I$ in $A$, the $A$-module
$I=\operatorname*{fib}(A\rightarrow A/I)$ gets a natural structure of a non-
unital derived commutative algebra in $A$-modules. A homotopy coherent version
of this construction is provided by the functor
$\mathrm{DAlg}_{R}^{\Delta^{1}_{\vee}}\rightarrow\mathrm{DAlg}_{R}^{\\{0,1\\},\vee}$
induced by the lax symmetric monoidal functor
$\mathrm{Mod}_{R}^{\Delta^{1}_{\vee}}\rightarrow\mathrm{Mod}_{R}^{\\{0,1\\},\vee}$
from Construction 3.1.9. As we observed in Remark 3.1.23, the
$\infty$-category $\mathrm{DAlg}_{R}^{\\{0,1\\},\vee}$ is equivalent to the
$\infty$-category of augmented derived $R$-algebras, i.e. pairs $(A,A^{+})$
where $A\in\mathrm{DAlg}_{R}$ and $A^{+}\in\mathrm{DAlg}_{A}^{\mathrm{nu}}$.
In Definition 3.2.9 we explained how to define non-unital derived divided
power algebras over a not necessarily connective base ring. This enables us to
give the following definition.

######  Definition 3.2.11.

We define an $\infty$-category
$\mathrm{DAlg}^{\\{0,1\\},\vee,\mathrm{pd}}_{R}$ as the $\infty$-category of
pairs $(A,A^{+})$ where $A^{+}$ has the structure of a non-unital divided
power $A$-algebra in the sense of Definition 3.2.9. There is a non-
conservative forgetful functor
$\mathrm{DAlg}^{\\{0,1\\},\vee,\mathrm{pd}}_{R}\rightarrow\mathrm{DAlg}_{R}$
(projection on the first component), and a conservative forgetful functor
$\mathrm{forget}_{\mathrm{pd}}:\mathrm{DAlg}_{R}^{\\{0,1\\},\vee,\mathrm{pd}}\rightarrow\mathrm{DAlg}_{R}^{\\{0,1\\},\vee}$,
forgetting the divided power structure on the second component. The left
adjoint of the latter
$\operatorname{Env}^{\mathrm{pd}}:\mathrm{DAlg}_{R}^{\\{0,1\\},\vee}\rightarrow\mathrm{DAlg}_{R}^{\\{0,1\\},\vee,\mathrm{pd}}$
sends a pair $(A,A^{+})$ to $(A,\operatorname{Env}^{\mathrm{pd}}(A^{+}))$,
where $\operatorname{Env}^{\mathrm{pd}}(A^{+}$ is the non-unital pd envelope
of the non-unital $A$-algebra $A^{+}$. The forgetful functor
$\mathrm{forget}_{\mathrm{pd}}$ is monadic, and so is the forgetful functor
$\mathrm{DAlg}^{\\{0,1\\},\vee,\mathrm{pd}}_{R}\rightarrow\mathrm{Mod}_{R}^{\\{0,1\\},\vee}$.
We denote
$(\mathrm{LSym},\mathrm{L\Gamma}^{+}):\mathrm{Mod}_{R}^{\\{0,1\\},\vee}\rightarrow\mathrm{Mod}_{R}^{\\{0,1\\},\vee}$
the corresponding monad, which satisfies the formula

$(\mathrm{LSym},\mathrm{L\Gamma}^{+})(X,Y)\simeq(\mathrm{LSym}_{R}(X),\mathrm{L\Gamma}^{+}_{\mathrm{LSym}_{R}(X)}(Y\otimes\mathrm{LSym}_{R}(X))).$

If $R$ is discrete, this monad is right-left extended from its restriction to
the full subcategory
$\operatorname{Ind}(\mathrm{Mod}^{\\{0,1\\},\vee}_{R,0})$, where
$\mathrm{Mod}_{R,0}^{\\{0,1\\},\vee}$ is the additive subcategory of
$\mathrm{Mod}_{R}^{\\{0,1\\},\vee}$ consisting of objects of the form $(X,Y)$,
where both $X$ and $Y$ are finitely generated free $R$-modules.

######  Definition 3.2.12.

Let $(I\rightarrow A)\in\mathrm{DAlg}_{R}^{\Delta^{1}_{\vee}}$. A divided
power (pd) structure on $I$ is a lift of the non-unital derived commutative
$A$-algebra $I$ to a non-unital derived divided power $A$-algebra. In this
situation, we call $(I\rightarrow A)$ a divided power (pd) pair, and $I$ a
divided power Smith ideal. They form an $\infty$-category
$\mathrm{DAlg}_{R}^{\Delta^{1}_{\vee},\mathrm{pd}}$ defined as the pull-back

$\textstyle{\mathrm{DAlg}_{R}^{\Delta^{1}_{\vee},\mathrm{pd}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{DAlg}_{R}^{\Delta^{1}_{\vee}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\operatorname{Fil}\rightarrow\mathrm{Gr})}$$\textstyle{\mathrm{DAlg}_{R}^{\\{0,1\\},\vee,\mathrm{pd}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{forget}_{\mathrm{pd}}}$$\textstyle{\mathrm{DAlg}_{R}^{\\{0,1\\},\vee},}$

where both functors involved in the diagram are forgetful functors.

We also say that an arrow $(A\rightarrow B)\in\mathrm{DAlg}_{R}^{\Delta^{1}}$
has the structure of a divided power map if the fiber
$\operatorname*{fib}(A\rightarrow B)$ is given the structure of a non-uinital
divided power $A$-algebra. We let $\mathrm{DAlg}_{R}^{\Delta^{1},\mathrm{pd}}$
be the $\infty$-category of divided power maps. By construction, there is an
equivalence

$\textstyle{\operatorname*{cofib}:\mathrm{DAlg}_{R}^{\Delta^{1}_{\vee},\mathrm{pd}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{\mathrm{DAlg}_{R}^{\Delta^{1},\mathrm{pd}}.}$

######  Proposition 3.2.13.

The $\infty$-category $\mathrm{DAlg}_{R}^{\Delta^{1}_{\vee},\mathrm{pd}}$ is
presentable, and the forgetful fuctor
$\mathrm{DAlg}_{R}^{\Delta^{1}_{\vee},\mathrm{pd}}\rightarrow\mathrm{DAlg}_{R}^{\Delta^{1}_{\vee}}$
preserves limits and sifted colimits.

###### Proof.

The $\infty$-category $\mathrm{DAlg}_{R}^{\Delta^{1}_{\vee},\mathrm{pd}}$ is
presentable being a pull-back of presentable $\infty$-categories along
accessible functors. Let
$\mathcal{D}\rightarrow\mathrm{DAlg}_{R}^{\Delta^{1}_{\vee},\mathrm{pd}},(i\in\mathcal{D})\longmapsto(A_{i},I_{i})$
be a diagram. Let $(I\rightarrow
A):=\underset{i\in\mathcal{D}}{\lim}(A_{i},I_{i})$ be the limit taken in
$\mathrm{DAlg}_{R}^{\Delta^{1}_{\vee}}$. The Smith ideal $I=\lim I_{i}$ has
the structure of a non-unital divided power $A$-algebra as the forgetful
functor
$\mathrm{DAlg}^{\mathrm{pd}}_{A}\rightarrow\mathrm{DAlg}_{A}^{\mathrm{nu}}$
creates limits. Therefore, $(I\rightarrow
A)\in\mathrm{DAlg}_{R}^{\Delta^{1}_{\vee},\mathrm{pd}}$. Let us show that the
object $(I\rightarrow A)$ satisfies the universal property. Assume
$(B,J)\rightarrow(A_{i},I_{i})$ is a collection of maps into the diagram
$\mathcal{D}$. We have a pull-back diagram

$\textstyle{\operatorname{Map}_{\mathrm{DAlg}_{R}^{\Delta^{1}_{\vee},\mathrm{pd}}}((B,J),(I\rightarrow
A))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Map}_{\mathrm{DAlg}_{R}^{\Delta^{1}_{\vee}},\inf}((B,J),(I\rightarrow
A))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Map}_{\mathrm{DAlg}_{R}^{\\{0,1\\},\vee,\mathrm{pd}}}((B,J),(I\rightarrow
A))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Map}_{\mathrm{DAlg}_{R}^{\\{0,1\\},\vee}}((B,J),(I\rightarrow
A)),}$

each term of which is equivalent to the limit
$\underset{\mathcal{D}}{\lim}\operatorname{Map}((B,J),(A_{i},I_{i}))$ computed
in the corresponding $\infty$-category. Therefore, we have

$\operatorname{Map}_{\mathrm{DAlg}_{R}^{\Delta^{1}_{\vee},\mathrm{pd}}}((B,J),(I\rightarrow
A))\simeq\underset{\mathcal{D}}{\lim}\operatorname{Map}_{\mathrm{DAlg}_{R}^{\Delta^{1}_{\vee},\mathrm{pd}}}((B,J),(A_{i},I_{i}))$

as well, and $(I\rightarrow A)$ is the limit of the diagram in
$\mathrm{DAlg}_{R}^{\Delta^{1}_{\vee},\mathrm{pd}}$, as desired. The same
argument shows that the forgetful functor also commutes with sifted colimits.

∎

######  Definition 3.2.14.

We define the functor of derived divided power (pd) envelope
$\operatorname{Env}^{\mathrm{pd}}:\mathrm{DAlg}_{R}^{\Delta^{1}_{\vee}}\rightarrow\mathrm{DAlg}_{R}^{\Delta^{1}_{\vee},\mathrm{pd}}$
as the left adjoint of the forgetful functor
$\mathrm{DAlg}_{R}^{\Delta^{1}_{\vee},\mathrm{pd}}\rightarrow\mathrm{DAlg}_{R}^{\Delta^{1}_{\vee}}$.
By commutation with sifted colimits and conservativity, it follows that the
forgetful functors
$\mathrm{DAlg}_{R}^{\Delta^{1}_{\vee}}\rightarrow\mathrm{DAlg}_{R}^{\Delta^{1}_{\vee}}$
and
$\mathrm{DAlg}_{R}^{\Delta^{1}_{\vee},\mathrm{pd}}\rightarrow\mathrm{Mod}_{R}^{\Delta^{1}_{\vee}}$
are monadic.

######  Proposition 3.2.15.

The derived divided power envelope functor
$\operatorname{Env}^{\mathrm{pd}}:\mathrm{DAlg}_{R}^{\Delta^{1}_{\vee}}\rightarrow\mathrm{DAlg}_{R}^{\Delta^{1}_{\vee},\mathrm{pd}}$
sends the full subcategory $\mathrm{DAlg}_{R,\geq
0}^{\Delta^{1}_{\vee}}\subset\mathrm{DAlg}_{R}^{\Delta^{1}_{\vee}}$ to
$\mathrm{DAlg}_{R,\geq
0}^{\Delta^{1}_{\vee},\mathrm{pd}}\subset\mathrm{DAlg}_{R}^{\Delta^{1}_{\vee},\mathrm{pd}}$

###### Proof.

By passing to right adjoints, the statement is equivalent to the statement
that for a divided power pair $(I\rightarrow A)$, the connective cover
$(\tau_{\geq 0}A,\tau_{\geq 0}I)$ is again a divided power pair. This comes
down to the connective cover $\tau_{\geq 0}I$ being defined as a non-unital
divided power $\tau_{\geq 0}A$-algebra. This is the content of Construction
3.2.8. ∎

We gave a rather abstract definition of the $\infty$-category of divided power
Smith ideals, without using any explicit compact projective generators. As a
result, it is apriori unclear what free algebras in this theory look like. The
next Proposition affirms that nothing “surprising” happens.

######  Proposition 3.2.16.

There are equivalences of divided power Smith ideals:

  * •

$\operatorname{Env}^{\mathrm{pd}}\mathrm{LSym}_{\mathbb{Z}}^{\Delta^{1}_{\vee}}(\operatorname{Id}:\mathbb{Z}\rightarrow\mathbb{Z})\simeq\mathbb{Z}\langle
x\rangle^{+}\rightarrow\mathbb{Z}\langle x\rangle,$

  * •

$\operatorname{Env}^{\mathrm{pd}}\mathrm{LSym}_{\mathbb{Z}}^{\Delta^{1}_{\vee}}(0\rightarrow\mathbb{Z})\simeq(0)\rightarrow\mathbb{Z}[x],$

where $\mathbb{Z}\langle x\rangle$ is as defined in Example 3.2.3.

###### Proof.

We will prove the first statement only, as the proof of the second is similar.
The proof goes by observing that the objects
$\operatorname{Env}^{\mathrm{pd}}\mathrm{LSym}_{\mathbb{Z}}^{\Delta^{1}_{\vee}}(\operatorname{Id}:\mathbb{Z}\rightarrow\mathbb{Z})$
lifts to the $\infty$-category $\mathrm{DAlg}_{R}^{\mathrm{nu},\mathrm{pd}}$
of non-unital divided power algebras by means of the following commutative
diagram

$\textstyle{\mathrm{Mod}_{\mathbb{Z}}^{\Delta^{1}_{\vee}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{Env}^{\mathrm{pd}}\mathrm{LSym}_{\mathbb{Z}}^{\Delta^{1}_{\vee}}}$$\textstyle{\mathrm{DAlg}_{\mathbb{Z}}^{\Delta^{1}_{\vee}.\mathrm{pd}}}$$\textstyle{\mathrm{Mod}_{\mathbb{Z}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{L\Gamma}^{+}_{\mathbb{Z}}}$$\textstyle{\mathrm{DAlg}^{\\{0,1\\},\vee,\mathrm{pd}}_{\mathbb{Z}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces,}$$\scriptstyle{\mathrm{aug}}$

where the left vertical functor sends $X\in\mathrm{Mod}_{\mathbb{Z}}$ to
$(\operatorname{Id}:X\rightarrow
X)\in\mathrm{Mod}_{\mathbb{Z}}^{\Delta^{1}_{\vee}}$, and the right vertical
functor sends a non-unital derived divided power algerba $A^{+}$ to the pd
Smith ideal $A^{+}\rightarrow\mathbb{Z}\oplus A^{+}$. Commutativity of this
diagram follows by passing to right adjoints, as the right adjoints of
horizontal functors are forgetful functors, the right adjoint of the left
vertical is the functor $\operatorname{ev}^{1}$ evaluating at the source of
the arrow, and the right adjoint of the right vertical is the functor
$(\operatorname{Fil}\rightarrow\mathrm{Gr})$ which takes a Smith ideal
$(I\rightarrow A)$ to the underlying non-unital derived pd algebra of $I$. As
we know that the free non-unutal divided power algebra on one generator is
$\mathbb{Z}\langle x\rangle^{+}$, and the corresponding pd Smith ideal is
$\mathbb{Z}\langle x\rangle^{+}\rightarrow\mathbb{Z}\langle x\rangle$, the
statement follows. ∎

######  Proposition 3.2.17.

Let $X\rightarrow Z$ be an object of $\mathrm{Mod}_{R,\geq 0}^{\Delta^{1}}$,
and $Y=\operatorname*{fib}(X\rightarrow Z)$. Let
$\mathrm{LSym}_{R}(X)\rightarrow\mathrm{LSym}_{R}(Z)$ be the free object in
$\mathrm{DAlg}_{R}^{\Delta^{1}}$ on $X\rightarrow Z$. There is an equivalence

$\operatorname{Env}^{\mathrm{pd}}\Bigl{(}\mathrm{LSym}_{R}(X)\rightarrow\mathrm{LSym}_{R}(Z)\Bigr{)}\simeq\mathrm{L\Gamma}_{R}(Y)\underset{\mathrm{LSym}_{R}(Y)}{\otimes}\mathrm{LSym}_{R}(X)\rightarrow\mathrm{LSym}_{R}(Z)$

as objects of $\mathrm{DAlg}_{R}^{\Delta^{1}}$.

###### Proof.

Note that there is a commutative diagram

$\textstyle{\mathrm{L\Gamma}_{R}(Y)\underset{\mathrm{LSym}_{R}(Y)}{\otimes}\mathrm{LSym}_{R}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{\mathrm{LSym}_{R}(Z)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{Id}}$$\textstyle{\mathrm{LSym}_{R}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{LSym}_{R}(Z),}$

where the map $f$ is induced by the Norm map
$\mathrm{L\Gamma}_{R}(Y)\rightarrow\mathrm{LSym}_{R}(Y)$. To produce a natural
map from the derived divided power envelope, we need to show that the fiber of
the map

$\varepsilon:\mathrm{L\Gamma}_{R}(Y)\underset{\mathrm{LSym}_{R}(Y)}{\otimes}\mathrm{LSym}_{R}(X)\rightarrow\mathrm{LSym}_{R}(Z)$

has the structure of a divided power Smith ideal. This can be done by
observing that formation of the map $\varepsilon$ is functorial in the fiber
sequence $Y\rightarrow X\rightarrow Z$ and commutes with sifted colimits, and
we ran resolve the fiber sequence $Y\rightarrow X\rightarrow Z$ as a sifted
colimit of split fiber sequences of the form $Y_{i}\rightarrow Y_{i}\oplus
Z_{i}\rightarrow Z_{i}$. It then suffices to show that the map

$\mathrm{L\Gamma}_{R}(Z_{i})\underset{\mathrm{LSym}_{R}(Z_{i})}{\otimes}\mathrm{LSym}_{R}(Y_{i}\oplus
Z_{i})\rightarrow\mathrm{LSym}_{R}(Y_{i})$

is naturally a divided power map. But we can identify it with the map
$\mathrm{L\Gamma}_{R}(Z_{i})\otimes\mathrm{LSym}_{R}(Y_{i})\rightarrow\mathrm{LSym}_{R}(Y_{i})$
which is the tensor product of maps $\mathrm{L\Gamma}_{R}(Z_{i})\rightarrow R$
and
$\operatorname{Id}:\mathrm{LSym}_{R}(Y_{i})\rightarrow\mathrm{LSym}_{R}(Y_{i})$,
so the fiber of the map
$\mathrm{L\Gamma}_{R}(Z_{i})\otimes\mathrm{LSym}_{R}(Y_{i})\rightarrow\mathrm{LSym}_{R}(Y_{i})$
is $\mathrm{L\Gamma}_{R}^{\geq 1}(Z_{i})\otimes\mathrm{LSym}_{R}(Y_{i})$ which
is a non-unital divided power algebra. Therefore, the map $\varepsilon$ is a
divided power map, and therefore there exists a map

$\textstyle{\operatorname{Env}^{\mathrm{pd}}\Bigl{(}\mathrm{LSym}_{R}(X)\rightarrow\mathrm{LSym}_{R}(Z)\Bigr{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{L\Gamma}_{R}(Y)\underset{\mathrm{LSym}_{R}(Y)}{\otimes}\mathrm{LSym}_{R}(X)\rightarrow\mathrm{LSym}_{R}(Z)}$

by the universal property of derived divided power envelope. To check that
this is an equivalence, since $X$ and $Z$ are connective, we can resolve the
map $X\rightarrow Z$ by finitely generated free $R$-modules, and by base
change deduce the statement from Remark 3.2.16. ∎

######  Proposition 3.2.18.

The forgetful functor
$\mathrm{DAlg}_{R}^{\Delta^{1}_{\vee},\mathrm{pd}}\rightarrow\mathrm{DAlg}_{R}^{\Delta^{1}_{\vee}}$
commutes with all colimits.

###### Proof.

To show that it commutes with finite coproducts. Here again, we argue as in
[R, Proposition 4.2.27]. Using formulas of Proposition 3.2.17, we see that the
maps

$\textstyle{\operatorname{Env}^{\mathrm{pd}}\bigl{(}\mathrm{LSym}_{R}^{\Delta^{1}_{\vee}}(0:0\rightarrow
X)\bigr{)}\otimes\operatorname{Env}^{\mathrm{pd}}\bigl{(}\mathrm{LSym}^{\Delta^{1}_{\vee}}_{R}(\operatorname{Id}:Z\rightarrow
Z)\bigr{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Env}^{\mathrm{pd}}\mathrm{LSym}_{R}^{\Delta^{1}_{\vee}}\bigl{(}(0:0\rightarrow
X)\oplus(\operatorname{Id}:Z\rightarrow Z)\bigr{)}}$

are equivalences for all $X,Z\in\mathrm{Mod}_{R}^{\mathrm{fg},\mathrm{free}}$.
From this, we conclude that
$\operatorname{Env}^{\mathrm{pd}}\mathrm{LSym}_{R}^{\Delta^{1}_{\vee}}$
transforms directs sums of compact projective generators of
$\mathrm{Mod}_{R,\geq 0}^{\Delta^{1}_{\vee}}$ to tensor products, and
commutation of the forgetful functor with coproducts follows. ∎

######  Definition 3.2.19.

It follows from Proposition 3.2.18 that there is an adjunction

$\textstyle{\mathrm{DAlg}_{R}^{\Delta^{1}_{\vee},\mathrm{pd}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{forget}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathrm{DAlg}_{R}^{\Delta^{1}_{\vee}},}$$\scriptstyle{\operatorname{Env}^{\mathrm{pd}}}$$\scriptstyle{\mathrm{coEnv}^{\mathrm{pd}}}$

where we call the left adjoint the derived divided power (pd) envelope, and
the right adjoint derived divided power (pd) coenvelope.

######  Remark 3.2.20.

The adjunction of Construction 3.1.27 lifts to an adjunction

$\textstyle{\mathrm{DAlg}_{R}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{ins}^{0}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathrm{DAlg}_{R}^{\Delta^{1}_{\vee},\mathrm{pd}}.}$$\scriptstyle{\operatorname{gr}^{0}}$$\scriptstyle{\operatorname{ev}^{0}}$

In particular, there is a commutative square

$\textstyle{\mathrm{DAlg}_{R}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{Id}}$$\scriptstyle{\mathrm{ins}^{0}}$$\textstyle{\mathrm{DAlg}_{R}^{\Delta^{1}_{\vee},\mathrm{pd}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{forget}}$$\textstyle{\mathrm{DAlg}_{R}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{ins}^{0}}$$\textstyle{\mathrm{DAlg}_{R}^{\Delta^{1}_{\vee}}.}$

Passing to left and right adjoints, we obtain commutative squares

$\textstyle{\mathrm{DAlg}_{R}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathrm{DAlg}_{R}^{\Delta^{1}_{\vee},\mathrm{pd}}\;\;\;\;\;\;\;\;\;\;\;\;}$$\scriptstyle{\operatorname{gr}^{0}}$$\textstyle{\mathrm{DAlg}_{R}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{Id}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathrm{DAlg}_{R}^{\Delta^{1}_{\vee}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces,}$$\scriptstyle{\operatorname{gr}^{0}}$$\scriptstyle{\operatorname{Env}^{\mathrm{pd}}}$$\textstyle{\mathrm{DAlg}_{R}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathrm{DAlg}_{R}^{\Delta^{1}_{\vee},\mathrm{pd}}}$$\scriptstyle{\operatorname{ev}^{0}}$$\textstyle{\mathrm{DAlg}_{R}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{Id}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathrm{DAlg}_{R}^{\Delta^{1}_{\vee}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces.}$$\scriptstyle{\operatorname{ev}^{0}}$$\scriptstyle{\mathrm{coEnv}^{\mathrm{pd}}}$

######  Remark 3.2.21.

Let $\mathrm{DAlg}_{R,\geq
0}^{\Delta^{1}_{\vee},\mathrm{pd}}\subset\mathrm{DAlg}_{R}^{\Delta^{1}_{\vee},\mathrm{pd}}$
be the inclusion of the full subcategory of all (divided power) Smith ideals
consisting of objects $(I\rightarrow A)$ with both $A$ and $I$ being
connective. Then there is a commutative diagram

$\textstyle{\mathrm{DAlg}_{R,\geq
0}^{\Delta^{1}_{\vee}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{Env}^{\mathrm{pd}}}$$\textstyle{\mathrm{DAlg}_{R,\geq
0}^{\Delta^{1}_{\vee},\mathrm{pd}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{DAlg}_{R}^{\Delta^{1}_{\vee}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{Env}^{\mathrm{pd}}}$$\textstyle{\mathrm{DAlg}_{R}^{\Delta^{1}_{\vee},\mathrm{pd}},}$

where the vertical functors are inclusions. In other words, divided power
envelopes preserve connective Smith ideals. Commutativity of the diagram
follows by passing to right adjoints as the right adjoint of inclusion is
connective cover which manifestly commutes with forgetful functors.

We let $\mathrm{DAlg}^{\mathrm{pd}}_{A/}$ be the $\infty$-category of divided
power maps $A\rightarrow B$ from $A$, and $\mathrm{DAlg}^{\mathrm{pd}}_{A/}$
be the $\infty$-category of divided power maps $C\rightarrow A$ to $A$.
Concretely, they are defined as fiber products:

$\textstyle{\mathrm{DAlg}^{\mathrm{pd}}_{R//A,\geq
0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{DAlg}_{R,\geq
0}^{\Delta^{1},\mathrm{pd}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{ev}^{1}}$$\textstyle{\\{A\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{DAlg}_{R,\geq
0}.}$$\textstyle{\mathrm{DAlg}^{\mathrm{pd}}_{A/,\geq
0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{DAlg}_{R,\geq
0}^{\Delta^{1},\mathrm{pd}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{ev}^{0}}$$\textstyle{\\{A\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{DAlg}_{R,\geq
0}}$

######  Proposition 3.2.22.

There exist adjunctions

$\textstyle{\mathrm{DAlg}_{A/}^{\mathrm{pd}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{forget}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathrm{DAlg}_{A/}\;\;,}$$\scriptstyle{\mathrm{coEnv}_{A/}^{\mathrm{pd}}}$
$\textstyle{\mathrm{DAlg}_{/A}^{\mathrm{pd}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{forget}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathrm{DAlg}_{/A}.}$$\scriptstyle{\operatorname{Env}_{/A}^{\mathrm{pd}}}$

The functor $\mathrm{coEnv}^{\mathrm{pd}}_{A/}$ preserves connective objects.
If $B\rightarrow A\in\mathrm{DAlg}_{A/,\geq 0}$ is surjective on $\pi_{0}$,
then the divided power envelope $\operatorname{Env}^{\mathrm{pd}}_{/A}(B)$ is
connective again.

###### Proof.

Both statement follow from Remark 3.2.20. ∎

######  Proposition 3.2.23.

For a derived pair $I\rightarrow A$, the divided power coenvelope
$\mathrm{coEnv}^{\mathrm{pd}}(I\rightarrow A)$ is equivalent to the pair
$I^{\sharp}\rightarrow A$, where $I^{\sharp}$ is as in Remark 3.2.4.

######  Definition 3.2.24.

In the particular case $I=A$ considered as the tautological Smith ideal
$\operatorname{Id}:A\rightarrow A$, the divided power ideal $I^{\sharp}$ will
be called divided power (pd) radical of $R$.

We will finish this subsection with the following easy observation relating
our approach to derived divided power Smith ideals with the $\infty$-category
of animated pd pairs of Z.Mao [Mao21].

######  Proposition 3.2.25.

Let $\mathrm{C}^{0}$ be the full subcategory of all divided power Smith ideals
consisting of finite tensor products of the objects $(0)\subset\mathbb{Z}[x]$
and $\mathbb{Z}\langle x\rangle^{+}\rightarrow\mathbb{Z}\langle x\rangle$ (see
Proposition 3.2.16). Then there is an equivalence of $\infty$-categories

$\mathrm{DAlg}^{\Delta^{1}_{\vee},\mathrm{pd}}_{\mathbb{Z},\geq
0}\simeq\mathcal{P}_{\Sigma}(\mathrm{C}^{0}).$

###### Proof.

Indeed, by monadicity of
$\mathrm{DAlg}^{\Delta^{1}_{\vee},\mathrm{pd}}_{\mathbb{Z},\geq 0}$ over
$\mathrm{Mod}_{\mathbb{Z},\geq 0}^{\Delta^{1}_{\vee}}$ and preservation of
sifted colimits by the monad, it follows that compact projective generators of
$\mathrm{DAlg}^{\Delta^{1}_{\vee},\mathrm{pd}}_{\mathbb{Z},\geq 0}$ are the
free objects on finite coproducts of $0\rightarrow\mathbb{Z}$ and
$\operatorname{Id}:\mathbb{Z}\rightarrow\mathbb{Z}$ (the latter being compact
projective generators of $\mathrm{Mod}_{\mathbb{Z},\geq
0}^{\Delta^{1}_{\vee}}$). Since coproducts in
$\mathrm{DAlg}^{\Delta^{1}_{\vee},\mathrm{pd}}_{\mathbb{Z},\geq 0}$ are the
same as tensor products, the statement follows from the computation of
Proposition 3.2.16. ∎

####  3.2.3 Filtered divided power algebras.

In this subsection we will define filtered divided power algebras. A typical
derived filtered divided power algebra $F^{\geq\star}A$ looks like a filtered
derived algebra together with a divided power structure on the “filtered Smith
ideal” $F^{\geq
1}A:=\operatorname*{fib}(F^{\geq\star}A\rightarrow\operatorname{gr}^{0}A)$. To
make this precise, a necessary step is to work out the definition of filtered
derived power non-unital algebras over a general derived ring. As an
intermediate step for this, we will introduce an auxilary construction in the
world of ordinary filtered derived rings.

######  Construction 3.2.26.

Let $\operatorname{Fil}^{\geq 0}\mathrm{DAlg}_{\mathbb{Z}}$ be the
$\infty$-category of filtered derived rings. Given any object
$F^{\geq\star}A$, the derived ring $F^{\geq 0}A$ acts on all higher stages of
the filtration $F^{\geq i}A$. Assume that the map $F^{\geq
0}A\rightarrow\operatorname{gr}^{0}A$ admits a derived algebra splitting
$\operatorname{gr}^{0}A\rightarrow F^{\geq 0}A$. Then $F^{\geq 0}A$ becomes an
augmented derived ring $F^{\geq 0}A\simeq\operatorname{gr}^{0}A\oplus F^{\geq
1}A$ endowed with a filtered augmentation ideal $F^{\geq 1}A$, on which
$\operatorname{gr}^{0}A$ acts via the section
$\operatorname{gr}^{0}A\rightarrow F^{\geq 0}A$. Let $\operatorname{Fil}^{\geq
0}\mathrm{DAlg}^{\mathrm{aug}}_{\mathbb{Z}}$ be the $\infty$-category of
filtered derived rings $F^{\geq 0}A$ endowed with a splitting
$\operatorname{gr}^{0}A\rightarrow F^{0}A$. Sending $F^{\geq\star}A$ to the
pair $(\operatorname{gr}^{0},\operatorname{Fil}^{\geq 1}A)$, we get an
equivalence of this $\infty$-category and the $\infty$-category of pairs
$(B,F^{\geq 1}B)$ where $B$ is a derived ring, and $F^{\geq 1}B$ is a
positively filtered non-unital derived $B$-algebra. The inverse functor
associates to a pair $(B,F^{\geq 1}B)$ the augmented filtered algebra
$...\rightarrow F^{\geq 1}B\rightarrow B\oplus F^{\geq 1}B$. When $R$ is any
connective derived ring, the left hand side can be redefined as the
$\infty$-category of under objects

$\operatorname{Fil}^{\geq
0}\mathrm{DAlg}_{R}^{\mathrm{aug}}:=(\operatorname{Fil}^{\geq
0}\mathrm{DAlg}_{\mathbb{Z}}^{\mathrm{aug}})_{R/},$

where $R$ is considered as a filtered ring inserted in degree $0$101010Note
that if we consider $R$ as a filtered ring with constant filtration, then its
associated graded ring is $0$. Therefore, any filtered ring receiving a map
from $R$ considered with constant filtration, would need to have associated
graded ring $0$ too, and therefore constant filtration is not the right
approach..

######  Construction 3.2.27.

Let $R$ be a derived commutative ring.

  * •

Assume $R=\mathbb{Z}$. Then the $\infty$-category
$\operatorname{Fil}\mathrm{Mod}_{\mathbb{Z}}$ has a derived filtered non-
unital divided power algebra
$\mathrm{L\Gamma}^{+,\geq\star}_{\mathbb{Z}}:\operatorname{Fil}\mathrm{Mod}_{\mathbb{Z}}\rightarrow\operatorname{Fil}\mathrm{Mod}_{\mathbb{Z}}$
using Construction 2.3.13 and Construction 2.4.21. We let
$\operatorname{Fil}\mathrm{DAlg}^{\mathrm{nu},\mathrm{pd}}:=\operatorname{Alg}_{\mathrm{L\Gamma}_{\mathbb{Z}}^{+,\geq\star}}(\operatorname{Fil}\mathrm{Mod}_{\mathbb{Z}})$.
The subcategory $\operatorname{Fil}^{\geq
1}\mathrm{Mod}_{\mathbb{Z}}\subset\operatorname{Fil}\mathrm{Mod}_{\mathbb{Z}}$
is closed under formation of free
$\mathrm{L\Gamma}_{\mathbb{Z}}^{+,\geq\star}$-algebras. Therefore, there is an
$\infty$-category

$\operatorname{Fil}^{\geq
1}\mathrm{DAlg}^{\mathrm{pd}}_{\mathbb{Z}}:=\operatorname{Fil}\mathrm{DAlg}^{\mathrm{pd}}_{\mathbb{Z}}\underset{\operatorname{Fil}\mathrm{Mod}_{\mathbb{Z}}}{\times}\operatorname{Fil}^{\geq
1}\mathrm{Mod}_{\mathbb{Z}}$

of derived filtered non-unital divided power $R$-algebras, which is monadic
over $\operatorname{Fil}^{\geq 1}\mathrm{Mod}_{\mathbb{Z}}$, and the monad
right-left extended from the subcategory
$\operatorname{Ind}(\operatorname{Fil}^{\geq 1}\mathrm{Mod}_{\mathbb{Z},0})$
by Remark 2.4.22.

  * •

If $R$ is not necessarily discrete, or even connective, we can define the
$\infty$-category $\operatorname{Fil}^{\geq 1}\mathrm{DAlg}^{\mathrm{pd}}_{A}$
using Construction 3.2.26. There is a forgetful functor
$\operatorname{Fil}^{\geq
1}\mathrm{DAlg}^{\mathrm{pd}}_{\mathbb{Z}}\rightarrow\operatorname{Fil}^{\geq
1}\mathrm{DAlg}_{\mathbb{Z}}$, and a forgetful functor
$\operatorname{Fil}^{\geq
0}\mathrm{DAlg}^{\mathrm{aug}}_{\mathbb{Z}}\rightarrow\operatorname{Fil}^{\geq
1}\mathrm{DAlg}_{\mathbb{Z}}$. Then we define the $\infty$-category of
augmented filtered derived divided power rings as the pull-back

$\textstyle{\operatorname{Fil}^{\geq
0}\mathrm{DAlg}_{\mathbb{Z}}^{\mathrm{pd},\mathrm{aug}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Fil}^{\geq
0}\mathrm{DAlg}_{\mathbb{Z}}^{\mathrm{aug}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Fil}^{\geq
1}\mathrm{DAlg}_{\mathbb{Z}}^{\mathrm{pd}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Fil}^{\geq
1}\mathrm{DAlg}_{\mathbb{Z}},}$

i.e. an augmented filtered divided power ring is an augmented filtered derived
ring endowed with a divided power structure on the filtered augmentation
ideal. Let $\operatorname{Fil}^{\geq
0}\mathrm{Mod}_{\mathbb{Z}}^{\mathrm{aug}}$ be the $\infty$-category of
augmented filtrations, i.e. object $F^{\geq\star}\in\operatorname{Fil}^{\geq
0}\mathrm{Mod}_{\mathbb{Z}}$ endowed with a splitting of the map $F^{\geq
0}\rightarrow\operatorname{gr}^{0}_{F}$. We think of objects of
$\operatorname{Fil}^{\geq 0}\mathrm{Mod}_{\mathbb{Z}}^{\mathrm{aug}}$ as pairs
$(X,F^{\geq 1}X)$. The $\infty$-category $\operatorname{Fil}^{\geq
0}\mathrm{DAlg}_{\mathbb{Z}}^{\mathrm{pd},\mathrm{aug}}$ is monadic over
$\operatorname{Fil}^{\geq 0}\mathrm{Mod}^{\mathrm{aug}}_{\mathbb{Z}}$. Let
$\textstyle{(\mathrm{LSym},\mathrm{L\Gamma}^{\geq\star}):\operatorname{Fil}^{\geq
0}\mathrm{Mod}_{\mathbb{Z}}^{\mathrm{aug}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Fil}^{\geq
0}\mathrm{Mod}_{\mathbb{Z}}^{\mathrm{aug}}}$ be the resulting monad. It
satisfies the formula

$(\mathrm{LSym},\mathrm{L\Gamma}^{\geq\star})(X,F^{\geq
1}X)\simeq(\mathrm{LSym}_{\mathbb{Z}}(X),\mathrm{L\Gamma}^{\geq\star}_{\mathrm{LSym}_{\mathbb{Z}}(X)}(F^{\geq
1}X\otimes\mathrm{LSym}_{\mathbb{Z}}(X))).$

By definition, the fiber of the forgetful functor
$\operatorname{ev}^{0}:\operatorname{Fil}^{\geq
0}\mathrm{DAlg}_{\mathbb{Z}}^{\mathrm{pd},\mathrm{aug}}\rightarrow\mathrm{DAlg}_{\mathbb{Z}}$
over some $A$ is the $\infty$-category of non-unital filtered derived divided
power $A$-algebras111111Compare this with Definition 3.2.9.. We denote
$\mathrm{L\Gamma}_{A}^{\geq\star}:\operatorname{Fil}^{\geq
1}\mathrm{Mod}_{A}\rightarrow\operatorname{Fil}^{\geq 1}\mathrm{Mod}_{A}$ he
free algebra monad. Similarly, if $R$ is a connective derived ring, we can
consider $R$ as a filtered derived ring inserted in degree $0$, and let

$\operatorname{Fil}^{\geq
0}\mathrm{DAlg}_{R}^{\mathrm{pd},\mathrm{aug}}:=\operatorname{Fil}^{\geq
0}\mathrm{DAlg}_{\mathbb{Z}}^{\mathrm{pd},\mathrm{aug}}\underset{\operatorname{Fil}^{\geq
0}\mathrm{DAlg}_{\mathbb{Z}}^{\mathrm{aug}}}{\times}\operatorname{Fil}^{\geq
0}\mathrm{DAlg}_{R}^{\mathrm{aug}}.$

Similarly, we have an $\infty$-category of graded derived divided power rings
$\mathrm{Gr}^{\geq 0}\mathrm{DAlg}^{\mathrm{pd}}_{R}$ whose fiber over any
$A\in\mathrm{DAlg}_{R}$ is the $\infty$-category of non-unital graded derived
divided power $A$-algebras, and the free algebra monad is denoted as
$\mathrm{L\Gamma}^{+,\bullet}_{A}:\mathrm{Gr}^{\geq
0}\mathrm{Mod}_{A}\rightarrow\mathrm{Gr}^{\geq 1}\mathrm{Mod}_{A}.$

For future purposes, we specifically fix the following notation.

######  Notation 3.2.28.

Let $\mathrm{DAlg}_{R}\mathrm{Mod}$ be the $\infty$-category of pairs $(A,M)$,
where $A\in\mathrm{DAlg}_{R}$ and $M\in\mathrm{Mod}_{A}$. The functor of free
graded divided power algebra on a module is the functor:

$\textstyle{\mathrm{DAlg}_{R}\mathrm{Mod}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{L\Gamma}^{\bullet}}$$\textstyle{\mathrm{Gr}^{\geq
0}\mathrm{DAlg}_{R}^{\mathrm{pd}}}$

which sends a pair $(A,M)\in\mathrm{DAlg}_{R}\mathrm{Mod}$ to
$(A,\mathrm{L\Gamma}^{+,\bullet}_{A}(M))$

The right adjoint of this functor is the functor

$\textstyle{\mathrm{Gr}^{\geq
0}\mathrm{DAlg}_{R}^{\mathrm{pd}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(-)^{\\{0,1\\}}\circ\>\mathrm{forget}_{\mathrm{pd}}}$$\textstyle{\mathrm{DAlg}_{R}\mathrm{Mod}.}$

In the same vein, we define a free graded derived algebra on a module
$\mathrm{LSym}^{\bullet}_{A}(M)$ for any pair
$(A,M)\in\mathrm{DAlg}_{R}\mathrm{Mod}$.

######  Example 3.2.29.

The free filtered non-unital divided power algebra $\Gamma^{\geq
1}_{\mathbb{Z}}(M)$ on a finitely gerated free $\mathbb{Z}$-module $M$ (put in
filtered weight $1$) has filtration with stages

$\Gamma_{\mathbb{Z}}^{\geq
n}(M):=\bigoplus_{i=n}^{\infty}\Gamma^{i}_{\mathbb{Z}}(M),$

where $n\geq 1$. Concretely, the $n$-th stage of filtration consists of
elements of the form:

$\Gamma^{\geq n}_{\mathbb{Z}}(M)=\\{\sum_{i_{1}+...i_{k}\geq
n}\gamma^{i_{1}}(x_{1})...\gamma^{i_{k}}(x_{k})|x_{1},...,x_{k}\in M,k\geq
1\\}.$

######  Construction 3.2.30.

We define a functor $\operatorname{Fil}^{\geq
0}\mathrm{DAlg}_{R}\rightarrow\operatorname{Fil}^{\geq
0}\mathrm{DAlg}_{R}^{\mathrm{aug}}$ by sending an object $F^{\geq\star}A$ to
the augmented filtered derived algebra $...\rightarrow F^{\geq 1}A\rightarrow
F^{\geq 0}A\oplus F^{\geq 1}A$ so that the map $F^{\geq 0}A\oplus F^{\geq
1}A\rightarrow F^{\geq 0}A$ has a (zero) section121212This is analogous to the
functor
$(\operatorname{Fil}\rightarrow\mathrm{Gr}):\mathrm{DAlg}^{\Delta^{1}_{\vee}}_{R}\rightarrow\mathrm{DAlg}^{\\{0,1\\},\vee}_{R}\simeq\mathrm{DAlg}^{\Delta^{1}_{\vee},\mathrm{aug}}_{R}$
which sends a Smith ideal $(I\rightarrow A)$ to the augmented Smith ideal
$(I\rightarrow A\oplus I)$..

######  Definition 3.2.31.

We define an $\infty$-category $\operatorname{Fil}^{\geq
0}\mathrm{DAlg}^{\mathrm{pd}}_{R}$ of filtered derived divided power (pd)
algebras as the fiber square

$\textstyle{\operatorname{Fil}^{\geq
0}\mathrm{DAlg}^{\mathrm{pd}}_{R}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Fil}^{\geq
0}\mathrm{DAlg}_{R}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Fil}^{\geq
0}\mathrm{DAlg}_{R}^{\mathrm{pd},\mathrm{aug}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Fil}^{\geq
0}\mathrm{DAlg}^{\mathrm{aug}}_{R},}$

where the lower horizontal functor is the forgetful functor, and the right
vertical functor is as defined in Construction 3.2.30.

######  Remark 3.2.32.

Before the next construction, we make a trivial observation. The functor
$\operatorname{ev}^{[0,1]}:\operatorname{Fil}^{\geq
0}\mathrm{DAlg}_{R}\rightarrow\mathrm{DAlg}^{\Delta^{1}_{\vee}}_{R}$
constructed in Construction 3.1.24 lifts to a functor

$\operatorname{ev}^{[0,1]}:\operatorname{Fil}^{\geq
0}\mathrm{DAlg}_{R}^{\mathrm{aug}}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\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{}$}}}}}}}\ignorespaces}}}}\ignorespaces\mathrm{DAlg}^{\Delta^{1}_{\vee},\mathrm{aug}}_{R}\simeq\mathrm{DAlg}_{R}^{\\{0,1\\},\vee}$

as for any object $F^{\geq\star}A\in\operatorname{Fil}^{\geq
0}\mathrm{DAlg}_{R}^{\mathrm{aug}}$, the Smith ideal $F^{\geq 1}A\rightarrow
F^{0}A$ has a splitting.

######  Construction 3.2.33.

We will construct a functor
$\operatorname{ev}^{[0,1]}:\operatorname{Fil}^{\geq
0}\mathrm{DAlg}_{R}^{\mathrm{pd},\mathrm{aug}}\rightarrow\mathrm{DAlg}^{\\{0,1\\},\vee,\mathrm{pd}}_{R}$
for any connective derived commutative ring $R$, lifting the functor
$\operatorname{ev}^{[0,1]}:\operatorname{Fil}^{\geq
0}\mathrm{DAlg}^{\mathrm{aug}}_{R}\rightarrow\mathrm{DAlg}^{\\{0,1\\},\vee}_{R}$
provided by Remark 3.2.32. This is sufficient to do in the case
$A=\mathbb{Z}$. In this case, the derived divided power algebra monad is
right-left extended. We observe that there is a commutative diagram

$\textstyle{\operatorname{Ind}(\operatorname{Fil}^{\geq
0}\mathrm{Mod}^{\mathrm{aug}}_{\mathbb{Z},0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\mathrm{LSym},\mathrm{L\Gamma}^{\geq\star})}$$\scriptstyle{\operatorname{ev}^{[0,1]}}$$\textstyle{\operatorname{Ind}(\mathrm{Mod}_{\mathbb{Z},0}^{\\{0,1\\},\vee})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\mathrm{LSym},\mathrm{L\Gamma}^{+})}$$\textstyle{\operatorname{Ind}(\operatorname{Fil}^{\geq
0}\mathrm{Mod}^{\mathrm{aug}}_{\mathbb{Z},0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{ev}^{[0,1]}}$$\textstyle{\operatorname{Ind}(\mathrm{Mod}_{\mathbb{Z},0}^{\\{0,1\\},\vee})}$

Therefore, by right-left extension we obtain a functor
$\operatorname{ev}^{[0,1]}:\operatorname{Fil}^{\geq
0}\mathrm{DAlg}_{\mathbb{Z}}^{\mathrm{pd},\mathrm{aug}}\rightarrow\mathrm{DAlg}^{\\{0,1\\},\vee,\mathrm{pd}}_{\mathbb{Z}}.$

For a commutative algebra $A$ with a divided power ideal $I\subset A$, we can
construct the filtration by ideals $...\subset I^{[3]}\subset I^{[2]}\subset
I\subset A$ where $I^{[n]}$ is spanned by elements
$x_{1}^{[i_{1}]}...x_{n}^{[i_{n}]}$ with $i_{1}+...+i_{n}\geq n$. The
associated graded algebra for this filtration is the free graded divided power
algebra $\Gamma_{A/I}(I/I^{2})=\bigoplus_{n\geq 0}I^{[n]}/I^{[n+1]}$.

######  Construction 3.2.34.

The functor $\operatorname{ev}^{[0,1]}$ lifts to a limit preserving functor
$\operatorname{ev}^{[0,1]}:\operatorname{Fil}^{\geq
0}\mathrm{DAlg}^{\mathrm{pd}}\rightarrow\mathrm{DAlg}_{R}^{\Delta^{1}_{\vee},\mathrm{pd}}$
on the $\infty$-categories of derived divided power algebras. For that, using
Definitions 3.2.31 and 3.2.12, we need to show that the functor
$\operatorname{ev}^{1}:\operatorname{Fil}^{\geq
1}\mathrm{DAlg}^{\mathrm{pd}}_{A}\rightarrow\mathrm{DAlg}^{\mathrm{nu}}_{A}$
lifts to a functor $\operatorname{ev}^{1}:\operatorname{Fil}^{\geq
1}\mathrm{DAlg}^{\mathrm{pd}}_{A}\rightarrow\mathrm{DAlg}^{\mathrm{pd},\mathrm{nu}}_{A}$
naturally for any $A\in\mathrm{DAlg}_{R}$, which is done in Construction
3.2.33. We let the functor of derived divided power (pd) adic filtration

$\textstyle{\mathrm{adic}_{\mathrm{pd}}:\mathrm{DAlg}_{R}^{\Delta^{1}_{\vee},\mathrm{pd}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Fil}^{\geq
0}\mathrm{DAlg}^{\mathrm{pd}}_{R}}$

to be left adjoint of $\operatorname{ev}^{[0,1]}:\operatorname{Fil}^{\geq
0}\mathrm{DAlg}^{\mathrm{pd}}_{R}\rightarrow\mathrm{DAlg}_{R}^{\Delta^{1}_{\vee},\mathrm{pd}}$.

Similarly, there is a functor
$\operatorname{ev}^{[0,1]}:\operatorname{Fil}^{\geq
0}\mathrm{DAlg}_{R}\rightarrow\mathrm{DAlg}_{R}^{\Delta^{1}_{\vee}}$ and its
left adjoint derived adic filtration functor
$\mathrm{adic}:\mathrm{DAlg}_{R}^{\Delta^{1}_{\vee}}\rightarrow\operatorname{Fil}^{\geq
0}\mathrm{DAlg}_{R}$.

In the next section we will compute associated gradeds of the (pd) adic
filtration. We will also identify the first truncation of the (pd) adic
filtraton with the universal square-zero extension.

###  3.3 Two-step filtrations and square-zero extensions.

In this subsection we will develop the formalism of square-zero extensions
from the point of view of two-step filtrations. Our motivation for the
constructions of this subsection comes from the following situation. Let
$\mathrm{L\Omega}^{\geq\star}_{A/R}$ be the derived De Rham cohomology of $A$
over $R$. We can truncate it by filtered pieces of weights $\geq 2$ to get a
“derived square-zero extension” of $A$ by the shift of the cotangent complex
$\mathbb{L}_{A/R}[-1]$. It is expectable that this square-zero extension is
controlled by the _universal derivation_. The formalism developed in this
subsection gives a simple proof of this fact which has the advantage of being
simple to generalize to other settings.

######  Definition 3.3.1.

Let $\\{0,1\\}$ be the subset of $\mathbb{Z}^{\mathrm{ds}}$ consisting of two
objects $0,1\in\mathbb{Z}$, considered as a discrete category. Similarly, let
$(0<1)$ be the full subcategory of $\mathbb{Z}_{\leq*}$ spanned by these two
objects. Fix a connective derived commutative ring $R$.

  * •

We define the $\infty$-category of two-step graded objects as follows:
$\mathrm{Gr}^{\\{0,1\\}}(\mathrm{Mod}_{R}):=\operatorname{Fun}(\\{0,1\\}^{\mathrm{op}},\mathrm{Mod}_{R})$.
An object of this $\infty$-category is the same as a pair of objects
$(X^{0},X^{1})$. This $\infty$-category is a localization of the
$\infty$-category $\mathrm{Gr}^{\geq 0}\mathrm{Mod}_{R}$. We let
$(-)^{\\{0,1\\}}:\mathrm{Gr}^{\geq
0}\mathrm{Mod}_{R}\rightarrow\mathrm{Gr}^{\\{0,1\\}}\mathrm{Mod}_{R}$ to be
the localization functor. It sends a positively graded object
$\\{X^{i}\\}_{i\geq 0}\in\mathrm{Gr}^{\geq 0}\mathrm{Mod}_{R}$ to the two-step
graded object $(X^{0},X^{1})\in\mathrm{Gr}^{\\{0,1\\}}\mathrm{Mod}_{R}$. The
$\infty$-category $\mathrm{Gr}^{\\{0,1\\}}\mathrm{Mod}_{R}$ has a symmetric
monoidal structure such that the localization functor is symmetric monoidal.

  * •

Similarly, we define the $\infty$-category of two-step filtrations
$\operatorname{Fil}^{[0,1]}(\mathrm{Mod}_{R}):=\operatorname{Fun}((0<1)^{\mathrm{op}},\mathrm{Mod}_{R})$.
An object of this $\infty$-category is the same as a map $X^{1}\rightarrow
X^{0}$ in $\mathrm{Mod}_{R}$. As with two-step graded objects, this
$\infty$-category is a localization of $\operatorname{Fil}^{\geq
0}\mathrm{Mod}_{R}$. The localization functor $(-)^{\leq
1}:\operatorname{Fil}^{\geq
0}\mathrm{Mod}_{R}\rightarrow\operatorname{Fil}^{[0,1]}\mathrm{Mod}_{R}$ is
given by the formula

$(...\rightarrow X^{2}\rightarrow X^{1}\rightarrow X^{0})^{\leq
1}\simeq(X^{1}/X^{2}\rightarrow X^{0}/X^{2}).$

The $\infty$-category $\operatorname{Fil}^{[0,1]}\mathrm{Mod}_{R}$ has a
symmetric monoidal structure making the localization functor symmetric
monoidal.

Let $\mathrm{DAlg}_{R}\mathrm{Mod}$ be the $\infty$-category of pairs $(A,M)$
with $A\in\mathrm{DAlg}_{R}$ and $M\in\mathrm{Mod}_{A}$, and
$\mathrm{Gr}^{\\{0,1\\}}\mathrm{Mod}_{R}\subset\mathrm{Gr}^{\geq
0}\mathrm{Mod}_{R}$ be the full subcategory of non-negatively filtered
$R$-module spectra consisting of objects concentrated in two degrees. There is
a symmetric monoidal structure on $\mathrm{Gr}^{\\{0,1\\}}\mathrm{Mod}_{R}$
such that the functor $(-)^{\\{0,1\\}}:\mathrm{Gr}^{\geq
0}\mathrm{Mod}_{R}\rightarrow\mathrm{Gr}^{\\{0,1\\}}\mathrm{Mod}_{R}$, left
adjoint to the inclusion, is symmetric monoidal. The following observation is
due to A.Raksit, see [R].

######  Proposition 3.3.2.

There is an equivalence of $\infty$-categories

$\textstyle{\mathrm{Gr}^{\\{0,1\\}}\mathrm{DAlg}_{R}\simeq\mathrm{DAlg}_{R}\mathrm{Mod}.}$

######  Remark 3.3.3.

As the functor
$\operatorname{Fil}^{[0,1]}\mathrm{Mod}_{R}\rightarrow\operatorname{Fil}^{\geq
0}\mathrm{Mod}_{R}$ (respectively,
$\mathrm{Gr}^{\\{0,1\\}}\mathrm{Mod}_{R}\rightarrow\mathrm{Gr}^{\geq
0}\mathrm{Mod}_{R}$) is an inclusion, we can say that a derived commutative
algebra in $\operatorname{Fil}^{[0,1]}\mathrm{Mod}_{R}$ (respectively, in
$\mathrm{Gr}^{\\{0,1\\}}\mathrm{Mod}_{R}$) is the same as a derived
commutative algebra in $\operatorname{Fil}^{\geq 0}\mathrm{Mod}_{R}$
($\mathrm{Gr}^{\geq 0}\mathrm{Mod}_{R}$) whose underlying linear object is in
$\operatorname{Fil}^{[0,1]}\mathrm{Mod}_{R}$
($\mathrm{Gr}^{\\{0,1\\}}\mathrm{Mod}_{R}$). We will use notations
$\operatorname{Fil}^{[0,1]}\mathrm{DAlg}_{R}$ for the $\infty$-category of
two-step filtered derived algebras and
$\mathrm{Gr}^{\\{0,1\\}}\mathrm{DAlg}_{R}$ for the $\infty$-category of two-
step graded derived algebras respectively.

We mentioned an equivalence
$\mathrm{Gr}^{\\{0,1\\}}\mathrm{DAlg}_{R}\simeq\mathrm{DAlg}_{R}\mathrm{Mod}$
before. Let us now compute the free two-step graded derived commutative
algebra $\mathrm{LSym}_{R}^{\\{0,1\\}}(M^{0}\oplus M^{1})$ on an object
$M^{0}\oplus M^{1}\in\mathrm{Gr}^{\\{0,1\\}}\mathrm{Mod}_{R}$. For a derived
commutative ring $A$ and an $A$-module, we let $A\oplus M$ be the derived
commutative algebra in $\mathrm{Gr}^{\\{0,1\\}}\mathrm{Mod}_{R}$ formed by
taking a trivial square zero extension of $A$ by $M$ put in weight 1.

######  Proposition 3.3.4.

There is an equivalence of derived commutative rings in
$\mathrm{Gr}^{\\{0,1\\}}\mathrm{Mod}_{R}$:

$\mathrm{LSym}_{R}^{\\{0,1\\}}(M^{0}\oplus
M^{1})\simeq\mathrm{LSym}_{R}(M^{0})\oplus M^{1}.$

###### Proof.

Indeed, let $\mathrm{LSym}_{R}^{\bullet}(M^{0}\oplus M^{1})$ be the free
graded derived commutative ring on $(M^{0},M^{1})$ considered as an object of
$\mathrm{Gr}^{\geq 0}\mathrm{Mod}_{R}$. There is a map of derived commutative
algebras $\mathrm{LSym}^{\bullet}_{R}(M^{0}\oplus
M^{1})\rightarrow\mathrm{LSym}_{R}(M^{0})\oplus M^{1}$ obtained by moding out
all higher symmetric powers of $M^{1}$. Applying symmetric monoidal truncation
functor $(-)^{\leq 1}:\mathrm{Gr}^{\geq
0}\mathrm{Mod}_{R}\rightarrow\mathrm{Gr}^{\\{0,1\\}}\mathrm{Mod}_{R}$, we
obtain a map $\mathrm{LSym}_{R}^{\\{0,1\\}}(M^{0}\oplus
M^{1})\simeq\mathrm{LSym}^{\bullet}_{R}(M^{0}\oplus M^{1})^{\leq
1}\rightarrow\mathrm{LSym}_{R}(M^{0})\oplus M^{1}$. By definition, truncation
functor kills all pieces in weights $\geq 2$. Since $n$-th derived symmetric
power of an object in weight $1$ is in weight $n$, it follows that truncation
of a free derived commutative algebra on an object $M^{0}\oplus M^{1}$ kills
all higher symmetric powers of $M^{1}$ of degree $n\geq 2$. Therefore, the
constructed map is an equivalence. ∎

The following remark enables us to define a symmetric monoidal
$\infty$-category of $\mathbb{D}_{-}$-module objects in
$\mathrm{Gr}^{\\{0,1\\}}\mathrm{Mod}_{R}.$

######  Construction 3.3.5.

The algebra $\mathbb{D}_{-}$ has the structure of a _derived cocommutative
bialgebra_ in $\mathrm{Gr}^{\\{0,1\\}}\mathrm{Mod}_{R}$. To see this, take the
free graded associative algebra $\mathrm{T}(R(1)[-1])$ on the shift of the
object $R(1)[-1]\in\mathrm{Gr}^{\geq 0}\mathrm{Mod}_{R}$. It has a structure
of a derived cocommutative bialgebra in $\mathrm{Gr}^{\geq
0}\mathrm{Mod}_{R}$. Applying symmetric monoidal truncation functor $(-)^{\leq
1}$, wee see that the trivial square zero extension algebra
$\mathrm{T}(R(1)[-1])^{\leq 1}\simeq R\oplus R(1)[-1]\simeq\mathbb{D}_{-}$
gets the structure of a derived cocommutative bialgebra in
$\mathrm{Gr}^{\\{0,1\\}}\mathrm{Mod}_{R}$.

######  Definition 3.3.6.

Using Construction 3.3.5, we define an $\infty$-category of differential two-
step graded modules

$\mathrm{DG}^{\\{0,1\\}}_{-}\mathrm{Mod}_{R}:=\mathrm{Mod}_{\mathbb{D}_{-}}(\mathrm{Gr}^{\\{0,1\\}}\mathrm{Mod}_{R})$

as $\mathbb{D}_{-}$-modules in $\mathrm{Gr}^{\\{0,1\\}}\mathrm{Mod}_{R}$ with
the symmetric monoidal structure coming from the cocommutative bialgebra
structure on $\mathbb{D}_{-}$. In particular, the forgetful functor
$\mathrm{DG}^{\\{0,1\\}}_{-}\mathrm{Mod}_{R}\rightarrow\mathrm{Gr}^{\\{0,1\\}}\mathrm{Mod}_{R}$
is symmetric monoidal. We will also consider the $\infty$-category

$\mathrm{DG}^{\\{0,1\\}}\mathrm{DAlg}_{R}:=\mathrm{DAlg}(\mathrm{DG}^{\\{0,1\\}}_{-}\mathrm{Mod}_{R})$

of derived commutative algebra objects in this symmetric monoidal
$\infty$-category, which is the same as derived differential graded algebras
concentrated in degrees $0,1$. These are referred to as derived differential
two-step graded algebras in $\mathrm{Mod}_{R}$.

The filtered cousine of differential two-step graded algebras is the
$\infty$-category $\operatorname{Fil}^{[0,1]}\mathrm{DAlg}_{R}$ of non-
negatively filtered derived algebras which are concentrated in degrees $0,1$.
We can also understand them as derived commutative algebras in the symmetric
monoidal $\infty$-category $\operatorname{Fil}^{[0,1]}\mathrm{Mod}_{R}$ (with
the symmetric monoidal structure making truncation functor $(-)^{\leq
1}:\operatorname{Fil}^{\geq
0}\mathrm{Mod}_{R}\rightarrow\operatorname{Fil}^{[0,1]}\mathrm{Mod}_{R}$
symmetric monoidal. Note that the functor
$\operatorname{gr}:\operatorname{Fil}^{\geq
0}\mathrm{Mod}_{R}\rightarrow\mathrm{DG}_{-}^{\geq 0}\mathrm{Mod}_{R}$
restricts to a symmetric monoidal functor
$\operatorname{gr}:\operatorname{Fil}^{[0,1]}\mathrm{Mod}_{R}\rightarrow\mathrm{DG}_{-}^{\\{0,1\\}}\mathrm{Mod}_{R}$.

######  Proposition 3.3.7.

The functor

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

is an equivalence of symmetric monoidal $\infty$-categories.

###### Proof.

Indeed, this follows from the fact that
$\operatorname{gr}:\widehat{\operatorname{Fil}^{\geq
0}\mathrm{Mod}}\rightarrow\mathrm{DG}_{-}^{\geq 0}\mathrm{Mod}$ is an
equivalence of symmetric monoidal $\infty$-categories, and the fact that any
two-step filtration is automatically complete. ∎

######  Corollary 3.3.8.

The functor $\operatorname{gr}$ lifts to an equivalence of $\infty$-categories

$\textstyle{\operatorname{gr}:\operatorname{Fil}^{[0,1]}\mathrm{DAlg}_{R}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{\mathrm{DG}_{-}^{\\{0,1\\}}\mathrm{DAlg}_{R}.}$

The category of discrete two-step graded commutative algebras
$\operatorname{Fil}^{[0,1]}\operatorname{CAlg}_{R,\heartsuit}$ can be seen to
be equivalent to the full subcategory of the category of algebras with a Smith
ideal spanned by objects $I\rightarrow A$ for which the multiplication map
$I\otimes_{A}I\rightarrow I$ is zero. In other words, this is the category of
“generalized square-zero extensions”. It is a classical fact in derived
algebraic geometry that square-zero extensions are controlled by derivations.
The equivalence of Corollary 3.3.8 suggests that the right hand side should
have a description in terms of derivations, and the left hand-side in terms of
derived square-zero Smith ideals. Below we will show that this is indeed the
case.

In what follows, we fix a connective derived commutative ring $R$. Recall from
Construction 3.1.9 that there is a lax symmetric monoidal structure on the
functor
$(\operatorname{Fil}\rightarrow\mathrm{Gr}):\mathrm{Mod}_{R}^{\Delta^{1}_{\vee}}\rightarrow\mathrm{Mod}_{R}^{\\{0,1\\},\vee}$.
We observe the following proposition.

######  Construction 3.3.9.

The tautological functor
$\mathrm{Gr}^{\\{0,1\\}}\mathrm{Mod}_{R}\rightarrow\mathrm{Mod}_{R}^{\\{0,1\\},\vee}$
has a lax symmetric monoidal structure. We can realize it as a composition of
two lax symmetric monoidal functors. First, we use lax symmetric monoidal
inclusion
$\mathrm{Gr}^{\\{0,1\\}}\mathrm{Mod}_{R}\hookrightarrow\mathrm{Gr}^{\geq
0}\mathrm{Mod}_{R}$. Next, we consider the functor
$\operatorname{ev}^{\\{0,1\\}}:\mathrm{Gr}^{\geq
0}\mathrm{Mod}_{R}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\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{}$}}}}}}}\ignorespaces}}}}\ignorespaces\mathrm{Mod}_{R}^{\\{0,1\\},\vee}$
defined as a composition of lax symmetric monoidal functors

$\textstyle{\mathrm{Gr}^{\geq
0}\mathrm{Mod}_{R}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{ev}^{\\{0,1\\}}}$$\scriptstyle{\mathrm{split}}$$\textstyle{\mathrm{Mod}^{\\{0,1\\},\vee}_{R}}$$\textstyle{\operatorname{Fil}^{\geq
0}\mathrm{Mod}_{R}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{ev}^{[0,1]}}$$\textstyle{\mathrm{Mod}_{R}^{\Delta^{1}_{\vee}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces.}$$\scriptstyle{(\operatorname{Fil}\rightarrow\mathrm{Gr})}$

The composition of the inclusion
$\mathrm{Gr}^{\\{0,1\\}}\mathrm{Mod}_{R}\hookrightarrow\mathrm{Gr}^{\geq
0}\mathrm{Mod}_{R}$ with $\operatorname{ev}^{\\{0,1\\}}$ thus has a lax
symmetric monoidal structure.

######  Construction 3.3.10.

We will construct a commutative square in the $\infty$-category
$\operatorname{CAlg}(\Pr^{L})_{\mathrm{lax}}$ of symmetric monoidal