###### Contents

  1. I Introduction
    1. I.1 De Rham cohomology
    2. I.2 Crystalline cohomology
    3. I.3 De Rham-Witt complex
    4. I.4 Witt vectors, $\delta$-rings and Frobenius lifts
    5. I.5 Derived geometry and cohomology theories
    6. I.6 Content of the thesis
      1. I.6.1 Frobenius lifts and graded loopspaces
      2. I.6.2 Graded functions on graded stacks
      3. I.6.3 Notations
  2. II Categorical notions
    1. II.1 $\infty$-categories
    2. II.2 Lax and colax fixed points
    3. II.3 Objects with endomorphisms
    4. II.4 Nerve and conerve constructions
    5. II.5 Simplicial sets
    6. II.6 Chain complexes
    7. II.7 Graded mixed chain complexes
    8. II.8 The bar construction
  3. III Reminders on algebra
    1. III.1 Simplicial and cosimplicial rings
      1. III.1.1 Simplicial algebras
      2. III.1.2 Cosimplicial algebras
      3. III.1.3 Dold-Kan correspondance
    2. III.2 $\delta$-structures
    3. III.3 The ring of Witt vectors
  4. IV Preliminaries in derived geometry
    1. IV.1 Stacks and derived stacks
    2. IV.2 $\mathcal{O}$-modules and quasi-coherent sheaves
    3. IV.3 Construction of derived stacks
    4. IV.4 Postnikov towers
    5. IV.5 Functions on a derived stack
    6. IV.6 Affine stacks
    7. IV.7 Cotangent complex and deformation theory
    8. IV.8 Group actions
    9. IV.9 Graded and filtered objects
    10. IV.10 Graded stacks
    11. IV.11 Line bundles and graded modules
    12. IV.12 The graded circle
  5. V Graded function on linear stacks
    1. V.1 Adjoint of the linear stack functor
    2. V.2 The equivalence
    3. V.3 General weights
  6. VI Overview of de Rham and crystalline cohomology
    1. VI.1 Crystalline cohomology
    2. VI.2 De Rham-Witt complex
    3. VI.3 De Rham Witt from Bhatt and Lurie
  7. VII Graded loopspaces in mixed characteristics
    1. VII.1 Frobenius lift structures
      1. VII.1.1 Classical and derived Frobenius lifts
      2. VII.1.2 Derived Witt functor
      3. VII.1.3 Modules on an algebra endowed with a Frobenius lift
    2. VII.2 Mixed graded Dieudonné complexes and algebras
      1. VII.2.1 The graded circle
      2. VII.2.2 Mixed graded Dieudonné complexes
      3. VII.2.3 Mixed graded Dieudonné algebras
      4. VII.2.4 Graded loopspace
    3. VII.3 Comparison theorems
      1. VII.3.1 Mixed structures classification : the non-Dieudonné case
      2. VII.3.2 Mixed structures classification : the Dieudonné case
      3. VII.3.3 De Rham-Witt complex
    4. VII.4 Saturation
      1. VII.4.1 The décalage functor
      2. VII.4.2 Saturated Dieudonné algebra
  8. VIII Future perspectives
    1. VIII.1 Improvements
    2. VIII.2 Symplectic forms
    3. VIII.3 Foliations
  9. A Categorical results
  10. B Geometrical results

##  I Introduction

###  I.1 De Rham cohomology

De Rham cohomology is a cohomology theory offering a purely algebraic
construction, based on differential forms, which recovers the Betti cohomology
of a smooth algebraic variety of characteristic zero. Precisely, the algebraic
de Rham cohomology groups $H^{*}_{DR}(X/\mathbb{C})$ are isomorphic to the
singular cohomology groups with complex coefficients of the underlying complex
manifold $X(\mathbb{C})$ when $X$ is a smooth complex algebraic variety. When
$X$ is the affine scheme $Spec(A)$, the de Rham complex is based on the
$A$-module $\Omega_{A/k}$ of Kähler differentials associated to $A$. This
module represents the functor of derivations :

$Der_{k}(A,M)\coloneqq Hom_{A-Mod}(\Omega_{A/k},M)\cong
Hom_{CRing_{k/A}}(A,A\oplus M)$

The universal property of the Kähler module defines the universal differential

$d:A\to\Omega_{A/k}$

which extends to a chain complex

$\Omega_{A/k}^{0}\xrightarrow{d}\Omega_{A/k}^{1}\xrightarrow{d}\Omega_{A/k}^{2}\xrightarrow{d}...$

where $\Omega_{A/k}^{k}\coloneqq\Lambda_{A}^{k}\Omega_{A/k}$. The algebraic de
Rham cohomology $H^{*}_{DR}(X/k)$ is then defined as the hypercohomology of
the de Rham complex.

The HKR theorem, originally proven in [HKR62], connects de Rham forms and
Hochschild cohomology by giving functorial isomorphisms

$\Omega_{A/k}^{i}\cong HH^{i}(A)$

for a smooth commutative $k$-algebra $A$ and $k$ a field of characteristic
zero. The de Rham cohomology is not as well-behaved for non-smooth algebras :
it needs to be replaced by a derived variant using the cotangent complex
$\mathbb{L}_{A/k}$ instead of Kähler differentials. The comparision between de
Rham complex and Hochschild cohomology is recovered for arbitrary algebras $A$
over a field of zero characteristic, see [TV09]. In fact, we get an
equivalence of commutative differential graded algebras

$DR(A)=Sym_{A}(\mathbb{L}_{A/k}[1])\simeq
HH(A)=A\otimes^{\mathbb{L}}_{A\otimes^{\mathbb{L}}_{k}A}A$

In characteristic zero, the equivalence of $\infty$-categories

$\epsilon-cdga\simeq S^{1}-SCR$

between graded mixed algebras and $S^{1}$-equivariant simplicial algebras
identifies the mixed structure on $Sym_{A}(\mathbb{L}_{X/k}[1])$, given by the
de Rham differential, and the circle action on

$A^{S^{1}}\simeq A\otimes^{\mathbb{L}}_{A\otimes^{\mathbb{L}}_{k}A}A$

These observations rely heavily on $\infty$-categorical notions, such as
derived tensor products of simplicial rings or cotensorization of a simplicial
ring by a space. We will use derived geometry in this thesis to formulate and
prove our results. The de Rham algebra

$DR(A)=Sym_{A}(\mathbb{L}_{A/k}[1])$

has been central in the study of notions and theorems imported from
differential geometry to the world of algebraic geometry or derived geometry.
It is used to define shifted symplectic forms in order to study deformation
quantization in [CPTVV] and it also appears in the definition of foliations in
[TV20]. Using the langage of derived geometry, the de Rham algebra can be
described as a mapping stack from a derived stack called the graded circle
$S^{1}_{gr}$ :

$\textbf{Map}(S^{1}_{gr},Spec(A))\simeq Spec(Sym_{A}(\mathbb{L}_{A/k}[1]))$

The graded circle $S^{1}_{gr}$ is a formal version of the topological circle
$S^{1}$ which has a canonical grading, see Definition IV.12.3 and [MRT20] for
details. By analogy with the circle $S^{1}$, a mapping stack from the graded
circle $S^{1}_{gr}$ will be called a graded loopspace.

The main purpose of this thesis is to find a similar description of the de
Rham-Witt complex, as a graded loopspace in the appropriate category.

###  I.2 Crystalline cohomology

Crystalline cohomology was born to fill a gap in the theory of $l$-adic
cohomologies for algebraic varieties over a field of characteristic $p$. When
$l\neq p$, $l$-adic cohomology is a well-behaved Weil cohomology theory, but
it is badly behaved for $l=p$. Initially, an analog of $l$-adic cohomology for
$l=p$ was motivated by the Riemann hypothesis for algebraic varieties.

When $X$ is a smooth and proper algebraic variety over $\mathbb{F}_{p}$, the
zêta function of $X$ is given by the formal series

$Z(X,t)\coloneqq exp\left(\sum_{n\geq
1}Card(X(\mathbb{F}_{p^{n}}))\frac{t^{n}}{n}\right)\in 1+t\mathbb{Z}[[t]]$

Pulling back to the algebraic closure of $\mathbb{F}_{p}$, $\bar{X}$ admits an
action of the Frobenius $Fr$ and we can identify fixed points of $Fr^{n}$ with
$\mathbb{F}_{p^{n}}$-points of $X$. To show the function $Z(X,-)$ is rational
using the Lefschetz fixed points formula, we need a sufficiently well-behaved
cohomology theory for varieties over a finite field which takes value in a
finite dimensional vector space over a field of characteristic zero.

Grothendieck’s idea for a definition of such a cohomology theory was as
follow. Let $X$ be a smooth proper scheme over a perfect field $k$ of positive
characteristic $p$. A natural candidate for the ring of coefficients of this
cohomology theory is the ring of Witt vectors $W(k)$. If $X$ lifts to a smooth
proper scheme $\widetilde{X}$ over $W(k)$, taking the hypercohomology of the
de Rham complex of $\widetilde{X}$ over $W(k)$ is independent of the lift.
This definition, although allowing easy computations for well-behaved schemes,
is not satisfactory and does not define a cohomology theory for a large class
of schemes.

So as to give an intrinsic definition, the notion of divided powers of an
ideal in a ring was used by Berthelot to define crystalline cohomology in
[Ber74]. He introduced a ringed site of divided power thickenings over the
Zariski site. Cohomology of the structural sheaf defines crystalline
cohomology. This powerful approach to build a cohomology theory allowed for
easy functorial properties, and a similar type of construction was used more
recently by Bhatt and Scholze to define a more general cohomology theory for
$p$-adic schemes called prismatic cohomology, see [BS22].

###  I.3 De Rham-Witt complex

The definition of the de Rham-Witt complex was motivated by the study of the
Hodge filtration on crystalline cohomology. The de Rham complex being endowed
with a natural filtration, the associated de Rham cohomology acquires a
canonical filtration called the Hodge filtration. However, when defining
crystalline cohomology of a smooth scheme as de Rham cohomology of a smooth
lift to Witt vectors, the canonical filtration is dependant on the chosen
lift. The de Rham-Witt complex bridges this gap as it is endowed with a
canonical "slope" filtration. Illusie introduced the de Rham-Witt complex
$W_{n}\Omega_{X}^{\bullet}$ in [Ill79]. This complex was defined as an initial
object in a category of elements having the same structure as the de Rham
complex and also the structure naturally arising on Witt vectors. Such
elements are called pro-$V$-complex, they are given by families of commutative
differential graded $W_{n}(k)$-algebras endowed with a Verschiebung map $V$
and a Frobenius morphism $F$ satisfying compatibility conditions. A natural
element in this category is $\Omega_{W_{n}(X)}^{\bullet}$, the de Rham complex
of the Witt vectors, therefore there is a canonical map

$\Omega_{W_{n}(X)}^{\bullet}\to W_{n}\Omega^{\bullet}_{X}$

As the category of pro-$V$-complexes is modeled from the properties of
$\Omega_{W_{n}(X)}^{\bullet}$. The de Rham-Witt complex
$W_{n}\Omega^{\bullet}_{X}$ can be thought of as applying a left adjoint
functor to $\Omega_{W_{n}(X)}^{\bullet}$, that is taking quotients and
completion procedures. We will denote $W_{n}\Omega^{\bullet}_{X}$ the limit of
the tower $W\Omega^{\bullet}_{X}$, we will call it the de Rham-Witt complex of
$X$.

The construction of the de Rham-Witt complex has being recently revisited by
Bhatt, Lurie and Matthew in [BLM22]. In this paper, the authors give a
simplified construction of this complex and we will review the main
definitions and some of the results which motivated this thesis.

######  Definition I.3.1.

A Dieudonné complex $(M,d,F)$ is a cochain complex of abelian groups $(M,d)$
and $F$ a morphism of graded abelian groups $F:M\to M$ such that

$dF=pFd$

The category of Dieudonné complexes is denoted DC.

Following [BLM22], a Dieudonné algebra $(A,d,F)$ is given by $(A,d)$ a
commutative differential graded algebra and $F:A\to A$ is a morphism of graded
rings satisfying compatibilities analogous to the ones defining a Dieudonné
complex.

Morphisms between Dieudonné algebras are ring morphisms compatible with
differentials and Frobenius structures. This defines the category of Dieudonné
algebras, which we denote DA.

Forgetting the multiplication defines the forgetful functor
$\textbf{DA}\to\textbf{DC}$. The following construction defines what we may
call the de Rham complex of a $p$-torsion-free ring endowed with a Frobenius
lift.

Let $R$ be a $p$-torsion-free commutative ring and $\phi:R\to R$ a classical
Frobenius lift. Then there is a unique ring morphism
$F:\Omega_{R}^{\bullet}\to\Omega_{R}^{\bullet}$ such that

  * •

$F(x)=\phi(x)$ for $x\in R=\Omega_{R}^{0}$.

  * •

$F(dx)=x^{p-1}dx+d(\frac{\phi(x)-x^{p}}{p})$ for $x\in R$.

In this case, $(\Omega_{R}^{\bullet},d,F)$ is a Dieudonné algebra.

The main purpose of this thesis is to give a geometrical interpretation of
these complexes in terms of graded loopspaces, which will give a
generalization of this proposition for $R$ a non-necessarily $p$-torsion-free
simplicial algebra. The de Rham complex has the following universal property,
which we will generalize.

Let $R$ be a $p$-torsion-free commutative ring and $\phi:R\to R$ be a
classical Frobenius lift. For $A$ a $p$-torsion-free Dieudonné algebra, the
construction given before defines a functorial bijection

$Hom_{\textbf{DA}}(\Omega_{R}^{\bullet},A)\xrightarrow{\sim}Hom_{Fr}(R,A^{0})$

where $Hom_{Fr}(R,A^{0})$ is the set of morphisms $R\to A^{0}$ compatible with
Frobenius structures.

In this thesis, we have generalized the previous constructions to simplicial
rings with a Frobenius lift. The $\infty$–category of simplicial rings with a
Frobenius lift being a "derived enhancement" of the $1$-category of
commutative rings endowed with a classical Frobenius lift, this thesis aims at
constructing a derived version of the de Rham-Witt complex.

Our constructions extend the ones previously defined in the following ways :

  * •

Working in simplicial rings allows for derived tensor products and homotopy
limits and colimits.

  * •

The constructions will be $\infty$-categorical and the functors will be
$\infty$-functors.

  * •

The assumption of the ring to be $p$-torsion-free is dropped.

  * •

We outline definitions and theorems for global objects, thus defining the de
Rham-Witt complex for schemes or derived stacks.

Dieudonné complexes and algebras can have the property of being saturated,
defining full subcategories of DC and DA. Furthermore there is a functorial
construction called saturation making a Dieudonné complex or algebra
saturated. We will define these constructions in the context of derived
Dieudonné complexes and algebras.

###  I.4 Witt vectors, $\delta$-rings and Frobenius lifts

We give a short presentation of some results using $\delta$-rings or Frobenius
lifts. A central notion for this thesis is that of homotopy Frobenius lift. It
seems to be the appropriate structure to study $p$-adic schemes.

The most recent results on $p$-adic cohomology theories are based on the
notion of $\delta$-rings, see [BS22]. We also note that the section 3.7 of
[BLM22] gives generalizations of their definitions of de Rham-Witt complex for
rings with $p$-torsion using $\delta$-rings.

######  Definition I.4.1.

([BLM22, Definition 3.7.4]) A $\delta$-cdga is a commutative differential
graded algebra $(A,d)$ endowed with a morphism of graded rings $F:A\to A$ such
that

  * •

The groups $A^{n}$ vanishes for $n<0$, that is $A$ is coconnective since we
are using cohomological conventions where the differential increases the
degree.

  * •

The pair $(A_{0},\delta)$ is a $\delta$-ring.

  * •

For $x\in A^{0}$, we have $F(x)=x^{p}+p\delta(x)$.

  * •

For $x\in A^{n}$, we have $dF(x)=pF(dx)$.

  * •

For $x\in A^{0}$, we have $F(dx)=x^{p-1}dx+d\delta(x)$

######  Proposition I.4.2.

([BLM22, Definition 3.7.6]) Let $(R,\delta)$ be a $\delta$-ring, the de Rham
complex $\Omega_{R}^{\bullet}$ is canonically a $\delta$-cdga, this defines a
functor

$\Omega_{-}:\delta-CRing\to\delta-cdga$

which is left adjoint to the forgetful functor

$\delta-cdga\to\delta-CRing$

The theory of prismatic cohomology is also based on the notion of
$\delta$-rings through the definition of prisms.

######  Definition I.4.3.

([BS22, Definition 3.2]) A prism is a pair $(A,I)$ where $A$ is a
$\delta$-ring and $I\subset A$ is an ideal defining a Cartier divisor on
$Spec(A)$ such that $A$ is derived $(p,I)$-complete and $p\in I+\phi(I)A$
where $\phi:A\to A$ is the Frobenius morphism induced by the
$\delta$-structure

We recall a classical theorem about the de Rham-Witt complex.

######  Theorem I.4.4.

([BLM22, Theorem 1.1.2]) Let $k$ be a perfect ring of characteristic $p$. Let
$\widetilde{X}$ be a smooth formal scheme over $Spf(W(k))$ with special fiber
$X$. The data of a Frobenius lift $F:\widetilde{X}\to\widetilde{X}$ of the
canonical Frobenius $Fr:X\to X$, if it exists, induces a natural quasi-
isomorphism

$f_{\widetilde{X},F}:\Omega^{\bullet}_{\widetilde{X}/W(k)}\xrightarrow{\sim}W\Omega_{X}^{\bullet}$

This theorem defines a natural transformation between

$\widetilde{X}\in
fSch^{sm}_{W(k)}\mapsto\Omega^{\bullet}_{\widetilde{X}/W(k)}\in\mathcal{D}(X)$

and

$\widetilde{X}\in fSch^{sm}_{W(k)}\mapsto
W\Omega_{X}^{\bullet}\in\mathcal{D}(X)$

where $fSch^{sm}_{W(k)}$ denotes the $1$-category of smooth formal schemes
over $Spf(W(k))$ and $\mathcal{D}(X)$ is the derived $1$-category of quasi-
coherent sheaves on $X$. Therefore working on categories of schemes or
commutative rings with Frobenius lifts seems fundamental.

###  I.5 Derived geometry and cohomology theories

In the past few years, the use of homotopy algebraic arguments to study
cohomology theories of schemes has not stopped growing. We give a brief
summary of some work related to derived algebra, or derived geometry, and the
cohomology of schemes.

In the papers [TV09], [MRT20] and [Rak20], a universal property is given
connecting graded mixed structures with the de Rham functor.

######  Definition I.5.1.

We define the category of mixed graded simplicial algebras $\epsilon-SCR^{gr}$
as the full subcategory of

$S^{1}_{gr}-dSt^{op}$

on $S^{1}_{gr}$-equivariant derived stacks which are affine schemes.

######  Theorem I.5.2.

The de Rham functor defines a functor to graded mixed algebras

$SCR\to\epsilon-SCR^{gr}$

which is left adjoint to taking the $0$-weighted part of a mixed graded
simplicial algebra.

An explicit connection is also given between Hochschild cohomology and De Rham
cohomology in [MRT20]. Since each of these cohomology theories have
geometrical avatars in derived geometry, we can obtain a global comparision
between Hochschild cohomology and de Rham cohomology of a derived scheme.

Let $X$ be a derived scheme over $\mathbb{Z}_{(p)}$, the loopspace of $X$ is
given by the mapping stack

$\mathcal{L}(X)\coloneqq\textbf{Map}(S^{1},X)$

It admits a natural filtration which has as its associated graded stack the
shifted tangent stack

$T[-1]X\coloneqq Spec_{X}Sym_{\mathcal{O}_{X}}(\mathbb{L}_{X}[1])$

Taking global functors defines a filtration of Hochschild cohomology of $X$
which admits as its associated graded the de Rham algebra of $X$.

We will use in this thesis the geometrical interpretation of graded derived
stacks as $\mathbb{G}_{m}$-equivariant derived stacks and filtered derived
stacks as $\mathbb{A}^{1}/\mathbb{G}_{m}$-equivariant derived stacks, see
[Mou19].

In the world of $p$-adic schemes, derived crystalline cohomology has proven to
be better-behaved than naive crystalline cohomology. Several papers have
compared de Rham and derived crystalline cohomology, such as [BDJ11] and
[Mon21b]. We recall a theorem of the latter.

######  Theorem I.5.3.

([Mon21b, Theorem 5.0.1]) Let

$dR:\mathbb{E}_{\infty}Alg^{sm}_{\mathbb{F}_{p}}\to
CAlg(Mod_{\mathbb{F}_{p}})$

be the algebraic de Rham cohomology functor defined on the category of smooth
$\mathbb{E}_{\infty}-\mathbb{F}_{p}$-algebras. Given $(A,m)$ an Artinian local
ring with residue field $\mathbb{F}_{p}$, the functor $dR$ admits a unique
deformation

$dR^{\prime}:\mathbb{E}_{\infty}Alg^{sm}_{\mathbb{F}_{p}}\to CAlg(Mod_{A})$

This deformation $dR^{\prime}$ is unique up to unique isomorphism.
Furthermore, this deformation is given by crystalline cohomology.

We also note the importance of derived $p$-completions instead of classical
$p$-completions used in [BS22] and [BDJ11].

##### Related work

We would like to mention connections between this thesis and work in progress
of Antieau. The following principle seems to central : graded objects defined
by geometric constructions should be naturally associated graded to canonical
HKR-type filtrations defined geometrically. As the graded circle is the
associated graded of the affinization of the topological circle, any
geometrical object defined with the graded circle $S^{1}_{gr}$ should satisfy
this heuristic. This philosophy will probably be correct for our construction
of de Rham-Witt complex as a graded loopspace and also to the definition of
foliations which relies on the graded circle and graded mixed complexes.
Indeed, in Section VIII, we will outline the construction of an analogue of
the Hochschild cohomology complex, based on a crystalline topological circle,
which should admit a filtration which has as its associated graded the de
Rham-Witt complex. This Hochschild cohomology complex seems to be a new and
interesting object to study.

###  I.6 Content of the thesis

We present the results of this thesis and some intuitions behind the main
definitions and theorems. We will work over the base commutative ring
$k=\mathbb{Z}_{(p)}$, therefore simplicial algebras will be simplicial
$\mathbb{Z}_{(p)}$-algebras and derived stacks will be derived stacks over
$Spec(\mathbb{Z}_{(p)})$.

####  I.6.1 Frobenius lifts and graded loopspaces

######  Definition I.6.1.

We define the category of derived stacks with a derived, or homotopy,
Frobenius lift $dSt^{Fr}$ as the pullback of categories

$dSt^{endo}\times_{dSt_{\mathbb{F}_{p}}^{endo}}dSt_{\mathbb{F}_{p}}$

where the functor $dSt_{\mathbb{F}_{p}}\to dSt_{\mathbb{F}_{p}}^{endo}$ is the
canonical functor adjoining the Frobenius endomorphism to a derived stack on
$\mathbb{F}_{p}$. Similarly, we define the category of graded derived stacks
endowed with a derived Frobenius lift $dSt^{gr,Fr}$ as the pullback of
categories

$dSt^{gr,endo}\times_{dSt_{\mathbb{F}_{p}}^{endo}}dSt_{\mathbb{F}_{p}}$

where the forgetful functor $dSt^{gr,endo}\to dSt_{\mathbb{F}_{p}}^{endo}$ is
given by taking $0$-weights, that is $\mathbb{G}_{m}$-fixed points, and taking
the fiber on $\mathbb{F}_{p}$.

######  Remark I.6.2.

Similarly, we define the category of simplicial algebras endowed with a
derived Frobenius lift $SCR^{Fr}$ and the category of graded simplicial
algebras endowed with a derived Frobenius lift $SCR^{gr,Fr}$.

As the classical definition of Dieudonné module encodes the structure of the
de Rham complex of a smooth commutative ring $A$ endowed with a classical
Frobenius lift, our notion of mixed graded Dieudonné modules is modeled on

$DR(A)=Sym_{A}\mathbb{L}_{A}[1]$

which is naturally a mixed graded complex. The usual compatiblity

$dF=pFd$

becomes

$\epsilon F=pF\epsilon$

A central object of our construction is the "crystalline graded circle"
denoted $S^{1}_{gr}$ which is a group in $dSt^{gr,Fr}$. It is the affine stack

$S^{1}_{gr}\simeq Spec^{\Delta}(\mathbb{Z}_{(p)}\oplus\mathbb{Z}_{(p)}[-1])$

where $\mathbb{Z}_{(p)}\oplus\mathbb{Z}_{(p)}[-1]=\mathbb{Z}_{(p)}[\eta]$ is
the denormalized cosimplicial algebra of a square zero extension commutative
differential graded algebra. Sending $\eta$ to $p\eta$ defines the Frobenius
structure on this stack. For details on the functor $Spec^{\Delta}(-)$ and
affine stacks, see [To06]. Now interpreting mixed graded modules as quasi-
coherent sheaves in

$QCoh(BS^{1}_{gr})$

the forgetful functor along $[p]$ is given by

$(M,d,\epsilon)\in QCoh(BS^{1}_{gr})\mapsto(M,d,p\epsilon)\in
QCoh(BS^{1}_{gr})$

Therefore we will see a graded mixed Dieudonné complex as a colax fixed points
of $[p]^{*}$, that is a pair $(M,f)$ where $M$ is a mixed graded complex and
$f$ is morphism of mixed graded complexes

$f:[p]^{*}M\to M$

######  Remark I.6.3.

The category of quasi-coherent complexes $QCoh(BS^{1}_{gr})$ is interpreted
with $S^{1}_{gr}$ seen as a graded stack, therefore this category is
implicitly defined as

$QCoh(B(S^{1}_{gr}\rtimes\mathbb{G}_{m}))$

See Remark IV.12.11 for details.

######  Definition I.6.4.

We define the category of mixed graded Dieudonné complexes, also called
derived Dieudonné complexes, by

$\epsilon-D-Mod^{gr}\coloneqq CFP_{[p]^{*}}(\epsilon-Mod^{gr})$

where $CFP_{[p]^{*}}(\epsilon-Mod^{gr})$ denotes the category of colax fixed
points in $\epsilon-Mod^{gr}$.

The category of mixed graded Dieudonné complexes admits the following
alternative description.

######  Proposition I.6.5.

(Proposition VII.2.12) We have a natural identifications

$\epsilon-D-Mod^{gr}\simeq(k[\epsilon],[p])-Mod^{gr,endo}$

where $[p]$ is the canonical endomorphism on $k[\epsilon]$, given by

$\epsilon\in k[\epsilon]\mapsto p\epsilon\in k[\epsilon]$

The object $(k[\epsilon],[p])$ is seen here as a commutative algebra object in
$Mod^{gr,endo}$.

######  Definition I.6.6.

Motivated by the previous proposition, we define mixed graded Dieudonné
stacks, also called derived Dieudonné stacks, by

$\epsilon-D-dSt^{gr}\coloneqq S^{1}_{gr}-dSt^{gr,Frob}$

This notion gives a definition of Dieudonné structures on global objects, such
as derived schemes or derived stacks.

We also define mixed graded Dieudonné simplicial algebras, also called
Dieudonné simplicial algebras, by

$\epsilon-D-SCR^{gr}\coloneqq(S^{1}_{gr}-dAff^{gr,Frob})^{op}$

In this definition, by abuse of notation, $S^{1}_{gr}-dAff^{gr,Frob}$ is the
category of elements of $S^{1}_{gr}-dSt^{gr,Frob}$ which have as an underlying
derived stack an affine derived scheme. An element of
$S^{1}_{gr}-dAff^{gr,Frob}$ can be thought of as a morphism $X\to
BS^{1}_{gr}$, in the topos $dSt^{gr,Frob}$, which is relatively affine.

A description of the category of derived Dieudonné stacks in terms of strict
models, for example using a model category, seems difficult. Therefore, the
use of derived stacks and the language of $\infty$-categories is essential in
order to define these objects. Similarly, in [Rak20], the use of
$\infty$-monads and homotopical algebra is essential to the description of the
universal property of the de Rham complex.

The definition of derived Dieudonné stack is fundamental for the study of the
multiplicative structure on the de Rham-Witt complex, however it adds a
considerable amount of difficulty by requiring the use of the
$\infty$-categories of simplicial rings and derived stacks.

The graded stack $S^{1}_{gr}$ admits an action from $\mathbb{G}_{m}$, which is
identified with the grading structure on $S^{1}_{gr}$, it also has a natural
action of $\mathbb{N}$ induced by its Frobenius lift structure, therefore we
can form the semi-direct product
$(S^{1}_{gr}\rtimes(\mathbb{N}\times\mathbb{G}_{m}))$. We notice that the
category $S^{1}_{gr}-dSt^{gr,endo}$ identifies with
$(S^{1}_{gr}\rtimes(\mathbb{N}\times\mathbb{G}_{m}))$-equivariant derived
stacks, that is derived stacks over
$B(S^{1}_{gr}\rtimes(\mathbb{N}\times\mathbb{G}_{m}))$. Furthermore, the
category $QCoh(B(S^{1}_{gr}\rtimes(\mathbb{N}\times\mathbb{G}_{m}))$
identifies with $\epsilon-D-Mod$. These identifications can be seen as
geometrical interpretations of Dieudonné algebras and complexes.

We will then construct the functor of functions on a Dieudonné derived stack

$C:\epsilon-D-dSt^{gr,op}\to\epsilon-D-Mod^{gr}$

refining the usual functor of functions

$C:dSt\to Mod$

######  Definition I.6.7.

Let $X$ a derived scheme endowed with a Frobenius lift. We define the
Frobenius graded loopspaces on $X$ as the mapping stack internal to the
category $dSt^{Fr}$ :

$\mathcal{L}^{gr,Fr}(X)\coloneqq\textbf{Map}_{dSt^{Fr}}(S^{1}_{gr},X)$

where $S^{1}_{gr}$ is endowed with its canonical Frobenius action. It is a
graded derived stack endowed with a Frobenius lift.

The canonical point $*\to S^{1}_{gr}$ defined by the augmentation
$\mathbb{Z}_{(p)}[\eta]\to\mathbb{Z}_{(p)}$, is a morphism of graded derived
stacks with Frobenius structures and induces a morphism of graded derived
stacks with Frobenius structures

$\mathcal{L}^{gr,Fr}(X)\to X$

The following proposition and corollary asserts that the crystalline graded
circle is the correct analogue of the graded circle in order to recover the de
Rham-Witt complex.

######  Proposition I.6.8.

(Proposition VII.2.36) Let $(X,F)$ be a affine derived scheme endowed with a
Frobenius structure. We write $X=Spec(C)$. The graded Frobenius loopspace’s
underlying graded stack identifies with the shifted linear stack:

$U\mathcal{L}^{gr,Fr}(X)\simeq
Spec_{X}Sym_{\mathcal{O}_{X}}\left(\mathbb{L}_{(X,F)}^{tw}[1]\right)$

where $Sym$ denotes the free $(C,F)$-module construction of Proposition
VII.1.26.

The $(C,F)$-module $\mathbb{L}\coloneqq\mathbb{L}_{(X,F)}^{tw}$ fits in a
triangle of $(C,F)$-modules

$\bigoplus_{\mathbb{N}}\mathbb{L}_{(C,F)}\otimes\mathbb{F}_{p}\to\mathbb{L}\to\mathbb{L}_{(C,F)}$

######  Corollary I.6.9.

(Corollary VII.2.38) With the notation of the previous proposition, the
Beilinson truncated derived stack associated to $\mathcal{L}^{gr,Fr}(X)$ is
given, after moding out by the $p$-torsion, by

$Spec(Sym(\Omega_{C}[1]))$

where the induced endomorphism is induced by

$\frac{dF}{p}:\Omega_{C}[1]\to\Omega_{C}[1]$

To construct the Dieudonné de Rham functor, we will need a classification
theorem of Dieudonné structures on free simplicial algebras. This result is
inspired by Proposition 2.3.1 in [To20], which was stated with a partial
proof.

######  Theorem I.6.10.

(Theorem VII.3.1) Let $A$ be smooth commutative $\mathbb{Z}_{(p)}$-algebra,
$M$ a projective $A$-module of finite type. We define $X$ as the derived
affine scheme $Spec(Sym_{A}(M[1]))=\mathbb{V}(M[1])$ endowed with its natural
grading. The classifying space of mixed graded structures on $X$ compatible
with its grading is discrete and in bijection with the set of commutative
differential graded algebra structures on the graded commutative
$\mathbb{Z}_{(p)}$-algebra $\bigoplus_{i}\wedge^{i}_{A}M[-i]$.

This key theorem will be proven using an analogue of the usual cellular
decomposition of the topological space $BS^{1}\simeq\mathbb{C}P^{\infty}$.
Since $BS^{1}_{gr}$ is given by

$BS^{1}_{gr}\simeq Spec^{\Delta}(D(H^{*}(BS^{1},\mathbb{Z}_{(p)})))$

where $D$ is the denormalization functor, we will need a decomposition on the
denormalized cosimplicial algebra

$D(H^{*}(BS^{1},\mathbb{Z}_{(p)}))\simeq D(\mathbb{Z}_{(p)}[u]/u^{2})$

where $u$ is in degree $2$. This decomposition is not the one obtained from
the topological decomposition therefore the construction has to be made from
scratch. This is due to the fact that a pushout of topological spaces has a
cosimplicial chain complex which is given by a pullback of the cosimplicial
chain complexes of the spaces, however, this identification is not compatible
with the natural grading. We will introduce weight-shifted formal spheres
$S^{2n+1}(n+1)$ which will exhibit $D(H^{*}(BS^{1},\mathbb{Z}_{(p)}))$ as the
appropriate limit, as a graded cosimplicial algebra.

Another fundamental ingredient of the proof will be a Postnikov decomposition
to compute the various mapping stacks, using obtruction theory. We will first
show that we can reduce to the $4$-skeleton $(BS^{1}_{gr})_{\leq 4}$, this
stack will be given by the homotopy cofiber of

$S^{3}_{f}(2)\to S^{2}_{f}(1)$

where $S^{3}_{f}(2)$ and $S^{2}_{f}(1)$ are formal version of the topological
spheres $S^{3}$ and $S^{2}$ with a shift of the gradings. This map is an
analogue of the Hopf fibration and we will study it using Koszul duality.

We will extend the previous theorem to the case of Dieudonné mixed graded
structures. This theorem is the main result of this thesis.

######  Theorem I.6.11.

(Theorem VII.3.15) Let $A$ be smooth commutative $\mathbb{Z}_{(p)}$-algebra,
$M$ a projective $A$-module of finite type. We fix a derived Frobenius lift
structure on the graded simplicial algebra $Sym_{A}(M[1])$, with $M$ in weight
$1$. From Proposition VII.1.8, it is equivalent to a classical Frobenius lift
$F$ on $A$ and a linear map of $A$-modules $\phi:M\to M$. We define $X$ as the
derived affine scheme $Spec(Sym_{A}(M[1]))=\mathbb{V}(M[1])$ endowed with its
natural grading, we regard it as an element of $dSt^{gr,Frob}$. The
classifying space of Dieudonné mixed graded structures on $X$ compatible with
its grading and Frobenius structure is discrete and in bijection with the set
of Dieudonné algebra structures on the graded commutative
$\mathbb{Z}_{(p)}$-algebra $\bigoplus_{i}\wedge^{i}_{A}M[-i]$ endowed with its
natural canonical Frobenius lift structure.

######  Corollary I.6.12.

(Corollary VII.3.19) With the notations of Theorem VII.3.15, the graded
derived affine scheme $Spec(Sym_{A}(\Omega_{A}[1]))$ admits a unique Dieudonné
structure induced by the standard Dieudonné structure on $\bigoplus_{i\geq
0}\Lambda_{A}^{i}\Omega_{A}$.

######  Definition I.6.13.

Let $(A,F)$ be a simplicial algebra with a Frobenius lift, we define the
Dieudonné de Rham complex of $(A,F)$ as

$\mathcal{O}(\mathcal{L}^{gr,Fr}(Spec(A),Spec(F)))$

where $\mathcal{O}$ is the functor of "simplicial functions" on derived
Dieudonné stacks

$\epsilon-D-dSt^{gr,op}\to\epsilon-D-SCR^{gr}$

defined in the previous proposition. We denote the element we obtain
$DDR(A,F)$.

######  Theorem I.6.14.

(Theorem VII.3.22) When $A$ is a smooth discrete algebra, $DDR(A,F)$ is
naturally the derived Dieudonné algebra

$Spec(Sym_{A}(\Omega_{A}[1]))$

endowed with its canonical Frobenius lift and mixed graded structure.

We will prove a theorem analoguous to Theorem I.5.2 and Proposition I.4.2. It
asserts that the Dieudonné de Rham functor is left adjoint to forgetting a
graded mixed structure.

######  Theorem I.6.15.

(Theorem VII.3.23) The Dieudonné de Rham functor

$SCR^{Fr}\to\epsilon-D-SCR^{gr}$

is left adjoint to the forgetful functor

$(0):\epsilon-D-SCR^{gr}\to SCR^{Fr}$

We also compare more precisely our construction of Dieudonné de Rham functor
with the one defined in [BLM22]. We define a t-structure inspired by Beilinson
t-structure, see [Rak20, §3.3], and prove truncation with respect to this
t-structure of our Dieudonné de Rham functor recovers the one in [BLM22] for
non-necessarily smooth algebras.

######  Definition I.6.16.

We recall the definition of the Beilinson t-structure on graded mixed complex.
Let $M$ be a graded mixed complex, $M$ is said to be t-connective for the
t-structure when $H^{i}(M(n))=0$ for $i>-n$, $M$ is said to be t-coconnective
for the t-structure when $H^{i}(M(n))=0$ for $i<-n$.

A derived Dieudonné complex or algebra is said to be t-truncated when its
underlying graded mixed complex is. We will abuse terminology and call these
constructions "t-structures".

######  Proposition I.6.17.

(Proposition VII.2.17) The heart of $\epsilon-D-Mod$ identifies with the
abelian category of Dieudonné complexes of [BLM22].

######  Theorem I.6.18.

For $A$ a commutative algebra, non-necessarily smooth, the t-truncation of
$DDR(A)$, with respect to the Beilinson t-structure, is equivalent to the
classical Dieudonné complex. That is

$t_{\geq 0}(DDR(A))\simeq i(\Omega^{\bullet}_{A})$

where $\Omega^{\bullet}_{A}$ is endowed with its canonical classical Dieudonné
structure, see Proposition VI.3.14 and $i$ is the functor

$i:\textbf{DA}\to\epsilon-D-SCR^{gr}$

identifying the category of classical Dieudonné algebra with the heart of
$\epsilon-D-SCR^{gr}$.

We will then define and study the notion of of saturation of a mixed graded
Dieudonné algebra.

######  Definition I.6.19.

We define the décalage $\eta_{p}$ functor as an endofunctor of $1$-categories
$\bf{\epsilon-Mod^{gr}}$ sending $M$ to the sub-graded mixed complex of
$M\times M$ on elements $(x,y)$ such that $\epsilon x=py$ and $\epsilon y=0$.

The following proposition defines and characterizes the saturation
$\infty$-functor.

######  Proposition I.6.20.

(Proposition VII.4.3) We have an adjunction of $1$-categories

$[p]^{*}:\bf{\epsilon-Mod^{gr}}\rightleftarrows\bf{\epsilon-
Mod^{gr}}:\eta_{p}$

In fact, this adjunction is a Quillen adjunction for the injective model
structures, therefore it induces an adjunction of $\infty$-categories.

A derived Dieudonné algebra is said to be saturated if its underlying
Dieudonné complex is. We will verify that the definition of saturated
Dieudonné complexes coincides with the one in [BLM22].

####  I.6.2 Graded functions on graded stacks

This thesis also contains the results presented in [Mon21] on graded functions
on linear stacks. We fix $X=Spec(A)$ an affine derived scheme.

######  Definition I.6.21.

Let $E$ be a quasi-coherent complex on $X$, the linear stack associated with
$E$ is the stack over $X$ given by

$u:Spec(B)\to X\mapsto Map_{N(B)-mod}(u^{*}E,N(B))\in SSet$

with $N$ the normalization functor. It is denoted $\mathbb{V}(E)$.

######  Proposition I.6.22.

(Proposition V.1.1) The functor

$\mathbb{V}:QCoh(X)^{op}\to\mathbb{G}_{m}-dSt_{/X}$

has a left adjoint given by $\mathcal{O}_{gr}(-)(1)$.

######  Theorem I.6.23.

(Theorem V.2.3) For $E\in QCoh^{-}(X)$, ie a bounded above complex over $X$,
the natural map constructed above

$E\to\mathcal{O}_{gr}(\mathbb{V}(E))(1)$

is an equivalence.

######  Corollary I.6.24.

(Corollary V.2.7) The functor
$\mathbb{V}:QCoh(X)^{op}\to\mathbb{G}_{m}-dSt_{/X}$ is fully faithful when
restricted to $QCoh^{-}(X)$.

This corollary is a crucial result for the study of foliations as it allows
the study of foliations of non-zero characteristic with cotangent complexes
which are not necessarily connective. We will combine it with the notion of
Dieudonné foliations so as to construct examples of Dieudonné foliations.

######  Proposition I.6.25.

(Proposition V.3.1) If $k$ is a $\mathbb{Q}$-algebra and $E$ is perfect
complex on $X$, $\mathcal{O}_{gr}(\mathbb{V}(E))(p)$ is canonically identified
with $Sym_{\mathbb{E}_{\infty}}^{p}(E)$, where the monad
$Sym_{\mathbb{E}_{\infty}}$ is the one associated to the formation of free
$\mathbb{E}_{\infty}$-$k$-algebras, or equivalently commutative differential
graded algebras.

####  I.6.3 Notations

######  Notation I.6.26.

In this thesis, all higher categorical notations are borrowed from [Lur09,
Lur17]. We fix $k$ to be a discrete commutative ring.

  * •

"$\infty$-category" will always mean $(\infty,1)$-categories, modeled for
example by quasicategories developed in [Lur09]. Everything is assumed to be
$\infty$-categorical, e.g "category" will mean $\infty$-category. We will
refer to the $1$-categories and $1$-categorical notions as "classical" or
"discrete", for example the $1$-category of discrete commutative rings will be
the usual category of commutative rings, denoted $\bf{CRing}$.

  * •

We will denote $1$-categories and model categories with bold text, e.g
$\bf{dg-mod_{k}}$ will be the $1$-category of chain complexes on $k$. The
model structure on a $1$-category will be denoted in subscripts or omitted,
e.g $\bf{dg-mod_{k,proj}}$ or $\bf{dg-mod_{k}}$ will be the model category of
chain complexes on $k$ with the projective model structure. All
$\infty$-categories will be denoted without bold text, e.g $Mod_{k}$ or $Mod$
the $\infty$-category of chain complexes on $k$.

  * •

We will use cohomological conventions : all chain complexes with have
increasing differentials, hence negative chain complexes will be connective.

  * •

$Mod_{k}$ is the $\infty$-category of chain complexes of $k$-modules,
$Mod_{k}^{\leq 0}$ or $Mod^{cn}$ is the full sub-$\infty$-category of
connective complexes and $Mod_{k}^{\geq 0}$ or $Mod^{ccn}$ is the full
sub-$\infty$-category of coconnective complexes. Similarly, for $A$ a
simplicial commutative $k$-algebra, $Mod_{A}$ will denote the usual
$\infty$-category obtained by inverting quasi-isomorphisms in the $1$-category
of differential graded modules $\bf{N(A)-dg-Mod}$ on the normalization of $A$.

  * •

$CAlg(\mathcal{C})$ is the $\infty$-category of commutative algebra objects in
a symmetric monoidal $\infty$-category, see [Lur17].

  * •

$\mathbb{E}_{\infty}Alg_{k}=CAlg(Mod_{k})$ is the $\infty$-category of
$\mathbb{E}_{\infty}$-algebras on $k$, see [Lur17].

  * •

$\bf{CRing_{k}}$ is the $1$-category of discrete commutative algebras on $k$.

  * •

$SCR_{k}$ is the $\infty$-category of simplicial commutative rings over $k$

  * •

$LSym$ or simply $Sym$ denotes the monad on $Mod_{k}^{\leq 0}$ associated with
the forgetful functor $SCR_{k}\to Mod_{k}^{\leq 0}$

  * •

$St_{k}$ and $dSt_{k}$ will denote respectively the $\infty$-category of
stacks over the étale site of discrete commutative $k$-algebras and the
$\infty$-category of derived stacks, also on the étale site.

  * •

Following [Lur18, 6.2.2.1, 6.2.2.7, 6.2.3.4], $Qcoh(X)$ will be the
$\infty$-category of quasi-coherent sheaves on a derived stack $X$, ie
$QCoh(X)$ is informally given as the limit of stable $\infty$-categories

$QCoh(X)\coloneqq lim_{Spec\text{ }A\to X}Mod_{A}$

  * •

A simplicial set or a derived stack $X$ will be $n$-truncated when
$\pi_{i}(X)=0$ for $i>n$, it will be $n$-connected when $\pi_{i}(X)=0$ for
$i\leq n$.

  * •

All tensor products are assumed to be derived.

  * •

The subscript notation $\mathcal{C}_{p}$ will denote the category of objects
over $k\otimes\mathbb{F}_{p}$ when $\mathcal{C}$ is a category of "algebra
type objects" over $k$. For example, when $SCR$ denoted the category of
simplicial algebras over $k$, $SCR_{p}$ is the category of simplicial algebras
over $k\otimes\mathbb{F}_{p}$.

  * •

The letters $\epsilon$, $\rho$ and $\eta$ will be reserved for generators of
commutative differential graded algebras in specific degrees : $\epsilon$ will
be in degree $-1$, $\rho$ in degree $0$ and $\eta$ in degree $1$. As
commutative differential graded algebras, we have identifications

$H^{*}(S^{1})\simeq k[\eta]$

and

$H_{*}(S^{1})\simeq k[\epsilon]$

These three generators will be in weight $1$ when they generate a graded
commutative differential graded algebra.

This thesis was supported by the Nedag project ERC-2016-ADG-741501.

##  II Categorical notions

In order to define our crystalline circle and to describe the de Rham-Witt
complex, in mixed characteristics, as a graded loopspace

$\textbf{Map}(S^{1}_{gr},X)$

we will need to review some results on $\infty$-categories and derived
geometry.

###  II.1 $\infty$-categories

This section gives the $\infty$-categorical notations and results which will
be used in this thesis. We ferer to [Lur09] for more details.

######  Notation II.1.1.

We will use the following notations and conventions regarding
$\infty$-categories :

  * •

We will use quasi-categories as models for our $\infty$-categories.

  * •

We will use Kan complexes as models for our $\infty$-groupoids.

  * •

The $\infty$-category of $\infty$-categories is denoted $Cat_{\infty}$, and
the sub-$\infty$-category of Kan complexes is denoted $Gpd_{\infty}$.

  * •

The inclusion $Cat_{\infty}\to Gpd_{\infty}$ admits a left adjoint denoted
$\widetilde{(-)}$.

  * •

The $\infty$-category of spaces is denoted $\mathcal{S}$.

  * •

When $\mathcal{C}$ is an $\infty$-category, and $x$ and $y$ are elements of
$\mathcal{C}$, $Map_{\mathcal{C}}(x,y)$ denotes the mapping space of morphisms
from $x$ to $y$. The set of connected components of morphisms from $x$ and $y$
is denoted $Hom_{\mathcal{C}}(x,y)$ or $[x,y]_{\mathcal{C}}$.

  * •

For $K$ a simplicial set and $\mathcal{C}$ an $\infty$-category,
$Fun(K,\mathcal{C})=\mathcal{C}^{K}$ is the $\infty$-category of functors from
$K$ to $\mathcal{C}$. In particular,
$Arrow(\mathcal{C})\coloneqq\mathcal{C}^{\Delta^{1}}$ is the $\infty$-category
of morphisms in $\mathcal{C}$.

  * •

The $\infty$-category of presheaves on an $\infty$-category $\mathcal{C}$ is
$\mathcal{C}\coloneqq Fun(\mathcal{C}^{op},\mathcal{S})$

  * •

In an $\infty$-category $\mathcal{C}$, with $X\in\mathcal{C}$ and $K$ a
simplicial set, the tensorization of $X$ by $K$ denoted $K\otimes X$ is, when
it exists, e.g when $\mathcal{C}$ is presentable, an object representing the
functor

$Z\in\mathcal{C}\mapsto Map_{\mathcal{C}}(X,Z)^{K}$

and the cotensorization of $X$ by $K$ denoted $X^{K}$ is, when it exists, e.g
when $\mathcal{C}$ is presentable, an object representing the functor

$Z\in\mathcal{C}^{op}\mapsto Map_{\mathcal{C}}(Z,X)^{K}$

  * •

A localization is a full sub-$\infty$-category
$\mathcal{C}_{0}\subset\mathcal{C}$ admitting a left adjoint.

######  Theorem II.1.2 (Adjoint functor theorem).

Let $F:\mathcal{C}\to\mathcal{D}$ be a functor between presentable
$\infty$-categories.

  * •

The functor $F$ admits a right adjoint if and only if it preserves small
colimits.

  * •

The functor $F$ admits a left adjoint if and only if it is accessible and
preserves small limits.

###### Proof.

See [Lur09, Corollary 5.5.2.9]. ∎

###  II.2 Lax and colax fixed points

######  Definition II.2.1.

Let $\mathcal{C}$ be a category and $F$ an endofunctor of $\mathcal{C}$. We
define the category of lax fixed points with respect to $F$ as the fiber
product of categories

$LFP_{F}(\mathcal{C})=\mathcal{C}^{\Delta^{1}}\times_{\mathcal{C}\times\mathcal{C}}\mathcal{C}$

where $\mathcal{C}\to\mathcal{C}\times\mathcal{C}$ is given by the composition
:

$\mathcal{C}\xrightarrow{\Delta}\mathcal{C}\times\mathcal{C}\xrightarrow{Id\times
F}\mathcal{C}\times\mathcal{C}$

We define the category of fixed points with respect to $F$ as the fiber
product of categories

$FP_{F}(\mathcal{C})=\mathcal{C}^{\widetilde{\Delta^{1}}}\times_{\mathcal{C}\times\mathcal{C}}\mathcal{C}$

The inclusion $\Delta^{1}\subset\widetilde{\Delta^{1}}$ defines an inclusion
$FP_{F}(\mathcal{C})\subset LFP_{F}(\mathcal{C})$.

######  Remark II.2.2.

The elements of the lax fixed points category $LFP_{F}(\mathcal{C})$ are given
by pairs $(x,f)$ with $x\in\mathcal{C}$ and $f:x\to Fx$. Then the category of
fixed points $FP_{F}(\mathcal{C})$ is the full subcategory of
$LFP_{F}(\mathcal{C})$ on pairs $(x,f)$ where $f$ is an equivalence.

######  Remark II.2.3.

Colax fixed points are defined similarly : $CFP_{F}(\mathcal{C})$ has elements
of the form $(x,f)$ with $x\in\mathcal{C}$ and $f:Fx\to x$.

######  Proposition II.2.4.

We have natural identifications

$CFP_{F}(\mathcal{C})^{op}\simeq LFP_{F}(\mathcal{C}^{op})$

###  II.3 Objects with endomorphisms

######  Definition II.3.1.

Let $\mathcal{C}$ be a category, we define the category of objects of
$\mathcal{C}$ endowed with an endomorphism as $Fun(B\mathbb{N},\mathcal{C})$,
where $B\mathbb{N}$ is the $1$-category with one object and $\mathbb{N}$ as
its monoid of arrows. We denote the associated category $\mathcal{C}^{endo}$.

The base point $*\to B\mathbb{N}$ gives by restriction the forgetful functor

$(X,\alpha)\in\mathcal{C}^{endo}\mapsto X\in\mathcal{C}$

######  Proposition II.3.2.

When $\mathcal{C}$ has $\mathbb{N}$-indexed coproducts, the forgetful functor

$\mathcal{C}^{endo}\to\mathcal{C}$

admits a left adjoint $L$, given by

$L:X\mapsto(\bigsqcup_{\mathbb{N}}X,S)$

with the shift $S$ given by the inlusion in the second factor

$\bigsqcup_{\mathbb{N}}X\xrightarrow{i_{2}}X\sqcup\bigsqcup_{\mathbb{N}}X\cong\bigsqcup_{\mathbb{N}}X$

When $\mathcal{C}$ has $\mathbb{N}$-indexed products, the forgetful functor

$\mathcal{C}^{endo}\to\mathcal{C}$

admits a right adjoint $R$, given by

$R:X\mapsto(\prod_{\mathbb{N}}X,S)$

with the shift $S$ induced by the projection map

$\prod_{\mathbb{N}}X\cong(\prod_{\mathbb{N}}X)\Pi(X)\xrightarrow{p_{1}}\prod_{\mathbb{N}}X$

######  Definition II.3.3.

Let $(X,\alpha)$ and $(Y,\beta)$ be two objects in $\mathcal{C}^{endo}$, we
define, when it exists, the morphism object with endomorphism
$\underline{Hom}_{\mathcal{C}^{endo}}((X,\alpha),(Y,\beta))$ as an element of
$\mathcal{C}^{endo}$ representing :

$(T,\gamma)\mapsto Map_{\mathcal{C}^{endo}}((X\times
T,\alpha\times\gamma),(Y,\beta)):\mathcal{C}^{endo}\to\mathcal{S}$

######  Proposition II.3.4.

We assume $\mathcal{C}$ has internal morphism objects and equalizers. Let
$(X,\alpha)$ and $(Y,\beta)$ be two objects in $\mathcal{C}^{endo}$, the
morphism object with endomorphism
$\underline{Hom}_{\mathcal{C}^{endo}}((X,\alpha),(Y,\beta))$ can be computed
as the equalizer :

$eq(\underline{Hom}_{\mathcal{C}}((X,\alpha),(Y,\beta))^{\mathbb{N}}\rightrightarrows\underline{Hom}_{\mathcal{C}}((X,\alpha),(Y,\beta))^{\mathbb{N}})$

where one of the arrows is given by postcomposing by $\beta$ and the other is
defined as by applying the shift $((f_{i})\mapsto(f_{i+1}))$ and precomposing
by $\alpha$. Its endomorphism is given by the shift $(f_{i})\mapsto(f_{i+1})$.

Explicitly, an object of
$\underline{Hom}_{\mathcal{C}^{endo}}((X,\alpha),(Y,\beta))$ can be thought of
as a sequence of morphisms $f_{i}:X\to Y$ such that $\beta\circ f_{i}$ is
identified with $f_{i+1}\circ\alpha$.

###  II.4 Nerve and conerve constructions

######  Definition II.4.1.

Let $\mathcal{C}$ be a presentable category endowed with the cartesian
symmetric monoidal structure. We denote $Gp(\mathcal{C})$ the category of
group objects in $\mathcal{C}$, it can be described by a full subcategory of
$Fun(\Delta^{op},\mathcal{C})$ on objects satisfying the Segal condition. See
[Lur09] for details.

The colimit functor, or geometric realization, will be denoted

$B=|-|:Gp(\mathcal{C})\to\mathcal{C}_{*}$

where $\mathcal{C}_{*}$ is the category of pointed objects in $\mathcal{C}$.

######  Proposition II.4.2.

The functor $B$ admits a right adjoint sending $*\to X$ to its Čech nerve.

######  Remark II.4.3.

Informally the nerve of $U\to X$ has $n$-simplices given by

$U\times_{X}U\times_{X}...\times_{X}U$

with $n+1$ copies of $U$. See [Lur09, Proposition 6.1.2.11] for details on
Čech nerves.

######  Definition II.4.4.

Similarly, we denote $coGr(\mathcal{C})$ the category of cogroup objects in
$\mathcal{C}$, it can be described by a full subcategory of
$Fun(\Delta,\mathcal{C})$ on objects satisfying the Segal condition.

The limit functor, or totalization, will be denoted

$lim_{\Delta}=Tot(-):coGr(\mathcal{C})\to\mathcal{C}_{/*}$

where $\mathcal{C}_{/*}$ is the category of copointed objects in
$\mathcal{C}$.

######  Proposition II.4.5.

The functor $lim_{\Delta}$ admits a left adjoint where $X\to*$ is sent to its
conerve.

###  II.5 Simplicial sets

######  Definition II.5.1.

Let $X$ be a simplicial set, the cosimplicial algebra of cohomology of $X$ has
$n$-simplices given by

$(k^{X})_{n}\coloneqq k^{X_{n}}$

where faces and degeneracies are induced by the ones on $X$.

###  II.6 Chain complexes

######  Theorem II.6.1 (The projective model structure on chain complexes).

For $f:M\to N$ a chain complex morphism in $\bf{dg-mod_{k}}$, $f$ is said to
be :

  * •

a weak equivalence if it is a quasi-isomorphism, ie it induces an isomorphism
on homology groups $H^{i}(M)\to H^{i}(N)$.

  * •

a fibration if it is a levelwise surjection.

This data makes $\bf{dg-mod_{k}}$ into a model category called the "projective
model structure on chain complexes". We will denote this model category
$\bf{dg-mod_{k,proj}}$

###### Proof.

See [Ho91]. ∎

######  Theorem II.6.2 (The injective model structure on chain complexes).

For $f:M\to N$ a chain complex morphism in $\bf{dg-mod_{k}}$, $f$ is said to
be :

  * •

a weak equivalence if it is a quasi-isomorphism, ie it induces an isomorphism
on homology groups $H^{i}(M)\to H^{i}(N)$.

  * •

a cofibration if it is a levelwise injection.

This data makes $\bf{dg-mod_{k}}$ into a model category called the "injective
model structure on chain complexes". We will denote this model category
$\bf{dg-mod_{k,inj}}$

###### Proof.

See [Dun10]. ∎

######  Definition II.6.3.

We define the naive coconnective truncation of a chain complex on $k$ as the
left adjoint to inclusion of $\bf{dg-mod_{k}^{\geq 0}}\subset\bf{dg-mod_{k}}$
It can be described as

$(A_{-1}\rightarrow A_{0}\rightarrow A_{1}\rightarrow A_{2})\mapsto(0\to
A_{0}\rightarrow A_{1}\rightarrow A_{2})$

It does not preserve quasi-isomorphisms.

We define the coconnective truncation of a chain complex on $k$ as the right
adjoint to inclusion of $\bf{dg-mod_{k}^{\geq 0}}\subset\bf{dg-mod_{k}}$ It
can be described as

$(A_{-1}\rightarrow A_{0}\rightarrow A_{1}\rightarrow A_{2})\mapsto(0\to
coker(d)\rightarrow A_{1}\rightarrow A_{2})$

It does induces a functor of $\infty$-categories $Mod_{k}\to Mod_{k}^{\geq 0}$
which is right adjoint to the inclusion.

We define the naive connective truncation of a chain complex on $k$ as the
right adjoint to inclusion of $\bf{dg-mod_{k}^{\leq 0}}\subset\bf{dg-mod_{k}}$
It can be described as

$(A_{-2}\rightarrow A_{-1}\rightarrow A_{0}\rightarrow
A_{1})\mapsto(A_{-2}\rightarrow A_{-1}\rightarrow A_{0}\rightarrow 0)$

It does not preserve quasi-isomorphisms.

We define the connective truncation of a chain complex on $k$ as the left
adjoint to inclusion of $\bf{dg-mod_{k}^{\leq 0}}\subset\bf{dg-mod_{k}}$ It
can be described as

$(A_{-2}\rightarrow A_{-1}\rightarrow A_{0}\rightarrow
A_{1})\mapsto(A_{-2}\rightarrow A_{-1}\rightarrow Ker(d)\rightarrow 0)$

It does induces a functor of $\infty$-categories $Mod_{k}\to Mod_{k}^{\leq 0}$
which is left adjoint to the inclusion.

######  Definition II.6.4.

A commutative differential graded algebra is a chain complex $(A,d)$ endowed
with a graded algebra structure over $k$ such that

  * •

The multiplication is graded commutative, that is, for $x\in A_{n}$ and $y\in
A_{m}$, we have $xy=(-1)^{mn}yx$ in $A_{m+n}$.

  * •

If $x\in A_{2n+1}$ is in odd degree, $x^{2}=0$.

  * •

The differential $d$ satisfies the Leibniz rule

$d(xy)=(dx)y+(-1)^{m}x(dy)$

for $x\in A_{m}$.

######  Remark II.6.5.

The category of commutative differential graded algebras is identified with a
subcategory of the category of commutative algebra objects in the $1$-category
$\bf{dg-mod_{k}}$ on objects satisfying $x^{2}=0$ for $x\in A_{2n+1}$.

###  II.7 Graded mixed chain complexes

We recall here elementary results on graded mixed complexes. We recommend
[TV09] for definitions and usage of graded mixed complexes in derived
geometry.

######  Definition II.7.1.

We define the $1$-category of graded mixed complexes $\bf{\epsilon-
mod_{k}^{gr}}$ as the module category $\bf{k[\epsilon]-dg-mod^{gr}_{k}}$ where
$k[\epsilon]$ is the commutative algebra object in $\bf{dg-mod_{k}^{gr}}$
where $\epsilon$ is in weight $1$ and degree $-1$, therefore the action of
$\epsilon$ on a graded mixed module is a morphism $M\to M((1))[-1]$ : the
operator decreases the degree, contrary to the differential, and increases the
weight.

######  Remark II.7.2.

The object $k[\epsilon]$ as an algebra object in $\bf{dg-mod_{k}^{gr}}$ can be
alternatively defined as the cohomology algebra of the circle
$H^{*}(S^{1};k)$. In fact Proposition IV.12.9 makes this identification more
precise.

######  Definition II.7.3.

The object $k[\epsilon]$ is naturally a coalgebra in graded differential
graded modules using the comultiplication

$\Delta:\epsilon\in k[\epsilon]\mapsto 1\otimes\epsilon+\epsilon\otimes 1\in
k[\epsilon]\otimes_{k}k[\epsilon]$

and counit

$n:\epsilon\in k[\epsilon]\mapsto 0\in k$

This makes $\bf{\epsilon-dg-mod_{k}^{gr}}$ into a symmetric monoidal category.

######  Definition II.7.4.

The $\infty$-category of graded mixed complex is the localization of
$\bf{\epsilon-dg-mod_{k}^{gr}}$ at quasi-isomorphisms. It is naturally a
symmetric monoidal $\infty$-category.

######  Proposition II.7.5.

Defining fibrations and trivial fibrations in $\bf{\epsilon-dg-mod_{k}^{gr}}$
as the ones in $\bf{dg-mod_{k,proj}}$, makes it into a model category called
the projective model structure on $\bf{\epsilon-dg-mod_{k}^{gr}}$.

Defining cofibration and trivial cofibration in $\bf{\epsilon-dg-
mod_{k}^{gr}}$ as the ones in $\bf{dg-mod_{k,proj}}$, makes it into a model
category called the injective model structure on $\bf{\epsilon-dg-
mod_{k}^{gr}}$.

They both present the same $\infty$-category denoted $\epsilon-Mod^{gr}_{k}$.

###### Proof.

See [TV09] for details. ∎

###  II.8 The bar construction

We recall the following construction which is useful to construct resolutions,
see [Ill71, §I.1.5] for details.

######  Definition II.8.1.

Let $T$ be a comonad on a category $\mathcal{C}$. Let $c\in\mathcal{C}$, we
define an augmented simplicial object in $\mathcal{C}$ by $T^{\bullet+1}(c)\to
c$. We call this the bar construction and denote the simplicial object
$Bar_{C}(c)$.

##  III Reminders on algebra

###  III.1 Simplicial and cosimplicial rings

####  III.1.1 Simplicial algebras

######  Definition III.1.1.

We define $SCR_{k}$ the $\infty$-category of simplicial commutative algebra
over $k$ as the localization of the $1$-category of simplicial commutative
algebra over $k$ with respect to the model structure where we ask weak
equivalences to be isomorphisms on homology and fibrations to be level-wise
surjections. From [Lur18, §25.1.1], this category admits a useful alternative
description as the category generated from $Poly_{k}$ under sifted colimits,
where $Poly_{k}\subset CRing_{k}$ is the full sub-$1$-category of polynomial
rings on a finite set of variables. Explicitly this category can be defined by
the full subcategory $Fun(Poly_{k}^{op},\mathcal{S})$ on functors preserving
finite products.

######  Definition III.1.2.

The simplicial version of the Dold-Kan construction induces a functor

$\theta^{cn}:SCR_{k}\to\mathbb{E}_{\infty}Alg_{k}^{cn}$

where $\mathbb{E}_{\infty}Alg_{k}^{cn}=CAlg(Mod_{k}^{cn})$ is the
$\infty$-category of connective $\mathbb{E}_{\infty}$-algebra on $k$.

######  Proposition III.1.3.

The functor $\theta^{cn}$ previously defined preserves small limits and
colimits and is conservative.

###### Proof.

See [Lur18, Proposition 25.1.2.2]. ∎

######  Definition III.1.4.

Let $A$ be a simplicial algebra and $M$ a simplicial module over $A$. The
square zero extension $A\oplus M$ of $A$ by $M$ is given by a square zero
extension degreee wise :

$(A\oplus M)_{n}\coloneqq A_{n}\oplus M_{n}$

with the canonical faces and degeneracies maps, every element in $M_{n}$
squares to zero.

####  III.1.2 Cosimplicial algebras

######  Definition III.1.5.

Let $f:A\to B$ be a morphism of cosimplicial algebras.

  * •

The map $f$ is said to be a weak equivalence when it induces an equivalence on
all the homology groups.

  * •

The map $f$ is said to be a fibration if it is levelwise surjective

######  Proposition III.1.6.

The data of weak equivalences and fibrations defines a simplicial model
structure on $\bf{coSCR_{k}}$.

###### Proof.

See [To06, Theorem 2.1.2]. ∎

######  Definition III.1.7.

We define the $\infty$-category of cosimplicial commutative rings $coSCR_{k}$
over $k$ as the localization of $\bf{coSCR_{k}}=Fun(\Delta,\bf{CRing_{k}})$
with respect to the simplicial model structure defined above.

######  Definition III.1.8.

The cosimplicial version of the Dold-Kan construction induces a functor

$\theta^{ccn}:coSCR\to CAlg_{k}^{ccn}$

Furthermore, $\theta$ preserves small limits and colimits and is conservative.

######  Definition III.1.9.

We recall the cotensorization of a cosimplicial algebra $A$ by a simplicial
set $X$, it is the cosimplicial algebra which has as $n$-simplices the algebra

$(A^{X})_{n}\coloneqq\prod_{X_{n}}A_{n}$

The mapping space of cosimplicial algebras, by requiring the usual universal
property, is given by the following construction. Let $A,B\in coSCR_{k}$,

$Map_{coSCR_{k}}(A,B)_{n}\coloneqq Hom_{coSCR_{k}}(A,B^{\Delta^{n}})$

####  III.1.3 Dold-Kan correspondance

######  Definition III.1.10.

For $\mathcal{A}$ an abelian category, the normalization construction is a
functor :

$\bf{\mathcal{A}^{\Delta^{op}}}\to\bf{Ch^{\leq 0}(\mathcal{A})}$

where we denote by $\bf{Ch^{\leq 0}(\mathcal{A})}$ the category of connective
chain complexes in $\mathcal{A}$. A simplicial object $E$ is sent to a chain
complex having in degree $-n$ the element $\bigcap_{i=1}^{n}d^{n}_{i}$,
meaning the intersection of the $i$-th face maps for $i>0$ : the induced
differential is given by the remaining $0$-face map.

######  Proposition III.1.11.

The normalization functor $N$ admits a right adjoint $D$ called
denormalization. And they are equivalences of categories. Let $A$ be a
simplicial ring, there is a Quillen equivalence

$\bf{NA-dg-mod^{\leq 0}}\simeq\bf{A-Mod}$

between the category of connective dg-modules over $NA$ and the category of
simplicial modules over $A$.

######  Remark III.1.12.

The natural isomorphism $ND\simeq Id$ is naturally monoidal with respect to
the shuffle and Alexander-Whitney maps and the isomorphism $DN\simeq Id$
admits a non-symmetriccal monoidal structure.

The normalization is lax symmetric monoidal but $D$ is only lax monoidal.

###### Proof.

See : [SS03] ∎

######  Definition III.1.13.

The normalization functor

$N:\bf{cosMod_{k}}\to\bf{dg-mod_{k}^{\geq 0}}$

from cosimplicial modules to differential graded modules induces the notion of
homology group of a cosimplicial algebra $A$ by composing the forgetful
functor

$\bf{coSCR_{k}}\to\bf{cosMod_{k}}$

with the normalization functor and the cohomology functor of a chain complex
$H^{i}$.

###  III.2 $\delta$-structures

######  Definition III.2.1.

Let $A$ be a discrete commutative ring, a classical Frobenius lift on $A$ is
given by a ring endomorphism $F:A\to A$ such that $F$ induces the Frobenius
morphism on $A/p$, ie $F(a)-a^{p}$ is $p$-divisible for any $a\in A$.

######  Definition III.2.2.

Let $A$ be a discrete commutative ring, a $\delta$-structure on $A$ is given
by a function $A\to A$ such that $\delta(0)=0$, $\delta(1)=0$, and for any
$a,b\in A$,

$\delta(a+b)=\delta(a)+\delta(b)-\sum_{i=1}^{p-1}\frac{1}{p}\binom{p}{i}a^{i}b^{p-i}$
$\delta(ab)=a^{p}\delta(b)+b^{p}\delta(a)+p\delta(a)\delta(b)$

######  Remark III.2.3.

Any $\delta$-structure induces a classical Frobenius lift
$F(a)=a^{p}+p\delta(a)$. This defines a bijection between $\delta$-structures
and classical Frobenius lifts when $A$ is $p$-torsion free. In fact, Theorem
III.3.8 identifies $\delta$-structures with derived Frobenius lifts .

###  III.3 The ring of Witt vectors

As seen in [Ser62], when $k$ is a field of positive characteristic $p$, there
is a unique complete discrete valuation ring of characteristic $0$ having $k$
as a residual field and $(p)$ as its maximal ideal, this is the ring of Witt
vectors on $k$. The ring of Witt vectors admits various generalizations when
$k$ need not be a field of positive characteristic. For details on Witt
vectors, see [Haz08] and [Rab14].

######  Definition III.3.1.

We define the ring of $p$-typical Witt vectors $W(A)$ associated to a
commutative ring $A$. As a set $W(A)$ is given by $A^{\mathbb{N}}$.

We define the ghost map as function :

$gh:(a_{n})\in W(A)\mapsto(w_{n})\in A^{\mathbb{N}}$

where $w_{n}\coloneqq\sum_{i=0}^{n}p^{i}a_{i}^{p^{n-i}}$.

The set $W(A)$ admits a unique functorial ring structure making $gh$ into a
natural transformation of commutative rings. An element $x=(a_{n})\in W(A)$
will have $(a_{n})$ as its "Witt coordinates" and $(w_{n})$ as its "ghost
coordinates".

The ghost morphism can be seen as a morphism of commutative rings in affine
schemes

$W\to\mathbb{G}_{a}^{\mathbb{N}}$

######  Proposition III.3.2.

The ghost map :

$gh:W(A)\to A^{\mathbb{N}}$

is injective when $A$ is $p$-torsion free. In this case, the ghost coordinates
can be use without ambiguity.

######  Definition III.3.3.

We define the $p$-typical Witt vectors of length $m$ as the quotient ring

$W_{m}(A)=W(A)/I$

where $I$ is the ideal of elements $(a_{n})$ where $a_{n}=0$ for $n<m$.
Elements of $W_{m}(A)$ can be written $(a_{0},...,a_{n-1})$.

######  Definition III.3.4.

The Verschiebung map is a natural transformation

$V:W(A)\to W(A)$

defined as $(a_{i})\mapsto(0,a_{0},a_{1},...)$. In ghost coordinates, it is
expressed as

$(w_{n})\mapsto(0,pw_{0},pw_{1},...)$

It is an additive morphism and induces a morphism of truncated Witt vectors :

$V:W_{m}(A)\to W_{m+1}(A)$

######  Definition III.3.5.

The Frobenius map is a natural transformation

$F:W(A)\to W(A)$

defined implicitly in ghost coordinates as $(w_{n})\mapsto(w_{1},w_{2},...)$
and by requiring functoriality.

The Frobenius $F$ is a ring morphism and induces a morphism of truncated Witt
vectors :

$F:W_{m}(A)\to W_{m-1}(A)$

######  Proposition III.3.6.

We have the usual identities :

$FV=p$

and

$V(F(x)y)=xV(y)$

###### Proof.

Both results can be proven by functoriality and reducing to the case of $A$
being without $p$-torsion where ghost coordinates can be used. ∎

######  Definition III.3.7.

The projection morphism $W(A)\to A$ admits a multiplicative section
$a\mapsto[a]\coloneqq(a,0,...)$, $[a]$ is called the Teichmüller
representative of $a$.

######  Theorem III.3.8.

The ghost coordinates induce a homotopy pullback diagram :

${W_{2}(A)}$${A}$${A}$${A\otimes_{\mathbb{Z}}^{\mathbb{L}}\mathbb{F}_{p}}$$\scriptstyle{w_{0}}$$\scriptstyle{w_{1}}$${\lrcorner}$$\scriptstyle{Fr_{p}\circ
pr}$$\scriptstyle{pr}$

As sections of the projection morphism $w_{0}:W_{2}(A)\to A$ are given by
$\delta$-structures, we can identify $\delta$-structures with derived
Frobenius lifts on $A$.

###### Proof.

See [BS22, Remark 2.5]. ∎

######  Proposition III.3.9.

The Witt vectors admits a natural $\delta$-structure, the forgetful functor

$\bf{\delta-CRing_{k}}\to\bf{CRing_{k}}$

admits $W$ as a right adjoint. From the previous proposition, we see the Witt
vector functor as the construction of adding cofreely a $\delta$-structure on
a commutative ring.

We also recall a theorem of Almkvist for the reference, see [A78], which we
will not use. For a commutative ring $R$, we define $\bf{Proj(R)}$ the
category of projective finitely generated $R$-modules. We write the category
of endomorphisms in $\bf{Proj(R)}$ :

$\bf{End(Proj(R))}\coloneqq\bf{Fun(B\mathbb{N},Proj(R))}$

as a functor category where $\bf{B\mathbb{N}}$ is the category with one object
and $\mathbb{N}$ as its monoid of arrows.

######  Definition III.3.10.

We define $K_{0}(\bf{End(Proj(R))})$ as the free abelian group generated by
isomorphism classes of objects in $\bf{End(Proj(R))}$ modulo the subgroup
generated by elements of the form

$[(M,f)]-[(M^{\prime},f^{\prime})]-[(M^{\prime\prime},f^{\prime\prime})]$

when there is a morphism of short exact sequences :

${0}$${M^{\prime}}$${M}$${M^{\prime\prime}}$${0}$${0}$${M^{\prime}}$${M}$${M^{\prime\prime}}$${0}$$\scriptstyle{f^{\prime}}$$\scriptstyle{f}$$\scriptstyle{f^{\prime\prime}}$

The category $K_{0}(\bf{Proj(R)})$ is similarly defined. The tensor product of
modules endows $K_{0}(\bf{End(Proj(R))})$ and $K_{0}(\bf{Proj(R)})$ with the
structure of commutative rings.

The kernel of the canonical morphism

$[(M,f)]\in K_{0}(\textbf{End(Proj(R))})\mapsto[M]\in K_{0}(\textbf{Proj(R)})$

is denoted $\widetilde{K}_{0}(\bf{End(Proj(R))})$.

######  Theorem III.3.11.

There is a canonical ring isomorphism

$\widetilde{K}_{0}(\textbf{End(Proj(R))})\cong Rat(R)$

where $Rat(R)$ is the subring of the ring of big Witt vectors $W_{big}(R)$
which are rational functions. When $W_{big}(R)$ is identified with formal
series $(1+tR[[t]])$, $Rat(R)$ is the subring

$\left\\{\frac{1+a_{1}t+...+a_{m}t^{m}}{1+b_{1}+...+b_{n}t_{n}}:a_{i},b_{i}\in
R\right\\}\subset(1+tR[[t]])$

The morphism is given by taking the characteristic polynomial of the
endomorphism

$[(M,f)]\in\widetilde{K}_{0}(\textbf{End(Proj(R))})\mapsto\chi_{f}(t)\in(1+tR[[t]])$

######  Remark III.3.12.

Many construction on the Witt vectors can be described at the level of the
category End(Proj(R)), therefore defining a satisfying explanation for the
structure arising on Witt vectors and the compatibility between the various
maps.

For example, the Frobenius $F_{n}$ acts on End(Proj(R)) as follows

$[(M,f)]\mapsto[(M,f^{n})]$

The Verschiebung $V_{n}$ acts as

$[(M,f)]\mapsto[(M^{\oplus n},f_{n})]$

where $f_{n}$ sends the $i$-th copy of $M$ to the $(i+1)$-th copy of $M$ by
identity and sends the $n$-th copy of $M$ to the first copy of $M$ by $f$.

The $\Lambda$-structure can be seen as taking exterior products :
$[(M,f)]\mapsto[(\Lambda_{n}M,\Lambda_{n}f)]$

##  IV Preliminaries in derived geometry

In this section we recall notions of derived geometry. We refer to [To06b] and
[To13] for overviews on the subject. The various notions and results of this
section are borrowed from [TV06], [Lur17] and [Lur18].

###  IV.1 Stacks and derived stacks

######  Definition IV.1.1.

We will consider stacks and derived stacks with respect to the étale topology
on simplicial algebras. We denote $St_{k}$ the $\infty$-category of
hypercomplete stacks and $dSt_{k}$ the $\infty$-category of hypercomplete
derived stacks.

######  Proposition IV.1.2.

The inclusion of stacks into derived stacks : $St_{k}\subset dSt_{k}$ admits a
right adjoint called truncation, denoted $t_{0}$, any derived stack $X$ admits
a canonical morphism from its truncation $t_{0}(X)\to X$.

######  Remark IV.1.3.

On affine derived schemes, truncation is given by

$t_{0}(Spec(A))=Spec\textrm{ }\pi_{0}(A)$

######  Remark IV.1.4.

Seeing a derived scheme $(X,\mathcal{O}_{X})$ as a classical scheme
$(X,\pi_{0}(\mathcal{O}_{X}))$ with $\pi_{i}(\mathcal{O}_{X})$ being a quasi-
coherent $\pi_{0}(\mathcal{O}_{X})$-modules, the truncation
$t_{0}(X,\mathcal{O}_{X})$ is given by the underlying scheme
$(X,\pi_{0}(\mathcal{O}_{X}))$.

We include here a motivation for the importance of the Frobenius morphism.

######  Proposition IV.1.5.

Let $k$ be a commutative ring, there is an isomorphism of abelian groups

$End_{Gp_{k}}(\mathbb{G}_{a})\cong kId\oplus\bigoplus_{p}\bigoplus_{n\geq
1}k[p]F_{p^{n}}$

where $p$ belongs in the prime numbers and $k[p]\coloneqq\\{x\in k:px=0\\}$ is
the ideal of $p$-torsion elements of $k$.

The set $k[p]F_{p^{n}}$ corresponds with morphisms of the form

$b\in B=\mathbb{G}_{a}(B)\mapsto xb^{p^{n}}\in B=\mathbb{G}_{a}(B)$

where $x\in k[p]$. We notice that composition is characterized by

$xF_{p^{m}}\circ yF_{q^{n}}=0$

when $p\neq q$ and

$xF_{p^{m}}\circ yF_{p^{n}}=xy^{p^{m}}F_{p^{m+n}}$

###### Proof.

The derived stack $\mathbb{G}_{a}$ being a classical stack, the space of group
endomorphisms on $\mathbb{G}_{a}$ is a set. It identifies with the set of
polynomials $P\in k[X]$ satisfying

$P(0)=0$

and

$P(X+Y)=P(X)+P(Y)$

the second condition can be checked degreewise, meaning we have to find the
elements $x\in k$ such that

$x(X+Y)^{m}=xX^{m}+xY^{m}$

elementary computations using the Legendre formula find that $x$ is null when
$m$ admits two disctinct prime divisors and $x$ must satisfy $px=0$ when
$m=p^{n}$ with $n\geq 1$. ∎

######  Remark IV.1.6.

In particular, for $k$ a torsion-free commutative ring, morphisms of group
schemes of $\mathbb{G}_{a}\rightarrow\mathbb{G}_{a}$ are given by
multiplication by an element of $k$. That is

$End_{Gp_{k}}(\mathbb{G}_{a})\cong k$

For $k$ a field of characteristic $p$, the ring of morphisms is given by the
non-commutative algebra $k[F]$ with $F$ the Frobenius, with the relations
$F.a=a^{p}.F$, for $a\in k$.

###  IV.2 $\mathcal{O}$-modules and quasi-coherent sheaves

######  Definition IV.2.1.

For a derived stack $X$, we introduce the $\infty$-category of
$\mathcal{O}_{X}$-modules. We first define the structure stack
$\mathcal{O}_{X}$ of $X$ as the stack over the big étale site over $X$ sending
$Spec(B)\to X$ to the underlying $\mathbb{E}_{\infty}$-ring $\theta(B)$. It is
a stack of $\mathbb{E}_{\infty}$-rings. Then the category
$\mathcal{O}_{X}-Mod$ of $\mathcal{O}_{X}$-modules is the category of modules
over the commutative ring object in derived stack $\mathcal{O}_{X}$.

######  Remark IV.2.2.

Intuitively an $\mathcal{O}_{X}$-module a given by a $\theta(B)$-module for
every simplicial commutative algebra $B$, functorially in $B$ and satisfying
descent as a presheaf on the étale site over $X$. In this description the full
subcategory $QCoh(X)\subset\mathcal{O}_{X}-Mod$ consists of
$\mathcal{O}_{X}$-modules where transition maps between modules are
equivalences.

######  Proposition IV.2.3.

The inclusion $QCoh(X)\subset\mathcal{O}_{X}-Mod$ has a right adjoint, which
we will denote $Q$.

###### Proof.

The source and target categories are both presentable, and the inclusion
preserves limits. We conclude using [Lur17, 5.5.2.9]. ∎

######  Proposition IV.2.4 (Functoriality of modules and quasi-coherent
sheaves).

A morphism of derived stack $\pi:X\to Y$ induces an adjunction between their
category of $\mathcal{O}$-modules :

$\widetilde{\pi}^{*}:\mathcal{O}_{Y}-Mod\rightleftarrows\mathcal{O}_{X}-Mod:\widetilde{\pi}_{*}$

and between their category of quasi-coherent sheaves :

$\pi^{*}:QCoh(Y)\rightleftarrows QCoh(X):\pi_{*}$

We have compatibility between pullbacks and the inclusion of quasi-coherent
sheaves into $\mathcal{O}$-module :

${QCoh(Y)}$${QCoh(X)}$${\mathcal{O}_{Y}-Mod}$${\mathcal{O}_{X}-Mod,}$$\scriptstyle{\pi^{*}}$$\scriptstyle{\widetilde{\pi}^{*}}$

which means that the $\mathcal{O}$-module pullback of a quasi-coherent sheaf
is quasi-coherent. However, the analogous statement for pushforwards is
incorrect, the adjunctions allow the calculation of quasi-coherent
pushforwards as the composition of $\mathcal{O}$-module pushforward with the
adjoint of the inclusion $QCoh(Y)\subset\mathcal{O}_{Y}-Mod$, ie

$\pi_{*}=Q\circ\widetilde{\pi}_{*}.$

###### Proof.

See [Lur11c, Proposition 2.5.1] . ∎

######  Definition IV.2.5.

We will use the standard t-structure on quasi-coherent complexes, see [Lur18,
Remark 6.2.5.8] for details. Let $X$ be a derived stack, a quasi-coherent
complex is said to be connective when its pullback to every affine is
connective.

######  Proposition IV.2.6.

Let $X$ a derived stack and $\mathcal{F}\in QCoh(X)$, the following conditions
are equivalent: :

  * •

The quasi-coherent sheaf $\mathcal{F}$ is perfect.

  * •

The quasi-coherent sheaf $\mathcal{F}$ is dualizable in $QCoh(X)$.

###### Proof.

See [Lur11c, Proposition 2.7.28]. ∎

######  Proposition IV.2.7.

Let $X$ be a Deligne-Mumford stack, then :

  * •

The inclusion $QCoh(X)\subset\mathcal{O}_{X}-Mod$ preserves small colimits.

  * •

$QCoh(X)$ is stable and presentable.

  * •

$QCoh(X)$ is both right and left t-complete, with respect to the standard
t-structure.

###### Proof.

See [Lur11c, Proposition 2.3.13, Proposition 2.3.18]. ∎

######  Remark IV.2.8.

Let $X$ is a stack represented as a simplicial scheme $Spec(A^{\bullet})$ with
$A$ an augmented cosimplicial commutative algebra such that $H^{i}(A)=k$ and
$k$ is $p$-torsion-free. Proposition B.8 provides with a cofibrant replacement
$QA$ such that boundary maps $QA^{m}\to QA^{n}$ are flat. Then proceeding as
in [MRT20, Notation 1.2.12] and [Lur18, §6.2.5] defines a left-complete
t-structure on $QCoh(X)$ compatible with the equivalence

$QCoh(X)\simeq lim_{n}Mod_{QA^{n}}$

Then

$QCoh(X)_{\geq 0}\simeq lim_{n}Mod_{QA^{n}}^{\geq 0}$

and

$QCoh(X)_{\leq 0}\simeq lim_{n}Mod_{QA^{n}}^{\leq 0}$

Furthermore $A-Mod$ admits a t-structure and the inclusion

$QCoh(A)\subset A-Mod$

identifies $QCoh(A)$ with the left completion of $A-Mod$ with respect to its
$t$-tructure. We will not prove the previous result, the proof in [Lur18,
§6.2.5] should carry out without difficulty.

###  IV.3 Construction of derived stacks

##### Linear stacks

######  Definition IV.3.1.

Let $E$ be a quasi-coherent complex on $X$, the linear stack associated to $E$
is the stack over $X$ given by

$u:Spec(B)\to X\mapsto Map_{N(B)-mod}(u^{*}E,N(B))\in SSet$

with $N$ the normalization functor. It is denoted $\mathbb{V}(E)$.

For example, if $E$ is connective, $\mathbb{V}(E)$ is simply the relative
affine scheme $Spec_{X}Sym_{\mathcal{O}_{X}}(E)$.

######  Definition IV.3.2.

We define the derived stack $\mathbb{G}_{m}$ associating to a simplicial
commutative algebra $B$ the simplicial set of autoequivalences of $N(B)$ :

$\mathbb{G}_{m}(B)\coloneqq Map^{eq}_{N(B)-mod}(N(B),N(B))$

In fact, $\mathbb{G}_{m}$ is representable by the discrete ring $k[X,X^{-1}]$,
and is equivalent to the usual derived stack classifying the group of units of
a simplicial algebra : $B\mapsto B^{*}$.

#####  $K(-,n)$ construction

######  Definition IV.3.3.

When $G$ is a group, we can define the simplicial set $E(G,1)$ which
$m$-simplices are $(G^{m+1})_{m}$ and faces and degeneracies are given by
projections and diagonals, $G$ acts on the left on $E(G,1)$ and $BG\coloneqq
K(G,1)$ is defined as the quotient simplicial set $E(G,1)/G=(G^{n})_{n}$.

This construction generalizes to a simplicial group $G$ where
$K(G,1)_{n}\coloneqq K(G_{n},1)_{n}$. The simplicial set $E(G,1)$ is simply
$G_{n}^{n+1}$

We generalize further to a group in derived stacks $G$, then $BG\coloneqq
K(G,1)$ and $E(G,1)$ are defined by functoriality.

Inductively, we define, when $G$ is abelian,

$E(G,m)\coloneqq E(K(G,m-1),1)$

and

$K(G,m)\coloneqq K(K(G,m-1),1)$

######  Proposition IV.3.4.

For $G$ a group in derived stacks, $BG$ is connected and its homotopy groups
are given by

$\pi_{i}(BG)\simeq\pi_{i-1}(G)$

Furthermore, when $A$ is an abelian group in derived stacks, $K(A,n)$
classifies cohomology of derived stacks and $Hom(F,K(A,n))$ will be denoted
$H^{n}(F,A)$. This construction recovers various cohomology theories, such as
sheaf cohomology and singular cohomology.

###### Proof.

See [TV06]. ∎

##### Mapping stack

######  Definition IV.3.5.

Let $X$ and $Y$ be derived stacks, the mapping stack $\textbf{Map}(X,Y)$ of
morphisms from $X$ to $Y$ is defined as a derived stack by

$\textbf{Map}(X,Y):B\in SCR\mapsto Map(X\times Spec(B),Y)$

###  IV.4 Postnikov towers

######  Definition IV.4.1.

The inclusion of $n$-truncated derived stacks into derived stacks admits a
left adjoint denote $t_{\leq n}$ and it will be called truncation. We have an
induced tower of morphisms

$F\to...t_{\leq n}(F)\to t_{\leq n-1}(F)\to...\to t_{\leq 0}(F)$

The induced morphism $F\to lim_{n}t_{\leq n}(F)$ is not necessarily an
equivalence.

###  IV.5 Functions on a derived stack

######  Proposition IV.5.1.

Let $\pi:X\to Y$ be a morphism of derived stacks. The pushforward of quasi-
coherent sheaves

$\pi_{*}:QCoh(X)\to QCoh(Y)$

is canonically lax monoidal.

###### Proof.

Its left adjoint $\pi^{*}$, the pullback of quasi-coherent sheaves, is
canonically symmetric monoidal, see [Lur17, Corollary 7.3.2.7]. ∎

######  Definition IV.5.2.

We recall the definition of the $\mathbb{E}_{\infty}$-algebra of functions on
a morphism of derived stack $\pi:X\to Y$. Since the structural sheaf
$\mathcal{O}_{F}$ of $F$ is an $\mathbb{E}_{\infty}$-algebra over $F$, meaning
an object of $CAlg(QCoh(F))$, $\pi_{*}(\mathcal{O}_{F})$ has a canonical
$\mathbb{E}_{\infty}$-algebra structure, see Proposition IV.5.1. We denote
this quasi-coherent algebra $C_{\mathbb{E}_{\infty}}(F)$ or simply $C(F)$ when
there is no ambiguity.

The pushforward functor also induces a morphism on $\mathbb{E}_{1}$-algebras :

$\mathbb{E}_{1}Alg(QCoh(X))\to\mathbb{E}_{1}Alg(QCoh(Y))$

It sends $\mathcal{O}_{X}$ to an $\mathbb{E}_{1}$-algebra on $Y$ denoted
$C_{\mathbb{E}_{1}}(X)$.

###  IV.6 Affine stacks

Following [To06], we recall the definition and basic properties of affine
stacks.

######  Definition IV.6.1.

Let $A$ a cosimplicial algebra, the affine stack associated to $A$ is the
stack

$Spec^{\Delta}(A):Aff\to SSet$

sending $B$ to the simplicial set $Hom_{\textbf{CRing}_{k}}(A,B_{\bullet})$

We can see $Spec^{\Delta}(A)$ as a derived stack by Kan extension. We define
the category of affine stacks as the essential image of $Spec^{\Delta}$, we
denote it $AffSt$.

######  Definition IV.6.2.

The cosimplicial algebra of functions on a stack $F$ is defined as

$C^{\bullet}_{\Delta}(F)=Hom(F_{\bullet},\mathbb{G}_{a})$

where we use the set of morphisms from $F_{n}$ to $\mathbb{G}_{a}$ seen as
presheaves.

######  Proposition IV.6.3.

The previous two functors give an adjunction

$C^{\bullet}_{\Delta}:St\rightleftarrows coSCR:Spec^{\Delta}$

and $Spec^{\Delta}$ is fully faithful.

###### Proof.

See [To06, Corollary 2.2.3]. ∎

######  Proposition IV.6.4.

The cosimplicial functor $C_{\Delta}(-)$ is an exhancement of the
$\mathbb{E}_{\infty}$-algebra functor $C_{\mathbb{E}_{\infty}}(-)$, meaning
that $C_{\mathbb{E}_{\infty}}(-)$ factors as the composition of taking the
normalization $\theta^{ccn}$ of the cosimplicial functions $C_{\Delta}(-)$.

######  Proposition IV.6.5.

The category of affine stacks is stable by small limits in stacks.

###### Proof.

See [To06, Proposition 2.2.7] ∎

######  Definition IV.6.6.

We say a derived stack is an affine stack if it lives in the essential image
of $Spec^{\Delta}$. We denote this image $AffSt_{k}$.

We define the affinization of a stack $F$ as $Spec^{\Delta}(C_{\Delta}(F))$.

######  Definition IV.6.7.

A morphism of derived stacks $X\to Y$ is said to be a relative affine stack
when the pullback functor

$dSt_{/Y}\to dSt_{/X}$

sends affine stacks to affine stacks.

######  Proposition IV.6.8.

Let $X$ be a space, ie a simplicial set, its stack affinization is given by

$AffSt(X)=Spec^{\Delta}k^{X}$

where $k^{X}\simeq C^{*}(X)$ is the cosimplicial algebra of cohomology of $X$.

######  Proposition IV.6.9.

Let $i$ be a positive integer, the stack $K(\mathbb{G}_{a},n)$ is an affine
stack and we have

$K(\mathbb{G}_{a},n)\simeq Spec^{\Delta}(D(k[n]))$

###### Proof.

See [To06, Lemma 2.2.5] ∎

###  IV.7 Cotangent complex and deformation theory

######  Definition IV.7.1.

Let $X$ be a derived stack, $X$ admits cotangent complex at
$x:SpecA\rightarrow X$ if there exists $f^{*}\mathbb{L}_{X}\in A-Mod$
representing the functor

$Hom_{SpecA/dSt}(Spec(A\oplus-),X)$

explicitly,

$Hom_{SpecA/dSt}(Spec(A\oplus M),X)=fib(X(A\oplus M)\rightarrow X(A))$

is given by the fiber over $x$.

We say $X$ has global cotangent complex if it has a cotangent complex at any
point and for every two points $x\in X(A)$ and $y\in X(B)$ and $u:A\rightarrow
B$ compatible with $x$ and $y$, we have an equivalence

$u^{*}x^{*}\mathbb{L}_{X}\simeq y^{*}\mathbb{L}_{X}$

######  Proposition IV.7.2.

When $X$ is a derived Artin stack locally of finite presentation, its
cotangent complex $\mathbb{L}_{X}$ is perfect, see [TV06, Corollary 2.2.5.3].
We denote $\mathbb{T}_{X}$ its dual in $QCoh(X)$. Then its shifted tangent
stacks $TX[-n]=\mathbb{V}(\mathbb{L}_{X}[n])$ are also derived Artin stack
locally of finite presentation.

######  Definition IV.7.3.

Let $X$ be a derived stack, $T=Spec(A)$ a derived affine scheme and $x:T\to X$
a $T$-point. Let $M$ be an $A$-module, we write the square zero extension
$A\oplus M$ and the associated derived affine scheme $T[M]$. The set of
derivations from $T$ to $M$ at $x$ is

$Def_{X}(T,M)\coloneqq Map_{T/dSt}(T[M],X)$

######  Remark IV.7.4.

When it exists, the cotangent complex $\mathbb{L}_{X,x}$ of $X$ at $x$
represents the functor $Der_{X}(T,-)$

######  Definition IV.7.5.

We define the cotangent stack of $X$ as the linear stack
$T^{*}X[n]=\mathbb{V}(\mathbb{L}_{X}[-n])$, for an integer $n$.

######  Definition IV.7.6.

Let $A$ be a simplicial algebra over $k$, $M$ an $A$-module and

$d:A\to A\oplus M[1]$

a derivation, we define the square zero extension induced by $d$ denoted
$A\oplus_{d}M$ by

${A\oplus_{d}M}$${A}$${A}$${A}$${\lrcorner}$$\scriptstyle{p}$$\scriptstyle{d}$$\scriptstyle{s}$

where $s$ is the "zero" section. The morphism $p:A\oplus_{d}M\to A$ will be
called the natural projection.

######  Proposition IV.7.7.

Let $X$ be a derived stack which has an obstruction theory, see [TV06,
Definition 1.4.2.2]. Let $A$ be a simplicial algebra, $M$ a simplicial
$A$-module, $d\in Der(A,M[1])$ a derivation with $A\oplus_{d}M$ the
corresponding square zero extension. We denote

$T\coloneqq Spec(A)\to T[M]\coloneqq Spec(A\oplus_{d}M)$

the morphism corresponding to the projection $A\oplus_{d}M\to A$. Let $x:T\to
X$ be an $A$-point.

  * •

There exists a natural obstruction

$\alpha(x)\in Map_{A-Mod}(\mathbb{L}_{X,x},M[1])$

vanishing if and only if $x$ extends to a morphism $x^{\prime}:T[M]\to X$.

  * •

If we assume $\alpha(x)=0$, then the space of lifts of $x$

$Map_{X/dSt}(T[M],X)$

is non canonically isomorphic to

$Map_{A-Mod}(\mathbb{L}_{X,x},M)$

###### Proof.

See [TV06, Proposition 1.4.2.5]. ∎

We will need the relative version of the previous proposition, so as to apply
obstruction theory to derived stacks endowed with a grading.

######  Proposition IV.7.8.

Let $f:X\to Y$ be a morphism of derived stack which has obstruction theory.
Let $A\in SCR$, $M\in A-Mod^{cn}$, $d\in Der(A,M[1])$ a derivation with
$A\oplus_{d}M$ the corresponding square zero extension. We denote

$T\coloneqq Spec(A)\to T[M]\coloneqq Spec(A\oplus_{d}M)$

the morphism corresponding to the projection $A\oplus_{d}M\to A$. Let $x:T\to
X$ be a point in $Map(T,X)\times_{Map(T[M],Y)}Map(T,Y)$. The fiber at $x$ of
the morphism

$Map(T[M],X)\to Map(T,X)\times_{Map(T[M],Y)}Map(T,Y)$

is denoted $L(x)$.

There is a natural point $\alpha(x)\in Map_{A-Mod}(\mathbb{L}_{X/Y,x},M[1])$
and a natural equivalence

$L(x)\simeq\Omega_{\alpha(x),0}Map_{A-Mod}(\mathbb{L}_{X/Y,x},M[1])$

##### The tangent bundle formalism

######  Definition IV.7.9.

Following [Lur07], let $\mathcal{C}$ be a presentable $\infty$-category, the
tangent bundle of $\mathcal{C}$ is a functor
$T_{\mathcal{C}}\to\mathcal{C}^{\Delta^{1}}$ which exhibits $T_{\mathcal{C}}$
as the stable envelope of $\mathcal{C}^{\Delta^{1}}\to\mathcal{C}$, the
evaluation at $\\{1\\}\subset\Delta^{1}$.

We can think of an element of $T_{\mathcal{C}}$ as a pair $(A,M)$ where
$A\in\mathcal{C}$ and $M$ an infinite loop space in $\mathcal{C}$.

######  Proposition IV.7.10.

Let $A$ be a simplicial algebra, we denote $Stab(SCR_{/A})$ the stable
category constructed as the stable enveloppe of $SCR_{A/}$. The functor
sending a non necessarily connected module $M$ over $A$ to the square zero
extension simplicial algebra $A\oplus M$, with its natural augmentation
defines an equivalence :

$A-Mod\xrightarrow{\sim}Stab(SCR_{/A})$

######  Proposition IV.7.11.

The cotangent complex formalism from [Lur07] defines a functor

$SCR^{\Delta^{1}}\to T_{SCR}$

which is identifies under the equivalence of the previous proposition with the
cotangent complex formation $A\to B\mapsto\mathbb{L}_{B/A}$.

We deduce by composition the absolute complex functor

$SCR\to T_{SCR}$

which is left adjoint to the forgetful functor $T_{SCR}\to SCR$ sending
$(A,M)$ to $A$.

###### Proof.

See [Lur07, Remark 1.2.3]. ∎

######  Remark IV.7.12.

This description of the cotangent complex makes it easier to understand and
prove the usual formulas of base change for the cotangent complex.

###  IV.8 Group actions

######  Definition IV.8.1.

Let $G$ be a group in derived stacks, the trivial action functor :

$dSt\to G-dSt$

admits a left adjoint denoted $(-)_{G}$ and a right adjoint denoted $(-)^{G}$,
called (homotopy) coinvariants and (homotopy) fixed points.

######  Definition IV.8.2.

In the case $G=\mathbb{G}_{m}$, $\mathbb{G}_{m}-dSt$ is the category of graded
derived stacks, then taking fixed points correspond to taking the $0$-weighted
associated derived stack.

######  Proposition IV.8.3.

Taking fixed points $(-)^{G}$ of a derived stack endowed with a $G$ action and
taking weight $0$ of a graded derived stack preserve small limits.

######  Proposition IV.8.4.

For $G$ a group in derived stacks, we have a natural equivalence :

$G-dSt_{k}\simeq dSt_{k/BG}$

###### Proof.

See [TV06, Proposition 1.3.5.3]. ∎

###  IV.9 Graded and filtered objects

In this section, we review basic definitions and properties of graded and
filtered objects in $\infty$-categories, following [Lur15, §3.1, §3.2]

######  Definition IV.9.1.

Let $\mathcal{C}$ be an $\infty$-category, we define the category of graded
objects in $\mathcal{C}$ :

$\mathcal{C}^{gr}\coloneqq
Fun(\mathbb{\textbf{Z}}^{disc},\mathcal{C})\simeq\prod_{\mathbb{Z}}\mathcal{C}$

where $\mathbb{\textbf{Z}}^{disc}$ is the discrete $1$-category on
$\mathbb{Z}$. We also define the category of filtered objects in $\mathcal{C}$
:

$Fil(\mathcal{C})\coloneqq Fun(\bf{(\mathbb{\textbf{Z}},\leq)},\mathcal{C})$

where $(\mathbb{\textbf{Z}},\leq)$ is the $1$-category associated to the usual
partial ordered structure on $\mathbb{Z}$. The inclusion
$\mathbb{\textbf{Z}}^{disc}\subset(\mathbb{\textbf{Z}},\leq)$ induces a
forgetful functor :

$Fil(\mathcal{C})\to\mathcal{C}^{gr}$

######  Remark IV.9.2.

In our definition, the structure morphisms $X_{n}\to X_{n+1}$ of a filtered
object need not be monomorphisms. However the two notions are shown to be
equivalent in [SS13].

######  Definition IV.9.3.

When $\mathcal{C}$ is a symmetric monoidal $\infty$-category,
$\mathcal{C}^{gr}$ and $Fil(\mathcal{C})$ are both symmetric monoidal
$\infty$-category when endowed with the Day product.

######  Definition IV.9.4.

The associated graded and underlying object functors are given by symmetric
monoidal functors :

$gr:Fil(\mathcal{C})\to\mathcal{C}^{gr}$

and

$colim:Fil(\mathcal{C})\to\mathcal{C}$

See [Lur15] for details on these construction.

We now recall the geometrical interpretation of graded modules and filtered
modules. See [Sim96] and [Mou19] for details.

######  Proposition IV.9.5.

There is an equivalences of stable symmetric monoidal $\infty$-categories :

$QCoh(\mathbb{A}^{1}/\mathbb{G}_{m})\simeq Fil(Mod_{k})$

and

$QCoh(B\mathbb{G}_{m})\simeq Mod_{k}^{gr}$

when $\mathbb{A}^{1}$ and $\mathbb{G}_{m}$ are the usual additive and
multiplicative groups in derived stacks over $k$.

Furthermore, pullback along the closed point
$0:B\mathbb{G}_{m}\to\mathbb{A}^{1}/\mathbb{G}_{m}$ and the open point
$1:*\simeq\mathbb{G}_{m}/\mathbb{G}_{m}\to\mathbb{A}^{1}/\mathbb{G}_{m}$
recover respectively the associated graded and the underlying object
constructions.

###### Proof.

See [Mou19, Theorem 1.1, Theorem 4.1] and [MRT20, Theorem 2.2.10]. ∎

###  IV.10 Graded stacks

######  Definition IV.10.1 (Graded derived stack).

A graded derived stack is defined as a derived stack endowed with an action of
$\mathbb{G}_{m}$, a morphism of graded derived stack is one compatible with
the actions. The category of derived affine schemes endowed with a
$\mathbb{G}_{m}$-action is naturally equivalent to the category of graded
simplicial algebras over $k$, see [Mou19]. The category of graded derived
stacks is denoted $\mathbb{G}_{m}$-dSt.

######  Remark IV.10.2.

A $\mathbb{G}_{m}$-action on a derived stack $X$ is equivalent to a morphism
of derived stacks $Y\to B\mathbb{G}_{m}$ and an identification of
$Y\times_{B\mathbb{G}_{m}}*$ with $X$, using Proposition IV.8.4. We call $X$
the total space associated to $Y\to B\mathbb{G}_{m}$.

######  Theorem IV.10.3.

From [Mou19], there is a symmetric monoidal equivalence of categories

$Mod_{B}^{gr}\simeq QCoh(B\mathbb{G}_{m}\times Spec(B))$

We will call graded quasi-coherent complex over $X$ an object of either
category.

######  Definition IV.10.4.

For $E$ a quasi-coherent complex of $X$, and $n$ an integer, $E((n))$ is
defined to be the graded quasi-coherent complex with $E$ in pure weight $n$.

######  Definition IV.10.5 (Relative global functions on graded derived
stacks).

Given a morphism of graded stacks $Y\to X$, we define graded functions on $Y$
relative to $X$ as

$\mathcal{O}_{gr,X}(Y)\coloneqq\pi_{*}(\mathcal{O}_{[Y/\mathbb{G}_{m}]})\in
CAlg(QCoh([X/\mathbb{G}_{m}]))$

with $\pi$ the canonical morphism $[Y/\mathbb{G}_{m}]\to[X/\mathbb{G}_{m}]$.

######  Definition IV.10.6.

We introduce graded $\mathcal{O}$-module function similarly :

$\widetilde{\mathcal{O}}_{gr,X}(Y)\coloneqq\widetilde{\pi}_{*}(\mathcal{O}_{[Y/\mathbb{G}_{m}]})\in
CAlg(\mathcal{O}_{[X/\mathbb{G}_{m}]}-Mod)$

using the pushforward
$\widetilde{\pi}_{*}:\mathcal{O}_{[Y/\mathbb{G}_{m}]}-Mod\to\mathcal{O}_{[X/\mathbb{G}_{m}]}-Mod$.
It is lax monoidal, as left adjoint of a symmetric monoidal functor, therefore
it induces a canonical morphism on commutative algebras in the respective
symmetric monoidal categories. We deduce that quasi-coherent graded functions
are simply $\mathcal{O}$-module graded functions after applying the left
adjoint of the forgetful functor from quasi-coherent complexes to
$\mathcal{O}$-modules.

######  Definition IV.10.7 (Graded stacks over a base).

Let $X$ be a graded derived stack, the $\infty$-category of graded derived
stacks over $X$ is the undercategory $\mathbb{G}_{m}-dSt_{/X}$, with the
trivial grading on $X$.

######  Example IV.10.8.

Linear stacks are naturally $\mathbb{G}_{m}$-graded, we can construct the
action

$\mathbb{G}_{m}\times\mathbb{V}(E)\to\mathbb{V}(E)$

as follows. For a given simplicial commutative $k$-algebra $B$,
$\mathbb{G}_{m}(B)$ is $Map^{eq}_{N(B)-mod}(N(B),N(B))$, it naturally operates
on

$\mathbb{V}(E)(B)=Map_{N(B)-Mod}(E,N(B))$

through the composition map

$Map_{N(B)-Mod}(N(B),N(B))\times Map_{N(B)-Mod}(E,N(B))\to
Map_{N(B)-Mod}(E,N(B))$

######  Definition IV.10.9.

Let $X$ be a graded derived stack, the associated derived stack of $0$-weights
is defined as the derived stack of fixed points by the action of
$\mathbb{G}_{m}$, we denote it

$X^{=0}\coloneqq X^{\mathbb{G}_{m}}$

######  Proposition IV.10.10.

The functor

$SCR\to SCR^{gr}$

endowing a simplicial algebra with its trivial grading admits a left adjoint,
which we denote $(-)^{=0}$.

###### Proof.

Using the adjoint functor theorem, see [Lur09, Corollary 5.5.2.9]. ∎

######  Remark IV.10.11.

For a general graded simplicial algebra $A$, $(A)^{=0}$ and $A(0)$ will not
coincide. We will call the former "$0$-weights" and the latter "naive
$0$-weights".

######  Remark IV.10.12.

When $A$ is a discrete commutative algebra, we notice that $(A)^{=0}$ is given
by the quotient ring of $A(0)$ where we quotient by elements of the form $ab$
where $a$ is in weight $n>0$ and $b$ in weight $-n$.

######  Remark IV.10.13.

When $A$ is a graded simplicial algebra, either positively graded or
negatively graded, then $(A)^{=0}$ coincides with $A(0)$.

######  Remark IV.10.14.

The two constructions of $0$-weights are compatible. Meaning that for $A$ a
graded simplicial algebra, we have a natural identification

$(Spec(A))^{=0}\simeq Spec(A^{=0})$

We verify it on a test simplicial algebra $R$

$(Spec(A))^{=0}(R)=Map_{\mathbb{G}_{m}}(Spec(R),Spec(A))\simeq
Map_{gr}(A,R)\simeq Map(A^{=0},R)$

###  IV.11 Line bundles and graded modules

In this section we discuss the relationship between $\mathbb{G}_{m}$-torsors
and invertible modules.

######  Definition IV.11.1.

If $B$ is a simplicial $k$-algebra, a morphism $Spec(B)\to B\mathbb{G}_{m}$ is
the data of an invertible $B$-module, we will denote both the morphism and the
module by $\mathcal{L}$. There is an ambiguity in the choice, the invertible
module associated to the morphism could be defined to be $\mathcal{L}^{\vee}$,
we choose $\mathcal{L}$ in such a way that the total space of the
$\mathbb{G}_{m}$-torsor associated to $Spec(B)\to B\mathbb{G}_{m}$ is
$Spec((Sym_{B}\mathcal{L}^{\vee})[(\mathcal{L}^{\vee})^{-1}])$, it is the
linear stack associated to $\mathcal{L}^{\vee}$ where we removed the zero
section, as a $B$-module. The function algebra
$(Sym_{B}\mathcal{L}^{\vee})[(\mathcal{L}^{\vee})^{-1}]$ is simply
$\bigoplus_{n\in\mathbb{Z}}(\mathcal{L}^{\vee})^{\otimes n}$. With this
convention $\mathcal{L}$ is functorial in the morphism $Spec(B)\to
B\mathbb{G}_{m}$.

######  Remark IV.11.2.

We can try and understand $[\mathbb{V}(E)/\mathbb{G}_{m}]$ over
$B\mathbb{G}_{m}\times X$ through its funtor of points. A morphism
$Y=Spec(B)\to[\mathbb{V}(E)/\mathbb{G}_{m}]$ over $B\mathbb{G}_{m}\times X$ is
the data of a morphism $Y\to B\mathbb{G}_{m}\times X$, ie a
$\mathbb{G}_{m}$-torsor $\widetilde{Y}\to Y$ with a map $Y\to X$, and a
$\mathbb{G}_{m}$-equivariant morphism $\widetilde{Y}\to\mathbb{V}(E)$ over
$X$. The torsor has total space
$\widetilde{Y}=Spec((Sym_{B}\mathcal{L}^{\vee})[(\mathcal{L}^{\vee})^{-1}])$,
therefore the $\mathbb{G}_{m}$-equivariant map $\widetilde{Y}\to\mathbb{V}(E)$
corresponds to a morphism of graded $N(B)$-modules
$u^{*}E((1))\to(Sym_{B}\mathcal{L}^{\vee})[(\mathcal{L}^{\vee})^{-1}]$ ie a
morphism of $N(B)$-module $u^{*}E\to\mathcal{L}^{\vee}$, with $u^{*}$ the
pullback morphism induced by the morphism $u:A\to B$ associated with $Y\to X$.
This is equivalent to having a morphism of $N(B)$-modules $\mathcal{L}\otimes
u^{*}E\to N(B)$, which is a $B$-point of $\mathbb{V}(\mathcal{L}\otimes
u^{*}E)$.

######  Remark IV.11.3.

We deduce an equivalence of $B$-modules

$[\mathbb{V}(E)/\mathbb{G}_{m}]\times_{B\mathbb{G}_{m}\times
X}Spec(B)\simeq\mathbb{V}(\mathcal{L}\otimes u^{*}E).$

The equivalence between graded complexes and quasi-coherent modules on
$B\mathbb{G}_{m}$ is given informally by sending a graded complex
$\bigoplus_{i}E_{i}$ to the $\mathcal{O}$-module stack

$Spec(B)\xrightarrow{\mathcal{L}}B\mathbb{G}_{m}\mapsto\bigoplus_{i}E_{i}\otimes\mathcal{L}^{\otimes
i}$

Therefore, for any integer $k$, $E((k))$ is the $\mathcal{O}$-module

$Spec(B)\xrightarrow{(\mathcal{L},u)}B\mathbb{G}_{m}\times
X\mapsto\mathcal{L}^{\otimes k}\otimes u^{*}E$

###  IV.12 The graded circle

######  Definition IV.12.1.

Let $A\coloneqq\mathbb{Z}[\frac{x^{n}}{n!}]\subset\mathbb{Q}[x]$, we define
$Ker$ the functor represented by $A$ pulled back to $k$.

######  Proposition IV.12.2.

The presheaf $Ker$ coincides with the intersection of the kernels of the
Frobenius morphisms on big Witt vectors

$F_{p}:W_{big}\to W_{big}$

therefore it has a natural abelian group structure. See [Haz08] and [Rab14]
for the construction of big Witt vectors.

######  Definition IV.12.3.

We define the graded circle $S^{1}_{gr}$ as

$S^{1}_{gr}\coloneqq BKer$

######  Proposition IV.12.4.

Recalling from [MRT20, Proposition 3.2.7], the stacks $S^{1}_{gr}$ and
$BS^{1}_{gr}$ are affine stacks.

######  Lemma IV.12.5.

Let $M$ be $\bigoplus_{m\geq 0}k[-2m]((m))\in Mod^{gr}$, the space of
$\mathbb{E}_{\infty}$-algebra structures on $M$ compatible with the grading is
equivalent to the set of classical commutative graded algebra structures on
its cohomology $H^{*}(M)$.

###### Proof.

We follow the proof of [MRT20, Lemma 3.4.11], the space of
$\mathbb{E}_{\infty}$-algebra structures considered is given by the mapping
space of $\infty$-operads :

$Map_{Op}(\mathbb{E}_{\infty}^{\otimes},End_{gr}(M)^{\otimes})$

where $\mathbb{E}_{\infty}^{\otimes}$ is the $\mathbb{E}_{\infty}$ operad and
$End_{gr}(M)^{\otimes}$ is the endomorphism operad of $M$ in the category of
graded modules.

The space of $n$-ary operations of $End_{gr}(M)^{\otimes}$ is given by
$Map_{Mod^{gr}}(M^{\otimes n},M)$.

We compute

$M^{\otimes n}=\left(\bigoplus_{m\geq 0}k[-2m]((m))\right)^{\otimes
n}=\bigoplus_{m\geq 0}k[-2m]((m))^{\oplus c_{m,n}}$

with $c_{m,n}=\binom{n+m-1}{n-1}$ is the cardinal of
$\\{(m_{1},m_{2},...,m_{n})\in\mathbb{N}^{n},\sum_{i=1}^{n}m_{i}=m\\}$, since
the differential is zero, $Map_{Mod^{gr}}(M^{\otimes n},M)$ is discrete.

We deduce that the space
$Map_{Op}(\mathbb{E}_{\infty}^{\otimes},End_{gr}(M)^{\otimes})$ is discrete.

Since $H^{*}$ is lax monoidal, we get a map :

$Map_{Op}(\mathbb{E}_{\infty}^{\otimes},End_{gr}(M)^{\otimes})\to
Map_{Op}(\mathbb{E}_{\infty}^{\otimes},End_{gr,cl}(H^{*}(M))^{\otimes})$

where $End_{gr,cl}(H^{*}(M))^{\otimes})$ is the classical operad of graded
endomorphisms on $M$. This map is an equivalence of spaces. ∎

######  Corollary IV.12.6.

The graded $\mathbb{E}_{1}$-algebra $k[u]$, with $u$ in degree $2$ and weight
$1$, admits a unique $\mathbb{E}_{\infty}$-algebra structure.

######  Proposition IV.12.7.

The cosimplicial functions of $BS^{1}_{gr}$ are given by

$C_{\Delta}(BS^{1}_{gr})\simeq k[u]$

and the equivalence is compatible with the augmentation, given by
$BS^{1}_{gr}\to*$. Here $k[u]$ denoted the denormalization of the free
commutative differential graded algebra on one generator in degree $2$ and
weight $1$. The identification is compatible with gradings.

###### Proof.

We start by the construction of a morphism $k[u]\to C_{\Delta}(BS^{1}_{gr})$.

Since

$H^{2}(BS^{1}_{gr})\simeq H^{2}(Tot(k\to k[\eta]\rightrightarrows
k[\eta]^{\otimes 2}...))$

the element $\eta$ defines an element $u\in H^{2}(BS^{1}_{gr})$ in weight $1$.
We deduce a morphism of $\mathbb{E}_{1}$-algebra $k[u]\to
C_{\mathbb{E}_{1}}(BS^{1}_{gr})$ as $k[u]$ is the free
$\mathbb{E}_{1}$-algebra on $k[-2]$. Now since the $\mathbb{E}_{1}$-algebra
$k[u]$ admits a unique $\mathbb{E}_{\infty}$-algebra structure, from Corollary
IV.12.6, we get a morphism of $\mathbb{E}_{\infty}$-algebra

$k[u]\to C_{\mathbb{E}_{\infty}}(BS^{1}_{gr})$

which is an equivalence after taking cohomology, therefore it is an
equivalence. We write it as

$\phi:\theta^{ccn}(k[u])\xrightarrow{\sim}\theta^{ccn}(C_{\Delta}(BS^{1}_{gr}))$

Now we prove it is an equivalence, following the proof of [MRT20, Theorem
3.4.17], we apply the double conerve functor to $\phi$ :

$coN^{2}(\theta^{ccn}(k[u]))\simeq
coN^{2}(\theta^{ccn}(C_{\Delta}(BS^{1}_{gr})))$

which is a morphism of bisimplicial $\mathbb{E}_{\infty}$-algebra. By
definition $S^{1}_{gr}\coloneqq BKer$, therefore we deduce an equivalence

$coN^{2}(\theta^{ccn}(C_{\Delta}(BS^{1}_{gr})))\simeq\theta^{ccn}(C_{\Delta}(Ker))^{\bullet\bullet}$

The stack $Ker$ being a classical affine scheme,
$\theta^{ccn}(C_{\Delta}(Ker))^{\bullet\bullet}$ is a bisimplicial object in
discrete $\mathbb{E}_{\infty}$ algebras. As
$\theta^{ccn}:coSCR\to\mathbb{E}_{\infty}-Alg$ restricts to an equivalence on
discrete objects, the equivalence

$\theta^{ccn}(coN^{2}(k[u]))\simeq\theta^{ccn}(C_{\Delta}(Ker))^{\bullet\bullet}$

lifts uniquely to an equivalence of bisimplicial cosimplicial algebras

$coN^{2}(k[u])\simeq C_{\Delta}(Ker)^{\bullet\bullet}$

Now totalization of the bisimplicial objects gives the required equivalence of
cosimplicial algebras :

$k[u]\simeq C_{\Delta}(BS^{1}_{gr})$

∎

######  Corollary IV.12.8.

The affine stack $BS^{1}_{gr}$ is given by

$BS^{1}_{gr}\simeq Spec^{\Delta}(k[u])$

where the identification is compatible with augmentations and gradings.

######  Proposition IV.12.9.

There is an equivalence of symmetric monoidal categories compatible with
grading, ie which commutes with the forgetful functors to graded complexes

$\epsilon-Mod_{k}\simeq Rep(S^{1}_{gr})=QCoh(BS^{1}_{gr})$

###### Proof.

See [MRT20, Proposition 4.2.3]. ∎

######  Definition IV.12.10.

We define the category of graded mixed derived stacks as the category of
$S^{1}_{gr}$-equivariant graded stacks

$\epsilon-dSt^{gr}\coloneqq S^{1}_{gr}-dSt^{gr}$

######  Remark IV.12.11.

Seeing $S^{1}_{gr}$ as a derived stack with a $\mathbb{G}_{m}$-action, it
admits a canonical morphism

$S^{1}_{gr}\to B\mathbb{G}_{m}$

which has as a total space the semi-direct product

$\mathcal{H}\coloneqq\mathbb{G}_{m}\ltimes S^{1}_{gr}$

Therefore we can see a graded mixed structure on a derived stack as a
$\mathbb{G}_{m}$ action and an action of $S^{1}_{gr}$ which are compatible.
Meaning that we have an identification

$\epsilon-dSt^{gr}\simeq\mathcal{H}-dSt$

Similarly $QCoh(BS^{1}_{gr})$ is defined with $S^{1}_{gr}$ considered as a
element of $dSt^{gr}$, that is $QCoh(BS^{1}_{gr})$ is implicitly defined as
$QCoh(B\mathcal{H})$.

##  V Graded function on linear stacks

This section is devoted to the computation of $1$-weighted graded functions on
a linear stack. This section is available online at [Mon21].

###  V.1 Adjoint of the linear stack functor

######  Proposition V.1.1.

The functor

$\mathbb{V}:QCoh(X)^{op}\to\mathbb{G}_{m}-dSt_{/X}$

has a left adjoint given by $\mathcal{O}_{gr}(-)(1)$.

###### Proof.

Let us consider the two functors

$Map_{\mathbb{G}_{m}-dSt_{/X}}(-,\mathbb{V}(E)),Map_{QCoh(X)}(E,\mathcal{O}_{gr}(-)(1)):\mathbb{G}_{m}-dSt_{/X}^{op}\to\mathcal{S}$

landing in the category of spaces. Both functors send all small colimits to
limits and are canonically identified on the subcategory of
$\mathbb{G}_{m}$-equivariant derived stack of the form
$Spec(B)\times\mathbb{G}_{m}$ with $\mathbb{G}_{m}$ acting by multiplication
on the right side : any $\mathbb{G}_{m}$-stack $Y$ can be recovered as a
colimit of objects of the form $Y\times\mathbb{G}_{m}$ with the action being
multiplication on the right side since $Y$ is canonically the colimit of a
diagram on $B\mathbb{G}_{m}$ sending the point to $Y\times\mathbb{G}_{m}$ and
sending elements of $\mathbb{G}_{m}$ to their diagonal action on
$Y\times\mathbb{G}_{m}$. Therefore the two functors are equivalent. The
equivalence being functorial in $E$, we deduce the adjunction.

∎

######  Remark V.1.2.

In the adjunction above, if we restrict to $\mathbb{G}_{m}$-equivariant
derived stack of the form $Spec(B)\times\mathbb{G}_{m}$ with $\mathbb{G}_{m}$
acting by multiplication on the right side, we deduce another adjunction :

$Map_{dSt}(Y,\mathbb{V}(E))\simeq Map_{QCoh(X)}(E,\mathcal{O}(Y)).$

###  V.2 The equivalence

######  Construction V.2.1.

We now construct a natural arrow $E((1))\to\mathcal{O}_{gr}(\mathbb{V}(E))$ in
$QCoh([X/\mathbb{G}_{m}])$, ie a map of graded complexes on $X$ a derived
stack. By adjunction, it just means we have to construct a map of graded
$\mathcal{O}$-modules $E((1))\to\mathcal{O}_{gr,\mathcal{O}}(\mathbb{V}(E))$.
For the construction, we go back and forth between quasi-coherent complexes
and $\mathcal{O}$-modules, exploiting the facts that quasi-coherent complexes
have better categorical properties and $\mathcal{O}$-modules are more
computable.

We assume $E$ to be cofibrant, we take $B$ a simplicial commutative
$k$-algebra and $u$ a morphism from $Spec(B)$ to $B\mathbb{G}_{m}\times X$. We
want to construct a map

$E((1))(B)\simeq\mathcal{L}\otimes
u^{*}E\to\mathcal{O}_{gr,\mathcal{O}}(\mathbb{V}(E))(B).$

However, using Remark IV.11.3, we have :

$\mathcal{O}_{gr,\mathcal{O}}(\mathbb{V}(E)))(B)\simeq\mathcal{O}_{\mathcal{O}}([\mathbb{V}(E)/\mathbb{G}_{m}]\times_{B\mathbb{G}_{m}\times
X}Spec(B))\simeq\mathcal{O}_{\mathcal{O}}(\mathbb{V}(\mathcal{L}\otimes
u^{*}E)).$

Therefore we are reduced to constructing a functorial map $\mathcal{L}\otimes
u^{*}E\to\mathcal{O}_{\mathcal{O}}(\mathbb{V}(\mathcal{L}\otimes u^{*}E))$. By
the adjunction property of quasi-coherent sheaves, it is equivalent to
constructing a functorial map

$\mathcal{L}\otimes u^{*}E\to\mathcal{O}(\mathbb{V}(\mathcal{L}\otimes
u^{*}E)).$

We can use the unit of the adjunction in Remark V.1.2 to construct this map.
From this map we deduce in weight $1$ a map :

$E\to\mathcal{O}_{gr}(\mathbb{V}(E)))(1)$

######  Proposition V.2.2.

If $E$ is connective, the natural map constructed above

$E\to\mathcal{O}_{gr}(\mathbb{V}(E)))(1)$

is an equivalence

###### Proof.

Since $E$ is connective, $\mathbb{V}(E)$ is simply the relative affine scheme
$Spec_{X}Sym_{\mathcal{O}_{X}}(E)$. The grading corresponds to having weight
$p$ of $Sym_{\mathcal{O}_{X}}(E)$ to be $Sym^{p}_{\mathcal{O}_{X}}(E)$. The
morphism

$E((1))\to\mathcal{O}_{gr}(Spec_{X}Sym_{\mathcal{O}_{X}}(E((1))))\simeq
Sym_{\mathcal{O}_{X}}(E((1)))$

corresponds to the inclusion of $E$ in weight $1$. Therefore it is an
equivalence when restricting to weight $1$.

∎

######  Theorem V.2.3.

Let $X$ be a derived affine scheme, for $E\in QCoh^{-}(X)$, ie a bounded above
complex over $X$, the natural map constructed above

$E\to\mathcal{O}_{gr}(\mathbb{V}(E))(1)$

is an equivalence.

###### Proof.

Let us proceed by induction on the Tor amplitude of $E$. Let $E$ be a complex
of Tor amplitude in $]-\infty,b]$, $b$ a positive integer. The case $b=0$ has
already been dealt with, so we assume $b>0$.

We know, as $X$ is assumed to be affine, there is a non canonical triangle

$0\to V[-b]\to E\to E^{\prime}\to 0$

with $V$ of tor amplitude $0$, ie a classical vector bundle on $X$, and
$E^{\prime}$ concentrated of Tor amplitude in $]-\infty,b-1]$. To construct
$E^{\prime}$, we can take a model of $E$ concentrated in degree smaller than
$b$ and cellular as an $N(A)$-module, then define $E^{\prime}$ as its naive
truncation in degree smaller than $b-1$. We deduce the morphism $E\to
E^{\prime}$.

Let us introduce the notation $R$ denoting
$\mathcal{O}_{gr}(\mathbb{V}(-))(1)$. By naturality of the construction above,
we have the following commutative diagram

${E}$${Tot(coN_{\bullet}(E\to E^{\prime}))}$${R(E)}$${Tot(R(coN_{\bullet}(E\to
E^{\prime})))}$

with $coN$ the conerve construction and $Tot$ the totalization of a
cosimplicial object. We will show these four maps are all equivalences.

The top arrow is an equivalence as $E$ and $E^{\prime}$ live in the stable
category of quasi-coherent sheaves on $X$, see [Lur17, Proposition 1.2.4.13].

We now show that the right arrow is an equivalence, in fact
$coN_{\bullet}(E\to E^{\prime})\to R(coN_{\bullet}(E\to E^{\prime}))$ is a
levelwise equivalence. We compule the conerve in the following lemma :

######  Lemma V.2.4.

For any natural integer $n$, we have a canonical equivalence

$coN_{n}(E\to E^{\prime})\simeq E^{\prime}\oplus V[-b+1]^{\oplus n}$

###### Proof.

The computation is standard, we proceed by induction on the degree $n$, in
zero degree it is obvious. Now $E^{\prime}\oplus_{E}E^{\prime}$ has a
canonical split projection to $E^{\prime}$, its kernel is the fiber of
$E^{\prime}\to K$, which is $V[-b]$. Therefore
$E^{\prime}\oplus_{E}E^{\prime}$ is identified with $E^{\prime}\oplus
V[-b+1]$. The general case follows. ∎

We are reduced to showing that the natural map

$E^{\prime}\oplus V[-b+1]^{\oplus n}\to R(E^{\prime}\oplus V[-b+1]^{\oplus
n})$

is an equivalence. Since $E^{\prime}$ and $V[-b+1]$ are in Tor amplitude in
$]-\infty,b-1]$, it follows from the induction hypothesis.

We now show the bottom arrow is an equivalence. We will use a lemma to compute
$\mathbb{V}(E)$ :

######  Lemma V.2.5.

$\mathbb{V}(E^{\prime})\to\mathbb{V}(E)$ is an epimorphism of derived stacks.

###### Proof.

Let us fix $B$ is a simplicial commutative $k$-algebra and $u$ a $B$-point of
$X$. We want to show that $\mathbb{V}(E^{\prime})(B)\to\mathbb{V}(E)(B)$ is
surjective after applying $\pi_{0}$. Let us fix $\alpha\in\mathbb{V}(E)(B)$ ie
$\alpha:u^{*}E\to N(B)$ a morphism of $B$-module.

Since there is an exact triangle

$0\to V[-b]\to E\to E^{\prime}\to 0$

therefore an exact triangle

$0\to u^{*}V[-b]\to u^{*}E\to u^{*}E^{\prime}\to 0$

we deduce that $\alpha$ lifts to $u^{*}E^{\prime}\to N(B)$ if and only if
composition $u^{*}V[-b]\to N(B)$ is zero. Since $u^{*}V[-b]$ is strictly
coconnective and $u^{*}N(B)$ is connective, the lift exists.

This concludes that $\mathbb{V}(E^{\prime})\to\mathbb{V}(E)$ is an epimorphism
of presheaves, hence is an epimorphism of derived stacks.

∎

######  Remark V.2.6.

The proof of the above gives a slightly stronger result, indeed we have shown
that the morphism

$\mathbb{V}(E^{\prime})\to\mathbb{V}(E)$

is a epimorphism of functors.

Using this lemma and the connections between effective epimorphisms in an
$\infty$-topos and simplicial resolutions of [Lur09, Corollary 6.2.3.5], we
have an equivalence

$|N(\mathbb{V}(E^{\prime})\to\mathbb{V}(E))|\xrightarrow{\sim}\mathbb{V}(E)$

with $|-|$ the geometric realization. $N$ is the nerve construction.

Since $\mathbb{V}$ and $\mathcal{O}_{gr}$ send colimits to limits, between
their categories of definition, we deduce

$\mathcal{O}_{gr}(\mathbb{V}(E))\xrightarrow{\sim}Tot(\mathcal{O}_{gr}(\mathbb{V}(coN(E\to
E^{\prime}))))$

We can then apply the weight $1$ functor $(-)(1)$, which commutes with limits
since limits can be computed levelwise for graded objects. Therefore

$R(E)\xrightarrow{\sim}Tot(R(coN_{\bullet}(E\to E^{\prime}))).$

We conclude that $E\to R(E)$ is an equivalence.

∎

######  Corollary V.2.7.

The functor $\mathbb{V}:QCoh(X)^{op}\to\mathbb{G}_{m}-dSt_{/X}$ is fully
faithful when restricted to $QCoh^{-}(X)$.

######  Remark V.2.8.

These claims are still correct for $X$ a general derived stack. We can check
it using a descent argument.

###  V.3 General weights

######  Proposition V.3.1.

If $k$ is a $\mathbb{Q}$-algebra and $E$ is a perfect complex on $X$,
$\mathcal{O}_{gr}(\mathbb{V}(E))(p)$ is canonically identified with
$Sym_{\mathbb{E}_{\infty}}^{p}(E)$, the monad here $Sym_{\mathbb{E}_{\infty}}$
is the one associated to the formation of free
$\mathbb{E}_{\infty}$-$k$-algebras, or equivalently commutative differential
algebras.

Before the proof of the proposition, we will establish a preliminary lemma.

######  Lemma V.3.2.

If $k$ is a $\mathbb{Q}$-algebra, $E$ is perfect complex on $X$ and
$E^{\bullet}$ is a cosimplicial object in perfect complexes on $X$. An
equivalence $E\xrightarrow{\sim}Tot(E^{\bullet})$ induces a canonical map
$Sym^{p}(E)\rightarrow Tot(Sym^{p}(E^{\bullet}))$, which is an equivalence.
Therefore $Tot(Sym^{p}(E^{\bullet}))$ is perfect.

###### Proof.

Starting from the equivalence $E\xrightarrow{\sim}Tot(E^{\bullet})$, we deduce
by taking duals an equivalence of perfect complexes :

$|(E^{\bullet})^{\vee}|\xrightarrow{\sim}E^{\vee}$

since dualization induces an equivalence of $\infty$-category on perfect
complexes. Knowing $Sym^{p}$ commutes with sifted colimits, we apply it on
both sides

$|Sym^{p}((E^{\bullet})^{\vee})|\xrightarrow{\sim}Sym^{p}(E^{\vee}).$

We can then dualize again and use the fact that $Sym^{p}$ and dualization
commute since we are in zero characteristic.

$Sym^{p}(E)\xrightarrow{\sim}Tot(Sym^{p}(E^{\bullet})).$

And it concludes the proof. ∎

###### Proof.

The proof of the proposition follows the same structure as the main theorem.
The canonical map

$E((1))\to\mathcal{O}_{gr}(\mathbb{V}(E))$

gives by adjunction a morphism of $\mathbb{E}_{\infty}$-algebras over $X$ :

$Sym(E((1)))\to\mathcal{O}_{gr}(\mathbb{V}(E))$

which is in degree $p$ :

$Sym^{p}(E)\to\mathcal{O}_{gr}(\mathbb{V}(E))(p).$

We obtain a commutative diagram

${Sym^{p}E}$${Tot(Sym^{p}(coN_{\bullet}(E\to
E^{\prime})))}$${R_{p}(E)}$${Tot(R_{p}(coN_{\bullet}(E\to E^{\prime})))}$

with $R_{p}=\mathcal{O}_{gr}(\mathbb{V}(-))(p)$

The right arrow is similarly shown to be an equivalence by induction and the
bottom one is an equivalence by the same reasoning above.

We now need to show the top arrow is an equivalence. Using the previous lemma
with

$E^{\bullet}\coloneqq coN_{\bullet}(E\to E^{\prime})$

concludes the proof.

∎

######  Remark V.3.3.

In general characteristic, and general weight,
$\mathcal{O}_{gr}(\mathbb{V}(E))(p)$ seems to correspond to $Sym_{M}^{p}(E)$,
the weight $p$ part of the symmetric algebra over a monad on complexes
generalizing simplicial algebras to non-necessarily connective complexes. This
monad is currently being investigated by Bhatt-Mathew, see [Rak20]. Hopefully,
this connection will be made explicit in a following paper.

######  Remark V.3.4.

We provide an example of a situation where functions on a linear stack are not
easily expressed in terms of the complex of sheaves. Working over a field of
characteristic zero, we can define a quasi-coherent sheaf on the point by
$E=k[2]\oplus k[-2]$. Then $Sym(E)\simeq Sym(k[2])\otimes Sym(k[-2])$ is a
free commutative differential graded $k$-algebra on two generators, one of
degree $2$, the other of degree $-2$, the differential is identically zero.

On the other hand
$\mathcal{O}(\mathbb{V}(E))=H^{*}(K(\mathbb{G}_{a},2),\mathcal{O}[u])$ and it
identifies with the ring $k[u][[v]]$ with $v$ a generator of
$H^{2}(K(\mathbb{G}_{a},2),\mathcal{O})$ : $u$ is in degree $-2$ and $v$ is a
degree $2$. In particular, in degree $0$, we get for $Sym(E)$ an infinite
product $\prod_{\mathbb{N}}k$ generated by the elements of the form
$u^{n}v^{n}$. Similarly, in degree $0$, $\mathcal{O}(\mathbb{V}(E))$ only
gives a direct sum $\bigoplus_{\mathbb{N}}k$, also generated by the elements
$u^{n}v^{n}$. Intuitively, the connective part of $E$ tends to give a
contribution by a free algebra (polynômial algebras) and the coconnective part
of $E$ gives a contribution by a completed free algebra (power series
algebras).

##  VI Overview of de Rham and crystalline cohomology

###  VI.1 Crystalline cohomology

We review the theory of classical crystalline cohomology, following [Ber74],
[CL91] and [LZ03]. For results on comparisions between de Rham and crystalline
cohomology, see [BDJ11] and [Mon21b].

Crystalline cohomology was motivated to fill a gap arising from $l$-adic
cohomology : $l$-adic cohomology for schemes over a field of characteristic
$p$ is a well-behaved Weil cohomology theory under the assumption that $l\neq
p$. However it is badly behaved when $l=p$. Crystalline cohomology was
constructed to give a Weil cohomology theory in the case $l=p$. Serre notices
in [Ser58] that there is no Weil cohomology theory with coefficients in
$\mathbb{Q}$ for schemes over a field of characteristic $p$. This motivates
the use of Witt vectors, which give the appropriate ring of coefficients,
being both a ring of characteristic zero and having a residue field of
characteristic $p$.

######  Definition VI.1.1.

For $X$ a classical scheme over a $\mathbb{F}_{p}$-algebra $k$, one defines a
crystalline site relative to $W_{n}$, the ring of Witt vector of length $n$ on
$k$, and a structural sheaf $\mathcal{O}_{X/W_{n}(k)}$ which defines
cohomology groups $H^{i}_{crys}(X/W_{n})$.

Crystalline cohomology is then the limit on $n$

$H^{i}_{crys}(X/W)\coloneqq\varprojlim H^{i}(X/W_{n})$

######  Remark VI.1.2.

On Witt vectors of length $1$, ie $W_{1}=k$, we recover De Rham cohomology,
therefore this defines a canonical morphism

$H^{i}_{crys}(X/W)\to H^{i}_{dR}(X/k)$

######  Proposition VI.1.3.

Crystalline cohomology is a Weil theory for smooth projective schemes over a
perfect field of characteristic $p$.

###### Proof.

See [Ber74]. ∎

###  VI.2 De Rham-Witt complex

This section is devoted to reviewing the de Rham-Witt complex, as developed in
[Ill79]. The motivation for defining the de Rham-Witt complex is twofold :

  * •

Giving an explicit tool to compute crystalline cohomology closer to the
definition of de Rham cohomology.

  * •

Comparing crystalline cohomology to other $p$-adic cohomology theories, étale
or de Rham cohomology, and studying other structures, such as the Hodge
filtration.

######  Construction VI.2.1.

Let $X$ be a scheme over $\mathbb{F}_{p}$, the de Rham-Witt pro-complex
$W_{\bullet}\Omega_{X}^{\bullet}$ is a projective system
$(W_{n}\Omega_{X}^{\bullet})$ of coconnective commutative differential graded
algebras such that

$W_{n}\Omega_{X}^{0}=W_{n}\mathcal{O}_{X}$

endowed with an additive morphism $V:W_{n}\Omega_{X}^{i}\to
W_{n+1}\Omega_{X}^{i}$ such that

$V(xdy)=VxdVy\text{ and }(d[x])Vy=V([x]^{p-1}d[x]y)$

There is a unique initial complex with theses properties and
$W_{n}\Omega^{\bullet}_{X}$ is a quotient of $\Omega_{W_{n}X}$ by the required
relations, it is a $W_{n}(k)$-algèbre, $W_{1}\Omega_{X}^{\bullet}$ is simply
the de Rham complex $\Omega_{X}^{\bullet}$.

We define $W\Omega^{i}_{X}=\varprojlim W_{n}\Omega_{X}^{i}$, it is a
$p$-torsion free commutative differential graded $W(k)$-algebra.

We recover the truncated de Rham-Witt complex $W_{n}\Omega_{X}$ by a quotient
:

$W_{n}\Omega_{X}^{i}=W\Omega^{i}_{X}/(V^{n}W\Omega_{X}^{i}+dV^{n}W\Omega_{X}^{i-1})$

Since $X$ is assumed to be smooth, there is a unique commutative differential
graded algebra morphism $F:W_{n}\Omega_{X}^{\bullet}\to
W_{n}\Omega_{X}^{\bullet}$ such that

$FdV=d\text{ and }F(d[x])=[x]^{p-1}d[x]$

######  Proposition VI.2.2.

A few properties of the de Rham-Witt vectors and its structure maps

$V:W_{n}\Omega_{X}^{\bullet}\to W_{n+1}\Omega_{X}^{\bullet}$
$F:W_{n}\Omega_{X}^{\bullet}\to W_{n}\Omega_{X}^{\bullet}$

are given by :

  * •

$W_{n}\Omega_{X}^{0}$ is given by $W_{n}\mathcal{O}_{X}$

  * •

$F$ and $V$ are additive.

  * •

$F$ coincides in degree $0$ with the Frobenius morphism and the Verschiebung
map on Witt vectors.

  * •

$FV=VF=p$

  * •

$FdV=d$

  * •

$dF=pFd$

  * •

$Fd[x]=[x^{p-1}]d[x]$ for $x\in\mathcal{O}_{X}$

  * •

$F(xy)=xy$

  * •

$xVy=VF(xy)$

  * •

$V(xdy)=V(x)dV(y)$

######  Remark VI.2.3.

The endomorphism deduced functorially by the Frobenius map of $X$ is on
$W_{\bullet}\Omega^{i}$ given by $p^{i}F$.

######  Theorem VI.2.4.

There is a canonical isomorphism compatible with the Frobenius action

$H^{*}_{crys}(X/W_{n}(k))\cong H^{*}(X,W_{n}\Omega_{X}^{\bullet})$

between crystalline cohomology and the hypercohomology of the de Rham-Witt
complex sheaf canonically defined.

######  Theorem VI.2.5.

Furthermore, there is even a more structured result : there is an equivalence
of $W_{n}(k)$-chain complexes on $X$

$Ru_{*}\mathcal{O}_{X/W_{n}(k)}\xrightarrow{\sim}W_{n}\Omega_{X}^{\bullet}\in\mathcal{D}(X,W_{n}(k))$

where $u$ is the forgetful functor from sheaves on the crystalline site of $X$
to sheaves on the Zariski site of $X$. We have used the notation
$\mathcal{D}(X,W_{n}(k))$ to denote the derived category of $W_{n}(k)$-modules
on $X$.

###  VI.3 De Rham Witt from Bhatt and Lurie

In this section, we recall the main notions and results of [BLM22].

######  Definition VI.3.1.

A Dieudonné complex $(M,d,F)$ is a cochain complex of abelian groups $(M,d)$
and $F$ a morphism of graded abelian groups $F:M\to M$ such that

$dF=pFd$

The $1$-category of Dieudonné complexes is denoted DC.

######  Definition VI.3.2.

Let $(M,d,F)$ be a Dieudonné complex, we define a Dieudonné sub-complex of $M$
denoted $(\eta_{p}M)$ as follows

$(\eta_{p}M)^{n}\coloneqq\\{x\in p^{n}M:dx\in p^{n+1}M^{n+1}\\}$

######  Definition VI.3.3.

Let $(M,d)$ be a $p$-torsion-free complex, the Frobenius morphism $F$ induces
a morphism of abelian groups

$\alpha_{n}:M^{n}\to(\eta_{p}M)^{n}=\\{x\in p^{n}M:dx\in p^{n+1}M^{n+1}\\}$

The complex $M$ is said to be saturated when $\alpha_{n}$ is an isomorphism
for all $n$. This defines the sub-$1$-category of saturated Dieudonné
complexes $\textbf{DC}_{sat}\subset\textbf{DC}$.

######  Definition VI.3.4.

Let $M$ be a saturated Dieudonné complex, for every $x\in M_{n}$, there is a
unique $V(x)$ such that $FV(x)=px$. This defines an additive morphism

$V:M\to M$

######  Proposition VI.3.5.

(See [BLM22, Proposition 2.3.1]) The inclusion of $1$-categories
$\textbf{DC}_{sat}\subset\textbf{DC}$ admits a left adjoint, called
saturation. It is denoted $Sat$.

######  Remark VI.3.6.

This saturation is constructed as the colimit of the diagram

$M\to\eta_{p}M\to\eta_{p}^{2}M...$

######  Definition VI.3.7.

For $M$ a saturated Dieudonné complex, we define $\mathcal{W}_{r}(M)$ as the
quotient complex of $M$ by the subcomplex $Im(V^{r})+Im(dV^{r})$.

The natural morphisms $\mathcal{W}_{n+1}(M)\to\mathcal{W}_{n}(M)$ defines a
diagram, which admits a limit called completion of $M$. We denote it
$\mathcal{W}(M)$.

######  Remark VI.3.8.

The maps $F$ and $V$ naturally extend to the completion of $M$.

We have a canonical map given by the projection :
$\rho_{M}:M\to\mathcal{W}(M)$.

######  Definition VI.3.9.

The Dieudonné complex $M$ is said to be strict when it is saturated and
$\rho_{M}$ is an isomorphism.

This defines a sub-$1$-category $\textbf{DC}_{str}\subset\textbf{DC}_{sat}$.

######  Proposition VI.3.10.

(See [BLM22, Proposition 2.7.8]) The inclusion
$\textbf{DC}_{str}\subset\textbf{DC}_{sat}$ admits a left adjoint given by
$\mathcal{W}$.

######  Proposition VI.3.11.

(See [BLM22, Notation 2.8.4]) The inclusion
$\textbf{DC}_{str}\subset\textbf{DC}$ admits a left adjoint given by
$\mathcal{W}Sat$.

######  Definition VI.3.12.

A Dieudonné algebra $(A,d,F)$ is given by $(A,d)$ a commutative differential
graded algebra and $F:A\to A$ is a morphism of graded rings such that

  * •

For $x\in A$, $dF(x)=pF(dx)$.

  * •

$A_{n}=0$ for $n<0$.

  * •

$Fx=x^{p}$ (mod $p)$, for $x\in A^{0}$.

Morphisms between Dieudonné algebras are required to be ring morphisms
compatible with differentials and Frobenius structures. This defines the
$1$-category of Dieudonné algebras DA. Forgetting the multiplication defines
the forgetful functor $\textbf{DA}\to\textbf{DC}$.

######  Proposition VI.3.13.

(See [BLM22, Proposition 3.2.1]) Let $R$ a $p$-torsion-free commutative ring
and $\phi:R\to R$ a classical Frobenius lift. Then there is a unique ring
morphism $F:\Omega_{R}^{\bullet}\to\Omega_{R}^{\bullet}$ such that

  * •

$F(x)=\phi(x)$ for $x\in R=\Omega_{R}^{0}$.

  * •

$F(dx)=x^{p-1}dx+d(\frac{\phi(x)-x^{p}}{p})$ for $x\in R$.

Furthermore, $(\Omega_{R}^{\bullet},d,F)$ is a Dieudonné algébra.

######  Proposition VI.3.14.

(See [BLM22, Proposition 3.2.3]) Let $R$ a $p$-torsion-free commutative ring
and $\phi:R\to R$ a classical Frobenius lift. For $A$ a $p$-torsion-free
Dieudonné algebra, the construction given before defines a functorial
bijection

$Hom_{\textbf{DA}}(\Omega_{R}^{\bullet},A)\xrightarrow{\sim}Hom_{CRing^{Fr}}(R,A^{0})$

where $Hom_{CRing^{Fr}}(R,A^{0})$ is the set of morphisms $R\to A^{0}$
compatible with Frobenius structures.

######  Definition VI.3.15.

A Dieudonné algebra is said to be saturated is it is as a Dieudonné complex.
This defines a sub-$1$-category $\textbf{DA}_{sat}\subset\textbf{DA}$.

######  Proposition VI.3.16.

(See [BLM22, Proposition 3.4.3]) The inclusion
$\textbf{DA}_{sat}\subset\textbf{DA}$ admits a left adjoint given by $Sat(A)$
endowed with the canonical Dieudonné algebra structure.

######  Proposition VI.3.17.

(See [BLM22, Proposition 3.5.5]) The constructions of $V$ and $\mathcal{W}$
extend naturally to Dieudonné algebras. For $A$ a Dieudonné algebra,
$\mathcal{W}(A)$ is a saturated Dieudonné algebra.

######  Definition VI.3.18.

A saturated Dieudonné algebra $A$ is said to be strict if it is as a Dieudonné
complex. This defines a sub-$1$-category
$\textbf{DA}_{str}\subset\textbf{DA}_{sat}$.

######  Proposition VI.3.19.

(See [BLM22, Proposition 3.5.9]) The inclusion
$\textbf{DA}_{str}\subset\textbf{DA}_{sat}$ admits a left adjoint given by
$\mathcal{W}$.

######  Proposition VI.3.20.

(See [BLM22, Proposition 3.5.10]) The inclusion
$\textbf{DA}_{str}\subset\textbf{DA}$ admits a left adjoint given by
$\mathcal{W}Sat$.

######  Proposition VI.3.21.

(See [BLM22, Proposition 4.1.4]) The functor $B\in\textbf{DA}_{str}\mapsto
B^{0}/VB^{0}\in CRing_{\mathbb{F}_{p}}$ admits a left adjoint denoted by
$R\mapsto\mathcal{W}\Omega_{R}^{\bullet}$ and called the saturated de Rham-
Witt complex of $R$. The complex can be constructed as

$\mathcal{W}\Omega_{R}^{\bullet}\coloneqq\mathcal{W}Sat(\Omega^{\bullet}_{W(R)})$

######  Definition VI.3.22.

Let $R$ be a commutative $\mathbb{F}_{p}$-algebra. An $R$-framed $V$-pro-
complex is given by

  * •

A chain of commutative differential graded algebras

$...\to A_{\bullet}(3)\to A_{\bullet}(2)\to A_{\bullet}(1)\to A_{\bullet}(0)$

with the limit denoted $A_{\bullet}(\infty)$.

  * •

A collection of maps $V:A(r)\to A(r+1)$ of graded abelian groups.

  * •

A ring morphism $\beta:W(R)\to A_{0}(\infty)$, where we denote $\beta_{r}$ the
composition

$W(R)\xrightarrow{\beta}A_{0}(\infty)\to A_{0}(r)$

such that

  * •

$A(r)=0$ when r=0 and $A_{i}(r)=0$ when $i<0$.

  * •

$V$ commutes with the restriction maps. Therefore $V$ induces a morphism on
$A(\infty)$ denoted $V$.

  * •

$\beta:W(R)\to A_{0}(\infty)$ is compatible with $V$.

  * •

$V:A(r)\to A(r+1)$ satisfies $V(xdy)=V(x)dV(y)$.

  * •

For $\lambda\in R$ and $x\in A(r)$, we have

$(Vx)d\beta_{r+1}[\lambda]=V(x(\beta_{r}[\lambda])^{p-1}d\beta_{r}[\lambda])$

A morphism of $R$-framed $V$-pro-complexes is a morphism of diagrams of
commutative differential graded algebras compatible with restriction,
Verschiebung and $\beta$ maps.

This defines a $1$-category $VPC_{R}$.

######  Proposition VI.3.23.

(See [BLM22, Proposition 4.4.4]) Let $R$ be a commutative
$\mathbb{F}_{p}$-algebra, the $1$-category $VPC_{R}$ has an initial object. We
denote this object $(W_{r}\Omega_{R}^{\bullet})$ and the limit of the induced
tower $(W\Omega_{R}^{\bullet})$, we call it the classical de Rham-Witt complex
of $R$.

######  Remark VI.3.24.

This construction gives the de Rham-Witt complex as defined in [Ill79]. See
[BLM22, Warning 4.4.8] for details.

######  Theorem VI.3.25.

(See [BLM22, Proposition 4.4.12]) There is a functorial morphism of
commutative differential graded algebra

$W\Omega_{R}^{\bullet}\to\mathcal{W}\Omega_{R}^{\bullet}$