which is an isomorphism when $R$ is a Noetherian $\mathbb{F}_{p}$-algebra. The
left hand side it the classical de Rham-Witt complex and the right hand side
is the de Rham-Witt complex defined by saturation and completion.

##  VII Graded loopspaces in mixed characteristics

In this section we will take $k=\mathbb{Z}_{(p)}$.

###  VII.1 Frobenius lift structures

####  VII.1.1 Classical and derived Frobenius lifts

######  Definition VII.1.1.

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 VII.1.2.

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 by
Definition IV.10.9 and taking the fiber on $\mathbb{F}_{p}$.

We will give a more general definition of derived Frobenius lifts.

######  Definition VII.1.3 (Homotopy coherent Frobenius lift).

Let $\mathcal{C}$ a category endowed with a functor $\mathcal{C}\to
dSt_{\mathbb{F}_{p}}^{endo}$. We define the category of Frobenius lifts on
$\mathcal{C}$ as

$\mathcal{C}^{Fr}\coloneqq\mathcal{C}\times_{dSt_{\mathbb{F}_{p}}^{endo}}dSt_{\mathbb{F}_{p}}$

where the $dSt_{\mathbb{F}_{p}}\to dSt_{\mathbb{F}_{p}}^{endo}$ is the
canonical functor adjoining the Frobenius endomorphism to a derived stack on
$\mathbb{F}_{p}$.

######  Example VII.1.4.

  * We give a series of examples for categories of "algebra-type objects"

  * •

with $\mathcal{C}=CRing_{\mathbb{Z}_{(p)}}$, we define the category of
Frobenius lifts on $CRing_{\mathbb{Z}_{(p)}}$ :

$CRing_{\mathbb{Z}_{(p)}}^{Fr}\coloneqq(CRing_{\mathbb{Z}_{(p)}}^{endo,op})^{Fr,op}$

The canonical morphism $CRing_{\mathbb{Z}_{(p)}}^{endo,op}\to
dSt^{endo}_{\mathbb{F}_{p}}$ is the derived affine functor modulo $p$.

  * •

with $\mathcal{C}=SCR_{\mathbb{Z}_{(p)}}$, we define the category of Frobenius
lifts on $SCR_{\mathbb{Z}_{(p)}}$ :

$SCR_{\mathbb{Z}_{(p)}}^{Fr}\coloneqq(SCR_{\mathbb{Z}_{(p)}}^{endo,op})^{Fr,op}$

The canonical morphism $SCR_{\mathbb{Z}_{(p)}}^{endo,op}\to
dSt^{endo}_{\mathbb{F}_{p}}$ is the derived affine functor modulo $p$.

  * •

with $\mathcal{C}=SCR_{\mathbb{Z}_{(p)}}^{gr}$, we define the category of
Frobenius lifts on $SCR_{\mathbb{Z}_{(p)}}^{gr}$ :

$SCR_{\mathbb{Z}_{(p)}}^{gr,Fr}\coloneqq(SCR_{\mathbb{Z}_{(p)}}^{gr,endo,op})^{Fr,op}$

The canonical morphism $SCR_{\mathbb{Z}_{(p)}}^{gr,endo,op}\to
dSt^{endo}_{\mathbb{F}_{p}}$ is the derived affine functor modulo $p$ after
taking the weight $0$ componant defined by Proposition IV.10.10. See Remark
IV.10.14 for details.

######  Remark VII.1.5.

Our definition requires a derived Frobenius lift to be homotopy equivalent on
$\mathbb{F}_{p}$ with the canonical Frobenius only on $0$-weights, we do not
require an homotopy to $0$ on the other weights.

######  Proposition VII.1.6.

Let $A$ be a $p$-torsion-free commutative algebra on ${\mathbb{Z}_{(p)}}$, the
space of Frobenius lifts on $A$ is discrete and in bijection with the set of
classical Frobenius lifts on $A$

$\\{\phi:A\to A:\phi_{p}:A/p\to A/p=Fr_{p}\\}$

###### Proof.

Given an endomorphism of $A$, the data of being a derived Frobenius lift is a
path in

$Map_{SCR_{\mathbb{F}_{p}}}(A\otimes^{\mathbb{L}}\mathbb{F}_{p},A\otimes^{\mathbb{L}}\mathbb{F}_{p})$

between the induced endomorphism and the canonical Frobenius. As $A$ is
$p$-torsion-free we have

$A\otimes^{\mathbb{L}}\mathbb{F}_{p}\simeq A/p$

which is a discrete commutative $\mathbb{F}_{p}$ algebra. Now, we deduce

$Map_{SCR_{\mathbb{F}_{p}}}(A\otimes^{\mathbb{L}}\mathbb{F}_{p},A\otimes^{\mathbb{L}}\mathbb{F}_{p})\simeq
Map_{SCR_{\mathbb{F}_{p}}}(A/p,A/p)\simeq
Hom_{CRing_{\mathbb{F}_{p}}}(A/p,A/p)$

which is a discrete space. Therefore the choice of a homotopy coherent
Frobenius lift is equivalent to the choice of a classical Frobenius lift. ∎

######  Definition VII.1.7.

Let $A$ be a $p$-torsion-free commutative algebra on $\mathbb{Z}_{(p)}$, let
$M$ and $N$ be $A$-module. An $(A,F)$-linear map is a morphism of
${\mathbb{Z}_{(p)}}$-modules $M\to N_{F}$ where $N_{F}$ is the
${\mathbb{Z}_{(p)}}$-module on $N$ endowed with the $A$-module structure
induced by composition by $F$.

######  Proposition VII.1.8.

Let $A$ be a $p$-torsion-free commutative algebra on $\mathbb{Z}_{(p)}$, $M$ a
projective $A$-module of finite type. The space of Frobenius lifts on the
graded simplicial algebra $Sym_{A}(M[n])$, with $M$ in weight $1$ and $n>0$ a
positive integer, is discrete and is made up by pairs $(F,\phi)$ with $F$ a
Frobenius lift of $A$ and $\phi:M\to M$ an $(A,F)$-linear map. Therefore once
the Frobenius $F$ is fixed, $\phi$ is the data of a classical $(A,F)$-module
structure on the $A$-module $M$.

###### Proof.

Taking weight $0$, a derived Frobenius lift on $Sym_{A}(M[n])$ induces a
derived Frobenius lift on $A$. From the previous proposition, a Frobenius
structure on $A$ is a classical Frobenius lift : we denote it $F$.

We start by considering the space of choices of endomorphisms on
$Sym_{A}(M[n])$ compatible with $F$, which are morphisms of simplicial
$A$-algebras from $Sym_{A}(M[n])$ to $Sym_{A}(M[n])$ with the former endowed
with the canonical $A$-algebra structure and the latter with the $A$-algebra
structure induced by $F$.

$\displaystyle End_{SCR^{gr}_{A}}(Sym_{A}(M[n]))$ $\displaystyle\simeq
Map_{A-Mod^{gr}}(M[n],Sym_{A}(M[n])_{F})$ $\displaystyle\simeq
Map_{A-Mod}(M[n],M_{F}[n])$ $\displaystyle\simeq Map_{A-Mod}(M,M_{F})$

where $M_{F}$ is $M$ endowed by the $A$-module structure induced by $F$.

Now the weight $0$ part of $Sym_{A}(M[1])$ is simply $A$. Therefore the
homotopy with the canonical Frobenius is uniquely determined by $F$. ∎

######  Proposition VII.1.9.

The space of derived Frobenius lifts on $\mathbb{F}_{p}$ is empty.

Before we move on to the proof of the proposition, we will need a lemma.

######  Lemma VII.1.10.

Let $A$ be a discrete commutative algebra and $a\in A$, there exists a
simplicial algebra $K(A,a)$ with the following universal property : let $B$ be
a simplicial $A$-algebra, there is a functorial equivalence

$Map_{A-SCR}(K(A,a),B)\to\Omega_{a,0}B$

where the space of paths $\Omega_{a,0}B$ from $a$ to $0$ in $B$ is defined as
the fiber of

$B^{\Delta^{1}}\xrightarrow{ev_{0},ev_{1}}B\times B$

over $(a,0)$.

###### Proof.

The category $SCR$ is presentable and the functor

$B\in SCR\to\Omega_{a,0}B$

preserves limits and is accessible, therefore using [Lur07, Proposition
5.5.2.7], it is representable. ∎

######  Remark VII.1.11.

We can explicitly construct $K(A,a)$ as $Sym_{A}(S^{1})$ where $Sym_{A}(-)$ is
the free simplicial $A$-algebra on a simplcial set and
$S^{1}=\Delta^{1}/\partial\Delta^{1}$ is the simplicial circle.

###### Proof of the proposition.

We first compute the mapping space $Map_{SCR}(\mathbb{F}_{p},\mathbb{F}_{p})$.
We know

$Map_{SCR_{\mathbb{Z}_{(p)}}}(\mathbb{F}_{p},\mathbb{F}_{p})=Hom_{CRing}(\mathbb{F}_{p},\mathbb{F}_{p})$

is contractible. The only endomorphism of $\mathbb{F}_{p}$ as a simplicial
algebra over $\mathbb{Z}_{(p)}$ is the identity up to contractible choice. We
take

$K(\mathbb{Z}_{(p)},p)$

as a cofibrant model of $\mathbb{F}_{p}$ and moding out by p :

$K(\mathbb{Z}_{(p)},p)\otimes_{\mathbb{Z}_{(p)}}\mathbb{F}_{p}\simeq
K(\mathbb{F}_{p},0)\simeq\mathbb{F}_{p}[\epsilon]$

is the free simplicial algebra on one generator in degree $1$, as the
construction $K$ is stable under base change. The identity induces in homology
the identity morphism, however, the Frobenius on $\mathbb{F}_{p}[\epsilon]$
sends the degree $1$ generator $\epsilon$ to $\epsilon^{p}=0$ in homology.
Therefore the identity does not have a derived Frobenius lift structure.

∎

######  Lemma VII.1.12.

The forgetful functor $U:dSt^{gr,Frob}\to dSt^{gr}$ preserves the classifying
space construction $B$.

###### Proof.

For a graded Frobenius derived stack in groups $G$, $BG$ is explicitly given
as a geometric realization of graded derived stacks of the form $G^{n}$. We
write $U$ as a composition

$dSt^{Fr}\to dSt^{endo}\to dSt$

the first one is the projection of a fiber product of categories, therefore it
preserves small limits. The second map also preserves small limits since in a
category of presheaves, they are computed pointwise. Hence $U$ preserves
finite products. The same decomposition show that $U$ preserves geometric
realizations, which concludes the proof, see the proof of Proposition VII.2.21
for a similar argument. ∎

######  Remark VII.1.13.

$B\mathbb{G}_{m}$ is the final element in $dSt^{gr}$ and it admits a unique
derived Frobenius lift structure, therefore it is also the final element in
$dSt^{gr,Frob}$.

######  Remark VII.1.14.

By definition

$SCR^{Fr}\coloneqq dAff^{Fr,op}$

We notice there is an equivalence of categories

$SCR^{Fr}\simeq SCR^{endo}\times_{SCR_{p}^{endo}}SCR_{p}$

compatible with the functor

$Spec:SCR\xrightarrow{\sim}dAff^{op}$

####  VII.1.2 Derived Witt functor

######  Proposition VII.1.15.

The forgetful functor

$U:SCR^{Fr}\to SCR$

admits a right adjoint adjoint, which we call the derived Witt ring functor.

###### Proof.

From the description of $SCR^{Fr}$ as the fiber product

$SCR^{Fr}\coloneqq SCR^{endo}\times_{SCR^{endo}_{p}}SCR_{p}$

the forgetful functor

$U:SCR^{Fr}\to SCR$

commutes with colimits. We conclude using the adjoint functor theorem [Lur09,
Corollary 5.5.2.9]. ∎

######  Remark VII.1.16.

When $A$ is discrete, $W(A)$ is the discrete commutative ring of Witt vectors.
See [Joy85] for details on the classical adjunctions. We can see the
simplicial algebra $W(A)$ is discrete by using the following identifications

$Map_{SCR}(R,W(A))\simeq Map_{SCR^{Fr}}(L(R),W(A))\simeq Map_{SCR}(L(R),A)$

for any simplicial algebra $R$, where $L$ is left adjoint to the forgetful
functor and $W$ is the right adjoint. The mapping space is discrete for every
simplicial algebra $R$ therefore $W(A)$ is discrete.

######  Construction VII.1.17.

Let $A$ be a graded simplicial algebra, which is positively or negatively
graded, we construct its associated graded simplicial algebra with Frobenius
lift of graded Witt vectors $W^{gr}(A)$ as follows. As a graded simplicial
algebra with an endomorphism, we define $W^{gr}(A)$ as the fiber product

$A^{\mathbb{N}}\times_{(A(0))^{\mathbb{N}}}W(A(0))$

where

$W(A(0))\to(A(0))^{\mathbb{N}}$

is the ghost morphism, see Remark IV.10.13 for the connection between naive
$0$ weights and $0$ weights. By construction, $W^{gr}(A)$ is positively or
negatively graded and its $0$ weights ring is given by taking naive $0$
weights

$W^{gr}(A)(0)\simeq W(A(0))$

The natural Frobenius structure on $W(A(0))$ endowes $W^{gr}(A)$ with the
structure of a graded simplicial algebra with a Frobenius lift.

This construction defines a functor

$W^{gr}:SCR^{gr}\to SCR^{gr,Fr}$

######  Proposition VII.1.18.

Let $A$ be a graded simplicial algebra, which is positively or negatively
graded. The functor

$R\in SCR^{gr,Fr}\mapsto Map_{SCR^{gr}}(B,A)\in\mathcal{S}$

is representable by $W^{gr}(A)$.

###### Proof.

Let $A$ be a graded simplicial algebra and $R$ be a graded simplicial algebra
with Frobenius lift. By the construction of $W^{gr}(A)$, a morphism of graded
simplicial algebras with Frobenius lifts

$R\to W^{gr}(A)$

is given by a morphism of graded simplicial algebras with endomorphisms

$R\to A^{\mathbb{N}}$

and a morphism of graded simplicial algebras with Frobenius lifts

$R\to W(A(0))$

with a compatibility between the two morphisms. The former morphism is simply
given by a morphism of graded simplicial algebras $A\to R$, the latter is
given by a morphism of simplicial algebras with Frobenius lifts

$R(0)\to W(A(0))$

which is simply a morphism of simplicial algebras

$R(0)\to A(0)$

which is fixed by requiring the compatibility. Therefore a morphism of graded
simplicial algebras with Frobenius lifts

$R\to W^{gr}(A)$

is uniquely determined by a morphism of graded simplicial algebras $A\to R$. ∎

####  VII.1.3 Modules on an algebra endowed with a Frobenius lift

######  Definition VII.1.19.

Let $(A,F)$ be a simplicial algebra endowed with an endomorphism. We define
the category of modules on $(A,F)$ as the stabilization of the category of
simplicial algebras over $(A,F)$, see section IV.7 for details of the tangent
category formalism. That is

$(A,F)-Mod^{endo}\coloneqq Stab(SCR^{endo}_{/(A,F)})$

Similarly, for $(A,F,h)$ a simplicial algebra endowed with a Frobenius lift,
we define its category of modules

$(A,F,h)-Mod^{Fr}\coloneqq Stab(SCR^{Fr}_{/(A,F,h)})$

######  Remark VII.1.20.

We can notice that a module on a simplicial algebra with endomorphism $(A,F)$
is simply a non-necessarily connective $A$-module endowed with an endophism
compatible with the action of $F$.

######  Remark VII.1.21.

The category $(A,F,h)-Mod$ can be identified with

$(A,F)-Mod^{endo}\times_{(A_{p},F_{p})-Mod^{endo}}A_{p}-Mod$

where $(A_{p},F_{p})$ is the simplicial algebra obtained by base changing
$(A,F)$ to $\mathbb{F}_{p}$. We deduce this identification from the following
description of $SCR^{Fr}_{/(A,F,h)}$ :

$SCR^{Fr}_{/(A,F,h)}\simeq
SCR^{endo}_{/(A,F)}\times_{SCR^{endo}_{p/(A_{p},F_{p})}}SCR_{p,/(A_{p},F_{p})}$

and the fact that stabilizations commute with the "endomorphism category
construction $(-)^{endo}$" and small limits.

Therefore we can identify an object of $(A,F,h)-Mod$ with a $(A,F)$-module and
an homotopy between $F_{p}$ and $0$.

######  Proposition VII.1.22.

Let $F:(A,F)-Mod^{endo}\to(A,F,h)-Mod^{Fr}$ be the functor sending the pair
$(M,\alpha)$ to $(M,p\alpha)$ with the canonical homotopy. The functor $F$ is
an equivalence of categories.

$F:(A,F)-Mod^{endo}\xrightarrow{\sim}(A,F,h)-Mod^{Fr}$

###### Proof.

We use the exact triangle

$M\xrightarrow{\times p}M\to M_{p}$

where $M_{p}$ denotes $M\otimes\mathbb{F}_{p}$. Given a pair $(M,\alpha)$ in
$(A,F)-Mod^{endo}$, promoting the pair to a $(A,F,h)$-module is given by the
data of a homotopy between

$M\xrightarrow{\alpha}M\to M_{p}$

and zero. Using the exact triangle, this is equivalent to specifying an
element $\alpha^{\prime}:M\to M$ and a homotopy between $p\alpha^{\prime}$ and
$\alpha$, that is to say, this is an element in $F^{-1}((M,\alpha))$. This
shows the essential surjectivity of the functor, and even the fully
faithfulness on mapping spaces with equal source and target. We proceed
similarly for general mapping spaces. ∎

######  Remark VII.1.23.

From the previous proposition, a triple in $(M,\alpha,t)$ in
$(A,F,h)-Mod^{Fr}$ may be seen as a pair $(M,\beta)$ in $(A,F)-Mod^{endo}$, we
will denote write $\frac{\alpha}{p}$ for $\beta$, this element is constructed
using the endomorphism $\alpha$ and the homotopy $t$.

######  Definition VII.1.24.

We can define the cotangent complex of a simplicial algebra endowed with a
endomorphism or a Frobenius lift using the formalism of [Lur07].

Let $(A,F)$ be a simplicial algebra endowed with an endomorphism. The
cotangent complex of $(A,F)$ is an $(A,F)-Mod$ representing the functor

$Map_{SCR^{endo}/(A,F)}((A,F),-)$

Let $(A,F,h)$ be a simplicial algebra endowed with a Frobenius lift. The
cotangent complex of $(A,F,h)$ is an $(A,F,h)-Mod$ representing the functor

$Map_{SCR^{endo}/(A,F,h)}((A,F,h),-)$

######  Remark VII.1.25.

The cotangent complex of $(A,F)$ is simply given by $(\mathbb{L}_{A},dF)$
where $dF$ is the endomorphism functorialy induced from $F$.

The forgetful functor $SCR^{Fr}\to SCR^{endo}$ induces a forgetful functor

$(A,F,h)-Mod^{Fr}\to(A,F)-Mod^{endo}$

######  Proposition VII.1.26.

Let $(A,F)$ be a simplicial algebra endowed with an endomorphism, the
forgetful functor

$SCR^{endo}_{(A,F)}\to(A,F)-Mod^{endo}$

admits a left adjoint denoted $Sym_{(A,F)}$.

###### Proof.

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

###  VII.2 Mixed graded Dieudonné complexes and algebras

####  VII.2.1 The graded circle

######  Definition VII.2.1.

The graded circle $S^{1}_{gr}$ as in [MRT20] is given by

$S^{1}_{gr}\coloneqq Spec^{\Delta}({\mathbb{Z}_{(p)}}[\eta])\simeq BKer$

It is endowed with an endomorphism induced from the multiplication by $p$
endomorphism on $Ker$, we will call this endomorphism $[p]$.The morphism $[p]$
sends $\eta$ to $p\eta$ in cohomology.

######  Proposition VII.2.2.

The endomorphism $[p]$ gives $S^{1}_{gr}$ the structure of a group in
$dSt^{Fr}$

###### Proof.

The stack $S^{1}_{gr}$ is given by applying the $B$ construction to the
derived stack with Frobenius lift $Ker$. ∎

######  Remark VII.2.3.

Defining a morphism
${\mathbb{Z}_{(p)}}[\frac{x^{n}}{n!}]\to{\mathbb{Z}_{(p)}}[\frac{x^{n}}{n!}]$
by sending $\frac{x^{n}}{n!}$ to $p^{n}\frac{x^{n}}{n!}$ is canonically a
morphism of graded classical affine schemes endowed with a Frobenius lift. As
$x$ is in weight $1$, the condition of being a Frobenius lift is trivial.

######  Remark VII.2.4.

We also notice that $[p]$ is compatible with the group structure, meaning
$\eta\mapsto p\eta$ is compatible with the coalgebra action $\eta\mapsto
1\otimes\eta+\eta\otimes 1$ and the counit map.

######  Definition VII.2.5 (Graded spheres).

We define variants of spheres as graded affine stacks

$S^{k}_{gr}(n)\coloneqq
Spec^{\Delta}({\raisebox{1.99997pt}{${\mathbb{Z}_{(p)}}[\eta_{k}]$}\left/\raisebox{-1.99997pt}{$\eta_{k}^{2}$}\right.})$

with $\eta_{k}$ of weight $n$ and degree $k$. We simply denote
$S^{k}_{gr}\coloneqq S^{k}_{gr}(1)$.

######  Remark VII.2.6.

The graded sphere $S^{1}_{gr}$ is the graded affine stack considered in
[MRT20]. The superior spheres $S^{n}_{gr}$ can be recovered from the
topological spheres as follow :

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

where $H^{*}(S^{n},{\mathbb{Z}_{(p)}})$ is the graded commutative differential
graded algebra of cohomology of the topological sphere $S^{n}$ with the zero
differential and $D$ denotes the denormalization functor.

######  Definition VII.2.7.

For $F$ a pointed graded stack, we define its graded homotopy groups :

$\pi_{k}^{(n)}(F)=Hom_{dSt^{gr,*}}(S^{k}_{gr}(n),F)=\pi_{0}Map_{dSt^{gr,*}}(S^{k}_{gr}(n),F)$

The notation $Hom$ denotes the $\pi_{0}$ set of the associated mapping space.

######  Proposition VII.2.8.

The graded circle endowed with its Frobenius structure can be recovered as the
following pushout

$S^{1}_{gr}\simeq*\sqcup_{Spec({\mathbb{Z}_{(p)}}[\rho])}*$

of graded affine stacks where $\rho$ is a generator in degree $0$ and weight
$1$ which squares to zero. Furthermore, the induced diagram obtained by taking
the product with $Spec(B)$ for $B$ a simplicial algebra

${Spec(\mathbb{Z}_{(p)}[\rho])\times
Spec(B)}$${Spec(B)}$${Spec(B)}$${S^{1}_{gr}\times Spec(B)}$

is a pushout diagram against derived affine schemes, meaning it induces a
pullback diagram on the simplicial algebras of functions.

###### Proof.

We follow the proof in [MRT20, Theorem 5.1.3]. We construct a commutative
diagram

${Spec({\mathbb{Z}_{(p)}}[\rho])}$${*}$${*}$${BKer}$

that is an element $Spec({\mathbb{Z}_{(p)}}[\rho])\to Ker\simeq\Omega BKer$.
We choose

$[\rho]=(\rho,0,0...)\in Ker({\mathbb{Z}_{(p)}}[\rho])$

It is an element of $Ker({\mathbb{Z}_{(p)}}[\rho])$ since the Frobenius $F$
acts on a Teichmüller element as $F[a]=[a^{p}]$.

We are reduced to showing the natural map

$C_{\Delta}(BKer)={\mathbb{Z}_{(p)}}\oplus{\mathbb{Z}_{(p)}}[-1]\to{\mathbb{Z}_{(p)}}\times_{{\mathbb{Z}_{(p)}}[\rho]}{\mathbb{Z}_{(p)}}$

is an equivalence of graded cosimplicial algebras. We only need to show it is
an equivalence on the underlying complexes which is obvious. The compatibility
with the endomorphism structures in straightforward and the additional
Frobenius structure is trivial since the weight $0$ part of $S^{1}_{gr}$ is
simply $Spec({\mathbb{Z}_{(p)}})$.

We move on to the second part of the proof. We want to show the following
diagram is a pushout diagram against derived affine schemes $X$.

${Spec(\mathbb{Z}_{(p)}[\rho])\times
Spec(B)}$${Spec(B)}$${Spec(B)}$${S^{1}_{gr}\times Spec(B)}$

We can reduce to the case of $X=\mathbb{A}^{1}$ using [Lur11, Proposition
4.1.9] since any derived affine scheme can be written as a limit of copies of
$\mathbb{A}^{1}$. Therefore we show the natural morphism

$\mathcal{O}(S^{1}_{gr}\times Spec(B))\to
B\times_{\mathbb{Z}_{(p)}[\rho]\otimes B}B$

is an equivalence. Since the functor of simplicial functions $\mathcal{O}$ is
given by the composition of the Dold-Kan functor with $C(-)$, the functor of
$\mathbb{E}_{\infty}$-functions, we simply show we have an equivalence

$C(S^{1}_{gr}\times Spec(B))\to B\times_{\mathbb{Z}_{(p)}[\rho]\otimes B}B$

of complexes. Now using the finite cohomology property of $S^{1}_{gr}$, see
[MRT20, Lemma 3.4.9] and the base change formula [HLP14, A.1.5-(2)], we have

$C(S^{1}_{gr}\times Spec(B))\simeq C(S^{1}_{gr})\otimes B$

Now as we know
$C(S^{1}_{gr})=H^{*}(S^{1}_{gr},\mathbb{Z}_{(p)})=\mathbb{Z}_{(p)}\oplus\mathbb{Z}_{(p)}[-1]$,
we deduce the result using the first part of the proof.

∎

####  VII.2.2 Mixed graded Dieudonné complexes

######  Definition VII.2.9.

The endomorphism $[p]:S^{1}_{gr}\to S^{1}_{gr}$ induces a pullback morphism

$[p]^{*}:QCoh(BS^{1}_{gr})\to QCoh(BS^{1}_{gr})$

which is an endofunctor of $\epsilon-Mod^{gr}$.

######  Remark VII.2.10.

Given $(M,d,\epsilon)$ a graded mixed complex, $[p]^{*}M$ is given by
$(M,d,p\epsilon)$.

######  Definition VII.2.11.

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 the category on the right hand side is the $\infty$-category of colax
fixed points of $[p]^{*}$ on $\epsilon-Mod^{gr}$, as defined in Remark II.2.3.

We see the colax fixed point morphism $[p]^{*}M\to M$ can be seen as a
morphism of graded mixed complexes

$(M,d,p\epsilon)\to(M,d,\epsilon)$

which is a morphism of graded complexes satisfying the usual Dieudonné
relation

$\epsilon F=pF\epsilon$

######  Proposition VII.2.12.

We have a natural identifications

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

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

###### Proof.

Given a mixed graded module $M\in\epsilon-Mod^{gr}$, promoting it to a
$(k[\epsilon],[p])$-module $(M,F)$ is equivalent to the data of a commutative
square

${k[\epsilon]\otimes M}$${M}$${k[\epsilon]\otimes
M}$${M}$$\scriptstyle{\textrm{[p]}\otimes F}$$\scriptstyle{F}$

which is equivalent to a colax fixed point structure

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

∎

##### The Beilinson t-structure

######  Definition VII.2.13.

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 $i<-n$.

######  Remark VII.2.14.

As a mixed graded complex, $Sym_{A}(\mathbb{L}_{A}[1])$ is t-connective, for
$A$ a simplicial algebra. It is in the heart of the t-structure when $A$ is a
smooth classical algebra.

######  Proposition VII.2.15.

The heart of $\epsilon-dg-mod^{gr}$ identifies with the $1$-category of
complexes $\bf{dg-mod_{\mathbb{Z}_{(p)}}}$. We associate to a complex $(M,d)$
the graded mixed complex being $M_{n}$ with trivial differential in weight
$-n$, the differential $d$ defines the mixed structure. This defines a functor

$i:\bf{dg-mod_{\mathbb{Z}_{(p)}}}\to\bf{\epsilon-dg-mod^{gr}}$

which induces an equivalence on the heart.

######  Definition VII.2.16 (Beilinson t-structure).

Following [Rak20, §3.3], we define a t-structure on the category of mixed
Dieudonné complexes by letting a graded mixed Dieudonné complexes be
t-connective, respectively t-connective, when its underlying graded mixed
complex is.

######  Proposition VII.2.17.

The heart of $\epsilon-D-Mod_{\mathbb{Z}_{(p)}}$ identifies with the abelian
category of Dieudonné complexes of [BLM22].

###### Proof.

This follows from the identification without Dieudonné structures, Proposition
VII.2.15. ∎

####  VII.2.3 Mixed graded Dieudonné algebras

We introduce the definition of the main objects we will study.

######  Definition VII.2.18.

Motivated by Proposition VII.2.12, we define mixed graded Dieudonné stacks,
also called derived Dieudonné stacks, by

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

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

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

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 relative affine.

######  Remark VII.2.19.

A derived Dieudonné stack can be seen as a derived stack endowed with an
endomorphism, which has a Frobenius lift structure, a grading and a compatible
$S^{1}_{gr}$-action and a compatibility condition between the Frobenius lift
and the action of $S^{1}_{gr}$. This compatibility condition is an analogue of
the Dieudonné complex equation $dF=pFd$ for derived stacks.

######  Remark VII.2.20.

We expect $\epsilon-D-SCR^{gr}$ to admit another description as the category
of modules over a monad $LSym$ on $\epsilon-D-Mod^{gr}$ as is done in [BM19]
and[Rak20].

######  Proposition VII.2.21.

The two $\infty$-categories $dSt^{gr,Frob}$ and $S^{1}_{gr}-dSt^{gr,Frob}$ are
$\infty$-topoi.

###### Proof.

The category $dSt^{gr,Frob}$ is, by definition, given by

$\tau^{endo}\times_{\tau_{p}^{endo}}\tau_{p}$

with $\tau=dSt^{gr}$ and $\tau_{p}=dSt^{gr}_{\mathbb{F}_{p}}$. Recalling
[Lur09, Proposition 6.3.2.3], to show that $dSt^{gr,Frob}$ is a topos, it is
enough to show that the projection morphisms $\tau^{endo}\to\tau_{p}^{endo}$
and $\tau_{p}\to\tau_{p}^{endo}$ are left adjoints and they preserve finite
limits, as $\tau^{endo}$, $\tau_{p}^{endo}$ and $\tau_{p}$ are already topoi.

Using [Lur09, Proposition 6.3.5.1], the morphism
$\tau^{endo}\to\tau_{p}^{endo}$ admits a right adjoint and commutes with
finite limits, since it admits a left adjoint : the forgetful functor.

The morphism $\tau_{p}\to\tau_{p}^{endo}$ commutes with finite limits, since
it admits a left adjoint, which is given by sending $(X,F)$ the homotopy
coequalizer of $F$ and the Frobenius on $X$ : $X_{F\simeq Fr}$ and admits a
right adjoint, which is given by sending $(X,F)$ the homotopy equalizer of $F$
and the Frobenius on $X$ : $X^{F\simeq Fr}$, see Proposition B.3.

Therefore, $S^{1}_{gr}-dSt^{gr,Frob}$ is also a topos as the classifying topos
of objects in $dSt^{gr,Frob}$ with a $S^{1}_{gr}$ action. ∎

######  Remark VII.2.22.

From the previous proposition, the previous categories can admit Postnikov
decomposition and obstruction theory, see [Lur07, Proposition 7.2.1.10] for
details.

##### Dieudonné functions on a Dieudonné stack

######  Construction VII.2.23.

We construct the functor of functions on a Dieudonné derived stack

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

by composition with the forgetful functor

$S^{1}_{gr}-dSt^{gr,Frob}\to S^{1}_{gr}-dSt^{gr,endo}$

we simply construct

$S^{1}_{gr}-dSt^{gr,endo,op}\to S^{1}_{gr}-Mod^{gr,endo}$

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
$B(S^{1}_{gr}\rtimes(\mathbb{N}\times\mathbb{G}_{m}))$. Indeed a
$\mathbb{N}\times\mathbb{G}_{m}$-action is given by a grading and a graded
endomorphism and an
$(S^{1}_{gr}\rtimes(\mathbb{N}\times\mathbb{G}_{m}))$-action is given by an
additional $S^{1}_{gr}$-action compatible with gradings and the endomorphisms.

Let us consider $X\in\epsilon-D-dSt^{gr}$, with its structure morphism

$\pi:X\to B(S^{1}_{gr}\rtimes(\mathbb{N}\times\mathbb{G}_{m}))$

Now the pushforward of the structure sheaf $\pi_{*}\mathcal{O}_{X}$ defines an
element of

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

The category $QCoh(B(S^{1}_{gr}\rtimes(\mathbb{N}\times\mathbb{G}_{m})))$
identifies with $S^{1}_{gr}-QCoh(B(\mathbb{N}\times\mathbb{G}_{m})))$, which
is $\epsilon-D-Mod$. We denote this element $\pi_{*}\mathcal{O}_{X}$ in
$\epsilon-D-Mod^{gr}$ as $C(X)$.

######  Remark VII.2.24.

The inclusion $\epsilon-D-SCR^{gr,op}\subset\epsilon-D-dSt^{gr}$ defines
functions on a Dieudonné simplicial algebra by composition

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

######  Remark VII.2.25.

Since the pushforward $\pi_{*}$ is canonically lax monoidal, $C(X)$ is in fact
an element of $CAlg(\epsilon-D-Mod)$. We may call $CAlg(\epsilon-D-Mod)$ the
category of mixed Dieudonnée $\mathbb{E}_{\infty}$-algebras. This notion could
give the definition of Dieudonné structures for spectral stacks but we will
not explore these notions.

######  Proposition VII.2.26.

The forgetful functor $\epsilon-D-SCR^{gr}\to\epsilon-D-Mod^{gr}$ commutes
with filtered colimits and small limits.

###### Proof.

Since the forgetful functor $\epsilon-D-Mod^{gr}\to Mod$ commutes with the
filtered colimits and small limits and is conservative, we show the forgetful
functor

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

preserves the necessary colimits and limits.

Now this functor factors as a composition

$\epsilon-D-SCR^{gr}\to
S^{1}_{gr}-SCR^{gr,endo}\simeq(\mathbb{N}\times\mathbb{G}_{m})\ltimes
S^{1}_{gr}-SCR\to Mod$

where both functors commute with the desired limits and colimits.

∎

######  Proposition VII.2.27.

The forgetful functor $U:dSt^{Fr}\to dSt$ admits a left adjoint denoted $L$.

######  Proposition VII.2.28.

We define the two pullback squares :

${SCR^{gr,Fr}}$${SCR^{gr,endo}}$${SCR}$${CRing^{p-tf,gr,Fr}}$${CRing^{p-tf,gr,endo}}$${CRing^{p-tf}}$${\urcorner}$${\urcorner}$

where $CRing^{p-ft}$ is the category of discrete commutative rings which are
$p$-torsion-free.

Then the morphism

$CRing^{p-tf,gr,Fr}\to CRing^{p-tf,gr,endo}$

is a fully faithful functor of $1$-categories.

######  Remark VII.2.29.

The previous proposition is often going to be used implicitly throughout this
thesis. In our constructions, the graded simplicial algebras will have
underlying weight $0$ simplicial algebras which are discrete and $p$-torsion
free. Therefore the choice of a Frobenius lift on such graded simplicial
algebras will simply be the choice of an endomorphism which is equal to the
canonical Frobenius after reduction modulo $p$.

######  Definition VII.2.30.

We define a subcategory $(\epsilon-D-SCR^{gr})_{\leq 0}\subset\epsilon-D-
SCR^{gr}$ on objects which are sent into $(\epsilon-D-Mod^{gr})_{\leq 0}$ when
applying the forgetful functor

$U:\epsilon-D-SCR^{gr}\to\epsilon-D-Mod^{gr}$

We define a subcategory $(\epsilon-D-SCR^{gr})_{\leq 0}\subset\epsilon-D-
SCR^{gr}$ on objects which are sent into $(\epsilon-D-Mod^{gr})_{\leq 0}$ when
applying the forgetful functor

$U:\epsilon-D-SCR^{gr}\to\epsilon-D-Mod^{gr}$

We will abuse terminology and call these objects coconnective, respectively
connective, for a "t-structure" on the non-stable category $\epsilon-D-
SCR^{gr}$.

Let us denote $(\epsilon-D-SCR^{gr})^{\heartsuit}$ the subcategory on
connective and coconnective objects. We will call this category the heart of
$\epsilon-D-SCR^{gr}$.

######  Remark VII.2.31.

These notions might be formalized using the notion of connectivity structure
on an $\infty$-category, developed in [BL21].

######  Proposition VII.2.32.

The inclusion

$(\epsilon-D-SCR^{gr})_{\leq 0}\subset\epsilon-D-SCR^{gr}$

admits a left adjoint denoted $t_{\leq 0}$.

The left adjoint $t_{\leq 0}$ commutes with the forgetful functor

$U:\epsilon-D-SCR^{gr}\to\epsilon-D-Mod^{gr}$

###### Proof.

The category $(\epsilon-D-SCR^{gr})_{\leq 0}$ is given by the pullback of
categories

${(\epsilon-D-SCR^{gr})_{\leq 0}}$${\epsilon-D-SCR^{gr}}$${(\epsilon-D-
Mod^{gr})_{\leq 0}}$${\epsilon-D-
Mod^{gr}}$$\scriptstyle{i}$$\scriptstyle{U_{0}}$${\lrcorner}$$\scriptstyle{U}$$\scriptstyle{j}$

Using the adjoint functor theorem, see [Lur09, Corollary 5.5.2.9], we are
reduced to proving that $i$ commutes with filtered colimits and small limits,
since $\epsilon-D-SCR^{gr}$ is presentable and $(\epsilon-D-SCR^{gr})_{\leq
0}$ is presentable as a limit of presentable categories. The functor $i$ is a
projection associated to a fiber product, hence we deduce the result from the
fact that $U$ and $j$ commute with the required colimits and limits. ∎

######  Remark VII.2.33.

The subcategory of $(\epsilon-D-SCR^{gr})^{\heartsuit}$ on objects which are
of $p$-torsion-free identify with the $1$-category of classical Dieudonné
algebra, see Remark VI.3.12. This defines a functor

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

######  Remark VII.2.34.

As seen before, the graded derived stack with Frobenius lift $S^{1}_{gr}$
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 Dieudonné structure on a derived stack as
a $\mathbb{G}_{m}$ action and an action of $S^{1}_{gr}$ which are compatible.
We have identifications

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

####  VII.2.4 Graded loopspace

######  Definition VII.2.35.

Let $X$ a derived stack endowed with a Frobenius lift. We define the category
of Frobenius graded loopspaces on $X$ as

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

where $S^{1}_{gr}$ is endowed with its canonical Frobenius action.

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

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

We also define

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

where $S^{1}_{gr}$ is endowed with the trivial endomorphism structure given by
identity.

We recall the forgetful functor

$U:dSt^{gr,Fr}\to dSt^{gr}$

######  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

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

###### Proof.

Using the description of the graded circle as a graded derived stack with a
Frobenius lift of Proposition VII.2.8:

$*\sqcup_{Spec({\mathbb{Z}_{(p)}}[\rho])}*\xrightarrow{\sim}S^{1}_{gr}$

which induces an equivalence

$\displaystyle\mathcal{L}^{gr,Fr}(X)$
$\displaystyle\xrightarrow{\sim}X\times_{\textbf{Map}_{dSt^{Fr}}(Spec({\mathbb{Z}_{(p)}}[\rho]),X)}X$

AS $U$ preserves limits, we deduce an equivalence of underlying graded derived
stacks

$\displaystyle U\mathcal{L}^{gr,Fr}(X)$
$\displaystyle\xrightarrow{\sim}UX\times_{U\textbf{Map}_{dSt^{Fr}}(Spec({\mathbb{Z}_{(p)}}[\rho]),X)}UX$

Let us compute the target of this map. We take $B$ a graded simplicial algebra
endowed with an Frobenius structure and $Spec(B)\to X$ a $B$-point of $X$.
Using Corollary B.5, we can reduce to points on derived affine schemes with a
free Frobenius lift. We may assume $B$ to be negatively graded as
$\mathcal{L}^{gr,Fr}(X)$ is negatively graded. We compute the former stack at
$D\coloneqq L(Spec(B))$ over $X$, where $L$ denotes the free graded derived
stack with Frobenius lift construction :

$\displaystyle
Map_{dSt^{gr,St}}(L(Spec(B)),\textbf{Map}_{dSt^{Fr}}(Spec({\mathbb{Z}_{(p)}}[\rho]),X))$
$\displaystyle\simeq
Map_{dSt^{gr,Fr}_{D/}}(Spec({\mathbb{Z}_{(p)}}[\rho])\times D,X)$
$\displaystyle\simeq
Map_{SCR^{gr,Fr}_{/W^{gr}(B)}}(C,W^{gr}(B)\otimes\mathbb{Z}_{(p)}[\rho])$

where we have used the following identification

$\mathcal{O}(Spec({\mathbb{Z}_{(p)}}[\rho])\times
D)\simeq\mathcal{O}(Spec({\mathbb{Z}_{(p)}}[\rho]))\times\mathcal{O}(D)$

which can be seen on the underlying modules where it follows from base change,
see [HLP14, A.1.5-(2)].

Therefore we want to compute the fiber of the canonical morphism from the
source space

$Map_{SCR^{gr,endo}}(C,R)\times_{Map_{SCR_{p}^{gr,endo}}(C_{p},R(0)_{p})}Map_{SCR_{p}^{gr}}(C_{p},R(0)_{p})$

where $R$ denotes $W^{gr}(B)\otimes\mathbb{Z}_{(p)}[\rho]$, to the target
space

$Map_{SCR^{gr,endo}}(C,R^{\prime})\times_{Map_{SCR_{p}^{gr,endo}}(C_{p},R^{\prime}(0)_{p})}Map_{SCR_{p}^{gr}}(C_{p},R^{\prime}(0)_{p})$

where $R^{\prime}$ denoted $W^{gr}(B)$. This fiber is simply given by the
fiber of

$Map_{SCR^{gr,endo}}(C,R)\to Map_{SCR^{gr,endo}}(C,R^{\prime})$

which is

$Map_{SCR^{gr,endo}_{/W^{gr}(B)}}(C,W^{gr}(B)\otimes\mathbb{Z}_{(p)}[\rho])\simeq
Map_{SCR^{endo}_{/W^{gr}(B)(0)}}(C,(W^{gr}(B)\otimes\mathbb{Z}_{(p)}[\rho])(0))$

as $C$ is concentrated in degree $0$. Explicitly, we have

$(W^{gr}(B)\otimes\mathbb{Z}_{(p)}[\rho])(0)\simeq W(B(0))\oplus
B_{-1}^{\mathbb{N}}$

where $W(B(0))$ is endowed by its canonical endomorphism and
$B_{-1}^{\mathbb{N}}$ is endowed by its twisted endomorphism $pS$, by taking
into account the endomorphism of $\mathbb{Z}[\rho]$ sending $\rho$ to $p\rho$.
Therefore, the fiber is computed as

$Map_{SCR^{endo}_{/W^{(}B(0))}}(C,W(B(0))\oplus B_{-1}^{\mathbb{N}})\simeq
Map_{(C,F)-Mod}(\mathbb{L}_{(C,F)},(B_{-1}^{\mathbb{N}},pS))$

######  Lemma VII.2.37.

Let $(M,u)$ and $(N,v)$ be $(C,F,h)$-modules, the fiber of

$Map_{(A,F,h)-Mod^{Fr}}((M,u),(N,v))\to Map_{(A,F)-Mod^{endo}}((M,u),(N,v))$

is given by

$Map_{C-Mod}(M_{p},N)$

###### Proof.

Using Remark VII.1.21, the mapping space $Map_{(A,F,h)-Mod^{Fr}}((M,u),(N,v))$
is given

$Map_{(A,F)-Mod^{endo}}((M,u),(N,v))\times_{Map_{(A_{p},F_{p})-Mod^{endo}}((M_{p},u_{p}),(N_{p},v_{p}))}Map_{A_{p}-Mod}(M_{p},N_{p})$

Therefore the fiber of the map

$Map_{(A,F,h)-Mod^{Fr}}((M,u),(N,v))\to Map_{(A,F)-Mod^{endo}}((M,u),(N,v))$

coincides with the fiber of

$Map_{A_{p}-Mod}(M_{p},N_{p})\to
Map_{(A_{p},F_{p})-Mod^{endo}}((M_{p},u_{p}),(N_{p},v_{p}))$

Now since $M$ and $N$ have $(A,F,h)$-structures, $u_{p}$ and $v_{p}$ are
homotopic to the zero endomorphisms. Therefore we have an identification

$Map_{(A_{p},F_{p})-Mod^{endo}}((M_{p},u_{p}),(N_{p},v_{p}))\simeq
Map_{A_{p}-Mod}(M_{p},N_{p})\times Map_{A_{p}-Mod}(M_{p}[1],N_{p})$

We deduce the fiber is given by

$\Omega Map_{A_{p}-Mod}(M_{p}[1],N_{p})\simeq Map_{A-Mod}(M_{p}[1],N)$

using the identification

$\underline{Hom}_{A-Mod}(\mathbb{F}_{p},N)\simeq N[-1]$

∎

We deduce that the fiber of

$Map_{(C,F,h)-Mod}((\mathbb{L}_{(C,F)},dF),(B_{-1}^{\mathbb{N}},pS))\to
Map_{(C,F)-Mod}(\mathbb{L}_{(C,F)},(B_{-1}^{\mathbb{N}},pS))$

is given by

$Map_{C-Mod}(\mathbb{L}_{(C,F)}\otimes\mathbb{F}_{p}[1],B_{-1}^{\mathbb{N}})$

which is also given by

$Map_{(C,F)-Mod}((\bigoplus_{\mathbb{N}}\mathbb{L}_{(C,F)}\otimes\mathbb{F}_{p}[1],S),(B_{-1}^{\mathbb{N}},S))$

Using Proposition VII.1.22, we have a natural identification

$Map_{(C,F,h)-Mod}((\mathbb{L}_{(C,F)},dF),(B_{-1}^{\mathbb{N}},pS))\simeq
Map_{(C,F)-Mod}((\mathbb{L}_{(C,F)},\frac{dF}{p}),(B_{-1}^{\mathbb{N}},S))$

Therefore there exists $\mathbb{L}$ which fits in a triangle of
$(C,F)$-modules

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

such that

$Map_{(C,F)-Mod}(\mathbb{L}_{(C,F)},(B_{-1}^{\mathbb{N}},pS))\simeq
Map_{(C,F)-Mod}(\mathbb{L},(B_{-1}^{\mathbb{N}},S))$

Then we deduce the equivalence

$\mathcal{L}^{gr,Fr}(X)\xrightarrow{\sim}X\times_{\textbf{Map}_{dSt^{Fr}}(Spec({\mathbb{Z}_{(p)}}[\rho]),X)}X\simeq
Spec(Sym_{(C,F)}(\mathbb{L}[1]))$

∎

######  Corollary VII.2.38.

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

$Spec(Sym(\mathbb{L}_{C}[1]))$

where the induced endomorphism is induced by

$\frac{dF}{p}:\mathbb{L}_{C}[1]\to\mathbb{L}_{C}[1]$

###### Proof.

The underlying derived stack of $\mathcal{L}^{gr,Fr}(X)$ is given by

$Spec(Sym(\mathbb{L}[1]))$

From Remark VII.1.25, the cotangent complex $\mathbb{L}_{(C,F)}$ has
$\mathbb{L}_{C}$ as an underlying $C$-module. In the description of the
twisted cotangent complex in a triangle of $(C,F)$-modules

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

where $\mathbb{L}_{C}$ is endowed with the endomorphism $\frac{dF}{p}$, we
notice this sequence is split since the first morphism is null. This concludes
the proof. ∎

The proof of Proposition VII.2.36 can easily be adapted to prove the following
result.

######  Remark VII.2.39.

The trivial graded endomorphism loopspace identifies with the shifted tangent
stack of $X$ :

$\mathcal{L}^{gr,endo}_{triv}(X)\simeq
Spec_{X}Sym_{\mathcal{O}_{X}}((\mathbb{L}_{(X,F)},dF)[1])$

where $X$ is no longer required to have a $p$-torsion free cotangent complex.

###  VII.3 Comparison theorems

####  VII.3.1 Mixed structures classification : the non-Dieudonné case

In this section, we recall a theorem of [To20, Proposition 2.3.1], which was
announced with an outlined proof. We give a detailed proof so as to generalize
to a Dieudonné variant of this theorem.

######  Theorem VII.3.1.

Let $A$ be smooth commutative ${\mathbb{Z}_{(p)}}$-algebra, $M$ be 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 $\bigoplus_{i}\wedge^{i}_{A}M[-i]$.

###### Proof.

The classifying space of mixed structures is given by the fiber of the
forgetful functor

$(\mathbb{G}_{m}\ltimes S^{1}_{gr})-dSt\to\mathbb{G}_{m}-dSt$

over $X$.

Since $(\mathbb{G}_{m}\ltimes S^{1}_{gr})-dSt$ identifies naturally with
$S^{1}_{gr}-(\mathbb{G}_{m}-dSt)$, the classifying space is given by the
mapping space

$Map_{Mon(dSt^{gr})}(S^{1}_{gr},\textbf{End}_{gr}(X))$

which, by connexity of $S^{1}_{gr}$, is also

$Map_{Mon(dSt^{gr})}(S^{1}_{gr},\textbf{End}^{0}_{gr}(X))$

where we define

$\textbf{End}^{0}_{gr}(X)\coloneqq
fib(\textbf{End}_{gr}(X)\to\pi_{0}(\textbf{End}^{0}_{gr}(X)))$

the fiber over the identity, meaning $\textbf{End}^{0}_{gr}(X)$ is the
substack on $\textbf{End}_{gr}(X)$ on endomorphisms that are homotopy
equivalent to identity. This space is equivalent to the mapping space of
pointed stacks

$Map_{dSt^{gr,*}}(BS^{1}_{gr},B\textbf{End}^{0}_{gr}(X))$

Since $BS^{1}_{gr}$ is a stack, we may consider $B\textbf{End}^{0}(X)$ as a
stack in $St\subset dSt$. Therefore, we are reduced to the computation of

$Map_{St^{gr,*}}(BS^{1}_{gr},B\textbf{End}^{0}_{gr}(X))$

We will need a Postnikov decomposition on $B\textbf{End}^{0}_{gr}(X)$,
therefore we have to compute its homotopy groups. To study the behaviour on
the base scheme $S=Spec(A)$, we notice that we have a fiber sequence of graded
derived stacks :

${\textbf{End}_{gr,S}^{0}(X)}$${B\mathbb{G}_{m}}$${\textbf{End}^{0}_{gr}(X)}$${\textbf{Hom}^{0}_{gr}(X,S)}$${\lrcorner}$

where $B\mathbb{G}_{m}$ is the final object in
$St^{gr}={\raisebox{1.99997pt}{$St$}\left/\raisebox{-1.99997pt}{$B\mathbb{G}_{m}$}\right.}$,
$\textbf{Hom}_{gr,S}(X)$ is the graded derived stack of endomorphisms of $X$
over $S$ and $\textbf{Hom}_{S,gr}^{0}(X)$ is the substack of maps that are
equivalent to the canonical projection $X\to S$. In this diagram, the bottom
arrow is the composition with the projection $X\to S$ and the left arrow is
the inclusion.

Being a relative linear stack, $X\to S$ has a section : the "zero" section.
Therefore

$\textbf{End}^{0}(X)\to\textbf{Hom}^{0}(X,S)$

is surjective on all homotopy groups and the long exact sequence of homotopy
groups splits as short exact sequences of graded sheaves of abelian groups on
$S$

$0\to\pi_{k}\textbf{End}_{S}^{0}(X)\to\pi_{k}\textbf{End}^{0}(X)\to\pi_{k}\textbf{Hom}^{0}(X,S)\to
0$

##### Computation of $\pi_{k}\textbf{End}^{0}_{S}(X)$

Since all the derived stacks are now truncated, we can take $B$ a test graded
commutative ${\mathbb{Z}_{(p)}}$-algebra and $Spec(B)\to S$ a $B$-point of
$S$, which is an $A$-algebra structure on $B$, ie $B$ is a relative affine
scheme over $S\times B\mathbb{G}_{m}$. We compute, for $k>0$ :

$\displaystyle\textbf{End}_{S,gr}(X)(B)$
$\displaystyle=Map_{SCR^{gr}_{A}}(Sym_{A}(M[1]),Sym_{A}(M[1])\otimes_{A}B)$
$\displaystyle=Map_{A-Mod^{gr}}(M[1],Sym_{A}(M[1])\otimes_{A}B)$

From which we deduce

$\displaystyle\pi_{k}\textbf{End}^{0}_{S,gr}(X)$
$\displaystyle=\pi_{k}\textbf{End}_{S,gr}(X)$
$\displaystyle=Hom_{A-Mod^{gr}}(M[k+1],Sym_{A}(M[1])\otimes_{A}B)$
$\displaystyle=Hom_{A-Mod^{gr}}(M,\wedge^{k+1}M\otimes_{A}B)$
$\displaystyle=Hom_{A-Mod^{gr}}(M\otimes_{A}(\wedge^{k+1}M)^{\vee},B)$

Therefore
$\pi_{k}\textbf{End}^{0}_{S}(X)\simeq\mathbb{V}(M\otimes(\wedge^{k+1}M)^{\vee})$

##### Computation of $\pi_{k}Hom^{0}(X,S)$

We have on $B$-points :

$\textbf{Hom}^{0}(X,S)(B)=Map_{SCR}^{0}(A,Sym_{A}(M[1])\otimes B)$

where the $0$ exponent denotes the connected component of the canonical map

$A\to Sym_{A}(M[1])\to Sym_{A}(M[1])\otimes B$

We can recover the homotopy groups from the Postnikov decomposition :

$\pi_{k}\textbf{Hom}^{0}(X,S)(B)[k]=fib(t_{\leq k}\textbf{Hom}^{0}(X,S)(B)\to
t_{\leq k-1}\textbf{Hom}^{0}(X,S)(B))$

As $Map(A,-)$ preserves Postnikov decompositions,
$\pi_{k}\textbf{Hom}^{0}(X,S)(B)[k]$ is given by the fiber

$fib(Map_{SCR}^{0}(A,Sym_{A}^{\leq k}(M[1])\otimes B)\to
Map_{SCR}^{0}(A,Sym_{A}^{\leq k-1}(M[1])\otimes B))$

Which is simply $Map_{SCR_{A}}^{0}(\Omega_{A}^{1},\Lambda^{k}M[k]\otimes B)$

We find
$\pi_{k}\textbf{Hom}^{0}(X,S)=\mathbb{V}(\Omega_{A}^{1}\otimes_{A}(\Lambda^{k}M)^{\vee})$

When $k$ is strictly greater than the rank of $M$ as an $A$-module,
$\Lambda^{k}M$ and $\Lambda^{k+1}M$ both vanish, therefore $\pi_{k}=0$, we
deduce $B\textbf{End}^{0}(X)$ is $(n+1)$-truncated.

We will, in fact, need a more precise description of
$\pi_{k}\coloneqq\pi_{k}\textbf{End}^{0}_{gr}(X)$ :

######  Proposition VII.3.2.

For $k\geq 1$, the sheaf of groups $\pi_{k}$ is given on a commutative algebra
$B$ by the discrete set of pairs $(d_{0},d_{1})$ with
$d_{0}:A_{B}\to\bigwedge^{k}_{A_{B}}M_{B}$ a derivation and
$d_{1}:M_{B}\to\bigwedge^{k+1}_{A_{B}}M_{B}$ a ${\mathbb{Z}_{(p)}}$-linear
map, with the compatibility condition

$d_{1}(am)=ad_{1}(m)+d_{0}(a)\wedge m$

for $a\in A_{B}$ and $m\in M_{B}$.

In this proposition, we have defined $A_{B}\coloneqq
A\otimes_{\mathbb{Z}_{(p)}}B$ and $M_{B}\coloneqq
M\otimes_{\mathbb{Z}_{(p)}}B$.

###### Proof.

On a $B$ point,

$Map_{SCR}(Sym_{A}(M[1]),Sym_{A}(M[1]))(B)\simeq
Map_{SCR_{B}}(Sym_{A_{B}}(M_{B}[1]),Sym_{A_{B}}(M_{B}[1]))$

We want to compute

$\pi_{k}Map_{SCR_{B}}^{0}(Sym_{A_{B}}(M[1]),Sym_{A_{B}}(M_{B}[1]))=\pi_{k}Map_{SCR_{B}}(Sym_{A_{B}}(M_{B}[1]),Sym_{A_{B}}(M_{B}[1]))$

which is given by the fiber over the identity of

$\pi_{0}(Map(S^{k},Map_{SCR_{B}}(C_{B},C_{B}))\to Map_{SCR_{B}}(C_{B},C_{B}))$

where $C_{B}$ denotes $Sym_{A_{B}}(M_{B}[1])$.

Since $Map(S^{k},Map_{SCR_{B}}(C_{B},C_{B}))$ can be computed as
$Map_{SCR_{B}}(C_{B},C_{B}^{S^{k}})$, we can rewrite

$\pi_{k}Map_{SCR_{B}}^{0}(Sym_{A_{B}}(M_{B}[1]),Sym_{A_{B}}(M_{B}[1]))\simeq
Hom_{SCR_{B}/C_{B}}(C_{B},C_{B}^{S^{k}})$

We will use the following lemma.

######  Lemma VII.3.3.

Let $D$ be a simplicial commutative algebra, we have a natural identification

$\pi_{0}Map_{SCR}(Sym_{A}(M[1]),D)\simeq\left\\{(u,v):u:A\to\pi_{0}(D)\in
CRing,v:M\to\pi_{1}(D)\in A-Mod\right\\}$

where $\pi_{1}(D)$ is endowed with the $A$-module structure induced by $u$.

###### Proof.

We explicitly construct this bijection. An element

$f\in\pi_{0}Map_{SCR}(Sym_{A}(M[1]),D)$

induces by composition a morphism of commutative rings

$A\to Sym_{A}(M[1])\xrightarrow{f}D\to\pi_{0}(D)$

which we call $u$. It also induces

$M[1]\to Sym_{A}(M[1])\to D$

which defines $v$ after taking $\pi_{1}$.

On the other hand, let us give ourselves a pair $(u,v)$ satisfying the
required conditions. By smoothness of $A$, $u$ extends to a morphism of
simplicial rings

$a:A\to D$

Now giving a morphism $Sym_{A}(M[1])\to D$ under $A$ is equivalent to the data
of an $A$-module morphism

$M[1]\to D$

We check that these maps are mutually inverse.

∎

The lemma can be used for the base-changed versions of $A$ and $M$ : $A_{B}$
and $M_{B}$. Taking $D=Sym_{A_{B}}(M_{B}[1])^{S^{k}}$, we know that

$\pi_{0}(D)=A_{B}\oplus\Lambda^{k}M_{B}$

and

$\pi_{1}(D)=M_{B}\oplus\Lambda^{k+1}M_{B}$

Now the set

$Hom_{SCR_{B}/C_{B}}(C_{B},C_{B}^{S^{k}})$

is identified with the set of pairs of $(\delta,d)$ with $\delta$ being a
morphism of simplicial algebras

$\delta:A_{B}\to A_{B}\oplus\Lambda^{k}M_{B}$

and $d$ being a morphism of $A$-modules

$d:M_{B}\to M_{B}\oplus\Lambda^{k+1}M_{B}$

such that they induce the identity $C_{B}\to C_{B}$ after composition.
Therefore $\delta$ is a derivation

$\delta(a)=a+d_{0}(a)$

and $d$ is given by

$d(m)=m+d_{1}(m)$

Requiring $d$ the be $A$-linear gives the compatibility condition on
$(d_{0},d_{1})$.

∎

##### Cell decomposition on $BS^{1}_{gr}$

: We now define a variation of cellular decomposition for $BS^{1}_{gr}$.

The classical cellular decomposition of the topological space
$BS^{1}\simeq\mathbb{C}P^{\infty}$, from its CW complex description, gives a
tower of topological spaces

$(BS^{1})_{\leq 0}=*\to(BS^{1})_{\leq 1}=*\to(BS^{1})_{\leq 2}\simeq
S^{2}\to(BS^{1})_{\leq 3}\simeq S^{2}\to(BS^{1})_{\leq 4}\to...$

The tower is given by iterated homotopy pushouts :

${(BS^{1})_{\leq 2n}}$${(BS^{1})_{\leq
2n+2}}$${S^{2n+1}}$${D^{2n+2}\simeq*}$${\llcorner}$

We want a similar decomposition for $BS^{1}_{gr}$ as a graded affine stack.

######  Definition VII.3.4.

Let us define a sequence of affine stacks $(BS^{1}_{gr})_{\leq n}$ for $n\geq
0$ :

$(BS^{1}_{gr})_{\leq 2n}=(BS^{1}_{gr})_{\leq 2n+1}\coloneqq
Spec^{\Delta}({\mathbb{Z}_{(p)}}[u]/(u^{n+1}))$

where $u$ is in weight $1$ and degree $2$.

The canonical projection
${\mathbb{Z}_{(p)}}[u]/(u^{n+2})\to{\mathbb{Z}_{(p)}}[u]/(u^{n+1})$ induces a
morphism of derived stack $(BS^{1}_{gr})_{\leq 2n}\to(BS^{1}_{gr})_{\leq
2n+2}$.

######  Proposition VII.3.5.

There is a homotopy pushout diagram of graded affine stacks

${(BS^{1}_{gr})_{\leq 2n}}$${(BS^{1}_{gr})_{\leq
2n+2}}$${S^{2n+1}_{gr}(n+1)}$${*}$${\llcorner}$

where $S^{2n+1}_{gr}(n+1)$ is defined as
$Spec^{\Delta}({\mathbb{Z}_{(p)}}\oplus{\mathbb{Z}_{(p)}}[-2n-1]((n+1)))$.

######  Remark VII.3.6.

The diagram is shown to be a pushout diagram in affine stacks, not in derived
stacks.

###### Proof.

This pushout diagram is equivalent to a pullback diagram in graded
cosimplicial algebras :

${{\mathbb{Z}_{(p)}}[u]/(u^{n+1})}$${{\mathbb{Z}_{(p)}}[u]/(u^{n+2})}$${{\mathbb{Z}_{(p)}}\oplus{\mathbb{Z}_{(p)}}[-2n-1]((n+1))}$${{\mathbb{Z}_{(p)}}}$${\llcorner}$

We define a diagram of commutative differential graded algebras :

${{\mathbb{Z}_{(p)}}[u,v]/(u^{n+2},dv=u^{n+1})}$${{\mathbb{Z}_{(p)}}[u]/(u^{n+1})}$${{\mathbb{Z}_{(p)}}[u]/(u^{n+2})}$${{\mathbb{Z}_{(p)}}\oplus{\mathbb{Z}_{(p)}}[-2n-1]((n+1))}$${{\mathbb{Z}_{(p)}}}$$\scriptstyle{f_{1}}$$\scriptstyle{\sim}$$\scriptstyle{f_{2}}$

where $f_{1}$ is a quasi-isomorphism sending $u$ to $u$ and $v$ to $0$ and $g$
sends $u$ to $0$ and $v$ to $\epsilon_{2n-1}$, $v$ being in degree $2n-1$ and
weight $n+1$.

Let us denote $T$ the fiber of

${\mathbb{Z}_{(p)}}[u,v]/(u^{n+2},dv=u^{n+1})\to{\mathbb{Z}_{(p)}}\oplus{\mathbb{Z}_{(p)}}[-2n-1]((n+1))$

over ${\mathbb{Z}_{(p)}}$. We check that sending $u$ to $u$ defines a quasi-
isomorphism

${\mathbb{Z}_{(p)}}[u]/(u^{n+2})\xrightarrow{\sim}T$

which recovers the canonical projection

${\mathbb{Z}_{(p)}}[u]/(u^{n+2})\to{\mathbb{Z}_{(p)}}[u]/(u^{n+1})$

after composing with $f_{1}$.

Therefore this diagram in homotopy pullback diagram on the underlying
complexes : it is strictly a pullback diagram and $f_{2}$ is a fibration.

Since denormalization $D$ respects quasi-isomorphisms, applying $D$ yields the
required diagram, which is a homotopy pullback diagram since it is on the
underlying complexes and the forgetful functor from cosimplicial algebras to
complexes is conservative.

∎

######  Lemma VII.3.7.

The diagram on $(\mathbb{N},\leq)$ defined by the tower $(BS^{1}_{gr})_{\leq
n}$ admits $BS^{1}_{gr}$ as a colimit in affine stacks.

###### Proof.

The canonical projections
${\mathbb{Z}_{(p)}}[u]\to{\mathbb{Z}_{(p)}}[u]/(u^{n+1})$, where $u$ is in
weight $1$ and cohomological degree $2$, are morphisms of graded cosimplicial
algebras. This defines a morphism of graded cosimplicial algebras

${\mathbb{Z}_{(p)}}[u]\to lim_{n}{\mathbb{Z}_{(p)}}[u]/(u^{n+1})$

which is an equivalence on the underlying complexes, therefore it is an
equivalence.

Passing to $Spec^{\Delta}$ yields the result, using Proposition IV.12.8.

∎

As we want to compute
$Map_{St^{gr,*}}(BS^{1}_{gr},B\textbf{End}^{0}_{gr}(X))$, we first need a
result :

######  Proposition VII.3.8.

The commutative diagram, denoted $(\star)$,

${(BS^{1}_{gr})_{\leq 2n}}$${(BS^{1}_{gr})_{\leq
2n+2}}$${S^{2n+1}_{gr}(n+1)}$${*}$${\llcorner}$

is a pushout against $B\textbf{End}_{gr}^{0}(X)$, meaning

${Map_{St^{gr,*}}((BS^{1}_{gr})_{\leq
2n},B\textbf{End}_{gr}^{0}(X))}$${Map_{St^{gr,*}}((BS^{1}_{gr})_{\leq
2n+2},B\textbf{End}_{gr}^{0}(X))}$${Map_{St^{gr,*}}(S^{2n+1}_{gr}(n+1),B\textbf{End}_{gr}^{0}(X))}$${*}$${\llcorner}$

is a pullback diagram.

The result is not obvious as $B\textbf{End}_{gr}^{0}(X)$ need not be an affine
stack.

######  Lemma VII.3.9.

The diagram $QCoh(\star)$ is a pullback diagram, ie the following diagram is a
pullback diagram.

${QCoh((BS^{1}_{gr})_{\leq 2n})}$${QCoh((BS^{1}_{gr})_{\leq
2n+2})}$${QCoh(S^{2n+1}_{gr}(n+1))}$${QCoh(*)}$${\llcorner}$

###### Proof.

We first show the canonical morphism

$QCoh(S^{2n+1}_{gr}(n+1))\to QCoh((BS^{1}_{gr})_{\leq
2n})\times_{QCoh(S^{2n+1}_{gr}(n+1))}QCoh(*)$

is an equivalence on bounded complexes.

We use the canonical t-structure on $QCoh(A)$ for $A$ an affine stack defined
in Remark IV.2.8. With this dévissage argument, we are reduced to showing the
morphism is an equivalence on hearts and on extension groups between objects
in the heart.

The $1$-connectivity of the stacks, see Proposition B.8, identifies their
quasi-coherent complexes with $Mod_{\mathbb{Z}_{(p)}}$. Therefore we have an
equivalence on the heart.

We invoke Proposition B.1, we deduce

$Map_{QCoh((BS^{1}_{gr})_{\leq 2n})}(M,N)\simeq C((BS^{1}_{gr})_{\leq
2n})\otimes Map_{Mod}(M,N)$

Similar results for $QCoh(S^{2n+1}_{gr}(n+1))$, $QCoh(S^{2n+1}_{gr}(n+1))$ and
$Mod$ hold. We deduce the result from the pullback diagram of $C(\star)$. Now,
adding small limits gives the equivalence on eventually coconnective
complexes, then left-completing the category with respect to the t-structure
recovers the categories of quasi-coherent complexes and yields the
equivalence, see [MRT20, Notation 1.2.12] and [Lur18, §6.2].

∎

######  Lemma VII.3.10.

Let $F$ be a stack such that

  * •

$F$ is $1$-connective, ie $\pi_{0}(F)=*$ and $\pi_{1}(F)=*$.

  * •

$F$ is $n$-truncated for some $n$.

  * •

$\pi_{i}(F)$ is quasi-coherent for all $i$.

then the pushout diagram $(\star)$ is a pushout diagram against $F$.

###### Proof.

The Postnikov tower of $F$ exhibits $F_{\leq i+1}$ as the fiber of

$F_{\leq i}\to K(\pi_{i+1},i+2)$

Therefore, we can assume $F$ be an Eilenberg-MacLane stack $K(\pi,i)$. We
have, for $Y$ a stack,

$Map_{St}(Y,F)=Map_{QCoh(Y)}(\mathcal{O}_{Y},p^{*}\pi[i])$

where $p:Y\to Spec({\mathbb{Z}_{(p)}})$ is the canonical projection. Now using
Lemma VII.3.9 and the fact that mapping spaces of fiber products of categories
are given by fiber products of the mapping spaces, we deduce that

$Map_{St}(-,F)$

sends $\star$ to a pullback diagram. ∎

###### Proof of Proposition VII.3.8 .

We combine VII.3.9, VII.3.10 and the fact that $B\textbf{End}_{gr}^{0}(X)$ is
truncated. ∎

We now move on to the computation of
$Map_{St^{gr,*}}(BS^{1}_{gr},B\textbf{End}_{gr}^{0}(X))$ using Proposition
VII.3.8. We will compute
$Map_{St^{gr,*}}(S^{2n+1}_{f}(n+1),B\textbf{End}^{0}_{gr}(X))$ using the
Postnikov decomposition of $B\textbf{End}_{gr}^{0}(X)$. By convention,
$\pi_{n}$ is $\pi_{n}(\textbf{End}_{gr}^{0}(X))$, it is a graded abelian sheaf
on $\textbf{End}_{gr}^{0}(X)$. We will denote $\Gamma(\pi_{n})$ the graded
abelian group of global sections and $\Gamma^{\mathbb{G}_{m}}(\pi_{n})$ the
abelian group of weight $0$ functions.

We need a lemma to compute
$Map_{St^{gr,*}}(S^{1}_{gr},B\textbf{End}^{0}_{gr}(X))$.

######  Lemma VII.3.11.

Using the previous notations, we have an equivalence of spaces

$Map_{St^{gr,*}}(S^{2n+1}_{f}(n+1),B\textbf{End}^{0}_{gr}(X))\simeq\left\\{\begin{array}[]{ll}B\Gamma^{\mathbb{G}_{m}}(\pi_{1})&\text{if
}n=0\\\ \Gamma^{\mathbb{G}_{m}}(\pi_{2})&\text{if }n=1\\\ 0&\text{if
}n>1\end{array}\right.$

###### Proof.

The Postnikov tower exhibits the graded stack
$(B\textbf{End}_{gr}^{0}(X))_{\leq m}$ as a homotopy fiber

$(B\textbf{End}_{gr}^{0}(X))_{\leq m}\to(B\textbf{End}_{gr}^{0}(X))_{\leq
m-1}\to K(\pi_{m-1},m+1)$

We deduce a triangle of topological spaces

${Map_{St^{gr,*}}(S^{2n+1}_{f}(n+1),(B\textbf{End}_{gr}^{0}(X))_{\leq
m})}$${Map_{St^{gr,*}}(S^{2n+1}_{f}(n+1),(B\textbf{End}_{gr}^{0}(X))_{\leq
m-1})}$${Map_{St^{gr,*}}(S^{2n+1}_{f}(n+1),K(\pi_{m-1},m+1))}$

Now,

$\pi_{k}Map_{St^{gr,*}}(S^{2n+1}_{f}(n+1),K(\pi_{m-1},m+1))$

can be computed, following [To06, §1.3] as

$\pi_{k}\Gamma(B\mathbb{G}_{m},p_{*}p^{*}\pi_{m-1}[m+1])$

where $p:S^{2n+1}_{f}(n+1)\to B\mathbb{G}_{m}$ is the structure morphism. The
later is simply given by

$H_{\mathbb{G}_{m}}^{m+1-k}(S^{2n+1}_{f}(n+1),\pi_{m-1})$

which is trivial when $n+1\neq m-1$ as $\pi_{m-1}$ is in weight $m-1$. When
$m=n+2$,

$H_{\mathbb{G}_{m}}^{n+3-k}(S^{2n+1}_{f}(n+1),\pi_{n+1})$

is non-trivial only when $k=2-n$, then it is simply
$\Gamma^{\mathbb{G}_{m}}(\pi_{n+1})$.

We deduce

$Map_{St^{gr,*}}(S^{1}_{f}(1),K(\pi_{1},3))\simeq
K(\Gamma^{\mathbb{G}_{m}}(\pi_{1}),2)$

$Map_{St^{gr,*}}(S^{3}_{f}(2),K(\pi_{2},4))\simeq
K(\Gamma^{\mathbb{G}_{m}}(\pi_{2}),1)$

$Map_{St^{gr,*}}(S^{5}_{f}(3),K(\pi_{3},5))\simeq\Gamma^{\mathbb{G}_{m}}(\pi_{3})$

Using the fact that $B\textbf{End}^{0}_{gr}(X)$ is truncated, therefore its
Postnikov tower converges, we have

$Map_{St^{gr,*}}(S^{2n+1}_{f}(n+1),B\textbf{End}^{0}_{gr}(X))\simeq\Omega_{*}(Map_{St^{gr,*}}(S^{2n+1}_{f}(n+1),K(\pi_{n+1},n+3)))$

which yields the required result.

∎

######  Lemma VII.3.12.

For $n>1$,

$Map_{St^{gr,*}}((BS^{1}_{gr})_{\leq 2n+2},B\textbf{End}^{0}_{gr}(X))\to
Map_{St^{gr,*}}((BS^{1}_{gr})_{\leq 2n},B\textbf{End}^{0}_{gr}(X))$

is an equivalence.

###### Proof.

The result is easily deduced from Proposition VII.3.8 and Lemma VII.3.11. ∎

We need to compute the fiber product

${Map_{St^{gr,*}}((BS^{1}_{gr})_{\leq
2},B\textbf{End}^{0}_{gr}(X))}$${Map_{St^{gr,*}}((BS^{1}_{gr})_{\leq
4},B\textbf{End}^{0}_{gr}(X))}$${Map_{St^{gr,*}}(S^{3}(2),B\textbf{End}^{0}_{gr}(X))}$${*}$${\llcorner}$

that is

${\Gamma^{\mathbb{G}_{m}}(\pi_{1})}$${Map_{St^{gr,*}}((BS^{1}_{gr})_{\leq
4},B\textbf{End}^{0}_{gr}(X))}$${\Gamma^{\mathbb{G}_{m}}(\pi_{2})}$${*}$${\llcorner}$

Intuitively, we want
$\Gamma^{\mathbb{G}_{m}}(\pi_{1})\mapsto\Gamma^{\mathbb{G}_{m}}(\pi_{2})$ to
send a pair $(\delta,d)$, with $\delta:A\to M$ and $d:M\to\Lambda M$, to its
"composition with itself". The equation $d\circ d=0$ involves $Sym_{A}(M[1])$
as a complex, without the full simplicial algebra structure. We first make the
following reduction.

######  Lemma VII.3.13.

The canonical commutative diagram

${Map_{St^{gr,*}}((BS^{1})_{\leq
4},B\textbf{End}^{0}_{SCR}(X))}$${Map_{St^{gr,*}}((BS^{1})_{\leq
4},B\textbf{End}_{Mod}(X))}$${Map_{St^{gr,*}}(S^{2}_{f}(1),B\textbf{End}^{0}_{SCR}(X))}$${Map_{St^{gr,*}}(S^{2}_{f}(1),B\textbf{End}_{Mod}(X))}$

is a pullback diagram.

###### Proof.

We have the following commutative diagram

${Map_{St^{gr,*}}((BS^{1})_{\leq
4},B\textbf{End}^{0}_{SCR}(X))}$${Map_{St^{gr,*}}((BS^{1})_{\leq
4},B\textbf{End}_{Mod}(X))}$${Map_{St^{gr,*}}(S^{2}_{f}(1),B\textbf{End}^{0}_{SCR}(X))}$${Map_{St^{gr,*}}(S^{2}_{f}(1),B\textbf{End}_{Mod}(X))}$${Map_{St^{gr,*}}(S^{3}_{f}(2),B\textbf{End}^{0}_{SCR}(X))}$${Map_{St^{gr,*}}(S^{3}_{f}(2),B\textbf{End}_{Mod}(X))}$

showing a natural transformation of triangles of spaces. Formality of
$\textbf{End}_{Mod}(X)$ allows for a computation of
$Map_{St^{gr,*}}(S^{3}_{f}(2),B\textbf{End}_{Mod}(X))$ and
$Map_{St^{gr,*}}(S^{2}_{f}(1),B\textbf{End}_{Mod}(X))$ using similar methods
as before. We compute the homotopy groups of $\textbf{End}^{0}_{Mod}(X)$ :

$\Gamma^{\mathbb{G}_{m}}(\pi_{k}(\textbf{End}^{0}_{Mod}(X)))=\bigoplus_{n}Hom(\Lambda^{n}M,\Lambda^{n+k}M)$

A Postnikov decomposition argument yields

$Map_{St^{gr,*}}(S^{3}_{f}(2),B\textbf{End}_{Mod}(X))\simeq\Gamma^{\mathbb{G}_{m}}(\pi_{2}(\textbf{End}^{0}_{Mod}(X)))=\bigoplus_{n}Hom(\Lambda^{n}M,\Lambda^{n+2}M)$

is discrete. Similarly,

$Map_{St^{gr,*}}(S^{2}_{f}(1),B\textbf{End}_{Mod}(X))\simeq\Gamma^{\mathbb{G}_{m}}(\pi_{1}(\textbf{End}^{0}_{Mod}(X)))=\bigoplus_{n}Hom(\Lambda^{n}M,\Lambda^{n+1}M)$

is also discrete. To show the commutative diagram of the lemma is a pullback
diagram, we simply show

$Map_{St^{gr,*}}(S^{3}_{f}(2),B\textbf{End}^{0}_{SCR}(X))\to
Map_{St^{gr,*}}(S^{3}_{f}(2),B\textbf{End}_{Mod}(X))$

is injective. But this map is the inclusion

$\Gamma^{\mathbb{G}_{m}}(\pi_{2})\subset\bigoplus_{n}Hom(\Lambda^{n}M,\Lambda^{n+2}M)$

∎

We need to compute the fiber of the morphism

$Map_{St^{gr,*}}(S^{2}_{f}(1),B\textbf{End}_{Mod}(X))\to
Map_{St^{gr,*}}(S^{3}_{f}(2),B\textbf{End}_{Mod}(X))$

Taking $B$ a test commutative algebra,

$B\textbf{End}_{QCoh(Spec({\mathbb{Z}_{(p)}}))}(X)(B)\simeq
B\textbf{End}_{QCoh(B)}(Sym_{A_{B}}(M_{B}[1]))$

which is the connected component of the space $QCoh(B)$ at
$Sym_{A_{B}}(M_{B}[1])$.

This defines a morphism of stacks
$B\textbf{End}_{QCoh(Spec({\mathbb{Z}_{(p)}}))}(X)\to QCoh(-)$ that is fully
faithful, meaning it induces an equivalence on homotopy sheaves $\pi_{k}$, for
$k>0$.

This induces a commutative diagram

${Map_{St^{gr,*}}(S^{2}_{f}(1),B\textbf{End}_{Mod}(X))}$${Map_{St^{gr,*}}(S^{3}_{f}(2),B\textbf{End}_{Mod}(X))}$${Map_{St^{gr,*}}(S^{2}_{f}(1),QCoh(-))}$${Map_{St^{gr,*}}(S^{3}_{f}(2),QCoh(-))}$

Therefore we identify the fiber of the top arrow with the fiber of the bottom
arrow.

We now compute the fiber of

$QCoh(S^{2}_{f}(1))\simeq{\mathbb{Z}_{(p)}}[u]/u^{2}-Mod\to{\mathbb{Z}_{(p)}}[v]-Mod\simeq
QCoh(S^{3}_{f}(2))$

using a Koszul duality argument.

Using Remark IV.2.8, we can simply compute the fiber of the morphism

${\mathbb{Z}_{(p)}}[u]/u^{2}-Mod\to{\mathbb{Z}_{(p)}}[v]-Mod$

where $u$ is in degree $2$ and $v$ is in degree $3$.

######  Lemma VII.3.14 (Koszul duality).

We have equivalences

${\mathbb{Z}_{(p)}}\\{\alpha\\}-Mod\simeq
IndCoh({\mathbb{Z}_{(p)}}[u]/u^{2}-Mod)$

and

${\mathbb{Z}_{(p)}}\\{\beta\\}-Mod\simeq IndCoh({\mathbb{Z}_{(p)}}[v]-Mod)$

where ${\mathbb{Z}_{(p)}}[\alpha]$ is the free differential graded algebra on
one generator in degree $1$, ${\mathbb{Z}_{(p)}}[\beta]$ is the free
differential graded algebra on one generator in degree $2$ and $IndCoh(-)$
denotes the completion by filtered colimits applied to the quasi-coherent
complex construction $QCoh(-)$ .

Moreover, the induced morphism

${\mathbb{Z}_{(p)}}\\{\alpha\\}-Mod\to{\mathbb{Z}_{(p)}}\\{\beta\\}-Mod$

sends $(M,\alpha:M\to M[-1])$ to $(M,\alpha\circ\alpha:M\to M[-2])$, that is
the forgetful functor induced by

$\alpha\in{\mathbb{Z}_{(p)}}\\{\alpha\\}\mapsto\alpha\circ\alpha\in{\mathbb{Z}_{(p)}}\\{\beta\\}$

###### Proof.

The functor $Hom_{{\mathbb{Z}_{(p)}}[u]/u^{2}}({\mathbb{Z}_{(p)}},-)$ gives an
adjunction

$End_{{\mathbb{Z}_{(p)}}[u]/u^{2}}({\mathbb{Z}_{(p)}})-Mod\leftrightarrows
IndCoh({\mathbb{Z}_{(p)}}[u]/u^{2}-Mod)$

which is an equivalence of category using Schwede-Shiple-Lurie’s theorem, see
[Lur17, Theorem 7.1.2.1]. We can take ${\mathbb{Z}_{(p)}}$ as a compact
generator of $IndCoh({\mathbb{Z}_{(p)}}[u]/u^{2}-Mod)$.

Similarly, $Hom_{{\mathbb{Z}_{(p)}}[v]}({\mathbb{Z}_{(p)}},-)$ induces the
equivalence

$End_{{\mathbb{Z}_{(p)}}[v]}({\mathbb{Z}_{(p)}})-Mod\leftrightarrows
IndCoh({\mathbb{Z}_{(p)}}[v]-Mod)$

Now to conclude the proof of the lemma, we only need to show

$Ext^{2}_{{\mathbb{Z}_{(p)}}[v]}({\mathbb{Z}_{(p)}},{\mathbb{Z}_{(p)}})\simeq{\mathbb{Z}_{(p)}}\to{\mathbb{Z}_{(p)}}\simeq
Ext^{2}_{{\mathbb{Z}_{(p)}}[u]/u^{2}}({\mathbb{Z}_{(p)}},{\mathbb{Z}_{(p)}})$

is an isomorphism. The standard resolutions of ${\mathbb{Z}_{(p)}}$ yields the
required result. ∎

The fiber of
${\mathbb{Z}_{(p)}}\\{\alpha\\}-Mod\to{\mathbb{Z}_{(p)}}\\{\beta\\}-Mod$ is
given by $(M,\alpha:M\to M[-1])$ where $M$ is a complex and
$\alpha\circ\alpha\simeq 0$ : which is the relation we wanted.

∎

####  VII.3.2 Mixed structures classification : the Dieudonné case

######  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.

###### Proof.

The classifying space of mixed structures is given by the fiber of the
forgetful functor

$S^{1}_{gr}-dSt^{gr,Frob}\to dSt^{gr,Frob}$

over $X$.

The classifying space is given by the mapping space
$Map_{Mon(dSt^{gr,Fr})}(S^{1}_{gr},\underline{\textbf{End}}_{gr,Fr}(X))$, here
$\underline{\textbf{End}}_{gr,Fr}(X)$ is a monoid in graded derived stack
endowed with a Frobenius lift. By connexity of $S^{1}_{gr}$, the classifying
space is equivalent to

$Map_{Mon(dSt^{gr,Fr})}(S^{1}_{gr},\underline{\textbf{End}}^{0}_{gr,Fr}(X))$

which is equivalent to the mapping space of pointed stacks

$Map_{dSt^{gr,Fr,*}}(BS^{1}_{gr},B\underline{\textbf{End}}^{0}_{gr,Fr}(X))$

Here $\underline{\textbf{End}}^{0}_{gr,Fr}(X)$ is the full subcategory on
$\underline{\textbf{End}}_{gr,Fr}(X)$ on endomorphisms that are homotopy
equivalent to identity. Since $BS^{1}_{gr}$ is a stack, we may consider
$B\underline{\textbf{End}}^{0}_{gr,Fr}(X)$ as a stack in $St^{gr,Fr}\subset
dSt^{gr,Fr}$.

We first reduce the classification to the computation of mapping spaces
compatible with endomorphisms instead of Frobenius structures. The mapping
space
$Map_{Gp(dSt^{gr})^{Fr}}(S^{1}_{gr},\underline{\textbf{End}}_{gr,Fr}^{0}(X))$
is computed by the fiber product

$Map_{Gp(dSt^{gr})^{endo}}(S^{1}_{gr},Z)\times_{Map_{Gp(dSt_{p}^{gr})^{endo}}(S^{1}_{gr,p}(0),Z_{p}(0))}Map_{Gp(dSt_{p}^{gr})}(S^{1}_{gr,p}(0),Z_{p}(0))$

where $Z\coloneqq\underline{\textbf{End}}_{gr,Fr}^{0}(X)$ and the $p$ index
denotes base changing to $\mathbb{F}_{p}$.

We start with a lemma to compute $Z_{p}(0)$

######  Lemma VII.3.16.

We have

$\underline{\textbf{End}}^{0}_{gr,endo}(X)(0)\simeq*$

###### Proof.

By definition of $\textbf{End}^{0}$, we have a triangle of derived stacks

$\underline{\textbf{End}}^{0}_{gr,endo}(X)\to\underline{\textbf{End}}_{gr,endo}(X)\to\pi_{0}(\underline{\textbf{End}}_{gr,endo}(X))$

and since taking $0$-weighted invariants of a graded stack preserve limits
(see Proposition IV.8.3), we have an induced triangle

$\textbf{End}^{0}_{gr,endo}(X)(0)\to\textbf{End}_{gr,endo}(X)(0)\to\pi_{0}(\textbf{End}_{gr,endo}(X))(0)\simeq\pi_{0}(\textbf{End}_{gr,endo}(X)(0))$

We only need to see that $\textbf{End}_{gr,endo}(X)(0)$ is a discrete stack.
Now, by Proposition II.3.4, $\textbf{End}_{gr,endo}(X)$ is computed as an
equalizer

$\textbf{End}_{gr,endo}(X)\simeq
eq(\textbf{End}_{gr}(X)^{\mathbb{N}}\rightrightarrows\textbf{End}_{gr}(X)(0)^{\mathbb{N}})$

which gives on $0$ weighted invariants :

$\textbf{End}_{gr,endo}(X)(0)\simeq
eq(\textbf{End}_{gr}(X)(0)^{\mathbb{N}}\rightrightarrows\textbf{End}_{gr}(X)(0)^{\mathbb{N}})$

From the proof of Theorem VII.3.1, $\textbf{End}_{gr}(X)(0)$ is discrete,
which concludes the proof. ∎

######  Lemma VII.3.17.

If we denote $\phi$ the endomorphism of $\textbf{End}^{0}_{gr,Fr}(X)$ and
$h^{\prime}$ the homotopy between $\phi_{\mkern 1.0mu\vrule
height=6.02777pt\mkern 2.0mup}$ and $Fr_{p}$ and $\psi$ the endomorphism of
$\textbf{End}^{0}_{gr,endo}(X)$. We have an equivalence between the underlying
graded derived stacks endowed with endomorphisms

$(\underline{\textbf{End}}^{0}_{gr,Fr}(X),\phi)\simeq(\underline{\textbf{End}}^{0}_{gr,endo}(X),\psi)$

meaning that we forget the "being homotopic to the canonical Frobenius modulo
$p$" part on the left hand side.

###### Proof.

We will promote $(\underline{\textbf{End}}^{0}_{gr,endo}(X),\psi)$ from an
object with an endomorphism to an object with a Frobenius lift in an
essentially unique way, then we will show that this object has the universal
property that holds for $(\underline{\textbf{End}}^{0}_{gr,Fr}(X),\phi)$.

Promoting $(\textbf{End}^{0}_{gr,endo}(X),\psi)$ to a graded derived stack
with a Frobenius has

$Map_{\underline{\textbf{End}}^{0}_{gr,endo}(X)_{p}(0)}(\psi_{p},Fr_{p})$

as a space of choices. We know, by Lemma VII.3.16, that

$Map_{\underline{\textbf{End}}^{0}_{gr,endo}(X)_{p}(0)}(\psi_{p},Fr_{p})$

is discrete and equivalent to a point, therefore
$Map_{\underline{\textbf{End}}^{0}_{gr,endo}(X)_{p}(0)}(\psi_{p},Fr_{p})$ is
contractible and $(\textbf{End}^{0}_{gr,endo}(X),\psi)$ can be promoted to an
object with Frobenius in an essentially unique way :
$(\textbf{End}^{0}_{gr,endo}(X),\psi,h)$.

Let $(T,\lambda,h_{T})$ be an object of $dSt^{Fr}$ and we write $\mu$ the
endomorphism of $X$ and $h_{X}$ the homotopy between $\mu_{p}$ and $Fr_{p}$.

The data of a morphism

$(T,\lambda)\times(X,\mu)\to(X,\mu)$

is equivalent to the data of a morphism

$(T,\lambda)\to(\underline{\textbf{End}}^{0}_{gr,endo}(X),\psi)$

by the universal property of
$(\underline{\textbf{End}}^{0}_{gr,endo}(X),\psi)$, which is equivalent to the
data of a morphism of objects with Frobenius lifts

$(T,\lambda,h_{T})\to(\underline{\textbf{End}}^{0}_{gr,endo}(X),\psi,h)$

since the choice of compatibility of the morphism of endomorphism objects with
the Frobenius lift structures is given by the data of a commutative cube as
follows :

${Y_{p}(0)}$${Z_{p}(0)}$${T_{p}(0)}$${Z_{p}(0)}$${T_{p}(0)}$${Z_{p}(0)}$${Y_{p}(0)}$${Z_{p}(0)}$$\scriptstyle{id}$$\scriptstyle{\lambda_{p}}$${\sim(h_{T})}$$\scriptstyle{\phi_{p}}$$\scriptstyle{id}$${\sim(h)}$$\scriptstyle{Fr_{p}}$$\scriptstyle{Fr_{p}}$$\scriptstyle{id}$$\scriptstyle{id}$

where $Z\coloneqq\underline{\textbf{End}}^{0}_{gr,endo}(X)$. This data lives
in a $2$-simplex in $Map_{gr}(Y_{p}(0),Z_{p}(0))$ and since $Z_{p}(0)$ is
contractible, so is $Map_{gr}(Y_{p}(0),Z_{p}(0))$, therefore the morphism of
endomorphism objects extends in an essentially unique way to a morphism of
objects with Frobenius lifts.

Now a similar argument shows that any morphism

$(T,\lambda)\times(X,\mu)\to(X,\mu)$

extends in an essentially unique way to a morphism of objects with Frobenius
lifts

$(T,\lambda,h_{T})\times(X,\mu,h_{X})\to(X,\mu,h_{X})$

Therefore $(\underline{\textbf{End}}^{0}_{gr,endo}(X),\psi,h)$ satisfies the
universal property of
$(\underline{\textbf{End}}^{0}_{gr,Fr}(X),\phi,h^{\prime})$ and they are
equivalent. ∎

######  Lemma VII.3.18.

We have

$\underline{\textbf{End}}_{gr,Fr}^{0}(X)_{p}(0)\simeq*$

###### Proof.

It is enough to show that $\underline{\textbf{End}}_{gr,Fr}^{0}(X)(0)$ is
contractible. We simply combine Lemma VII.3.16 and Lemma VII.3.17.

∎

We deduce that

$Map_{Gp(dSt^{gr,Fr})}(S^{1}_{gr},\underline{\textbf{End}}_{gr,Fr}^{0}(X))\simeq
Map_{Gp(dSt^{gr,endo})}(S^{1}_{gr},\underline{\textbf{End}}_{gr,endo}^{0}(X))$

We have reduced the study of the Frobenius structure to a mere endomorphism
structure. The previous mapping space is given by the equalizer

$Map_{Gp(dSt^{gr})}(S^{1}_{gr},\underline{\textbf{End}}_{gr,endo}^{0}(X))\leavevmode\hbox
to16.67pt{\vbox to9.47pt{\pgfpicture\makeatletter\hbox{\hskip
3.33301pt\lower-6.13298pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{{}{}}}{ {}{}{}}{}{{}}{}{
{}{}{}}{}{{}}{}\hbox{\hbox{\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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{}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
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{}{}{}} {}{}{}{{ {\pgfsys@beginscope
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}{}{}{{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{9.60002pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{9.80002pt}{0.0pt}\pgfsys@invoke{
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}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
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{}{}{}}
{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{-3.533pt}\pgfsys@lineto{9.60002pt}{-3.533pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}Map_{Gp(dSt^{gr})}(S^{1}_{gr},\underline{\textbf{End}}_{gr,endo}^{0}(X))$

where the top map is given by precomposing by $[p]$ and the bottom map is
postcomposing by $S$ the endomorphism of
$\underline{\textbf{End}}_{gr,endo}^{0}(X)$.

Now, an element of
$Map_{Gp(dSt^{gr})}(S^{1}_{gr},\underline{\textbf{End}}_{gr,endo}^{0}(X))$ can
be described using Proposition II.3.4 as the set of maps

$f_{k}:S^{1}_{gr}\to\textbf{End}_{gr}^{0}(X)$

with $prec_{\phi_{X}}\circ f_{k+1}=post_{\phi_{X}}\circ f_{k}$, where $prec$
and $post$ are the precomposing and postcomposing morphisms. The equalizer is
then given by a family $(f_{k})$ such that $f_{k+1}=f_{k}\circ[p]$. This
amount to the data of

$f:S^{1}_{gr}\to\textbf{End}_{gr}^{0}(X)$

such that $prec_{\phi_{X}}\circ f\circ[p]=post_{\phi_{X}}\circ f_{k}$.

We now use the classification for classical mixed structures, Theorem VII.3.1,
to identify the element

$f:S^{1}_{gr}\to\textbf{End}_{gr}^{0}(X)$

with a couple $(\delta:A\to M,d:M\to\Lambda^{2}_{A}M)$ satisfying the usual
equations. Unwinding the action of the endomorphisms on $f$. We see that
precomposing with $\phi_{X}$ induces the postcomposition map

$(\delta,d)\mapsto(\phi_{M}\circ\delta,(\phi_{M}\wedge\phi_{M})\circ d)$

Similarly, postcomposing with $\phi_{X}$ induces the precomposition map

$(\delta,d)\mapsto(\delta\circ\phi_{A},d\circ\phi_{M})$

And finally the action of $[p]$ is multiplication by $p$

$(\delta,d)\mapsto(p\delta,pd)$

The equations are therefore

$\phi_{M}\circ\delta=p\delta\circ\phi_{A}$

and

$(\phi_{M}\wedge\phi_{M})\circ d=pd\phi_{M}$

which are the relations required for $(\delta,d)$ to define a Dieudonné
algebra structure on $\bigoplus_{i}\wedge^{i}_{A}M[-i]$. ∎

######  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 $\bigoplus_{i\geq
0}\Lambda_{A}^{i}\Omega_{A}$.

####  VII.3.3 De Rham-Witt complex

######  Construction VII.3.20.

The functor of "simplicial functions"

$\mathcal{O}:dSt^{op}\to SCR$

induces a functor from $G$-equivariant derived stacks, where $G$ is a group in
$dSt$ :

$\mathcal{O}:G-dSt^{op}\to G-SCR$

where $G-SCR$ is abusively defined as the category of relative affine derived
schemes over $BG$. Therefore we can see this construction as taking the
scheme-affinization of a derived stack, relatively to $BG$, using Proposition
IV.3.4.

By taking $G=S^{1}_{gr}\rtimes(\mathbb{G}_{m}\times\mathbb{N})$, we can check
that we obtain a functor

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

######  Definition VII.3.21.

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 VII.3.22.

Let $A$ be a smooth discrete $p$-torsion-free algebra, the Dieudonné de Rham
complex coincides with the de Rham complex endowed with its classical
Dieudonné structure defined in [BLM22]

###### Proof.

It is a simple application of Corollary VII.3.19. ∎

###### Proof.

It is an easy corollary of Theorem VII.3.15. ∎

######  Theorem VII.3.23.

The Dieudonné de Rham functor

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

is left adjoint to the forgetful functor

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

###### Proof.

This is an application of the definition of the various categories and the
Dieudonné de Rham functor. ∎

######  Theorem VII.3.24.

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
$\epsilon-D-SCR^{gr}$.

###### Proof.

As the t-truncation is compatible with the forgetful map

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

and the forgetful map is conservative, the natural morphism

$Sym_{A}(\Omega_{A}^{\bullet}[1])\to t_{\geq 0}(Sym_{A}(\mathbb{L}_{A}[1]))$

is an equivalence on underlying graded mixed Dieudonné complexes, therefore it
is an equivalence. ∎

###  VII.4 Saturation

In this section we study in more details the properties of saturation and
comparing it to the classical saturation for Dieudonné algebras.

####  VII.4.1 The décalage functor

######  Definition VII.4.1.

We define an endofunctor of $\epsilon-Mod^{gr}_{\mathbb{Z}_{(p)}}$ denoted
$[p]^{*}$ by forgetting along

$\epsilon\in{\mathbb{Z}_{(p)}}[\epsilon]\mapsto
p\epsilon\in{\mathbb{Z}_{(p)}}[\epsilon]$

######  Definition VII.4.2.

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

######  Proposition VII.4.3.

We have an adjunction of $1$-categories

$[p]^{*}:\bf{\epsilon-
Mod^{gr}_{\mathbb{Z}_{(p)}}}\rightleftarrows\bf{\epsilon-
Mod^{gr}_{\mathbb{Z}_{(p)}}}:\eta_{p}$

######  Proposition VII.4.4.

The adjunction is a Quillen adjunction

$[p]^{*}:\bf{(\epsilon-dg-
mod^{gr}_{\mathbb{Z}_{(p)}})_{inj}}\rightleftarrows\bf{(\epsilon-dg-
mod^{gr}_{\mathbb{Z}_{(p)}})_{inj}}:\eta_{p}$

Therefore this adjunctions induces an adjunction between the corresponding
$\infty$-categories.

###### Proof.

The injective model structure on $(\epsilon-dg-
mod^{gr}_{\mathbb{Z}_{(p)}})_{inj}$ is defined by transfer along the forgetful
functor $U:\bf{\epsilon-dg-mod^{gr}_{\mathbb{Z}_{(p)}}}\to\bf{dg-
mod_{\mathbb{Z}_{(p)}}}$. We need to verify that $F^{*}_{p}$ sends
cofibrations and trivial cofibrations to cofibrations and trivial
cofibrations. Since for any graded mixed complex $M$, the underlying complex
of $[p]^{*}M$ and $M$ are the same, the result is obvious. ∎

######  Proposition VII.4.5.

Under the equivalence $(\epsilon-
Mod^{gr}_{\mathbb{Z}_{(p)}})^{\heartsuit}\simeq Mod_{\mathbb{Z}_{(p)}}$ of
VII.2.15, the endofunctor $\eta_{p}$ induces an endofunctor on
$Mod_{\mathbb{Z}_{(p)}}$, which, when restricted to $p$-torsion free complex,
identifies with the décalage functor $\eta_{p}$ from [BLM22].

###### Proof.

When $M$ is a $p$-torsion free graded mixed complex, an element $x\in M_{n}$
such that $\epsilon x$ is $p$-divisible defines a unique $y\in M_{n+1}$ such
that $\epsilon x=py$, furthermore $\epsilon y=0$ is automatic since $p\epsilon
y=\epsilon^{2}x=0$. ∎

######  Remark VII.4.6.

From the definition of Dieudonné structures on complexes and the adjunction of
Proposition VII.4.3, the data of a Dieudonné structure on a graded mixed
complex is equivalent to the data of a lax fixed structure with respect to
$\eta_{p}$.

######  Proposition VII.4.7.

The functor $\eta_{p}:\epsilon-Mod^{gr}_{\mathbb{Z}_{(p)}}\to\epsilon-
Mod^{gr}_{\mathbb{Z}_{(p)}}$ commutes with filtered colimits.

###### Proof.

The forgetful functor $\epsilon-Mod^{gr}_{\mathbb{Z}_{(p)}}\to
Mod_{\mathbb{Z}_{(p)}}$ is conservative. Furthermore the composition
$U\circ\eta_{p}$ is given by forming a finite limit

$M\mapsto M\times_{M[1]}M[1]\times_{M[2]}0$

where the arrow are $\epsilon:M\to M[1]$, $-\times p:M[1]\to M[1]$,
$\epsilon:M[1]\to M[2]$ and $0:0\to M[2]$. Then $U\circ\eta_{p}$ commutes with
filtered colimits, hence so does $U$. ∎

######  Proposition VII.4.8.

The fully faithful inclusion of fixed points

$i_{FP}:FP_{\eta_{p}}(\epsilon-Mod^{gr}_{\mathbb{Z}_{(p)}})\subset
LFP_{\eta_{p}}(\epsilon-Mod^{gr}_{\mathbb{Z}_{(p)}})$

admits a left adjoint.

###### Proof.

The category $\epsilon-Mod^{gr}_{\mathbb{Z}_{(p)}}$ is presentable as a module
category. The functor $\eta_{p}$ commutes with filtered colimits (Proposition
VII.4.7) and with small limits (Proposition VII.4.3). We conclude using
Proposition A.3. ∎

######  Definition VII.4.9.

Identifying the category of mixed Dieudonné complexes with the category
$LFP_{\eta_{p}}(\epsilon-Mod^{gr}_{\mathbb{Z}_{(p)}})$, the subcategory of
saturated mixed Dieudonné complexes is defined as $FP_{\eta_{p}}(\epsilon-
Mod^{gr}_{\mathbb{Z}_{(p)}})$, we denote it $\epsilon-D-Mod^{gr,sat}$.

####  VII.4.2 Saturated Dieudonné algebra

######  Definition VII.4.10.

We define the décalage functor for Dieudonné stacks. Let $\pi:X\to
BS^{1}_{gr}$ be a Dieudonné stack, the décalé Dieudonné stack is simply $X$
endowed with the composed structure morphism :
$X\xrightarrow{\pi}BS^{1}_{gr}\xrightarrow{[p]}BS^{1}_{gr}$. This décalage
construction defines an endofunctor of $D-dSt$.

######  Remark VII.4.11.

A Dieudonné simplicial algebra, that is a Dieudonné derived stack $X\to
BS^{1}_{gr}$ which is relatively affine, has an associated décalé Dieudonné
derived stack, which is not necessarily a Dieudonné simplicial algebra.

######  Proposition VII.4.12.

The décalage construction has a right adjoint given by taking the fiber
product of the structure map along $[p]:BS^{1}_{gr}\to BS^{1}_{gr}$.

######  Definition VII.4.13.

We define $\epsilon-D-SCR^{gr,sat}$, the category of saturated mixed graded
Dieudonné simplicial algebras, as the subcategory of mixed graded Dieudonné
algebras having an underlying mixed graded Dieudonné complex which is
saturated. Meaning :

$\epsilon-D-SCR^{gr,sat}=\epsilon-D-SCR^{gr}\times_{\epsilon-D-
Mod^{gr}}\epsilon-D-Mod^{gr,sat}$

######  Proposition VII.4.14.

The inclusions

$i:\epsilon-D-Mod^{gr,sat}\subset\epsilon-D-Mod^{gr}$

and

$j:\epsilon-D-SCR^{gr,sat}\subset\epsilon-D-SCR^{gr}$

both admit a left adjoint.

###### Proof.

For the first inclusion, it follows from Remark VII.4.6 and Proposition A.3.

By definition of saturated Dieudonné algebras, we have a pullback diagram :

${\epsilon-D-SCR^{gr,sat}}$${\epsilon-D-SCR^{gr}}$${\epsilon-D-
Mod^{gr,sat}}$${\epsilon-D-
Mod^{gr}}$$\scriptstyle{U_{0}}$$\scriptstyle{j}$${\lrcorner}$$\scriptstyle{U}$$\scriptstyle{i}$

We show $j$ commutes with filtered colimits and small limits. Since $U$
commutes with these colimits and limits and it is reflective, we show $U\circ
j\simeq i\circ U_{0}$ preserves the required colimits and limits. The functor
$j$ preserves these colimits and limits from the proof of Proposition A.3 and
so does $U_{0}$ since it is the projection of a fiber product and fibered
product commutes with filtered colimits and small limits. We conclude with the
adjoint functor Theorem II.1.2.

∎

######  Remark VII.4.15.

We can notice the décalé Dieudonné derived stack of a Dieudonné stack has
functions given by the saturation of functions of the Dieudonné stack,
therefore this construction gives a geometrical interpretation of saturation.

##  VIII Future perspectives

###  VIII.1 Improvements

We will expose many possible directions for improvements and generalizations
of the results. In this thesis, we have defined the de Rham-Witt complex of a
simplicial $\mathbb{Z}_{(p)}$-algebra with a Frobenius lift. In order to
define de Rham-Witt of a simplicial $\mathbb{F}_{p}$-algebra R, we will need
to construct the functor Verschiebung $V$. We will consider a completed
version with respect to a $V$-filtration of the Dieudonné de Rham algebra of
$(W(R),F)$.

##### The $V$ functor and the $V$ filtration

Following the construction of the de Rham-Witt complex in [BLM22], we can
construct a Verschiebung map $V$. The fundamental compatibility

$FV=p$

is expected to hold. From which we deduce, assuming we work on $p$-torsion-
free saturated Dieudonné complexes

$p\epsilon V=V\epsilon$

That is, we want to construct a decomposition of multiplication by $p$ :

$M\xrightarrow{V}[p]^{*}M\xrightarrow{F}M$

The construction of such a map is obvious for $M$ a $p$-torsion-free saturated
derived Dieudonné complex. We assume to have constructed such a map $V$.

######  Definition VIII.1.1.

For $M$ a saturated derived Dieudonné complex, we denote $\mathcal{W}_{r}(M)$
the graded mixed complexes representing the functor

$N\in\epsilon-Mod^{gr}\mapsto Map^{V^{r}}(M,N)$

where $Map^{V^{r}}(M,N)$ is the mapping space of graded mixed morphisms $M\to
N$ such that the composition $M\xrightarrow{V^{r}}M\to N$ is homotopic to
zero. Meaning $Map^{V^{r}}(M,N)$ is the fiber of

$Map_{\epsilon-Mod^{gr}}(M,N)\xrightarrow{-\circ V^{r}}Map_{Mod}(M,N)$

We have a canonical restriction map

$res:M\to lim_{r}\mathcal{W}_{r}(M)$

We denote $\mathcal{W}(M)\coloneqq lim_{r}\mathcal{W}_{r}(M)$.

The Dieudonné complex $M$ is said to be strict when $res$ is an equivalence.

######  Remark VIII.1.2.

The usual definition of $\mathcal{W}_{r}(M)$ is given by

$\mathcal{W}_{r}(M)\coloneqq M/(Im(V^{r})+Im(dV^{r}))$

they coincide as taking the cofiber of $M\to M$ in the category of graded
mixed complexes kills $Im(dV^{r})$ automatically.

We recall an alternative description of the process of strictification.

######  Corollary VIII.1.3.

([BLM22, Corollary 2.8.2]) Let $M$ be a saturated Dieudonné complex. The
restriction map

$M\to\mathcal{W}(M)$

exhibits $\mathcal{W}(M)$ as a $p$-completion of $M$ in the derived category
of abelian groups.

This corollary motivates defining a derived Dieudonné complex to be strict
when is is derived $p$-complete.

We can then extend these definitions to derived Dieudonné algebras : a derived
Dieudonné algebra is said to be strict when it is as a derived Dieudonné
complex. However the description of the strictification process seems more
involved, in particular, defining the $A/VA$ as a simplicial algebra seems to
be a difficult task in this homotopy context.

We define the category of strict derived Dieudonné complexes and algebras,
respectively $\textbf{DC}_{str}$ and $\textbf{DA}_{str}$. We expect the
inclusion

$\textbf{DA}_{str}\subset\textbf{DA}$

to admit a left adjoint denoted $\mathcal{W}Sat(-)$.

Defining $f:B\in\textbf{DA}_{str}\mapsto B^{0}/VB^{0}\in
SCR_{\mathbb{F}_{p}}$, we can ask ourselves

######  Question VIII.1.4.

Does the functor

$f:\textbf{DA}_{str}\to SCR_{\mathbb{F}_{p}}$

admit a left adjoint given by

$R\mapsto\mathcal{W}Sat(DDR(R^{red}))$

where $R^{red}$ is the reduction of $R$ ?

######  Remark VIII.1.5.

The construction of the left adjoint might not need to consider the reduction
since our main results hold for simplicial algebras which are not $p$-torsion
free.

##### De Rham-Witt complex of a $\mathbb{F}_{p}$-algebra

To complete the construction of the de Rham-Witt complex, we need a universal
property of the Witt vectors. In [BLM22, Proposition 3.6.3], the Witt vectors
give the following identification

$Hom_{F}(W(R),B)\cong Hom_{\textbf{DA}}(R,B^{0}/VB^{0})$

where $B$ is a strict Dieudonné algebra and $R$ is a $\mathbb{F}_{p}$-algebra.
The set $Hom_{F}(W(R),B)$ consists of ring morphisms commuting with the
Frobenius morphisms.

We hope to have a similar result in the context of simplicial algebras with
Frobenius lifts.

######  Definition VIII.1.6.

We can define the Dieudonné algebra of a simplicial $\mathbb{F}_{p}$-algebra
$R$ as $\mathcal{W}Sat(\mathbb{L}_{W(R)})$.

##### A more general base

Our main results are given for schemes on $\mathbb{Z}_{(p)}$, we could develop
a theory of graded loopspaces on a derived scheme endowed with a Frobenius
over a general commutative ring $k$. However some precautions have to be
taken, for example, for the crystalline circle

$S^{1}_{gr}=Spec^{\Delta}(k\oplus k[1])$

to have a derived Frobenius lift, we need $k$ to admit a derived Frobenius
lift. When $k=\mathbb{Z}_{(p)}$, such a lift is essentially unique, for
$k=\mathbb{F}_{p}$, there is not such lift since a commutative ring of finite
$p$-torsion does not admit a $\delta$-ring structure. Therefore the base
commutative ring $k$ must be endowed with a Frobenius lift structure.

Another generalization would be to replace the base $\mathbb{Z}_{(p)}$ by
$\mathbb{Z}$. In this context, all prime numbers have to be considered for the
Frobenius lifts, therefore, we have to use the more general notion of
commutating Frobenius lifts. Furthermore big Witt vectors need to be used
instead of $p$-typical Witt vectors.

######  Construction VIII.1.7.

Let $\mathcal{C}$ be an $\infty$-category. We define the category of
commutating endomorphisms on $\mathcal{C}$

$\mathcal{C}^{\mathbb{N}^{*}endo}\coloneqq Fun(B\mathbb{N}^{*},\mathcal{C})$

where $(\mathbb{N}^{*},\times)\simeq\mathbb{N}^{(\mathbb{N})}$ is the free
abelian monoid on the set $\mathbb{N}$. Sending $1$ to the $i$-th prime number
$p_{i}$ induces a morphism of monoids

$\mathbb{N}\to\mathbb{N}^{*}$

which gives by restriction

$p_{i}:\mathcal{C}^{\mathbb{N}^{*}endo}\to\mathcal{C}^{endo}$

Taking $\mathcal{C}=SCR_{\mathbb{Z}_{(p)}}$, we define the category of
simplicial algebras with commutating Frobenius lifts as

$SCR^{\mathbb{N}^{*}Fr}\coloneqq\mathcal{C}^{\mathbb{N}^{*}endo}\times_{(SCR_{p}^{endo})^{\mathbb{N}}}(SCR_{p})^{\mathbb{N}}$

An element in this category is a simplicial algebra $A$ endowed with morphisms

$\phi_{p}:A\to A$

commutating up to coherent homotopy and the data of a homotopy between
$\phi_{p}|_{\mathbb{F}_{p}}$ and the canonical Frobenius $Fr_{p}$.

This definition is close to the notion of cyclic spectra developed in [NS18]
and some connections with topological cyclic homology might be found.

Combining these notions could allow us to work over any general discrete
commutative ring endowed with commutating Frobenius lifts.

##### Study of derived stacks with Frobenius lift

The theory of derived stacks with Frobenius lift has been quickly description
in this thesis but a deeper examination could be fruitful. In particular, a
theory of Postnikov towers could shorten many proofs.

We have proven that $dSt^{Fr}$ is an $\infty$-topos. Extending PropositionB.7,
we can ask ourselves the following question.

######  Question VIII.1.8.

Let $\mathcal{C}_{0}$ be the full subcategory of $dSt^{Fr}$ on objets which
are given by freely adjoining a Frobenius to a derived affine scheme, these
objects are of the form $L(Spec(C))$, with $C$ a simplicial algebra. Does the
category $\mathcal{C}_{0}$ admit a site structure such that the topos of
sheaves on $\mathcal{C}_{0}$ identifies with $dSt^{Fr}$ ? In this case, we
would have the identification :

$Sh(\mathcal{C}_{0})\simeq dSt^{Fr}$

A precise description of the contructions and results of this thesis on
derived stacks with endomorphisms instead of Frobenius lifts could be
illuminating. We expect to be able to recover the construction of the de Rham-
Witt complex when considering

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

when $X$ admits a Frobenius lift, instead of

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

##### HKR-type filtration

Inspired by [MRT20], we could construct a filtration analoguous to the one on
hochschild cohomology. In the classical case, the affinization of the circle

$Aff(S^{1})\simeq BFix$

admits a filtration which has $BKer$ as its associated graded stack.

The stack $BFix$ admits an endomorphism structure induced by the morphism

$\mathbb{Z}\left[\binom{X}{n}\right]=\\{Q\in\mathbb{Q}[X]:Q(\mathbb{Z})\subset\mathbb{Z}\\}\to\\{Q\in\mathbb{Q}[X]:Q(\mathbb{Z})\subset\mathbb{Z}\\}=\mathbb{Z}\left[\binom{X}{n}\right]$

sending $Q$ to $Q(pX)$.

This endomorphism is not a Frobenius lift. We denote it $[p]$.

######  Definition VIII.1.9.

Let $(X,F)$ be a derived scheme endowed with an endomorphism, we define the
loopspace of $X$ as

$\mathcal{L}^{endo}((X,F))\coloneqq\textbf{Map}_{dSt^{endo}}((BFix,[p]),(X,F))$

similarly, the graded loopspace is defined as

$\mathcal{L}^{endo}_{gr}((X,F))\coloneqq\textbf{Map}_{dSt^{endo}}((BKer,[p]),(X,F))$

######  Remark VIII.1.10.

The "multiplication by $p$" map of topological spaces

$S^{1}\to S^{1}$

defines an endomorphism of the stack $S^{1}$. And we can expect an
identification of graded affine stacks with endomorphisms

$Aff((S^{1},\times p))\simeq(BFix,[p])$

and also an equivalence

$\textbf{Map}_{dSt^{endo}}((S^{1},\times
p),(X,F))\simeq\textbf{Map}_{dSt^{endo}}((BFix,[p]),(X,F))$

######  Definition VIII.1.11.

When $A$ is a simplicial algebra over $\mathbb{Z}_{(p)}$ and $F$ is en
endomorphism of $A$, we define the Hochschild cohomology of $(A,F)$ as

$HH((A,F))\coloneqq\mathcal{O}_{SCR}(\mathcal{L}^{endo}((Spec(A),Spec(F))))$

and the de Rham algebra of $(A,F)$ as

$DR((A,F))\coloneqq\mathcal{O}_{SCR}(\mathcal{L}^{endo}_{gr}((Spec(A),Spec(F))))$

######  Remark VIII.1.12.

We expect $DR((A,F))$ to be very close to

$Sym_{A}((\mathbb{L}_{A}[1],\frac{dF}{p}))$

at least when $A$ is $p$-torsion free and the endomorphism can be promoted to
a Frobenius lift.

We can ask the following

######  Question VIII.1.13.

Does the Hochschild cohomology of a simplicial algebra with endomorphism
$(A,F)$ admit a natural filtration which has $DR(A,F)$ as its associated
graded simplicial algebra ?

Defining such a filtration can be achieved by constructing a filtered analogue
of our graded circle. However some precautions have to be taken since the
natural endomorphism on the affinization of the circle $Aff(S^{1})$ is not a
Frobenius lift. Therefore a careful comparision between Frobenius graded
loopspace and endomorphism graded loopspace will probably be necessary.

##### Prismatic circle

We expect our construction of graded circle to have an analogue over the
prismatic site. We would call this object the prismatic circle. Precisely, for
$(A,I)$ a prism, $A$ is a $\delta$-ring, therefore it has a canonical
Frobenius lift. We can then consider

$\textbf{Map}_{dSt^{Fr}}(S^{1}_{gr},Spec(A))$

We expect taking levelwise mapping space with the crystalline circle to define
a sheaf on the prismatic site. This sheaf could then recover prismatic
cohomology from a mixed graded complex.

######  Remark VIII.1.14.

The above definition is incomplete and needs to be modified to take into
account the ideal $I$ when the prism is not a crystalline prism.

###  VIII.2 Symplectic forms

One of the main motivations behind our work on the graded Dieudonné loopspaces
was to define a theory of shifted symplectic forms in mixed characteristic.
The theory of shifted symplectic structures and shifted Poisson structures is
carried out in [PTVV] and [CPTVV]. Furthermore they are helpful for the study
of quantization deformations.

We recall some definitions from [PTVV].

######  Definition VIII.2.1.

Let $F$ be a derived Artin stack over a commutative ring of characteristic
zero $k$. The space of $n$-shifted $p$-forms on $F$ is

$\mathcal{A}^{p}(F,n)\coloneqq Map_{Mod}(k(p)[-p-n],DR(X))$

and the space of closed $n$-shifted $p$-forms on $F$ is

$\mathcal{A}^{p,cl}(F,n)\coloneqq Map_{\epsilon-Mod^{gr}}(k(p)[-p-n],DR(X))$

We have a natural forgetful functor

$\mathcal{A}^{p,cl}(F,n)\to\mathcal{A}^{p}(F,n)$

and the differential induces

$d_{DR}:\mathcal{A}^{p}(F,n)\to\mathcal{A}^{p+1,cl}(F,n)$

######  Definition VIII.2.2.

Let $\omega$ be a closed $2$-form of degree $n$ on $F$, by adjunction it
induces a morphism of quasi-coherent sheaves on

$\Theta_{w}:\mathbb{T}_{F}\to\mathbb{L}_{F}[n]$

since $F$ is a derived Artin stack, therefore $\mathbb{L}_{F}$ is dualizable.
The form $w$ is said to be an $n$-shifted symplectic form when $\Theta_{w}$ is
an equivalence.

We generalize these notions to the Dieudonné de Rham complex.

######  Definition VIII.2.3.

Let $X$ be a derived scheme over $\mathbb{Z}_{(p)}$, endowed with a Frobenius
lift $F$. The space of $n$-shifted Dieudonné $p$-forms on $X$ as

$\mathcal{A}^{p}(X,F,n)\coloneqq Map_{Mod}(k(p)[-p-n],DDR(X,F))$

and the space of closed $n$-shifted Dieudonné $p$-forms on $X$ as

$\mathcal{A}^{p,cl}(X,F,n)\coloneqq Map_{\epsilon-
Mod^{gr}}(k(p)[-p-n],DDR(X,F))$

We have a natural forgetful functor

$\mathcal{A}^{p,cl}(X,F,n)\to\mathcal{A}^{p}(X,F,n)$

and the differential induces

$d_{DR}:\mathcal{A}^{p}(X,F,n)\to\mathcal{A}^{p+1,cl}(X,F,n)$

Furthermore, the endomorphism on $DDR(X,F)$, which we can see as
$\frac{dF}{p}$ on $\mathbb{L}_{X}$ induces natural transformations :

$F_{form}:\mathcal{A}^{p}(X,F,n)\to\mathcal{A}^{p}(X,F,n)$

and

$F_{cl}:\mathcal{A}^{p,cl}(X,F,n)\to\mathcal{A}^{p,cl}(X,F,n)$

We define a variation on the definition which takes the Frobenius structure
into account.

######  Definition VIII.2.4.

The space of $n$-shifted Dieudonné $p$-forms on $X$ as

$\mathcal{A}^{p}_{Fr}(X,F,n)\coloneqq Map_{Mod^{endo}}(k(p)[-p-n],DDR(X,F))$

where we endow $k(p)[-p-n]$ with the "multiplication by $p$" endomorphism and
the space of closed $n$-shifted Dieudonné $p$-forms on $X$ as

$\mathcal{A}^{p,cl}_{Fr}(X,F,n)\coloneqq Map_{\epsilon-
Mod^{gr,endo}}(k(p)[-p-n],DDR(X,F))$

We have a natural forgetful functor

$\mathcal{A}^{p,cl}_{Fr}(X,F,n)\to\mathcal{A}^{p}_{Fr}(X,F,n)$

and the differential induces

$d_{DR}:\mathcal{A}^{p}_{Fr}(X,F,n)\to\mathcal{A}^{p+1,cl}_{Fr}(X,F,n)$

######  Remark VIII.2.5.

We recall the definition of Frobenius-derivations used for Fedosov
quantization in [BK07]. Let $A$ be commutative ring and $M$ an $A$-module, a
Frobenius-derivation $D:A\to M$ is a morphism of abelian groups such that
$D(1)=0$ and

$D(ab)=a^{p}D(b)+b^{p}D(a)$

This definition suggest yet another notion of $n$-shifted $p$-forms, on a
simplicial algebra with Frobenius lift $(A,F)$, based on a notion of
Frobenius-twisted cotangent complex

$\mathbb{L}^{tw}_{(A,F)}\coloneqq\mathbb{L}_{A}\otimes_{A}A$

where the morphism $A\to A$ is $F$.

We expect to be able to define symplectic forms and use them to study
deformation quantizations for derived schemes in mixed characteristic.

The difference between the Frobenius-twisted cotangent complex and the
cotangent complex we study here $\mathbb{L}_{(A,F)}$ seems to be connected to
the difference between $p$-restricted and partition Lie algebra, see [BM19].

###  VIII.3 Foliations

We recall from [To20], the definition of derived foliations.

######  Definition VIII.3.1.

Let $X$ be a derived scheme over a commutative ring $k$. A derived foliation
on $X$ is the data of a graded mixed derived stack $\mathcal{F}$ and a
morphism of graded mixed derived stack $\mathcal{F}\to\mathcal{L}^{gr}(X)$
such that

  * •

The projection $\mathcal{F}\to X$ is relatively affine.

  * •

The quasi-coherent complex $\mathcal{O}_{\mathcal{F}}(1)$ is in Tor-amplitude
$]-\infty,m]$, for an integer $m$, where $\mathcal{O}_{\mathcal{F}}(1)$ is the
$1$-weighted part of functions of $\mathcal{F}$ relative to $X$.

  * •

The natural morphism of graded derived stack

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

is an equivalence.

In this definition, we were able to define Dieudonné foliations for possibly
non-connective cotangent complexes $\mathbb{L}_{\mathcal{F}}$ using Corollary
V.2.7. Indeed, we keep the expected property that

$\mathcal{F}\mapsto\mathbb{L}_{\mathcal{F}}$

is conservative.

We extend the definition of foliations to our framework.

######  Definition VIII.3.2.

Let $X$ be a derived scheme over $\mathbb{Z}_{(p)}$, endowed with a Frobenius
lift. A Dieudonné foliation on $(X,F)$ is the data of a derived Dieudonné
stack $F$ and a morphism of derived Dieudonné stack
$\mathcal{F}\to\mathcal{L}^{gr}(X,F)$ such that

  * •

The projection $\mathcal{F}\to X$ is relatively affine.

  * •

The quasi-coherent complex $\mathcal{O}_{\mathcal{F}}(1)$ is in Tor-amplitude
$]-\infty,-1]$ where $\mathcal{O}_{\mathcal{F}}(1)$ is the $1$-weighted part
of functions of $\mathcal{F}$ relative to $X$.

  * •

The natural morphism of graded derived stack endowed with a Frobenius lift

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

is an equivalence.

Our classification Theorem VII.3.15 gives a more precise description of
Dieudonné foliations.

######  Theorem VIII.3.3.

Let $A$ be a smooth commutative $k$-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 $\phi:M\to M$. The space of Dieudonné foliations
$\mathcal{F}$ over $A$ having $M$ as a cotangent complex is discrete and in
bijection with the set of Dieudonné algebra structures on the graded
commutative $k$-algebra $\bigoplus_{i}\wedge^{i}_{A}M[-i]$ endowed with its
natural canonical Frobenius lift structure.

This theorem gives an easier description of Dieudonné foliations, which have a
quite abstract definition.

######  Example VIII.3.4.

We outline the construction of a fundamental example of a Dieudonné foliation
using the previous theorem. Let $(X,F)$ be a smooth derived scheme endowed
with a Frobenius lift. We define the tangent complex $\mathbb{T}_{(X,F)}$ as
the dual of the cotangent complex $\mathbb{L}_{(X,F)}$ in the category of
$(X,F)$-modules, we note that $\mathbb{T}_{(X,F)}$ does not admit
$\mathbb{T}_{X}$ as an underlying $\mathcal{O}_{X}$-module : an element of
$\mathbb{T}_{(X,F)}$ can be thought of as a sequence of elements of
$\mathbb{T}_{X}$ which satisfy compatibility conditions with the Frobenius
morphism. We fix a sub-bundle $I$ of $\mathbb{T}_{(X,F)}$ which is stable by
the Lie bracket. The derived stack

$\mathcal{F}\coloneqq\mathbb{V}(I^{\vee}[1])$

has a canonical mixed graded structure and a Frobenius structure. The
canonical map

$I\to\mathbb{T}_{(X,F)}$

induces a morphism

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

which makes $\mathcal{F}$ into a Dieudonné foliation.

######  Example VIII.3.5.

Let $f:X\to Y$ be a morphism of derived schemes with Frobenius lifts. Pulling
back along $f$ should define a canonical foliation $f^{*}0_{Y}\in Fol(X)$.

Extending on the theory of Dieudonné foliations, we hope to construct, for a
fixed Dieudonné foliation, the de Rham-Witt complex along the leaves of the
foliation, from which we will deduce crystalline cohomology along the
foliation.

From these constructions, we can expect results on Dieudonné foliations
regarding the vanishing of cohomology classes. In [To20], for a crystal $E$ on
a foliation $\mathcal{F}$, cohomology classes $c_{i}(E(0))$ in
$H^{2i}_{DR}(X/S)$ are shown to vanish in $H^{2i}_{DR}(X/\mathcal{F})$ where
$\mathcal{F}$ is a foliation on $X$ relative to $S$. These classes are shown
to come from classes in crystalline cohomology and a similar vanishing theorem
is proven. These classes are, in fact, canonically lifted to rigid cohomology
classes and the theory of Dieudonné foliations could help better understand
the vanishing of the crystalline classes and give similar results for rigid
cohomology classes.

##  Appendix A Categorical results

######  Proposition A.1.

Let $\mathcal{C}$ be an $\infty$-topos, $G$ a group in $\mathcal{C}^{endo}$.
$G$ is naturally a group in $\mathcal{C}$ and it has a compatible endomorphism
$\alpha:G\to G$ which induces a forgetful functor $\alpha^{*}:G-\mathcal{C}\to
G-\mathcal{C}$. Then there is an equivalence :

$G-\mathcal{C}^{endo}\simeq CFP_{\alpha^{*}}(G-\mathcal{C})$

######  Proposition A.2.

Let $\mathcal{C}_{0}\to\mathcal{C}_{01}\leftarrow\mathcal{C}_{1}$ a pullback
diagram of $\infty$-categories and $G$ be group in
$\mathcal{C}_{0}\times_{\mathcal{C}_{01}}\mathcal{C}_{1}$. $G$ induces a group
$G_{0}$ in $\mathcal{C}_{0}$, $G_{1}$ in $\mathcal{C}_{1}$, $G_{01}$ in
$\mathcal{C}_{01}$. The canonical morphism is an equivalence :

$G-(\mathcal{C}_{0}\times_{\mathcal{C}_{01}}\mathcal{C}_{1})\xrightarrow{\sim}(G_{0}-\mathcal{C}_{0})\times_{G_{01}-\mathcal{C}_{01}}(G_{1}-\mathcal{C}_{1})$

######  Proposition A.3.

Let $\mathcal{C}$ a presentable category and $\eta$ an endofunctor of
$\mathcal{C}$ which commutes with filtered colimits and small limits. Then the
inclusion functor

$U:FP(\mathcal{C})\subset LFP(\mathcal{C})$

admits a left adjoint.

###### Proof.

The functor $U$ commutes with filtered colimits and with small limits as the
functor $\eta$ does. The categories $FP_{\eta}(\mathcal{C})$ and
$LFP(\mathcal{C})$ are presentable as limits of presentable categories, see
Proposition II.2.1. Using the adjoint functor theorem, Theorem II.1.2,
concludes the proof. ∎

##  Appendix B Geometrical results

######  Proposition B.1.

Let $F$ be an affine stack with $C(F)$ projective of finite type as a complex
on $k$. We denote $p:F\to*$ the canonical structure morphism. Let $M$ and $N$
be $k$-complexes.

$C(F)\otimes Map_{Mod}(M,N)\to Map_{QCoh(F)}(p^{*}M,p^{*}N)$

is an equivalence, where $Map_{Mod}(M,N)$ is considered with its $k$-module
structure.

###### Proof.

As a complex, $C(F)$ is a retract of a free complex, therefore tensoring with
$C(F)$ preserves all limits. Now, the morphism

$C(F)\otimes Map_{Mod}(M,N)\to Map_{QCoh(F)}(M,N)$

is compatible with colimits in $M$ and limits in $N$. We are reduced to
proving the result for $M=k$ and $N=k$, where the result is obvious. ∎

######  Proposition B.2.

Let $A$ be a simplicial algebra over $\mathbb{F}_{p}$. Endowing a simplicial
algebra over $A$ with its canonical Frobenius endomorphism defines a functor

$B\in SCR_{A}\mapsto(B,Fr_{B})\in SCR_{A}^{endo}$

which admits a right adjoint given by taking the homotopy equalizer

$(B,F)\in SCR_{A}^{endo}\mapsto B^{F\simeq Fr_{B}}\coloneqq eq(B\ext@arrow
0359\arrowfill@\mathrel{\mathchoice{\raise 1.66194pt\hbox
to0.0pt{$\displaystyle\relbar$\hss}\lower
1.66194pt\hbox{$\displaystyle\relbar$}}{\raise 1.66194pt\hbox
to0.0pt{$\textstyle\relbar$\hss}\lower
1.66194pt\hbox{$\textstyle\relbar$}}{\raise 1.66194pt\hbox
to0.0pt{$\scriptstyle\relbar$\hss}\lower
1.66194pt\hbox{$\scriptstyle\relbar$}}{\raise 1.66194pt\hbox
to0.0pt{$\scriptscriptstyle\relbar$\hss}\lower
1.66194pt\hbox{$\scriptscriptstyle\relbar$}}}\mathrel{\mathchoice{\raise
1.66194pt\hbox to0.0pt{$\displaystyle\relbar$\hss}\lower
1.66194pt\hbox{$\displaystyle\relbar$}}{\raise 1.66194pt\hbox
to0.0pt{$\textstyle\relbar$\hss}\lower
1.66194pt\hbox{$\textstyle\relbar$}}{\raise 1.66194pt\hbox
to0.0pt{$\scriptstyle\relbar$\hss}\lower
1.66194pt\hbox{$\scriptstyle\relbar$}}{\raise 1.66194pt\hbox
to0.0pt{$\scriptscriptstyle\relbar$\hss}\lower
1.66194pt\hbox{$\scriptscriptstyle\relbar$}}}\rightrightarrows{F}{Fr_{B}}B)\in
SCR_{A}$

and a left adjoint given by taking homotopy coinvariants

$(B,F)\in SCR_{A}^{endo}\mapsto B_{F\simeq Fr_{B}}\coloneqq coeq(B\ext@arrow
0359\arrowfill@\mathrel{\mathchoice{\raise 1.66194pt\hbox
to0.0pt{$\displaystyle\relbar$\hss}\lower
1.66194pt\hbox{$\displaystyle\relbar$}}{\raise 1.66194pt\hbox
to0.0pt{$\textstyle\relbar$\hss}\lower
1.66194pt\hbox{$\textstyle\relbar$}}{\raise 1.66194pt\hbox
to0.0pt{$\scriptstyle\relbar$\hss}\lower
1.66194pt\hbox{$\scriptstyle\relbar$}}{\raise 1.66194pt\hbox
to0.0pt{$\scriptscriptstyle\relbar$\hss}\lower
1.66194pt\hbox{$\scriptscriptstyle\relbar$}}}\mathrel{\mathchoice{\raise
1.66194pt\hbox to0.0pt{$\displaystyle\relbar$\hss}\lower
1.66194pt\hbox{$\displaystyle\relbar$}}{\raise 1.66194pt\hbox
to0.0pt{$\textstyle\relbar$\hss}\lower
1.66194pt\hbox{$\textstyle\relbar$}}{\raise 1.66194pt\hbox
to0.0pt{$\scriptstyle\relbar$\hss}\lower
1.66194pt\hbox{$\scriptstyle\relbar$}}{\raise 1.66194pt\hbox
to0.0pt{$\scriptscriptstyle\relbar$\hss}\lower
1.66194pt\hbox{$\scriptscriptstyle\relbar$}}}\rightrightarrows{F}{Fr_{B}}B)\in
SCR_{A}$

###### Proof.

We sketch the proof. Let $B\in SCR_{A}$ and $C\in SCR_{A}^{endo}$.

In the following diagram :

${B}$${C}$${C}$${B}$${C}$${C}$$\scriptstyle{Fr_{B}}$$\scriptstyle{f}$$\scriptstyle{Fr_{C}}$$\scriptstyle{Id}$$\scriptstyle{F}$$\scriptstyle{f}$$\scriptstyle{Id}$

the left diagram is a commutative diagram.

For a fixed morphism $f:B\to C$, a homotopy making the outer diagram commute
is equivalent to a homotopy between $Fr_{C}\circ f$ and $F\circ f$ in the
following diagram.

${B}$${C}$${C}$${C}$${C}$$\scriptstyle{f}$$\scriptstyle{Fr_{C}}$$\scriptstyle{Id}$$\scriptstyle{F}$$\scriptstyle{Id}$

The latter is then the data of $f$ factoring through

$C^{F\simeq Fr_{C}}\coloneqq eq(C\ext@arrow
0359\arrowfill@\mathrel{\mathchoice{\raise 1.66194pt\hbox
to0.0pt{$\displaystyle\relbar$\hss}\lower
1.66194pt\hbox{$\displaystyle\relbar$}}{\raise 1.66194pt\hbox
to0.0pt{$\textstyle\relbar$\hss}\lower
1.66194pt\hbox{$\textstyle\relbar$}}{\raise 1.66194pt\hbox
to0.0pt{$\scriptstyle\relbar$\hss}\lower
1.66194pt\hbox{$\scriptstyle\relbar$}}{\raise 1.66194pt\hbox
to0.0pt{$\scriptscriptstyle\relbar$\hss}\lower
1.66194pt\hbox{$\scriptscriptstyle\relbar$}}}\mathrel{\mathchoice{\raise
1.66194pt\hbox to0.0pt{$\displaystyle\relbar$\hss}\lower
1.66194pt\hbox{$\displaystyle\relbar$}}{\raise 1.66194pt\hbox
to0.0pt{$\textstyle\relbar$\hss}\lower
1.66194pt\hbox{$\textstyle\relbar$}}{\raise 1.66194pt\hbox
to0.0pt{$\scriptstyle\relbar$\hss}\lower
1.66194pt\hbox{$\scriptstyle\relbar$}}{\raise 1.66194pt\hbox
to0.0pt{$\scriptscriptstyle\relbar$\hss}\lower
1.66194pt\hbox{$\scriptscriptstyle\relbar$}}}\rightrightarrows{F}{Fr_{C}}C)$

The dual proof for the left adjoint goes similarly.

∎

A similar proof yield the following proposition.

######  Proposition B.3.

Let $A$ be a simplicial algebra over $\mathbb{F}_{p}$. Endowing a derived
stack over $A$ with its canonical Frobenius endomorphism defines a functor

$X\in dSt_{A}\mapsto(X,Fr_{X})\in dSt_{A}^{endo}$

which admits a right adjoint given by taking the homotopy equalizer

$(X,F)\in dSt_{A}^{endo}\mapsto X^{F\simeq Fr_{X}}\coloneqq eq(X\ext@arrow
0359\arrowfill@\mathrel{\mathchoice{\raise 1.66194pt\hbox
to0.0pt{$\displaystyle\relbar$\hss}\lower
1.66194pt\hbox{$\displaystyle\relbar$}}{\raise 1.66194pt\hbox
to0.0pt{$\textstyle\relbar$\hss}\lower
1.66194pt\hbox{$\textstyle\relbar$}}{\raise 1.66194pt\hbox
to0.0pt{$\scriptstyle\relbar$\hss}\lower
1.66194pt\hbox{$\scriptstyle\relbar$}}{\raise 1.66194pt\hbox
to0.0pt{$\scriptscriptstyle\relbar$\hss}\lower
1.66194pt\hbox{$\scriptscriptstyle\relbar$}}}\mathrel{\mathchoice{\raise
1.66194pt\hbox to0.0pt{$\displaystyle\relbar$\hss}\lower
1.66194pt\hbox{$\displaystyle\relbar$}}{\raise 1.66194pt\hbox
to0.0pt{$\textstyle\relbar$\hss}\lower
1.66194pt\hbox{$\textstyle\relbar$}}{\raise 1.66194pt\hbox
to0.0pt{$\scriptstyle\relbar$\hss}\lower
1.66194pt\hbox{$\scriptstyle\relbar$}}{\raise 1.66194pt\hbox
to0.0pt{$\scriptscriptstyle\relbar$\hss}\lower
1.66194pt\hbox{$\scriptscriptstyle\relbar$}}}\rightrightarrows{F}{Fr_{X}}X)\in
dSt_{A}$

and a left adjoint given by taking homotopy coinvariants

$(X,F)\in dSt_{A}^{endo}\mapsto X_{F\simeq Fr_{X}}\coloneqq coeq(X\ext@arrow
0359\arrowfill@\mathrel{\mathchoice{\raise 1.66194pt\hbox
to0.0pt{$\displaystyle\relbar$\hss}\lower
1.66194pt\hbox{$\displaystyle\relbar$}}{\raise 1.66194pt\hbox
to0.0pt{$\textstyle\relbar$\hss}\lower
1.66194pt\hbox{$\textstyle\relbar$}}{\raise 1.66194pt\hbox
to0.0pt{$\scriptstyle\relbar$\hss}\lower
1.66194pt\hbox{$\scriptstyle\relbar$}}{\raise 1.66194pt\hbox
to0.0pt{$\scriptscriptstyle\relbar$\hss}\lower
1.66194pt\hbox{$\scriptscriptstyle\relbar$}}}\mathrel{\mathchoice{\raise
1.66194pt\hbox to0.0pt{$\displaystyle\relbar$\hss}\lower
1.66194pt\hbox{$\displaystyle\relbar$}}{\raise 1.66194pt\hbox
to0.0pt{$\textstyle\relbar$\hss}\lower
1.66194pt\hbox{$\textstyle\relbar$}}{\raise 1.66194pt\hbox
to0.0pt{$\scriptstyle\relbar$\hss}\lower
1.66194pt\hbox{$\scriptstyle\relbar$}}{\raise 1.66194pt\hbox
to0.0pt{$\scriptscriptstyle\relbar$\hss}\lower
1.66194pt\hbox{$\scriptscriptstyle\relbar$}}}\rightrightarrows{F}{Fr_{X}}X)\in
dSt_{A}$

######  Proposition B.4.

Let $\mathcal{C}_{0}$ be the full subcategory of $dSt^{Fr}$ on objects which
are given by freely adjoining a Frobenius lift to a derived affine scheme,
these objects are of the form $L(Spec(C))$, with $C$ a simplicial algebra. See
Proposition VII.2.27 for the construction of $L$.

The subcategory $\mathcal{C}_{0}$ generates $dSt^{Fr}$ under colimits.

###### Proof.

The comonad $T$ induced from the adjunction

$L:dSt\rightleftarrows dSt^{Fr}:U$

induces a resolution of $X$ denoted $T^{\bullet+1}(X)$ using the bar
construction, by combining the fact that $T$ commutes with small colimits with
the Barr-Beck Theorem, see [Lur17, Theorem 4.7.3.5] and the proof of
Proposition VII.2.21. Therefore $X$ is given by the geometric realization of
$T^{\bullet+1}(X)$.

Now we consider $Y=L(Z)$ obtained by freely adjoining a Frobenius lift to a
derived stack. Writing $Z$ as a colimit of derived affine schemes, using the
co-Yoneda lemma

$Z\simeq colim_{i}Spec(A_{i})$

we deduce

$Y=L(colim_{i}Spec(A_{i}))\simeq colim_{i}L(Spec(A_{i}))$

∎

######  Corollary B.5.

The restriction of the Yoneda embedding gives a functor

$dSt^{Fr}\to Fun(\mathcal{C}_{0},\mathcal{S})$

which is fully faithful.

Similar arguments provide analoguous results for derived stacks endowed with
endomorphisms.

######  Proposition B.6.

Let $\mathcal{C}_{0}$ be the full subcategory of $dSt^{endo}$ on objets which
are given by freely adjoining an endomorphism to a derived affine scheme,
these objects are of the form $L(Spec(C))$, with $C$ a simplicial algebra.

The subcategory $\mathcal{C}_{0}$ generates $dSt^{endo}$ under colimits.

######  Corollary B.7.

The restriction of the Yoneda embedding gives a functor

$dSt^{endo}\to Fun(\mathcal{C}_{0},\mathcal{S})$

which is fully faithful.

######  Proposition B.8.

Let $A\to k$ be an augmented cosimplicial $k$-algebra such that $H^{0}(A)\to
k$ is an isomorphism, where $k$ is a principal torsion-free commutative ring.
Let $n\geq 0$ be such that

$H^{i}(A)=0$

for $1\leq i\leq n$. Note that this condition is empty for $n=0$. Then there
is a cofibrant replacement of $A$

$QA\xrightarrow{\sim}A$

such that

  * •

All coface maps $QA_{i}\to QA_{j}$ are flat.

  * •

All coface maps $QA_{i}\to QA_{0}$ are isomorphisms for $i\leq n-1$.

###### Proof.

We construct a tower of cosimplicial algebras by adding successive cells to
kill the various cohomology groups since the model structure on cosimplicial
algebras is cofibrantly generated. Let us define $X\coloneqq
Spec^{\Delta}(A)$. We assume that we have constructed a cofibrant model
$X^{m}=Spec^{\Delta}(A_{m})$ with a canonical morphism

$p_{m}:X\to X^{m}$

which is an isomorphism on all cohomology groups $H^{i}$ for $i\leq n$. We
have a natural identification

$H^{m+1}(A)\cong[X,K(\mathbb{G}_{a},m+1)]$

We define $I\coloneqq[X,K(\mathbb{G}_{a},m+1)]$ as a set, and the tautological
morphism

$X\to K(\mathbb{G}_{a},m+1)^{I}$

We define $X^{\prime m}(1)\coloneqq X^{m}\times
K(\mathbb{G}_{a},m+1)^{I}=Spec^{\Delta}(A^{\prime m}(1))$, where $A^{\prime
m}(1)$ is the tensor product of the free cosimplicial algebra on a coproduct
of $I$ copies of $k[-m-1]$ and $A^{m}$. The natural morphism

$A^{\prime m}(1)\to A$

is surjective on the $H^{m+1}$ group. From the isomorphisms

$H^{m+1}(A^{\prime m}(1))\cong H^{m+1}(K(\mathbb{G}_{a},m+1)^{I})\times
H^{m+1}(A^{m})\cong End_{Gp}(\mathbb{G}_{a})^{(I)}\times H^{m+1}(A^{m})$

We deduce

$H^{m+1}(A^{\prime m}(1))\cong k^{(I)}\times H^{m+1}(A^{m})$

because $k$ is torsion-free. The kernel of the surjective morphism

$k^{(I)}\times H^{m+1}(A^{m})\to H^{m+1}(A)$

is denoted $J$. We deduce a map

$X^{m}\to X^{\prime m}(1)\to K(\mathbb{G}_{a},m+1)^{J}$

which factors through

$X^{m}(1)\coloneqq X^{\prime
m}(1)\times_{K(\mathbb{G}_{a},m+1)^{J}}E(\mathbb{G}_{a},m+1)^{J}$

We iterate this construction to obtain a tower

$X^{m}\to...\to X^{m}(k)\to X^{\prime m}(k)\to...X^{m}(1)\to X^{\prime m}(1)$

which gives

$\alpha_{m}:X^{m}\to X^{m+1}$

where $X^{m+1}$ is the limit of the tower

$X^{m}(k)\to X^{\prime m}(k)\to...X^{m}(1)\to X^{\prime m}(1)$

We check that $\alpha_{m}$ is an isomorphism on the $H^{i}$ groups where
$i\leq m+1$ and we define $QA$ by requiring

$Spec^{\Delta}(QA)\coloneqq colim_{m}X^{m}$

where we take the colimit in the category of affine stacks.

The flatness of the transition morphisms come from the flatness of

$E(\mathbb{G}_{a},i)\to K(\mathbb{G}_{a},i)$

when $i>0$.

Now to get the last assertion, we choose a specific model for
$K(\mathbb{G}_{a},i)$ and $E(\mathbb{G}_{a},i)$. We take the following :

$K(\mathbb{G}_{a},i)\coloneqq Spec^{\Delta}(Free_{coSCR}(k[-m]))$

and

$K(\mathbb{G}_{a},i)\coloneqq
Spec^{\Delta}(Free_{coSCR}(k_{m-1}\xrightarrow{Id}k_{m}))$

As the associated cosimplicial algebras of these affine stacks satisfy

$Y_{i}\xrightarrow{\sim}Y_{0}$

where $i<m-1$, this concludes the proof. ∎

######  Remark B.9.

We notice in the proof that if $H^{n+1}(A)$ is a free $k$-module, $A^{\prime
n+1}(1)\to A$ is an isomorphism on $H^{m+1}$ groups, therefore we can take
$X^{n+1}(1)$ to be $X^{\prime n+1}(1)$ and we obtain a cofibrant model such
that $QA_{n}\to QA_{0}$ is also an isomorphism.

From the remark, we deduce :

######  Corollary B.10.

The formal $n$-sphere

$S^{n}_{f}\coloneqq Spec^{\Delta}(DH^{*}(S^{n},k))$

is $(n-1)$-connective.

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