# Infinitesimal derived foliations

Bertrand Toën and Gabriele Vezzosi

###### Abstract.

We introduce a notion of _infinitesimal derived foliation_. We prove it is
related to the classical notion of infinitesimal cohomology, and satisfies
some formal integrability properties. We also provide some hints on how
infinitesimal derived foliations compare to our previous notion of derived
foliations of [Toë20, TV].

###### Contents

  1. 1 Derived affine stacks
    1. 1.1 Cosimplicial-simplicial algebras
    2. 1.2 Spectrum of cosimplicial-simplicial rings
  2. 2 Infinitesimal derived foliations
  3. 3 Infinitesimal cohomology
  4. 4 Integrability by formal groupoids
  5. 5 Towards a comparison between infinitesimal derived foliations and derived foliations
    1. 5.1 Red shift
    2. 5.2 Infinitesimal structures on derived foliations

## Introduction

The object of this short note is to present a notion of _infinitesimal derived
foliations_ on general schemes and stacks over bases of arbitrary
characteristics (see Definition 2.1). This notion is slightly different from
the one exposed in [Toë20] but it is based on similar ideas. By definition, an
infinitesimal derived foliation on $X$ consists of a perfect complex $E$ (the
cotangent complex of the foliation) together with an action of
$G_{0}=\Omega_{0}\mathbb{G}_{a}$, the loop group of the additive group, on the
total space $\mathbb{V}(E[-1])$ of the shift $E[-1]$, compatible with the
natural gradings. This is very similar to the definition in [Toë20], except
that the group $BKer$ is replaced here by $G_{0}$, and the shift by $1$ is
replaced by a shift by $-1$. These changes may sound purely aesthetic choices,
but outside of characteristic zero these two notions of derived foliations
are, in fact, different.

A major advantage of infinitesimal derived foliations is that they satisfy
several formal integrability results (see Corollary 4.2) that do _not_ hold
for derived foliations. The key point here is that the natural cohomology
theory related to infinitesimal derived foliations is (Hodge-completed
derived) infinitesimal cohomology, whereas derived foliations are related to
(Hodge-completed derived) de Rham cohomology. This relation with infinitesimal
cohomology makes it possible to formally integrate any (smooth) infinitesimal
derived foliation by a smooth formal groupoid. In a way, infinitesimal derived
foliations can be thought of as derived foliations endowed with extra
conditions/structures. This resembles the difference between foliations and
_restricted foliations_ in characteristic $p>0$ (see for instance [Eke87,
Prop. 2.3] as well as [Miy87, Def. 3.6]).

The first section of this paper contains results on _derived affine stacks_ ,
that are a derived extension of affine stacks of [Toe06]. We present in
particular algebraic models for those, using a certain model category of
cosimplicial-simplicial commutative algebras, which might be of independent
interests. These models will be particularly useful in our last section where
we construct the red-shift construction for graded derived affine stacks (see
§5). The second section presents the general definition of infinitesimal
derived foliations and their basic properties. In section $\S 3$ we study the
cohomology of infinitesimal derived foliations and prove that it recovers the
usual notion of infinitesimal cohomology when applied to the tautological
foliation. This cohomology comparison is then used to construct, for any
smooth infinitesimal derived foliation $\mathcal{F}$ on a smooth scheme $X$, a
smooth formal groupoid $G_{\mathcal{F}}\to X$, and we show that
$\mathcal{F}\mapsto G_{\mathcal{F}}$ produces an equivalence of categories.
This result has to be related with [Eke87, Prop. 2.3], and in a way, is one
possible generalization of Ekedahl’s result. Finally, in the very last
section, we study the relations between infinitesimal derived foliations and
our original notion of derived foliations (see [Toë20, TV]). Unfortunately the
results here are not optimal, and we leave several open questions on the way.
We think that there should exists a forgetful functor from infinitesimal
derived foliations to derived foliations, obtained by some application of the
red-shift construction, but we could not find a completely satisfying
construction. The reader should consider this last part as a suggestion for
future research.

As a final comment, this note will appear in an extended form as a chapter of
a book in preparation on the subject ([TV]).

Acknowledgments. We thank J. Nuiten and J. Fu for several discussions on the
content of this paper. The idea of infinitesimal derived foliations has been
inspired by J. Fu’s current work on the structure of the Chevalley-Eilenberg
complex of partition Lie algebras.

##  1\. Derived affine stacks

###  1.1. Cosimplicial-simplicial algebras

We let denote by $sCR$ be the category of simplicial commutative rings. It is
endowed with its usual _model category_ structure for which equivalences and
fibrations are defined on the underlying simplicial sets.

Similarly, we denote by $csCR$ the category of cosimplicial objects in $sCR$,
or equivalently of cosimplicial-simplicial commutative rings. Its objects will
be denoted by $A^{*}_{*}$, where the superscript will refer to the
cosimplicial direction and subscript to the simplicial direction. The category
$csCR$ can be endowed with a _tensored_ and _cotensored_ structure over the
category $sSet$ of simplicial sets. We warn the reader here that there are
several natural such structures and that the choice we make here is important
for the sequel. For a simplicial set $K$ and an object $A$ of $csCR$, we
define the cotensored structure by $A^{K}$ by the formula

$(A^{K})^{q}_{p}:=(A^{q}_{p})^{K_{q}}$

for $[p]\in\Delta^{op}$ and $[q]\in\Delta$. The tensored structure $K\otimes
A$, for $A\in csCR$ and $K\in sSets$, is defined by the usual adjunction
formula $Hom(K\otimes A,B)\simeq Hom(A,B^{K})$, for arbitrary $B\in csCR$.
Similarly, we have a canonical simplicial enrichment for which the simplicial
Homs are denoted by $\underline{Hom}$ and are defined by

$\underline{Hom}(A,B)_{p}:=Hom_{csCR}(A,B^{\Delta^{p}}).$

By considering the natural embedding $Mod_{\mathbb{Z}}\subset C(\mathbb{Z})$,
from $\mathbb{Z}$-modules to cochain complexes, any object $A\in csCR$ can be
considered, by forgetting the multiplicative structure, as a cosimplicial-
simplicial object in $C(\mathbb{Z})$. We denote by $Tot^{\pi}(A)$ the complex
equals to $\prod_{q-p=n}A^{q}_{p}$ in cohomological degree $n$ and whose
differential is given by the alternated sums of the faces and cofaces of the
cosimplicial-simplicial diagram. Equivalently, $A$ can be first turned into a
bi-complex, by individual totalisations along the cosimplicial and the
simplicial directions, and $Tot^{\pi}(A)$ can then be realized as the product-
total complex of this bicomplex. Note also that $Tot^{\pi}(A)$ is a model for
the homotopy limit-colimit of $A$

$Tot^{\pi}(A)\simeq\mathrm{holim}_{[p]\in\Delta}\left(\mathrm{hocolim}_{[q]\in\Delta^{op}}A^{q}_{p}\right)\in
C(\mathbb{Z}).$

######  Definition 1.1.

A morphism $A\to B$ in $csCR$ is a _completed quasi-isomorphism_ if the
induced morphism $Tot^{\pi}(A)\to Tot^{\pi}(B)$ is a quasi-isomorphism of
complexes.

We warn the reader that even though the categories $csCR$ and $scCR$, of
cosimplicial-simplicial commutative rings and of simplicial-cosimplicial
rings, are equivalent, the notion of completed quasi-isomorphisms is not the
same as the notion of weak equivalences used for instance in [BCN21].

######  Theorem 1.2.

The category $csCR$ is endowed with a model category structure for which the
weak equivalences are the completed quasi-isomorphism, and the fibrations are
the epimorphisms. This model category structure is moreover cofibrantly
generated and is a simplicial model category for the above mentioned
simplicial enrichment.

Proof. As fibrations111Note that $f:A\to B$ in $csCR$ is surjective iff
$Tot^{\pi}(f)$ is. and equivalences are defined via the functor $Tot^{\pi}$,
the proof of the theorem consists of transferring the projective model
structure on $C(\mathbb{Z})$ to $csCR$ via the functor $Tot^{\pi}$. For this
we apply the path object argument (see for instance [BM03, §2.6]).

The functor $Tot^{\pi}$ is a right adjoint whose left adjoint sends a complex
$E\in C(\mathbb{Z})$ to the free commutative ring over the cosimplicial-
simplicial module given by

$(p,q)\mapsto(C^{*}(\Delta^{p})\otimes C_{*}(\Delta^{q})\otimes
E)^{0}/Im\,d^{-1},$

where $C^{*}(\Delta^{p})$ and $C_{*}(\Delta^{q})$ are the cohomology and
homology complexes of the standard simplex $\Delta^{p}$ and $\Delta^{q}$. This
left adjoint will be denoted by $L\phi$, as being the composition of the free
commutative ring functor $L:csMod\to csCR$, with the functor
$\phi:C(\mathbb{Z})\to csMod$ defined by the above formula. Clearly, the
functor $Tot^{\pi}$ preserves small objects (but does not commute with
filtered colimits in general). In order to apply [BM03, §2.6] we thus simply
have to prove that $csCR$ possesses a fibrant replacement functor and that all
fibrant object possesses a path object. But all objects are fibrant by
definition, and we are thus reduced to show the existence of path objects. For
this we use the following lemma.

######  Lemma 1.3.

For any simplicial set $K$ and any $A\in csCR$, there exists a canonical
quasi-isomorphism

$Tot^{\pi}(A^{K})\simeq Tot^{\pi}(A)^{K},$

where for $E\in C(\mathbb{Z})$ and $K\in sSet$ we define

$E^{K}:=\mathrm{holim}_{[p]\in\Delta}E^{K_{p}}.$

Proof of the lemma. We remind that for any functor
$F:\Delta\times\Delta\to\mathbf{dg}$ (or more generally any bi-cosimplicial
object in a complete $\infty$-category), the homotopy limit of $F$ can be
computed, up to a quasi-isomorphism, either component-wise, or diagonally

$\mathrm{holim}_{[p]\in\Delta}F(p,p)\simeq\mathrm{holim}_{[p]\in\Delta}\mathrm{holim}_{[q]\in\Delta}F(p,q).$

We apply this to the functor sending $([p],[q])$ to the complex
$N(A^{q}_{*})^{K_{p}}$, where $N:sMod\to dg$ is the normalization
construction. Computing the homotopy limit diagonally yields
$Tot^{\pi}(A^{K})$, whereas the double homotopy limit construction yields
$Tot^{\pi}(A)^{K}$. $\Box$

Lemma 1.3 now implies the existence of path objects. Indeed, for $A\in csCR$
an explicit path object is given by $A\to A^{\Delta^{1}}\to
A^{\partial\Delta^{1}}=A\times A$. The fact that the constant map morphism
$A\to A^{\Delta^{1}}$ is a completed quasi-isomorphism follows from the lemma
1.3. The fact that $A^{\Delta^{1}}\to A^{\partial\Delta^{1}}=A\times A$ is a
fibration simply follows from the fact that it is a levelwise epimorphism as
this can be checked easily using the explicit formula for $A^{K}$.

In order to finish the proof of the theorem, we are left to showing that the
simplicial enrichement is compatible with the model category structure. In
other words, we must show that for any cofibration of simplicial sets
$i:K\hookrightarrow L$ and any fibration $f:A\to B$ in $csCR$, the induced
morphism

$f^{i}:A^{L}\longrightarrow B^{L}\times_{B^{K}}A^{K}$

is a fibration, which is also an equivalence if $i$ is a trivial cofibration
or $f$ is a trivial fibration. The fact that the morphism $f^{i}$ is a
fibration is clear because fibrations are epimorphisms and because of the
explicit formula for the exponentiation by a simplicial set in $csCR$.
Finally, the fact that $f^{i}$ is also a weak equivalence when $i$ or $f$ is
so simply follows from Lemma 1.3. $\Box$

######  Definition 1.4.

The _$\infty$ -category of cs-rings_ is the $\infty$-category obtained from
$csCR$ by inverting the completed quasi-isomorphisms. It is denoted by
$\mathbf{csCR}$ (or by $\mathbf{csCR}_{\mathbb{U}}$ if one wants to specify
cosimplicial-simplicial rings belonging to a given universe $\mathbb{U}$).

######  Remark 1.5.

_We note also that the model structure on $csCR$ can be equivalently be
obtained by first constructing a model structure on $csMod_{\mathbb{Z}}$, the
category of cosimplicial-simplicial abelian groups, where again the
equivalences and fibrations are defined via the $Tot^{\pi}$ functor. The model
structure on $csCR$ can then be obtained by transferring along the forgetful
functor $csCR\to csMod_{\mathbb{Z}}$, with left adjoint given by the free
commutative cosimplicial-simplicial ring._

######  Remark 1.6.

_To finish, we mention that it is possible to a construct an $\infty$-functor
$Tot^{\pi}:\mathbf{csCR}\longrightarrow LSym-Alg$, from our $\infty$-category
of cosimplicial-simplicial commutative rings to the $\infty$-category of
_derived rings_ , namely modules over the $LSym$-monad. Here we denote by
$LSym$ the monad associated to the derived commutative operad as in [BCN21,
Ex. 3.71]. This $\infty$-functor simply sends $A$ to $\lim_{q}A^{q}_{*}$,
where the limit is taken in $LSym-Alg$ and $A^{q}_{*}$ is considered as a
connective $LSym$-algebra and thus as an object in $LSym-Alg$. This
$\infty$-functor is the right adjoint of an adjunction, and preserves free
objects over perfect complexes. It is likely that $Tot^{\pi}$ is an
equivalence of $\infty$-categories when restricted to nice enough objects._

###  1.2. Spectrum of cosimplicial-simplicial rings

For a fixed commutative simpicial ring $k\in sCR$, we can work relatively over
$k$ and define the model category $k-csCR$ of cosimplicial-simplicial
commutative $k$-algebras, and its associated $\infty$-category
$k-\mathbf{csCR}$. As usual, we have a natural equivalence of
$\infty$-categories

$k-\mathbf{csCR}\simeq k/\mathbf{csCR},$

where $k$ is considered as an object in $\mathbf{csCR}$ which is constant is
the cosimplicial direction.

We consider $dAff_{k}:=(k-sCR)^{op}$, the model category of derived affine
$k$-schemes, which is defined to be the opposite category of that of
simplicial commutative $k$-algebras. Remind from [TV08] that it can be endowed
with the fpqc model topology, and that we can consider the model category of
(hyper-complete) stacks $dAff_{k}^{\sim,fpqc}$. There are here some set-
theoretical issues, that can be solved, as usual, by fixing two universes
$\mathbb{U}\in\mathbb{V}$. By definition $k-sCR$ refers here to
$\mathbb{U}$-small simplicial $k$-algebras, and $dAff_{k}^{\sim,fpqc}$ is then
the category of functors $k-sCR\to sSet_{\mathbb{V}}$ to $\mathbb{V}$-small
simplicial sets.

We then consider a functor

$\mathbf{Spec}^{\Delta}:k-csCR_{\mathbb{V}}^{op}\longrightarrow
dAff_{k}^{\sim,fpqc},$

from $\mathbb{V}$-small cosimplicial-simplicial commutative $k$-algebras to
$dAff_{k}^{\sim,fpqc}$. It is defined by sending $A\in k-csCR_{\mathbb{V}}$ to
the functor $\mathbf{Spec}^{\Delta}\,A:sCR\to sSet_{\mathbb{V}}$ defined by

$\mathbf{Spec}^{\Delta}\,A:B\mapsto\underline{Hom}(A,B)=Hom_{csCR}(A,B^{\Delta^{\bullet}}),$

where $\underline{Hom}$ are the simplicial $Hom$’s of the simplicial
enrichement described in the previous section, and $B$ is considered as an
object in $k-csCR_{\mathbb{V}}$ by viewing it as constant in the cosimplicial
direction.

######  Propositon 1.7.

The functor $\mathbf{Spec}^{\Delta}:k-csCR_{\mathbb{V}}^{op}\longrightarrow
dAff_{k}^{\sim,fpqc}$ is right Quillen, and the induced functor on
$\infty$-categories

$\mathbf{Spec}^{\Delta}:k-\mathbf{csCR}_{\mathbb{U}}\longrightarrow\mathbf{dSt}_{k}$

is fully faithful. The essential image of $\mathbf{Spec}^{\Delta}$ is the
smallest full sub-$\infty$-category of $\mathbf{dSt}_{k}$ containing the
objects $K(\mathbb{G}_{a},n)$ for various $n$, and which is stable by
$\mathbb{U}$-small limits.

Proof. This is proven in a very similar manner than [Toe06, Cor. 2.2.3]. The
left adjoint to $\mathbf{Spec}^{\Delta}$, denoted by $\mathcal{O}$, sends a
representable presheaf $h^{B}=Hom(B,-):k-sCR\to sSet$ to $B\in k-csCR$, which
is considered as constant in the cosimplicial direction. It also sends an
object of the form $K\times h^{B}$, for a simplicial set $K\in sSet$, to
$B^{K}\in k-csCR$. Finally, it is uniquely defined by these properties
together with the requirement that it sends colimits in $dAff_{k}^{\sim,fpqc}$
to limits in $k-csCR$.

To prove that $\mathbf{Spec}^{\Delta}$ is right Quillen we use that
$dAff_{k}^{\sim,fpqc}$ is a left Bousfield localization of the levelwise
projective model structure on the category of simplicial presheaves
$Fun(k-sCR_{\mathbb{U}},sSet)$ obtained by inverting fpqc-local equivalences
and equivalences in $sCR$ (see [TV08]). Therefore, by general facts about
Bousfield localizations, it is enough to show that $\mathbf{Spec}^{\Delta}$ is
a right Quillen functor for the levelwise projective model structure, and
moreover that for any cofibrant $A\in k-csCR$, $\mathbf{Spec}^{\Delta}\,(A)$
is a fibrant object in $dAff_{k}^{\sim,fpqc}$. The first of these statements
easily follows from the fact that $k-csCR$ is a simplicial model category in
which every object is fibrant, and thus that if $A\to A^{\prime}$ is a
(trivial) cofibration in $k-csCR$, for any $C\in k-sCR$ the morphism
$\underline{Hom}(A^{\prime},C)\to\underline{Hom}(A,C)$ is a (trivial)
fibration of simplicial sets. For the second of these statements, let $A\in
k-csCR$ be a cofibrant object. Any equivalence $B\to B^{\prime}$ in $k-sCR$
obviously induces an equivalence in $k-csCR$ when considered both $B$ and
$B^{\prime}$ as constant in the cosimplicial direction, and thus
$\underline{Hom}(A,B)\to\underline{Hom}(A,B^{\prime})$ is an equivalence of
simplicial sets. This shows that $\mathbf{Spec}^{\Delta}(A)$ is fibrant when
the topology is the trivial topology. Finally, if $B=\mathrm{holim}_{i}B_{i}$
is a homotopy limit in $k-sCR$, it is also a homotopy limit in $k-csCR$, and
thus the natural morphism
$\underline{Hom}(A,B)\to\mathrm{holim}_{i}\underline{Hom}(A,B_{i})$ is an
equivalence. As the fpqc model topology is subcanonical, this implies that
$\mathbf{Spec}^{\Delta}(A)$ is indeed a stack for the fpqc topology, and thus
fibrant as an object in $dAff_{k}^{\sim,fpqc}$ (see [TV08]).

To finish the proof of the proposition, let $A\in k-csCR_{\mathbb{U}}$ and
$X=\mathbf{Spec}^{\Delta}(A)$. By definition of the simplicial structure on
$k-csCR$, the functor $X$ is the geometric realization of the simplicial
object $q\mapsto h^{A_{*}^{q}}$, where $h^{A_{*}^{q}}:k-sCR\to sSet$ is
corepresented by $A_{*}^{q}\in k-sCR$. In other words, we have
$\mathbf{Spec}^{\Delta}(A)\simeq\mathrm{hocolim}_{q}h^{A^{q}_{*}}$. Therefore,
the adjunction morphism

$A\longrightarrow\mathcal{O}(\mathbf{Spec}^{\Delta}(A))$

is the canonical morphism $A\to\mathrm{holim}_{q}A^{q}_{*}$ in
$Ho(k-csCR_{\mathbb{U}})$. This canonical morphism is obviously an
equivalence, showing the fully faithfulness property in the proposition. It is
then formal that the essential image of $\mathbb{R}\mathbf{Spec}^{\Delta}$ is
stable by limits. It also contains the objects $K(\mathbb{G}_{a},n)$, as these
are the images of $L\phi(\mathbb{Z}[-n])$, where $L\phi:C(\mathbb{Z})\to
k-csCR$ is the left adjoint to $Tot^{\pi}$. Finally, as $k-csCR$ is
cofibrantly generated, any object is equivalent to a $I$-cell object, where
$I$ is the image by $L\phi$ of the generating cofibrations in $C(\mathbb{Z})$.
The images of $I$ by $\mathbf{Spec}^{\Delta}$ are equivalent to morphisms of
the form $*\to K(\mathbb{G}_{a},n)$, so that any object is the essential image
of $\mathbb{R}\mathbf{Spec}^{\Delta}$ lies in the smallest
sub-$\infty$-category containing the $K(\mathbb{G}_{a},n)$ and stable by
limits. $\Box$

######  Definition 1.8.

The $\infty$-category of _( $k$-linear) derived affine stacks_ is the
essential image of
$\mathbb{R}\mathbf{Spec}^{\Delta}:k-\mathbf{csCR}_{\mathbb{U}}^{op}\hookrightarrow\mathbf{dSt}_{k}$.
It is denoted by $\mathbf{dChAff}_{k}$.

The above notions and results also have _graded versions_ , for which the
details are left to the reader. We denote by $k-csCR^{gr}$ the category of
graded cosimplicial-simplicial commutative $k$-algebras. We endow this
category with the model category structure for which equivalences and
fibrations are defined by the graded Tot functor to graded complexes
$Tot^{\pi}:k-scCR^{gr}\to C(\mathbb{Z})^{gr}$, defined by taking the
$Tot^{\pi}$ of each graded component. The proof of the existence of this model
category structure follows the same lines as in the non-graded case. The
corresponding $\infty$-category will be denoted by $k-\mathbf{csCR}^{gr}$.

We consider the strict multiplicative group
$\mathbb{G}_{m}^{st}\in\mathbf{dAff}_{k}^{\sim,fpqc}$, given by the functor
sending $B\in k-sCR$ to the group $B_{0}^{*}$ of invertible elements in the
ring $B_{0}$ of $0$-simplices in $B$. This is a group object in the model
category $\mathbf{dAff}_{k}^{\sim,fpqc}$, which is not a fibrant object (it
does not preservers equivalences in $B$), but its image in $\mathbf{dSt}_{k}$
is the usual multiplicative group scheme $\mathbb{G}_{m}$. Indeed,
$\mathbb{G}_{m}^{st}=h^{k[t,t^{-1}]}$ is corepresented by $k[t,t^{-1}]$, and,
as shown in [TV08], a fibrant model for $h^{A}$ is the derived scheme
$\mathbf{Spec}\,A:B\mapsto Map(A,B)$. In particular, we have that the model
category $\mathbb{G}_{m}^{st}-\mathbf{dAff}_{k}^{\sim,fpqc}$, of
$\mathbb{G}_{m}^{st}$-equivariant objects in $\mathbf{dAff}_{k}^{\sim,fpqc}$,
is a model for the $\infty$-category $\mathbb{G}_{m}-\mathbf{dSt}_{k}$, of
$\mathbb{G}_{m}$-equivariant derived stacks, called _graded derived
$k$-stacks_.

The graded version of the spec functor is the functor

$\mathbf{Spec}^{\Delta,gr}:(k-csCR^{gr})^{op}\longrightarrow\mathbb{G}_{m}^{st}-\mathbf{dAff}_{k}^{\sim,fpqc},$

sending $A\in k-csCR^{gr}$ to $\underline{Hom}(A,-)$, endowed with the natural
$\mathbb{G}_{m}^{st}$-action coming from the grading on $A$. More explicitly,
for $B\in k-sCR$, the set of q-simplices of $\mathbf{Spec}^{\Delta,gr}(A)(B)$
is the set of morphisms $Hom(A^{q},B)$, which is endowed with a
$B_{0}^{*}$-action as follows. For $b\in B_{0}^{*}$ and $f:A^{q}\to B$, we
define $bf:A^{q}\to B$ by the formula $(bf)(x)=b^{n}.f(x)$ for a homogeneous
element $x$ of degree $n$. As in the non-graded case,
$\mathbf{Spec}^{\Delta,gr}$ is a right Quillen functor, and the induced
$\infty$-functor

$\mathbb{R}\mathbf{Spec}^{\Delta,gr}:(k-\mathbf{csCR}^{gr}_{\mathbb{U}})^{op}\longrightarrow\mathbb{G}_{m}-\mathbf{dSt}_{k}$

is fully faithful. Its essential image is the smallest sub-$\infty$-category
containing all the objects $K(\mathbb{G}_{a}^{\chi},n)$, for $n\geq 0$ and
$\chi\in\mathbb{Z}$, where $\mathbb{G}_{m}$ acts on $\mathbb{G}_{a}$ with
weight $\chi\in\mathbb{Z}$, and which is stable by $\mathbb{U}$-small limits.

######  Definition 1.9.

The $\infty$-category of _graded derived affine $k$-stacks_ is the essential
image of
$\mathbb{R}\mathbf{Spec}^{\Delta,gr}:(k-\mathbf{csCR}_{\mathbb{U}}^{gr})^{op}\hookrightarrow\mathbb{G}_{m}-\mathbf{dSt}_{k}$.
It is denoted by $\mathbf{dChAff}_{k}^{gr}$.

We finish this section by recalling the notion of _linear derived stack_ ,
already introduced and studied in [Mon21]. We first notice that the Tot
functor $Tot^{\pi}:csMod_{k}\longrightarrow C(\mathbb{Z})$ admits a natural
lax monoidal structure. Therefore, it gives rise to an $\infty$-functor

$Tot^{\pi}:\mathsf{csMod}_{k}\longrightarrow\mathbf{dg}_{k},$

where $\mathsf{csMod}_{k}$ is the $\infty$-category of cosimplicial-simplicial
$k$-modules. This $\infty$-functor turns out to be an equivalence of
$\infty$-categories, and we denote by
$\phi:\mathbf{dg}_{k}\to\mathsf{csMod}_{k}$ its inverse.

For any object $E\in\mathbf{dg}_{k},$ we consider
$\phi(E)\in\mathsf{csMod}_{k}$, and define

$Sym^{\Delta}(E):=L\phi(E)\in csCR^{gr}$

the free cosimpicial-simplicial commutative ring generated by $\phi(E)$ (same
notation as in the proof of Theorem 1.2). Being a free commutative ring,
$Sym^{\Delta}(E)$ comes natural equipped with a graduation, and thus is
considered as an object in the $\infty$-category $\mathbf{csCR}^{gr}$.

######  Definition 1.10.

The _$k$ -linear derived stack associated to a complex $E\in\mathbf{dg}_{k}$_,
is the graded derived affine $k$-stack defined by

$\mathbb{V}(E):=\mathbb{R}\mathbf{Spec}^{\Delta,gr}\,(Sym^{\Delta}(E))\in\mathbb{G}_{m}-\mathbf{dSt}_{k}.$

Note that by construction the functor of points of $\mathbb{V}(E)$ sends $B\in
k-sCR$ to $Map_{\mathbf{dg}_{k}}(E,N(B))$, where $N(B)$ is the normalisation
of $B$ as a simplicial $k$-module. The $\mathbb{G}_{m}$-action is then the
natural action of invertible elements of $B$ on $B$ itself.

It is proven in [Mon21] that the $\infty$-functor sending $E$ to
$\mathbb{V}(E)$ is fully faithful when restricted to bounded above $k$-dg-
modules $E$, and in particular for perfect $k$-dg-modules $E$.

##  2\. Infinitesimal derived foliations

As before, we fix $k$ a base simplicial commutative ring.

We let $\mathbb{G}_{a}$ be the additive group scheme over $k$, and consider
$G_{0}:=\Omega_{0}\mathbb{G}_{a}$ its derived loop scheme based at the origin.
This is a derived affine group scheme over $k$, and as an affine derived
scheme is given by $G_{0}\simeq\mathbf{Spec}\,k[\epsilon_{-1}]$, where
$k[\epsilon_{-1}]$ is the free commutative simplicial $k$-algebra over one
generator $\epsilon$ in homotopical degree $1$. The standard
$\mathbb{G}_{m}$-action on $\mathbb{G}_{a}$ of weight $1$ induces a
$\mathbb{G}_{m}$-action on $G_{0}$ corresponding to the grading on
$k[\epsilon_{-1}]$ for which $\epsilon_{-1}$ is of weight $-1$. We then form
the semi-direct product $\mathcal{H}_{\pi}:=\mathbb{G}_{m}\ltimes G_{0}$,
which is a derived affine group scheme over $k$.

Before going further, let us mention that representations of the group
$\mathcal{H}_{\pi}$ are precisely the graded mixed complexes for which the
mixed structure is of cohomological degree $1$ (as opposed of the usual
conventions taken for instance in [PTVV13, Toë20, TV] where the cohomological
degree is $-1$). More precisely, we have an equivalence of symmetric monoidal
$\infty$-categories

$\mathsf{QCoh}(B\mathcal{H}_{\pi})\simeq+\epsilon-\mathbf{dg}^{gr}_{k},$

where $+\epsilon-\mathbf{dg}^{gr}_{k}$ is the symmetric monoidal
$\infty$-category of graded mixed complexes with mixed structure of
cohomological degree $1$. The $\infty$-category of usual graded mixed
complexes with mixed structure of degree $-1$ is denoted by
$\epsilon-\mathbf{dg}^{gr}_{k}$, and is itself given by representations of the
group stack $\mathcal{H}=\mathbb{G}_{m}\ltimes BKer$ of [MRT22]. It is well
known that there exists a symmetric monoidal equivalence of
$\infty$-categories

$\mathsf{RS}:\epsilon-\mathbf{dg}^{gr}_{k}\simeq+\epsilon-\mathbf{dg}^{gr}_{k}$

called the _red-shift self equivalence_. The $\infty$-functor $\mathsf{RS}$
sends a graded complex $\oplus_{n}E^{(n)}$ to $\oplus_{n}E^{(n)}[-2n]$, and
thus transforms a mixed structure $E^{(n)}\to E^{(n+1)}[-1]$ of cohomological
degree $-1$ to a mixed structure $E^{(n)}[-2n]\to E^{(n+1)}[-2n-1]\simeq
E^{(n+1)}[-2n+2][1]$ of cohomological degree $1$. To summarize, we have a
commutative square of symmetric monoidal equivalences

$\textstyle{\mathsf{QCoh}(B\mathcal{H})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$$\textstyle{\mathsf{QCoh}(B\mathcal{H}_{\pi})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{\epsilon-\mathbf{dg}^{gr}_{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathsf{RS}}$$\textstyle{+\epsilon-\mathbf{dg}^{gr}_{k}.}$

It is proven in [MRT22] that $\mathsf{QCoh}(B\mathcal{H})$ is equivalent, as a
symmetric monoidal $\infty$-category, to the $\infty$-category
$\widehat{\mathbf{dg}^{fil}_{k}}$ of complete filtered complexes. This
equivalence is given by the Tate realization $|-|^{t}$ sending a graded mixed
complex $\oplus_{n}E^{(n)}$ to $\oplus_{i\leq 0}\prod_{p\geq i}E^{(p)}[-2p]$
endowed with the total differential, sum of the mixed structure and the
cohomological differential. Similarly, $\mathsf{QCoh}(B\mathcal{H}_{\pi})$ is
equivalent to $\widehat{\mathbf{dg}^{fil}_{k}}$ by means of the corresponding
Tate realization $|-|^{t}$, sending $\oplus_{n}E^{(n)}$ to
$\mathrm{colim}_{i\leq 0}(\prod_{p\geq i}E^{(p)})$ with the total
differential.

For any derived stack $X\in\mathbf{dSt}_{k}$, we can form
$\mathcal{L}^{gr}_{\pi}(X/k):=\underline{\mathbf{Map}}(G_{0},X)$, the derived
Hom stack (relative to $k$) from $G_{0}$ to $X$. It comes equipped with a
natural action of $\mathcal{H}$, where $G_{0}$ acts on itself by translations
and $\mathbb{G}_{m}$ acts by the grading on $G_{0}$. Together with this
$\mathcal{H}_{\pi}$-action, $\mathcal{L}^{gr}_{\pi}(X)$ is the cofree
$\mathcal{H}_{\pi}$-equivariant derived stack generated by $X$ endowed with
the trivial $\mathbb{G}_{m}$-action.

######  Definition 2.1.

Let $X=\mathbf{Spec}\,A$ be a derived affine scheme over $k$. A (perfect)
_infinitesimal derived foliation on $X$ (relative to $k$)_ consists of a
$\mathcal{H}_{\pi}$-equivariant derived stack $\mathcal{F}$ together with a
$\mathcal{H}_{\pi}$-equivariant morphism $p:\mathcal{F}\to X$, such that, as a
$\mathbb{G}_{m}$-equivariant stack over $X$, $\mathcal{F}$ is a perfect linear
derived affine stack (i.e. of the form $\mathbb{V}(E)$ for $E$ a perfect
complex on $X$).

The infinitesimal derived foliations on $X$ (relative to $k$) form an
$\infty$-category $\mathcal{F}ol^{\pi}(X/k)$, which by definition is a full
sub-$\infty$-category of
$(\mathcal{H}_{\pi}-\mathbf{dSt}_{k})/\mathcal{L}^{gr}_{\pi}(X/k)$, the
$\mathcal{H}_{\pi}$-equivariant derived stacks over
$\mathcal{L}^{gr}_{\pi}(X/k)$. The construction
$X\mapsto\mathcal{F}ol^{\pi}(X/k)$ can be promoted to an $\infty$-functor
$\mathcal{F}ol^{\pi}(-/k):\mathbf{dAff}_{k}^{op}\to\mathbf{Cat}_{\infty}$, by
defining pull-backs as follows: for a morphism $f:X\to Y$ of affine derived
$k$-schemes, we define $f^{*}$ to be induced by the base change

$(\mathcal{H}_{\pi}-\mathbf{dSt}_{k})/\mathcal{L}^{gr}_{\pi}(Y/k)\longrightarrow(\mathcal{H}_{\pi}-\mathbf{dSt}_{k})/\mathcal{L}^{gr}_{\pi}(X/k),$

along the induced morphism
$\mathcal{L}^{gr}_{\pi}(f/k):\mathcal{L}^{gr}_{\pi}(X/k)\to\mathcal{L}^{gr}_{\pi}(Y/k)$.
As the $\infty$-functor $\mathcal{F}ol^{\pi}(-/k)$ clearly is a hyperstack for
the fpqc topology, it can be uniquely extended to a colimit preserving
$\infty$-functor

$\mathcal{F}ol^{\pi}(-/k):\mathbf{dSt}_{k}\to\mathbf{Cat}_{\infty}^{op}.$

Let $X$ be a derived affine $k$-scheme, and
$\mathcal{F}\in\mathcal{F}ol^{\pi}(X/k)$ be an infinitesimal derived foliation
on $X$. By [Mon21], the linear stack $\mathcal{F}\to X$ is of the form
$\mathbb{V}(\mathbb{L}_{\mathcal{F}/k}[-1])$, for some uniquely defined
perfect complex $\mathbb{L}_{\mathcal{F}/k}$ on $X$.

######  Definition 2.2.

With the notations above, the complex $\mathbb{L}_{\mathcal{F}/k}$ is called
the _cotangent complex of $\mathcal{F}$ (relative to $k$)_. It is also simply
denoted by $\mathbb{L}_{\mathcal{F}}$ if $k$ is clear from the context.

When $X$ is a general derived stack and
$\mathcal{F}\in\mathcal{F}ol^{\pi}(X/k)$, it is also possible to define the
cotangent complex $\mathbb{L}_{\mathcal{F}/k}$, as a quasi-coherent complex
over $X$. This construction is not straightforward, as cotangent complexes in
the sense above are not stable by pull-backs and thus do not glue naively on
stacks. When $X$ is moreover a derived Artin stack, the cotangent complex
$\mathbb{L}_{\mathcal{F}/k}$ is a perfect complex over $X$. This global
aspects of cotangent complexes will not be studied in details in this note,
and we refer to [TV] for more on the subject. However, the special case where
$X$ is a derived Deligne-Mumford stack presents no difficulties, as the
cotangent complex of Definition 2.2 are stable by étale pull-backs, and thus
will glue globally on $X$. This applies in particular to the case where $X$ is
any derived scheme.

Let $X$ be a derived affine scheme of finite presentation over $k$. The
$\infty$-category $\mathcal{F}ol^{\pi}(X/k)$ admits a final object, namely
$\mathcal{F}=\mathcal{L}^{gr}_{\pi}(X/k)$. It is denoted by $*_{X/k}$, and its
cotangent complex is naturally equivalent to $\mathbb{L}_{X/k}$, the cotangent
complex of $X$. Indeed, it is easy to see, simply by contemplating the
functors of points and using the very definition of the cotangent complex,
that $\underline{\mathbf{Map}}(G_{0},X)\simeq\mathbb{V}(\mathbb{L}_{X/k}[1])$.
Similarly, the initial object in $\mathcal{F}ol^{\pi}(X/k)$ exists, is denoted
by $0_{X/k}$, and its cotangent complex is $0$. It corresponds to the
constant-map morphism
$\mathcal{F}=X\longrightarrow\mathcal{L}^{gr}_{\pi}(X/k)$.

##  3\. Infinitesimal cohomology

Let $X$ be a derived Artin stack of finite presentation over $k$, and
$\mathcal{F}\in\mathcal{F}ol^{\pi}(X/k)$ be an infinitesimal derived foliation
on $X$. We define the $\infty$-category of quasi-coherent complexes along
$\mathcal{F}$ as follows. We first consider
$\mathsf{QCoh}([\mathcal{F}/\mathcal{H}_{\pi}])$, the $\infty$-category of
quasi-coherent complexes on the quotient stack
$[\mathcal{F}/\mathcal{H}_{\pi}]$ (which is also, by definition, the
$\infty$-category of $\mathcal{H}_{\pi}$-equivariant quasi-coherent complexes
on $\mathcal{F}$). The $\infty$-category $\mathsf{QCoh}^{\mathcal{F}}(X)$ is
defined as the full sub-$\infty$-category of
$\mathsf{QCoh}([\mathcal{F}/\mathcal{H}_{\pi}])$ whose objects $E$ are graded
free: the pull-back of $E$ on $[\mathcal{F}/\mathbb{G}_{m}]$ is of the form
$p^{*}(E_{0})$ where $p:[\mathcal{F}/\mathbb{G}_{m}]\to X$ is the natural
projection.

######  Definition 3.1.

With the notations above, the _$\infty$ -category of quasi-coherent complexes
along $\mathcal{F}$_ is $\mathsf{QCoh}^{\mathcal{F}}(X)$. For
$E\in\mathsf{QCoh}^{\mathcal{F}}(X)$, _the infinitesimal cohomology of $X$
along $\mathcal{F}$ with coefficients in $E$_ is defined to be

$\widehat{C}_{\mathrm{inf}}^{*}(\mathcal{F},E):=q_{*}(E)\in\mathbf{dg}_{k}$

where $q:[\mathcal{F}/\mathcal{H}_{\pi}]\longrightarrow\mathbf{Spec}\,k$ is
the structural map.

The definition above will be justified by our next result, stating that when
$\mathcal{F}=*_{X/k}$ and $E=\mathcal{O}_{X}$,
$\widehat{C}_{\mathrm{inf}}^{*}(\mathcal{F},E)$ coincides with (completed
derived) infinitesimal cohomology. For this, recall that if
$X\to\mathbf{Spec}\,k$ is a smooth affine scheme, we have the infinitesimal
stack $X_{\mathrm{inf}}$ (relative to $k$), defined by the functor sending
$A\in k-\mathsf{sCR}$ to $X(A_{red})$ (where $A_{red}=\pi_{0}(A)_{red}$ by
definition). In characteristic zero, the functor $X_{\mathrm{inf}}$ is often
denoted by $X_{DR}$ ([Sim96]). We prefer to avoid using the notation $X_{DR}$
in non-zero characteristic, and use $X_{\mathrm{inf}}$ instead, as the
cohomology of $X_{\mathrm{inf}}$ computes infinitesimal cohomology and not de
Rham cohomology. As any arbitrary derived stack, $X_{\mathrm{inf}}$ possesses
a cohomology with coefficients in the structure sheaf $\mathcal{O}$, denoted
by $C^{*}(X_{\mathrm{inf}},\mathcal{O})$.

######  Theorem 3.2.

Let $X\to\mathbf{Spec}\,k$ be a smooth affine scheme (and assume $k$ is
discrete for simplicity). There exists an equivalence in $\mathbf{dg}_{k}$

$\phi_{X}:C^{*}(X_{\mathrm{inf}},\mathcal{O})\simeq\widehat{C}_{\mathrm{inf}}^{*}(*_{X/k},\mathcal{O}_{X}).$

Proof. We first define the morphism $\phi_{X}$. For this we use the canonical
morphism $p:X\to X_{\mathrm{inf}}$, induced by $X(A)\to X(A_{red})$ coming
from the canonical projection $A\to A_{red}$. It induces a natural morphism of
$\mathcal{H}_{\pi}$-equivariant derived stacks
$\mathcal{L}^{gr}_{\pi}(X)\to\mathcal{L}^{gr}_{\pi}(X_{\mathrm{inf}})$. By the
very definition of $X_{\mathrm{inf}}$, and because
$G_{0}=\Omega_{0}\mathbb{G}_{a}$ is a derived affine scheme with trivial
reduced sub-scheme, we see that the canonical morphism
$X_{\mathrm{inf}}\to\mathcal{L}^{gr}_{\pi}(X_{\mathrm{inf}})$ is in fact an
($\mathcal{H}_{\pi}$-equivariant) equivalence of derived stacks. We thus
obtain a natural $\mathcal{H}_{\pi}$-equivariant morphism
$\mathcal{L}^{gr}_{\pi}(X)\to X_{\mathrm{inf}}$, or equivalently a morphism of
derived stacks

$q_{X}:[\mathcal{L}^{gr}_{\pi}(X)/\mathcal{H}_{\pi}]\to X_{\mathrm{inf}},$

which is clearly functorial in $X$. It induces, by pull-back, a morphism
between the cohomology complexes with coefficients in $\mathcal{O}$

$\phi_{X}:=q_{X}^{*}:C^{*}(X_{\mathrm{inf}},\mathcal{O})\to\widehat{C}_{\mathrm{inf}}(*_{X/k},\mathcal{O})$

which is again functorial in $X$.

To prove that $\phi_{X}$ is an equivalence, we proceed by descent along
$p:X\to X_{\mathrm{inf}}$. As $X$ is smooth, the morphism $p$ is an
epimorphism of derived stacks (infinitesimal criterion for smoothness). The
simplicial nerve $X_{*}$ of $p$, that comes equipped with its augmentation
$X_{*}\to X_{\mathrm{inf}}$, is an hypercovering. In particular, the canonical
morphism

$C^{*}(X_{\mathrm{inf}},\mathcal{O})\longrightarrow\lim_{[n]\in\Delta}\mathcal{O}(X_{n})$

is a quasi-isomorphism. Note that $X_{n}$ is naturally isomorphic to the
formal completion of $X^{n}$ along the diagonal embedding $X\hookrightarrow
X^{n}$, and thus $C^{*}(X_{n},\mathcal{O})$ is the discrete algebra
$\mathcal{O}(X_{n})$ of functions on the formal scheme $X_{n}$.

On the other hand, the fact that $X$ is smooth also implies that
$X\to[\mathcal{L}^{gr}_{\pi}(X)/\mathcal{H}_{\pi}]$ is an epimorphism, because
$\mathcal{L}^{gr}_{\pi}(X)\simeq\mathbb{V}(\Omega_{X/k}^{1}[1])=B\mathbb{T}_{X}$
is the classifying stack of the tangent bundle of $X$. We thus consider the
commutative diagram of derived stacks with $\mathcal{H}_{\pi}$-actions

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{id}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{L}^{gr}_{\pi}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q_{X}}$$\textstyle{X_{\mathrm{inf}}.}$

This diagram induces a morphism of simplicial objects on the nerves of the
vertical morphisms. The nerve of the morphism $X\to\mathcal{L}^{gr}_{\pi}(X)$
can be described as the simplicial object
$[n]\mapsto\mathcal{L}^{gr}_{\pi}(X/X^{n})$, where $X$ sits inside $X^{n}$ via
the diagonal embedding, and $\mathcal{L}^{gr}_{\pi}(X/X^{n})$ is the relative
mapping stack from $G_{0}$ to $X$ relative to $X^{n}$. At the simplicial level
$n$, the morphism induced on the nerve is thus an
$\mathcal{H}_{\pi}$-equivariant morphism of stacks over $S$

$q_{X,n}:\mathcal{L}^{gr}_{\pi}(X/X^{n})\longrightarrow X_{n},$

(where $X_{n}$ is the formal completion of $X^{n}$ along its diagonal).

######  Lemma 3.3.

The morphism $q_{X,n}$ induces by pull-back a quasi-isomorphism of complexes

$C^{*}(X_{n},\mathcal{O})\simeq
C^{*}([\mathcal{L}^{gr}_{\pi}(X/X^{n})/\mathcal{H}_{\pi}],\mathcal{O}).$

Proof of Lemma 3.3. By descent we can easily restrict to the case where
$X=\mathbf{Spec}\,A$ is a smooth affine scheme over $S=\mathbf{Spec}\,k$. In
this case, $C^{*}(X_{n},\mathcal{O})$ identifies canonically with
$\hat{A}^{\otimes_{k}n}$, the formal completion of $A^{\otimes_{k}n}$ along
the augmentation $A^{\otimes_{k}n}\to A$ induced by the multiplication map.
Similarly, $\mathcal{L}^{gr}_{\pi}(X/X^{n})$ is canonically identified with
the smooth affine scheme $\mathbf{Spec}\,Sym_{A}((\Omega_{A/k}^{1})^{n-1})$.
Therefore, the morphism $\phi_{X,n}$ is a morphism of discrete commutative
algebras

$\phi_{A,n}:\hat{A}^{\otimes_{k}n}\longrightarrow|Sym_{A}((\Omega_{A/k}^{1})^{n-1})|^{t},$

where the right hand side is the Tate realization of the graded mixed object
$Sym_{A}((\Omega_{A/k}^{1})^{n-1})$ induced by the $\mathcal{H}_{\pi}$-action
on $\mathcal{L}^{gr}_{\pi}(X/X^{n})$. The morphism $\phi_{A,n}$ is compatible
with the canonical augmentations to $A$, and thus is a morphism of complete
filtered commutative algebras. Therefore, in order to prove it is an
isomorphism it is enough to show that it induces an isomorphism

$gr\phi_{A,n}:Gr^{*}(\hat{A}^{\otimes_{k}n})\longrightarrow
Sym_{A}((\Omega_{A/k}^{1})^{n-1})$

on the associated graded objects. Since $A$ is a smooth $k$-algebra, the left
hand side is canonically isomorphic to $Sym_{A}(I_{n}/I_{n}^{2})$, where
$I_{n}$ is the augmentation ideal. Moreover, we have the usual natural
isomorphism of $A$-modules $I_{n}/I_{n}^{2}\simeq(\Omega_{A/k}^{1})^{n-1}$,
and thus the morphism $gr\phi_{A,n}$ can be identified with a graded
endomorphism of the graded algebra $Sym_{A}((\Omega_{A/k}^{1})^{n-1})$. As it
is compatible with augmentations to $A$ it is thus determined by an $A$-linear
endomorphism of $(\Omega_{A/k}^{1})^{n-1}$. By functoriality in $A$ and in the
simplicial direction $n$, we see moreover that it is determined by a
functorial endormorphism the $A$-module $\Omega_{A/k}^{1}$, and thus must be
the multiplication by an element $\lambda\in k$. To conclude the lemma, we
have to show that $\lambda=1$.

By functoriality, it is easy to reduce to the case $A=k[T]$, a polynomial
ring, and by base change we can even assume that $k=\mathbb{Z}$. In order to
check that $\lambda=1$, we can base change from $\mathbb{Z}$ to $\mathbb{Q}$,
and thus even reduce to the case $A=\mathbb{Q}[T]$ and $k=\mathbb{Q}$. As we
are now in characteristic zero,
$C^{*}([\mathcal{L}^{gr}_{\pi}(X/X^{n})/\mathcal{H}_{\pi}]$ identifies
canonically with the completed de Rham complex
$\widehat{C}_{DR}^{*}(X/X^{n})$. Indeed, the red shift equivalence sends the
graded mixed cdga $\mathcal{O}(\mathcal{L}^{gr}_{\pi}(X/X^{n}))$ to
$\mathcal{O}(\mathcal{L}^{gr}(X/X^{n}))$. In such terms, the morphism
$\phi_{X,n}$ is now the natural isomorphism relating derived de Rham
cohomology of a closed embedding with functions on the formal completion (see
[CPT+17]). $\Box$

Lemma 3.3 concludes the proof of Theorem 3.2, as we have a commutative diagram

$\textstyle{C^{*}(X_{\mathrm{inf}},\mathcal{O})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{X}}$$\scriptstyle{\sim}$$\textstyle{\widehat{C}_{\mathrm{inf}}(*_{X},\mathcal{O}_{X})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{\lim_{[n]}\mathcal{O}(X_{n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{\lim_{[n]}\widehat{C}_{\mathrm{inf}}(*_{X/X^{n}},\mathcal{O}_{X}).}$

$\Box$

As a direct corollary of Theorem 3.2, we obtain that, for a general derived
Artin stack $X$, $\widehat{C}_{\mathrm{inf}}(*_{X/k},\mathcal{O})$ computes
the _completed derived infinitesimal cohomology of $X$_ (relative to $k$).
Indeed, it is easy to see the $\infty$-functor
$X\mapsto\mathcal{O}(\mathcal{L}^{gr}_{\pi}(X/k))$, from affine derived
schemes to graded mixed complexes, is equivalent to its extension by sifted
colimits from smooth $X$’s. For $A\in\mathsf{sCR}$, where each $A_{n}$ is a
smooth algebra over $k$, we thus have

$\widehat{C}_{\mathrm{inf}}(*_{X},\mathcal{O}_{X})\simeq\mathrm{colim}_{[n]}C^{*}((Spec\,A_{n})_{\mathrm{inf}},\mathcal{O}),$

where the colimit is computed in the $\infty$-category of _complete_ filtered
complexes. The right hand side of this equivalence is, by definition, the
completed derived infinitesimal cohomology of $X$, and is denoted by
$\mathbb{R}\hat{C}^{*}(X_{\mathrm{inf}},\mathcal{O})$. This notion is extended
to any derived Artin stack locally of finite presentation over $k$ by the
usual gluing formula

$\mathbb{R}\hat{C}^{*}(X_{\mathrm{inf}},\mathcal{O}):=\lim_{U\to
X}\mathbb{R}\hat{C}^{*}(U_{\mathrm{inf}},\mathcal{O})$

where the limit runs over affine $U$ locally of finite presentation.

######  Corollary 3.4.

For a general derived Artin stack $X$ of finite presentation over $k$,
$\widehat{C}_{\mathrm{inf}}(*_{X/k},\mathcal{O})$ recovers the completed
derived infinitesimal cohomology of $X$ relative to $k$

$\widehat{C}_{\mathrm{inf}}(*_{X/k},\mathcal{O})\simeq\mathbb{R}\hat{C}^{*}(X_{\mathrm{inf}},\mathcal{O}).$

##  4\. Integrability by formal groupoids

We believe that infinitesimal derived foliations are formally integrable in a
very strong sense which we do not investigate in this paper in full
generality. Rather, we prove here a special case, showing that the
$\infty$-category of smooth infinitesimal derived foliations over a $k$-scheme
$X$ is in fact _equivalent_ to the category of _formally smooth formal
groupoids_ over $X$.

Let $X$ be a smooth $k$-scheme and $\mathcal{F}\in\mathcal{F}ol^{\pi}(X/k)$.
We say that $\mathcal{F}$ is _smooth_ if $\mathbb{L}_{\mathcal{F}}$ is a
vector bundle over $X$. In this case, we consider $X\times_{\mathcal{F}}X$,
endowed with its natural $\mathcal{H}_{\pi}$-action and with its canonical
projection to $X$. Note that, as a graded derived stack,
$X\times_{\mathcal{F}}X$ is of the form $\mathbb{V}(\mathbb{L}_{F})$, and thus
is a the total space of the tangent bundle $\mathbb{T}_{\mathcal{F}}$ of
$\mathcal{F}$. Also, the natural projection $X\times_{\mathcal{F}}X\to X$ is
$\mathcal{H}_{\pi}$-equivariant. As a result, $X\times_{\mathcal{F}}X$ is a
new object in $\mathcal{F}ol^{\pi}(X)$, that will be simply denoted by
$\Omega_{X}\mathcal{F}$, and called the _loop space_ of $\mathcal{F}$. Being
the fiber product of the canonical morphism $0_{X}\to\mathcal{F}$, the loop
space $\Omega_{X}\mathcal{F}$ comes equipped with a natural structure of a
groupoid object in $\mathcal{F}ol^{\pi}(X)$, acting on the object $0_{X}$.

######  Propositon 4.1.

The full sub-$\infty$-category of $\mathcal{F}ol^{\pi}(X/k)$ formed by objects
$\mathcal{F}$ whose cotangent complex is of the form $V[1]$ for $V$ a vector
bundle on $X$, is naturally equivalent to the category of formal schemes $Z$
with an identification $Z_{red}\simeq X$, which are locally equivalent on $X$
to $X\times\widehat{\mathbb{A}}^{d}$ where $d=rank(V)$.

Proof. To each $\mathcal{F}$ as in the proposition, we associate the sheaf of
infinitesimal cohomology $\widehat{C}_{\mathrm{inf}}(\mathcal{F})$ on $X$. By
assumptions on the cotangent complex of $\mathcal{F}$,
$\widehat{C}_{\mathrm{inf}}(\mathcal{F})$ is a sheaf of discrete complete
filtered commutative algebras, augmented to $\mathcal{O}_{X}$ and its
associated graded object is isomorphic to
$\mathcal{O}_{X}[[t_{1},\dots,t_{d}]]$. Therefore,
$Z_{\mathcal{F}}:=\mathsf{Spf}(\widehat{C}_{\mathrm{inf}}(\mathcal{F}))$ is a
formal scheme with a canonical identification $Z_{red}\simeq X$ which is
locally equivalent to $X\times\widehat{\mathbb{A}}^{d}$.

The fact that the construction $\mathcal{F}\mapsto Z_{\mathcal{F}}$ induces
the equivalence of the proposition simply follows from the fact that the Tate
realization induces an equivalence between graded mixed complexes and complete
filtered complexes (see [TV20, Prop. 1.3.1]). $\Box$

By Proposition 1.7, for any smooth infinitesimal derived foliation
$\mathcal{F}$, its loop space $\Omega_{X}\mathcal{F}$ can be realized as a
groupoid object in formal schemes acting on $X$

$Z_{\Omega_{X}\mathcal{F}}\longrightarrow X\times X.$

The construction $\mathcal{F}\mapsto Z_{\Omega_{X}\mathcal{F}}$ provides an
functor from the category of smooth infinitesimal derived foliations on $X$,
and the category of formally smooth formal groupoids over $X$. We can produce
a functor in the other direction, as follows. Starting from a formally smooth
formal groupoid $G\rightarrow X\times X$, we can consider its nerve $G_{*}\to
X$, which is a simplicial object in formal schemes. Taking functions on
$G_{*}$ provides a cosimplicial complete filtered commutative algebra
$\mathcal{O}(G_{*})$ whose associated graded is the cosimplicial algebra of
functions on the simplicial scheme $BV$, where $V=e^{*}(\Omega_{G/X}^{1})$ is
the vector bundle of invariant relative forms on $G$. Using the equivalence
between graded mixed complexes and complete filtered complexes of [TV20, Prop.
1.3.1], we find that $\mathcal{O}(G_{*})$ can be realized as a cosimplicial
object in the category of graded commutative algebras endowed with a
compatible action on $G_{0}$. Passing then to $Spec$, we get a simplicial
diagram of affine schemes endowed with a $\mathcal{H}_{\pi}$-action, whose
underlying simplicial diagram (obtained by forgetting the action) is the usual
simplicial scheme $BV$. Taking geometric realization we get a
$\mathcal{H}_{\pi}$-action on the derived stack $\mathbb{V}(V[1])$, which
defines a smooth infinitesimal derived foliations $\mathcal{F}$ on $X$.

We can then subsume the previous discussion in the following corollary.

######  Corollary 4.2.

The category of smooth infinitesimal derived foliations on $X$ is equivalent
to the category of formally smooth formal groupoids on $X$.

We note that the above corollary marks a major difference between derived
foliations of [Toë20, TV] and infinitesimal derived foliations. Indeed, any
finite dimensional Lie algebra $L$ over a field $k$, defines a smooth derived
foliation over $k$, by considering $Sym(L^{\vee}[1])$ endowed with the mixed
structure coming from the Chevalley-Eilenberg differential. However, it is not
true that any Lie algebra can be integrated to a smooth formal scheme in
general. On the contrary, the above corollary, when applied to
$X=\mathbf{Spec}\,k$ with $k$ a field, states that smooth infinitesimal
derived foliations over $k$ form a category equivalent to smooth formal groups
over $k$. Any such smooth infinitesimal derived foliation is of the form
$\mathbb{V}(V[-1])$ for $V$ a finite dimensional $k$-vector space $V$.
Therefore, the data of an infinitesimal derived foliation structure on
$\mathbb{V}(V[-1])$ seems to be related to that of a _partition Lie algebra
structure_ in the sense of [BCN21]. The precise comparison is out of the scope
of this note, but some recent results of J. Fu ([Fu]) make us strongly believe
this is possible.

##  5\. Towards a comparison between infinitesimal derived foliations and
derived foliations

###  5.1. Red shift

We have already mentioned the red-shift equivalence
$\mathsf{RS}:\epsilon-\mathbf{dg}^{gr}_{k}\simeq+\epsilon-\mathbf{dg}^{gr}_{k}$.
We will now lift this to an endofunctor on the level of derived affine stacks
endowed with actions of $\mathcal{H}$ and $\mathcal{H}_{\pi}$. For this, we
use our cosimplicial-simplicial models for graded derived affine stacks. We
treat the absolute case of derived affine stacks over $\mathbb{Z}$, the
realtive situation follows from considering graded affine stacks over a given
affine derived scheme $X=\mathbf{Spec}\,k$ (trivially graded). For technical
reasons we will actually invert $2$, i.e. most of the time work over
$\mathbb{Z}[1/2]$.

We denote by $\mathbb{Z}[u]$ the cosimplicial commutative ring obtained by
denormalization from the free commutative algebra over one generator in
homotopical degree $0$ (endowed with the $0$ differential and considered as a
non-negatively graded commutative dg-algebra). In [MRT22], it is proven that
$\mathbf{Spec}^{\Delta}(\mathbb{Z}[u])\simeq BS_{gr}^{1}$, the classifying
stack of the _graded circle_ , also called the _graded infinite projective
space_. We consider $\mathbb{Z}[u]$ as an object in $csCR^{gr}$ constant in
the simplicial direction where $u$ is of weight $1$. Similarly, we denote by
$\mathbb{Z}[v]$ the free simplicial commutative ring generated by one variable
in homotopical degree $2$. We consider $\mathbb{Z}[v]$ as an object in
$csCR^{gr}$ constant in the cosimplicial direction where $v$ is of weight
$-1$. Finally, we set

$\mathbb{Z}<u,v>:=\mathbb{Z}[v]\times_{\mathbb{Z}}\mathbb{Z}[u]\in csCR^{gr},$

which is a cosimplicial-simplicial commutative $\mathbb{Z}$-graded ring.

For two commutative graded rings $A$ and $B$, we define their _convolution
product_ $A\odot B$, which is a new commutative graded ring with

$(A\odot B)^{(n)}=A^{(n)}\otimes B^{(n)}$

with the natural componentwise multiplication. This can be extended to a
convolution product for two objects in $csCR^{gr}$ by applying this
construction levelwise, both in the simplicial and cosimplicial directions.

######  Definition 5.1.

The red shift endofunctor $\mathsf{RS}:csCR^{gr}\to csCR^{gr}$ is defined by

$\mathsf{RS}(A):=A\odot(\mathbb{Z}<u,v>).$

It is easy to see that $Tot^{\pi}(\mathsf{RS}(A))$ is a graded complex which
is naturally quasi-isomorphic to $\mathsf{RS}(Tot^{\pi}(A))$. Therefore,
$\mathsf{RS}$ preserves weak equivalence in $csCR^{gr}$, and thus induces a
well defined $\infty$-functor
$\mathsf{RS}:\mathbf{csCR}^{gr}\to\mathbf{csCR}^{gr}$ covering the red-shift
self equivalence by the functor $Tot^{\pi}$

$\textstyle{\mathbf{csCR}^{gr}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathsf{RS}}$$\scriptstyle{Tot^{\pi}}$$\textstyle{\mathbf{csCR}^{gr}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Tot^{\pi}}$$\textstyle{\mathbf{dg}^{gr}_{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathsf{RS}}$$\textstyle{\mathbf{dg}^{gr}_{k}.}$

We note however that $\mathsf{RS}$ does not induce a self equivalence of
$\mathbf{csCR}^{gr}$, and for instance does not preserve free objects, as
opposed for instance to the corresponding situation with
$E_{\infty}$-algebras. In fact the behaviour of $\mathsf{RS}$ with respect to
free objects is quite subtle, and the authors do not claim to fully understand
the situation.

Note also that $\mathsf{RS}(A)$ is only well behaved when $A$ is only
$\mathbb{Z}_{\geq}0$-graded (or $\mathbb{Z}_{\leq 0}$-graded). For instance,
$\mathsf{RS}(\mathbb{Z}[t,t^{-1}])$ is equivalent to $\mathbb{Z}<v,u>$, for
which $u$ is the image of $t$ and $v$ the image of $t^{-1}$. However, by
definition $uv=0$, and thus the multiplicative structure on
$\mathsf{RS}(\mathbb{Z}[t,t^{-1}])$ is somehow degenerate. Trying to use
$\mathbb{Z}[u,u^{-1}]$ is not a solution here, as the presence of divided
powers in the homotopy of commutative simplicial rings would force us to work
over $\mathbb{Q}$.

Using Proposition 4.1, we can then consider $\mathsf{RS}$ as an endofunctor of
the $\infty$-category of graded derived affine stacks
$\mathsf{RS}:\mathbf{dChAff}^{gr}\to\mathbf{dChAff}^{gr}$. The convolution
construction $-\odot B$ for $B\in csCR^{gr}$ is lax monoidal in the sense that
there is a natural morphism $(A\odot B)\otimes(A^{\prime}\odot B)\to(A\otimes
A^{\prime})\odot B$. This lax monoidal structure is monoidal if and only if
the multiplication maps $B^{(p)}\otimes B^{(q)}\to B^{(p+q)}$ are completed
quasi-isomorphisms of cosimplicial-simplicial modules (i.e. becomes quasi-
isomorphismes ater applying $Tot^{\pi}$). In the case $B=\mathbb{Z}<u,v>$ this
is never the case, as $uv=0$. However, if $A$ is $\mathbb{Z}_{\geq 0}$-graded,
we have $A\odot\mathbb{Z}<u,v>\simeq A\odot\mathbb{Z}[u]$, and clearly
$\mathbb{Z}[u]^{(p)}\otimes\mathbb{Z}[v]^{(q)}\to\mathbb{Z}[u]^{(p+q)}$ is a
completed quasi-isomorphism as its image by $Tot^{\pi}$ is equivalent to the
canonical isomorphism
$\mathbb{Z}[-2p]\otimes\mathbb{Z}[-2q]\simeq\mathbb{Z}[-2p-2q]$.

######  Lemma 5.2.

Let $X$ and $Y$ be two graded derived affine stacks which are both
$\mathbb{Z}_{\geq 0}$-graded. Then, the natural morphism $\mathsf{RS}(X\times
Y)\to\mathsf{RS}(X)\times\mathsf{RS}(Y)$ is an equivalence.

Another special case where $\mathsf{RS}$ preserves products is given in the
following Lemma. It follows from the easy observation that the image by
$Tot^{\pi}$ of the multiplication
$\mathbb{Z}[v]^{(-1)}\otimes\mathbb{Z}[v]^{(-1)}\to\mathbb{Z}[v]^{(-2)}$ is
equivalent to the multiplication by $2$ (because $\mathbb{Z}[v]$ has divided
powers)

$\times
2:\mathbb{Z}[2]\otimes\mathbb{Z}[2]\simeq\mathbb{Z}[4]\to\mathbb{Z}[4].$

######  Lemma 5.3.

Let $X$ and $Y$ be two graded derived affine stacks whose weights are
concentrated in $[-1,0]$. Then, the natural morphism $\mathsf{RS}(X\times
Y)\to\mathsf{RS}(X)\times\mathsf{RS}(Y)$ is an equivalence when restricted to
$\mathbf{Spec}\,\mathbb{Z}[1/2]$.

Remind from [MRT22, Toë20] the group object
$BKer=S^{1}_{gr}\in\mathbf{dChAff}^{gr}$. As a graded derived affine stack its
weights are concentrated in $[-1,0]$, so Lemma 5.3 applies, and we deduce that
that $\mathsf{RS}(BKer)$ is another group object in $\mathbf{dChAff}^{gr}$ (at
least over $\mathbf{Spec}\,\mathbb{Z}[1/2]$). As a graded affine stack, it is
clearly of the form $\mathbf{Spec}(\mathbb{Z}[1/2][\epsilon_{-1}])\simeq
G_{0}$. As the group structure on $G_{0}$ is essentially unique (see [MRT22]),
we moreover deduce that $\mathsf{RS}(BKer)\simeq G_{0}$ as group objects in
$\mathbf{dChAff}^{gr}$.

We have thus seen that the image of the group $BKer$ by $\mathsf{RS}$ is the
group $G_{0}$. We believe that the functor $\mathsf{RS}$ can also be promoted
to a functor on the level of equivariant objects. This is not a formal
statement, as $\mathsf{RS}$ does not commute with finite products in general.
The precise construction is outside of scope of this note and we leave this as
an open question for the future.

######  Question 5.4.

Can the $\infty$-functor $\mathsf{RS}$ be extended to

$\mathsf{RS}:BKer-\mathbf{dChAff}^{gr+}\longrightarrow
G_{0}-\mathbf{dChAff}^{gr+},$

in such a way that the cofree object $X^{BKer}$ is sent to
$\mathsf{RS}(X)^{G_{0}}$ (at least over $\mathbf{Spec}\,\mathbb{Z}[1/2]$) ?

######  Remark 5.5.

_The above question is a coherence problem. It is indeed possible to prove
easily that for any $X\in\mathbf{dChAff}^{gr+}$, there exists a canonical
morphism $\mathsf{RS}(X^{BKer})\to\mathsf{RS}(X)^{G_{0}}$ of graded derived
affine stacks. Therefore, a $BKer$-action on $X$, induces a morphism $X\to
X^{BKer}$, and therefore, via $\mathsf{RS}$, a morphism
$\mathsf{RS}(X)\to\mathsf{RS}(X^{Bker})\to\mathsf{RS}(X)^{G_{0}}$. The problem
here is to control the higher coherences in order to promote the previous
morphism to an action of $G_{0}$ on $\mathsf{RS}(X)$._

###  5.2. Infinitesimal structures on derived foliations

We have now everything we need in order to compare infinitesimal derived
foliations and our original definition of derived foliations, assuming that we
have a positive answer to Question 5.4.

Let $\mathcal{F}\to X$ be a derived foliation on a derived $k$-scheme $X$ in
the sense of [Toë20]. Remind that it consists of a linear derived stack
$\mathbb{V}(\mathbb{L}_{\mathcal{F}}[1])$, for $\mathbb{L}_{\mathcal{F}}$ a
perfect complex on $X$, together with an action of the graded group $BKer$. By
applying the red-shift functor, we get a graded derived affine stack
$\mathsf{RS}(\mathbb{V}(\mathbb{L}_{\mathcal{F}}[1]))$ together with an action
of $G_{0}$. The complex of functions on
$\mathsf{RS}(\mathbb{V}(\mathbb{L}_{\mathcal{F}}[1]))$ is the graded complex
$\mathsf{RS}(Sym_{\mathcal{O}_{X}}(\mathbb{L}_{\mathcal{F}}[1]))$ and thus
receive a canonical morphism
$\mathbb{L}_{\mathcal{F}}[-1]\to\mathsf{RS}(Sym_{\mathcal{O}_{X}}(\mathbb{L}_{\mathcal{F}}[1]))$.
By the universal property of the $Sym$ construction, we thus get a canonical
morphism of graded stacks
$\mathsf{RS}(\mathbb{V}(\mathbb{L}_{\mathcal{F}}[1]))\to\mathbb{V}(\mathbb{L}_{\mathcal{F}}[-1]))$.

######  Definition 5.6.

Let $\mathcal{F}$ be a derived foliation on $X$ as above. An _infinitesimal
structure on $\mathcal{F}$_ consists of an extension of the $G_{0}$-action
along the morphism
$\mathsf{RS}(\mathbb{V}(\mathbb{L}_{\mathcal{F}}[1]))\to\mathbb{V}(\mathbb{L}_{\mathcal{F}}[-1]))$.

By definition, an infinitesimal structure on $\mathcal{F}$ determines an
infinitesimal derived foliation $\mathcal{F}^{\pi}$, given by the
$G_{0}$-action on $\mathbb{V}(\mathbb{L}_{\mathcal{F}}[-1]))$. We can thus
define the infinitesimal cohomology of $\mathcal{F}$ by taking infinitesimal
cohomology of $\mathcal{F}^{\pi}$ in the sense of our Definition 3.1.
Therefore, the first consequence of the data of an infinitesimal structure is
the possibility to define infinitesimal cohomology. Note that with the
definition above, there is a canonical morphism

$\widehat{C}_{\mathrm{inf}}(\mathcal{F}^{\pi},\mathcal{O})\longrightarrow\widehat{C}_{DR}(\mathcal{F},\mathcal{O})$

from the infinitesimal cohomology of $\mathcal{F}^{\pi}$ to the derived de
Rham cohomology of $\mathcal{F}$ introduced in [Toë20]. This is a
generalization of the well known comparison morphism between infinitesimal and
crystalline cohomology. This can be enhanced to a pull-back functor, from
$\mathsf{QCoh}(\mathcal{F}^{\pi})$ to $\mathsf{QCoh}(\mathcal{F})$. Moreover,
the differential operators construction of [TV20] can also be performed for
$\mathcal{F}$ and $\mathcal{F}^{\pi}$. This provides two different sheaves of
filtered dg-algebras on $X$, the crystalline differential operators
$\mathcal{D}_{\mathcal{F}}$ along $\mathcal{F}$ and the Grothendieck
differential operators $\mathcal{D}^{\infty}_{\mathcal{F}^{\pi}}$ along
$\mathcal{F}^{\pi}$. These are two filtered dg-algebras, with a natural
morphism
$\mathcal{D}_{\mathcal{F}}\longrightarrow\mathcal{D}^{\infty}_{\mathcal{F}^{\pi}}$,
whose induced morphism on the associated graded objects is
$Sym(\mathbb{T}_{\mathcal{F}})\longrightarrow
Sym^{pd}(\mathbb{T}_{\mathcal{F}})$, from the symmetric algebra (in the sense
of cosimplicial-simplicial models of our §1) to its PD-completion.

We also believe that it should be possible to prove that a derived foliation
that admits an infinitesimal structure is automatically _formally integrable_.
In fact, it seems that when it exists, $\mathcal{F}^{\pi}$ completely recovers
$\mathcal{F}$ as follows. The graded derived affine stack
$\mathsf{RS}(\mathbb{V}(\mathbb{L}_{\mathcal{F}}[1]))$ can be obtained as the
PD completion of $\mathbb{V}(\mathbb{L}_{\mathcal{F}}[-1]))$ along its zero
section, and it is very reasonable to expect that the $\mathsf{RS}$
construction induces an equivalence between graded derived affine stacks and
graded derived affine stacks with PD structures. In particular, the
$G_{0}=\mathsf{RS}(BKer)$-action on
$\mathsf{RS}(\mathbb{V}(\mathbb{L}_{\mathcal{F}}[1]))$ should comes from a
unique $BKer$-action on $\mathbb{V}(\mathbb{L}_{\mathcal{F}}[1])$. What we are
describing here is probably the existence of a fogertful functor, from
infinitesimal derived foliations to derived foliations, even though we are not
able, at the moment, to provide the details of this construction. This
forgetful functor seems very closely related to the forgetful functor from
partition Lie algebras to Lie algebras, and recent results of J. Fu ([Fu])
indicate that infinitesimal derived foliations could be essentially the same
thing as (perfect) partition Lie algebroids. The fact that partition Lie
algebras are related to formal moduli problem then explains the fact that
infinitesimal derived foliations have nice formal integrability properties.

Bertrand Toën, IMT, CNRS, Université de Toulouse, Toulouse (France)
<EMAIL_ADDRESS>

Gabriele Vezzosi, DIMAI, Università di Firenze, Firenze (Italy)
<EMAIL_ADDRESS>

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