# The canonical global quantization of symplectic varieties in characteristic
$p$

Ekaterina Bogdanova  Harvard University, USA<EMAIL_ADDRESS>,
Dmitry Kubrak  Institut des Hautes Études Scientifiques, France
<EMAIL_ADDRESS>,  Roman Travkin  Skolkovo Institute of Science and
Technology, Russia<EMAIL_ADDRESS>and  Vadim Vologodsky
National Research University “Higher School of Economics”, Russia
<EMAIL_ADDRESS>
###### Abstract.

Let $X$ be a smooth symplectic variety over a field $k$ of characteristic
$p>2$ equipped with a restricted structure, which is a class $[\eta]\in
H^{0}(X,\Omega^{1}_{X}/d\mathcal{O}_{X})$ whose de Rham differential equals
the symplectic form. In this paper we construct a functorial in $(X,[\eta])$
formal quantization of the category $\mathrm{QCoh}(X)$ of quasi-coherent
sheaves on $X$. We also construct its natural extension to a quasi-coherent
sheaf of categories $\mathrm{QCoh}_{h}$ on the product
$X^{(1)}\times{\mathbb{S}}$ of the Frobenius twist of $X$ and the projective
line ${\mathbb{S}}=\mathbb{P}^{1}$, viewed as the one-point compactification
of $\mathrm{Spec}\ \\!k[h]$. Its global sections over $X^{(1)}\times\\{0\\}$
is the category of quasi-coherent sheaves on $X$. If $X$ is affine,
$\mathrm{QCoh}_{h}$, restricted to $X^{(1)}\times\mathrm{Spf}\ \\!k[[h]]$, is
equivalent to the category of modules over the distinguished “Frobenius-
constant” quantization of $(X,[\eta])$ defined in [BK].

###### Contents

  1. 1 Introduction
    1. 1.1 Plan of the paper
    2. 1.2 Properties of $\operatorname{{QCoh}}_{h}$.
    3. 1.3 Acknowledgments
  2. 2 Differential geometry on pseudo-smooth schemes
    1. 2.1 Pseudo-smooth schemes
    2. 2.2 Restricted Lie algebras and Jacobson’s formula
    3. 2.3 Crystalline differential operators
    4. 2.4 Differential operators on smooth schemes
    5. 2.5 Local structure of a pseudo-smooth scheme
    6. 2.6 De Rham complex and Cartier operation
    7. 2.7 Line bundles with connection and Milne’s exact sequence
    8. 2.8 Restricted Poisson structures
    9. 2.9 Restricted symplectic geometry
    10. 2.10 The group scheme $G_{0}$ and Darboux lemma
  3. 3 Sheaves of algebras $\mathcal{A}_{\mathbb{S}}$ and $\mathcal{A}_{\mathbb{S}}^{\flat}$ and their categories of modules
    1. 3.1 Rees construction over $\mathbb{S}\coloneqq\mathbb{P}^{1}$
    2. 3.2 Twistor differential operators $\mathcal{D}_{X,\mathbb{S}}$
    3. 3.3 Twistor reduced Weyl algebra $\mathcal{A}_{\mathbb{S}}$
    4. 3.4 Algebra $A_{0}$
    5. 3.5 $h$-separable algebras
    6. 3.6 Automorphisms of $\mathcal{A}_{\mathbb{S}}$
    7. 3.7 Algebra $\mathcal{A}_{\mathbb{S}}^{\flat}$ and the $G_{0}$-action on $\mathcal{A}_{\mathbb{S}}^{\flat}-\operatorname{{Mod}}$
  4. 4 Quasi-coherent sheaves of abelian categories
    1. 4.1 Sheaves of categories.
    2. 4.2 Groups acting on a category.
  5. 5 Action of $G_{0}$ on ${\mathcal{A}}_{\mathbb{S}}{\operatorname{\mathsf{-Mod}}}$.
    1. 5.1 Main Theorem.
    2. 5.2 The restriction of $({\mathcal{A}}_{\mathbb{S}}{\operatorname{\mathsf{-Mod}}})_{{\mathbb{S}}/G_{0}}$ to $\\{0\\}/G_{0}$.
    3. 5.3 A ${\mathbb{G}}_{m}\ltimes G_{0}$-equivariant structure on ${\mathcal{A}}_{\mathbb{S}}{\operatorname{\mathsf{-Mod}}}$.
    4. 5.4 Further remarks.
  6. 6 Canonical quantization of $\operatorname{{QCoh}}(X)$.
    1. 6.1 Construction of the canonical quantization and its uniqueness.
    2. 6.2 ${\mathbb{G}}_{m}$-equivariant quantizations.
    3. 6.3 Quantizations of $\operatorname{{QCoh}}(X)$ vs quantizations of ${\mathcal{O}}_{X}$
    4. 6.4 Canonical quantization of Lagrangian subschemes.

##  1\. Introduction

Let $k$ be a field of characteristic $p>2$. In [BK], Bezrukavnikov and Kaledin
initiated the study of quantizations of symplectic varieties over $k$. More
precisely, they introduced new notion of a restricted symplectic $k$-scheme:
namely, they consider pairs $(X,[\eta])$, where $[\eta]$ is a global section
of the sheaf
$\operatorname{{coker}}(\mathcal{O}_{X}\xrightarrow{d}\Omega^{1}_{X})$, such
that $\omega:=d([\eta])\in H^{0}(X,\Omega^{2}_{X})$ is a symplectic form111One
can think of $[\eta]$ as an algebraic analogue of a “contact form” on $X$.. In
the case $H^{i}(X,\mathcal{O}_{X})=0$ for $i=1,2,3$, they construct the a
distinguished “Frobenius-constant” quantization of $(X,[\eta])$. Roughly
speaking, such a quantization is a sheaf $O_{h}$ of associative
$k[[h]]$-algebras on $X$ equipped with two isomorphisms

  * •

$\mathcal{O}_{X}\xrightarrow{\sim}O_{h}/h$,

  * •

$\mathcal{O}_{X^{(1)}}[[h]]\xrightarrow{\sim}Z(O_{h})$,

that are also compatible with the restricted structure given by $[\eta]$ in a
certain way. Here $X^{(1)}$ denotes the Frobenius-twist of $X$ and $Z(O_{h})$
is the center of $O_{h}$. The prototypical example of such a quantization is
given by the following: namely, the $k[[h]]$-version of the Rees algebra
$\mathcal{D}_{Y,h}$ of differential operators on $Y$ gives the distinguished
Frobenius-constant quantization of $X=T^{*}Y$ with the restricted structure
given by the canonical 1-form $\eta$. An unfortunate feature of the
construction is that it is not functorial: there is no natural way to make the
distinguished quantization equivariant with respect to automorphisms of
$(X,[\eta])$, and moreover the quantization exists only if we require the
aforementioned cohomology vanishing for $X$. A solution proposed by Kontsevich
([K]) and studied further in ([VdB], [Y]) in characteristic $0$ context is to
replace the algebra $O_{h}$ by the corresponding category of modules
$O_{h}{\operatorname{\mathsf{-Mod}}}$. We develop a parallel picture in
characteristic $p$: namely, we show that the category
$O_{h}{\operatorname{\mathsf{-Mod}}}$ is functorial in the pair $(X,[\eta])$
and glues to a canonical quantization of the category
$\operatorname{{QCoh}}(X)$ of quasi-coherent sheaves on $X$ for any restricted
symplectic $k$-scheme $(X,[\eta])$.

Moreover, our construction allows to extend the range of the quantum parameter
$h$ from being a formal variable to a genuine coordinate on
${\mathbb{P}}^{1}$, where the corresponding category
$\operatorname{{QCoh}}_{h}$ carries a formal resemblance with the twistor
space construction appearing in Simpson’s correspondence (see Remark 1.1
below).

###  1.1. Plan of the paper

The idea of “formal geometry” in the sense of Gelfand-Kazhdan is to first deal
with differential-geometric questions in the basic case of a formal disc, and
then descend to smooth varieties via the corresponding torsor of formal
coordinates. As observed by Bezrukavnikov and Kaledin in [BK] in
characteristic $p$ one can replace formal disc by the corresponding Frobenius
neighborhood of zero, which makes things simpler. To be more precise, the key
idea is to consider a smooth $S$-scheme $X$ of dimension $d$ as a scheme over
its Frobenius twist $X^{(1)}$ via the relative Frobenius $F_{X/S}\colon
X\rightarrow X^{(1)}$. This is not only a finite locally free map, but in fact
a locally trivial bundle in flat topology: namely, Zariski-locally on $X$, the
fiber product $X\times_{X^{(1)}}X$ splits as
$X\times\operatorname{{Spec}}A_{0}$ where $\operatorname{{Spec}}A_{0}$ is the
Frobenius-neighborhood of 0 in $d$-dimensional affine space222Explicitly,
$A_{0}\simeq\mathbb{F}_{p}[x_{1},\ldots,x_{d}]/x_{1}^{p}=\ldots=x_{d}^{p}=0$..
One then can consider the canonical torsor $\mathcal{M}_{X}\rightarrow
X^{(1)}$ of “Frobenius frames”: a $T$-point $T\rightarrow\mathcal{M}_{X}$ is
given by an isomorphism $T\times_{X^{(1)}}X\simeq
T\times\operatorname{{Spec}}A_{0}$, or, in other words, an identification of
(pull-back of) $X$ with the Frobenius neiborhood of zero in an affine space of
dimension $d$ over $T$. The scheme $\mathcal{M}_{X}\rightarrow X^{(1)}$ is a
torsor over the group scheme $\underline{\operatorname{{Aut}}}(A_{0})$ of
automorphisms of $\operatorname{{Spec}}A_{0}$ and many basic objects of
differential-geometric nature such as333Strictly speaking one needs to
consider their pushforwards to $X^{(1)}$ under relative Frobenius.
differential operators $\mathcal{D}_{X}$, differential 1-forms
$\Omega^{1}_{X}$, vector fields $\mathcal{T}_{X}$, or just the structure sheaf
$\mathcal{O}_{X}$ all come via descent from some representations of
$\underline{\operatorname{{Aut}}}(A_{0})$. This also gives a way to define
differentiable structures on smooth varieties in characteristic $p$ by
considering the subgroup $H$ of $\underline{\operatorname{{Aut}}}(A_{0})$ that
stabilizes that “structure” in the case of $X=\operatorname{{Spec}}A_{0}$.
Prescribing such a structure on a given scheme $X$ then corresponds to a
reduction of the $\underline{\operatorname{{Aut}}}(A_{0})$-torsor
$\mathcal{M}_{X}\rightarrow X^{(1)}$ to $H$. We will discuss in some detail
how this picture looks in the case of restricted symplectic structure (see
Section 2.10).

The above picture motivates considering things like differential operators or
symplectic structures in the case of schemes like
$\operatorname{{Spec}}A_{0}$, or $X$ considered as an $X^{(1)}$-scheme. These
schemes are not smooth, but nevertheless they are pseudo-smooth: namely the
sheaf $\Omega^{1}_{X/S}$ of relative Kähler differentials is locally free. In
Section 2 we develop444In fact, in [BK] Bezrukavnikov and Kaledin freely
assume the relevant parts of the theory in the pseudo-smooth (or, in their
terminology, quasi-regular) setup, mostly without a proper justification which
sometimes leads them to small inaccuracies or even false claims. So we tried
to carefully setup the theory, mostly from scratch, only sometimes referencing
their results. some basic theory of pseudo-smooth schemes. In fact pseudo-
smoothness turns out to be enough to extend most of the standard features of
differential geometry of smooth schemes in characteristic $p$: e.g. the
algebra of differential operators is still an Azumaya algebra over the
Frobenius twist of the total space of cotangent bundle (Corollary 2.38); there
are also versions of Cartier isomorphism (Proposition 2.40) and Milne’s exact
sequence (Proposition 2.46). The only difference with the smooth setting is
that the notion of Frobenius twist should be slightly modified: namely,
instead of Frobenius twist $X^{(1)}\coloneqq X\times_{S,F_{S}}S$ one should
consider the schematic image $X^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}\hookrightarrow
X^{(1)}$ of the relative Frobenius $F_{X/S}\colon X\rightarrow X^{(1)}$ (see
Section 2.1 for more details). The corresponding map $F_{X/S}^{\\{\\!\\!\\!\
p\ \\!\\!\\!\\}}\colon X\rightarrow X^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}$ induced
by $F_{X/S}$ turns out to be finite locally free, and is again a locally
trivial bundle with fiber given by the Frobenius neighborhood of 0 in the
affine space (Lemma 2.67), however its dimension is now given by
$\operatorname{{rk}}\Omega^{1}_{X/S}$ and not the relative dimension of $X$
over $S$.

The key geometric input which allows to get a good control on pseudo-smooth
schemes is their local structure (Proposition 2.35) namely, Zariski locally
any pseudo-smooth $S$-scheme $X$ is given by the Frobenius-neighborhood of a
closed subscheme in a smooth scheme. This observation allows us to extend most
of the results from the smooth to pseudo-smooth setting almost for free.

Moreover, in Section 2.8 we recall the definition of restricted Poisson
structure by introducing a restricted Poisson algebra monad. This section is
an attempt to explain and motivate the definition of restricted Poisson
structure given in [BK]. In Section 2.9 we then discuss restricted symplectic
schemes (see Definition 2.60), where a restricted structure exteding a
symplectic form $\omega$ has an equivalent description in terms of a class
$[\eta]\in\operatorname{{coker}}(\mathcal{O}_{X}\rightarrow\Omega^{1}_{X})$
such that $d[\eta]=\omega$ (see Remark 2.61). Finally, in Section 2.10 we
endow $\operatorname{{Spec}}A_{0}$ with a restricted symplectic structure and
define the group $G_{0}\subset\underline{\operatorname{{Aut}}}(A_{0})$ of its
restricted Poisson automorphisms, that plays a very important role in most of
the constructions below.

Construction of $\operatorname{{QCoh}}_{h}$. Our goal in this paper will be to
construct a certain sheaf of categories $\operatorname{{QCoh}}_{h}$ on
$X^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}\times\mathbb{P}^{1}$, given a restricted
symplectic scheme $(X,[\eta])$. We want this construction to be functorial in
$(X,[\eta])$ and agree555Meaning that the restriction of
$\operatorname{{QCoh}}_{h}$ to the formal neighborhood of 0 in
$\mathbb{P}^{1}$ should be given by $O_{h}{\operatorname{\mathsf{-Mod}}}$.
with the distinguished quantization of Bezrukavnikov and Kaledin in the cases
when the latter exists. We will discuss other expected properties of
$\operatorname{{QCoh}}_{h}$ slightly later, in Section 1.2 below.

For the reader’s convenience let us note right away that in Section 4 we
remind the notion of a quasi-coherent sheaf of (abelian) categories on a
scheme or, more generally, an algebraic stack. We then also discuss actions of
group schemes on the sheaves of categories in Section 4.2.

Let $\mathbb{S}\coloneqq\mathbb{P}^{1}$, the idea behind the notation being
that $\mathbb{P}^{1}$ resembles a 2-dimensional sphere. In the case $X=T^{*}Y$
with the canonical 1-form $\eta$ one could expect $\operatorname{{QCoh}}_{h}$
to come as the category of modules over the sheaf of twistor differential
operators $\mathcal{D}_{Y,\mathbb{S}}$ . The latter comes as an instance of a
$\mathbb{P}^{1}$-version of the Rees construction (see Section 3.1): namely,
$\mathcal{D}_{Y,\mathbb{S}}$ corresponds to differential operators
$\mathcal{D}_{Y}$ endowed with the pair of Hodge and conjugate
filtrations666That are dual to Hodge and conjugate filtrations on the de Rham
complex in a certain sense., see Section 3.2 for more details. Below, we will
define $\operatorname{{QCoh}}_{h}$ by certain universal property, and then
check that it is indeed given by
$\mathcal{D}_{Y,\mathbb{S}}{\operatorname{\mathsf{-Mod}}}$ in the case of
$T^{*}Y$.

The starting point of our construction is the following. Namely, to a
restricted symplectic scheme $(X,[\eta])$ one canonically associates a
$G_{0}$-torsor $\mathcal{M}_{X,[\eta]}\rightarrow X^{{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}}$ of its “Darboux frames”: a $T$-point
$T\rightarrow\mathcal{M}_{X}$ is given by a restricted Poisson isomorphism

$T\times_{X^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}}X\simeq
T\times\operatorname{{Spec}}A_{0}$

(see Construction 2.68). This way a pair $(X,[\eta])$ produces a natural map

$\pi\colon X^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}\rightarrow BG_{0}.$

A natural way to define $\operatorname{{QCoh}}_{h}$ would be to take the pull-
back of a certain universal sheaf of categories over
$BG_{0}\times\mathbb{S}\simeq[\mathbb{S}/G_{0}]$ via the map
$\pi\times\operatorname{{Id}}\colon X^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}\times\mathbb{S}\rightarrow BG_{0}\times\mathbb{S}$. This is
exactly what we do.

We construct the universal sheaf of categories on $[\mathbb{S}/G_{0}]$ via
descent. Namely, we first produce a certain sheaf of algebras
$\mathcal{A}_{\mathbb{S}}$ over $\mathbb{S}$ (see Section 3.3) and then
construct an action of $G_{0}$ on the corresponding sheaf of categories over
$\mathbb{S}$ given by $\mathcal{A}_{\mathbb{S}}{\operatorname{\mathsf{-Mod}}}$
(Section 5). If $(X,[\eta])$ is $\operatorname{{Spec}}A_{0}$ equipped with the
natural restricted symplectic structure then $\operatorname{{QCoh}}_{h}$ is
equivalent $\mathcal{A}_{\mathbb{S}}{\operatorname{\mathsf{-Mod}}}$. We will
call $\mathcal{A}_{\mathbb{S}}$ the “twistor reduced Weyl algebra”.

Let us very briefly explain the construction of $\mathcal{A}_{\mathbb{S}}$.
Namely, in the even-dimensional case one can identify
$\operatorname{{Spec}}A_{0}$ with a subvariety in cotangent bundle of a
similar Frobenius neighborhood $\operatorname{{Spec}}C$, but of half the
dimension. More precisely, $\operatorname{{Spec}}A_{0}$ is just a Frobenius-
neighborhood of zero section in $T^{*}\operatorname{{Spec}}C$. The sheaf of
algebras $\mathcal{A}_{\mathbb{S}}$ is defined as the corresponding central
reduction of twistor differential operators
$\mathcal{D}_{\operatorname{{Spec}}C,\mathbb{S}}$, see Section 3.3 and
Definition 3.14 in particular. $\mathcal{A}_{\mathbb{S}}$ is a locally free
sheaf of algebras on $\mathbb{S}$ with the fiber at $\\{0\\}\in\mathbb{S}$ is
given by $A_{0}$, and with restriction to $\mathbb{S}\backslash\\{0\\}$ being
a split Azumaya algebra: this way one can view $\mathcal{A}_{\mathbb{S}}$ as a
certain order in a matrix algebra over $\mathbb{S}$.

In Section 5 we construct the desired $G_{0}$-action on
$\mathcal{A}_{\mathbb{S}}{\operatorname{\mathsf{-Mod}}}$. The key observation
is that imposing certain properties on the action automatically makes it
canonical. More precisely, in Theorem 1 we show that there is a unique
$G_{0}$-action on $\mathcal{A}_{\mathbb{S}}{\operatorname{\mathsf{-Mod}}}$
that restricts to the natural action of $G_{0}$ on
$A_{0}{\operatorname{\mathsf{-Mod}}}$ over $\\{0\\}\in\mathbb{S}$ and is
trivial over $\\{\infty\\}\in\mathbb{S}$. The proof of uniqueness crucially
uses the description of automorphisms of $\mathcal{A}_{\mathbb{S}}$ whose
restriction to $\\{0\\}\in\mathbb{S}$ is trivial, which we establish in
Section 3.6. The construction of the existence is quite elaborate and uses an
auxiliary sheaf of algebras $\mathcal{A}_{\mathbb{S}}^{\flat}$ with a natural
$G_{0}$-action (see Section 3.7), as well as a certain $G_{0}$-action on
$\mathcal{A}_{\mathbb{S}}\otimes_{\mathcal{O}_{\mathbb{S}}}\mathcal{A}_{\mathbb{S}}^{\flat,\operatorname{{op}}}{\operatorname{\mathsf{-Mod}}}$
constructed in [BV] in the formal neighborhood of 0. In Section 5.3 we also
show that the $G_{0}$-action on
$\mathcal{A}_{\mathbb{S}}{\operatorname{\mathsf{-Mod}}}$ extends to a
${\mathbb{G}}_{m}\ltimes G_{0}$-action equivariant structure where
$\mathbb{G}_{m}$ now acts non-trivially on $\mathbb{S}$. This allows to
descend $\mathcal{A}_{\mathbb{S}}{\operatorname{\mathsf{-Mod}}}$ further to a
sheaf of categories over $[\mathbb{S}/({\mathbb{G}}_{m}\ltimes G_{0})]$.

In Section 6 we discuss the proprties of the resulting sheaf of categories
$\operatorname{{QCoh}}_{h}$. In 6.1 we translate the defining properties of
our $G_{0}$-action on $\mathcal{A}_{\mathbb{S}}{\operatorname{\mathsf{-Mod}}}$
into defining properties for $\operatorname{{QCoh}}_{h}$. In Section 6.2 we
give a slightly finer construction in the presence of a
$\mathbb{G}_{m}$-action on $X$ that rescales the contact form $[\eta]$ with a
weight coprime to $p$. In this case one can descend
$\operatorname{{QCoh}}_{h}$ further to the quotient stack $[(X^{\\{\\!\\!\\!\
p\ \\!\\!\\!\\}}\times\mathbb{S})/\mathbb{G}_{m}]$ under the diagonal action,
where $\mathbb{G}_{m}$ acts on $\mathbb{S}$ by rescaling with the same weight
as above. In Section 6.3 we discuss the relation of
$\operatorname{{QCoh}}_{h}$ with the Frobenius-constant quantization $O_{h}$
constructed by Bezrukavnikov and Kaledin. Finally, in Section 6.4 we also
construct the canonical quantization of restricted Lagrangian subvarieties
extending the work of Mundinger [Mu].

###  1.2. Properties of $\operatorname{{QCoh}}_{h}$.

For simplicity, below we let $X$ be a smooth scheme over a field of
characteristic $p>2$. In this case $X^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}\simeq
X^{(1)}$, and $\operatorname{{QCoh}}_{h}$ is a sheaf of categories over
$X^{(1)}\times\mathbb{S}$; thus for each scheme $T\rightarrow
X^{(1)}\times\mathbb{S}$ over $X^{(1)}\times\mathbb{S}$ we obtain a category
$\operatorname{{QCoh}}_{h}(T)$. Let us summarize the nice properties of
$\operatorname{{QCoh}}_{h}$ that we obtain in Section 6.

  1. (1)

If we consider the $G_{0}$-torsor
$\pi\colon{\mathcal{M}}_{X,[\eta]}\rightarrow X^{(1)}$ of Darboux frames
corresponding to $[\eta]$ then the pull-back
$(\pi\times\operatorname{{id}})^{*}\operatorname{{QCoh}}_{h}$ is equivalent to
$(p_{2}^{*}{\mathcal{A}}_{\mathbb{S}}){\operatorname{\mathsf{-Mod}}}$, where
$p_{2}\colon{\mathcal{M}}\times\mathbb{S}\rightarrow\mathbb{S}$ is the
projection.

  2. (2)

The global sections $\operatorname{{QCoh}}_{h}(X^{\prime}\times\\{0\\})$ on
the fiber over $\\{0\\}\in\mathbb{S}$ are given by $\operatorname{{QCoh}}(X).$

  3. (3)

In the case $H^{i}(X,{\mathcal{O}}_{X})=0$ for $i=1,2,3$, we have an
equivalence
$\operatorname{{QCoh}}_{h}|_{X^{(1)}\times\widehat{\mathbb{S}}}\simeq
O_{h}{\operatorname{\mathsf{-Mod}}}$ where $O_{h}$ is the distinguished
Frobenius-constant quantization constructed in [BK].

  4. (4)

$\operatorname{{QCoh}}_{h}$ is functorial in $(X,[\eta])$. In particular, for
any map $T\rightarrow\mathbb{S}$ there is a natural $\mathcal{O}(T)$-linear
action of restricted Poisson automorphisms $\mathrm{Aut}((X,[\eta]))$ on the
global sections $\operatorname{{QCoh}}_{h}(X^{(1)}\times T)$.

  5. (5)

The restriction of $\operatorname{{QCoh}}_{h}$ to $X^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}\times({\mathbb{S}}\backslash\\{0\\})$ is equivalent to
${\mathcal{D}}_{X,\frac{[\eta]}{h}}{\operatorname{\mathsf{-Mod}}}$, where
${\mathcal{D}}_{X,\frac{[\eta]}{h}}$ is the central reduction of
$\mathcal{D}_{X,\mathbb{S}\backslash\\{0\\}}$ correponding to $[\eta]$ (see
Construction 3.12).

  6. (6)

The global sections $\operatorname{{QCoh}}_{h}(X^{\prime}\times\\{\infty\\})$
on the fiber over $\\{\infty\\}\in\mathbb{S}$ are given by
$\operatorname{{QCoh}}(X^{\prime}).$

  7. (7)

For a smooth Lagrangian subvariety $Y\subset X$ such that ${[\eta]|}_{Y}=0$
there exists a canonical flat object
${\mathcal{V}}_{Y^{(1)}\times\mathbb{S}}\in\operatorname{{QCoh}}_{h}(Y^{(1)}\times\mathbb{S})$
whose fiber over $Y^{(1)}\times\\{0\\}$ is given by $(\Omega_{Y}^{\dim
Y})^{\tfrac{1-p}{2}}$ (considered as a quasi-coherent sheaf on $X$).

  8. (8)

If $X$ is equipped with a $\mathbb{G}_{m}$-action which rescales $[\eta]$ with
a weight 1, then $\operatorname{{QCoh}}_{h}$ canonically descends further to
the quotient stack $(X^{(1)}\times\mathbb{S})/\mathbb{G}_{m}$ where
$\mathbb{G}_{m}$ acts diagonally.

######  Remark 1.1.

In the case $X=T^{*}Y$ and $\eta$ is the canonical 1-form on $T^{*}Y$, the
global sections $\operatorname{{QCoh}}_{h}(X^{(1)}\times\mathbb{S})$ can be
alternatively described as the category of modules over the sheaf of twisted
differential operators $\mathcal{D}_{X,\mathbb{S}}$.

Also, from the above picture one gets the following properties

  * •

$\operatorname{{QCoh}}_{h}(X^{(1)}\times\\{0\\})$ is given by the category of
Higgs fields on $Y$;

  * •

$\operatorname{{QCoh}}_{h}(X^{(1)}\times\\{1\\})$ can be identified777Using
the Morita equivalence ${\mathcal{D}}_{X,[\eta]}\sim{\mathcal{D}}_{Y}$ ([BB]).
with the category ${\mathcal{D}}_{Y}{\operatorname{\mathsf{-Mod}}}$ of
$D$-modules on $Y$;

  * •

$\operatorname{{QCoh}}_{h}(X^{(1)}\times\\{\infty\\})$ is the category of
Higgs fields on the Frobenius twist $Y^{(1)}$.

This resembles the construction of the twistor space due to Deligne and
Simpson: namely, given a smooth projective variety $Y$ over $\mathbb{C}$ they
construct a complex-analytic space $W\rightarrow{\mathbb{P}}^{1}$ whose fiber
over $\lambda\in{\mathbb{P}}^{1}({\mathbb{C}})$ is the “moduli space” of
$\lambda$-connections on $Y$. In particular, fibers $W_{0}$, $W_{1}$ and
$W_{\infty}$ over $0$, $1$ and $\infty\in{\mathbb{P}}^{1}({\mathbb{C}})$ are
given by

  * •

holomorphic Higgs fields on $Y$;

  * •

bundles with holomorphic flat connections on $Y$;

  * •

holomorphic Higgs fields on the complex conjugate $\overline{Y}$

correspondingly. This way the category $\operatorname{{QCoh}}_{h}$ can be
considered as a categorified characteristic $p$ analogue of their
construction. This motivates calling $\operatorname{{QCoh}}_{X,[\eta],h}$ the
twistor category associated to $(X,[\eta])$.

###  1.3. Acknowledgments

The authors are grateful to Dmitry Kaledin for his constant attention to this
work and to Alexander Efimov for fruitful discussions of the categorical
aspects of the paper. We would also like to thank Peter Scholze for bringing
up the question about an additional $\mathbb{G}_{m}$-equivariant structure on
$\operatorname{{QCoh}}_{h}$. The third author wishes to thank Boris Feigin for
introducing to him the ideas of Gelfand-Kazhdan “formal geometry” back in the
year of $2005$.

The work of the last author was supported in part by RNF grant No
$21-11-00153$.

##  2\. Differential geometry on pseudo-smooth schemes

###  2.1. Pseudo-smooth schemes

Let $B$ be an $\mathbb{F}_{p}$-algebra. Following [BK] we consider the
following notion.

######  Definition 2.1.

A finite-type $B$-algebra $A$ is called pseudo-smooth if $\Omega^{1}_{A/B}$ is
a locally free $A$-module. A qcqs morphism $X\rightarrow S$ locally of finite
type is called pseudo-smooth if $\Omega^{1}_{X/S}$ is a locally free sheaf of
finite type.

######  Remark 2.2.

In [BK], Bezrukavnikov and Kaledin call888Strictly speaking they also assume
that $A/B$ is flat and $B$ is Noetherian. such algebras quasi-regular.
However, since terms quasi-regular and quasi-smooth are quite broadly used
with a different meaning, we decided to change it to pseudo-smooth.

######  Example 2.3.

  1. (1)

Smooth schemes are pseudo-smooth.

  2. (2)

Let $A\coloneqq B[x]/x^{p}$. Then $\Omega^{1}_{A/B}=A\cdot dx$ and $A$ is
pseudo-smooth.

  3. (3)

Let $Z\hookrightarrow S$ be a closed subscheme. Then $\Omega^{1}_{Z/S}=0$ and
$Z$ is pseudo-smooth over $S$.

######  Remark 2.4.

One might count Example 2.3(3) as a pathological one and impose flatness
condition in Definition 2.1. This is relatively harmless for our goals:
namely, pseudo-smooth schemes that will appear in main applications (e.g.
quantizations of smooth schemes) will all be flat.

######  Remark 2.5.

Pseudo-smoothness is preserved under base change: namely, having a pseudo
smooth $S$-scheme $X\rightarrow S$ and a morphism $S^{\prime}\rightarrow S$
the base change $X_{S^{\prime}}=X\times_{S}S^{\prime}$ is pseudo smooth over
$S^{\prime}$. Indeed one has $\Omega^{1}_{X/S^{\prime}}=f^{*}\Omega^{1}_{X/S}$
where $f\colon X_{S^{\prime}}\rightarrow X$ is the natural map.

######  Construction 2.6 (Reduced Frobenius: algebras).

Let $A$ be a $B$-algebra. The absolute Frobenius map $F_{A}\colon A\rightarrow
A$ factors naturally through the Frobenius twist $A^{(1)}\coloneqq
A\otimes_{B,F_{B}}B$:

$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F_{A}}$$\scriptstyle{W_{A/B}}$$\textstyle{A}$$\textstyle{A^{(1)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F_{A/B}}$

with $W_{A/B}\colon a\mapsto a\otimes 1\in A^{(1)}$ and $F_{A/B}\colon
a\otimes b\mapsto a^{p}b\in A$. The map $F_{A/B}$ is usually called the
relative Frobenius; by construction it is $B$-linear (while $F_{A}$ and
$W_{A/B}$ are $F_{B}$-linear). However, if $B$ or $A$ are not reduced one can
factor $F_{A/B}$ even further. Namely, let $A^{{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}}\subset A$ be the image of $F_{A/B}$: this is the $B$-subalgebra
in $A$ generated by $p$-th powers $A^{p}$, in other words $A^{{\\{\\!\\!\\!\
p\ \\!\\!\\!\\}}}\coloneqq B\cdot A^{p}\subset A$. By definition, one has a
natural surjection $F_{A}^{{int}}\colon A^{(1)}\twoheadrightarrow
A^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}$. Moreover, we get a further factorization

$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F_{A}}$$\scriptstyle{W_{A/B}}$$\textstyle{A}$$\textstyle{A^{(1)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F_{A/B}}$$\scriptstyle{F_{A}^{int}}$$\textstyle{A^{\\{\\!\\!\\!\
p\
\\!\\!\\!\\}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F_{A}^{\\{\\!\\!\\!\
p\ \\!\\!\\!\\}}}$

We have $F_{A/B}=F_{A}^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}\circ F_{A}^{int}$ and
we can put $W_{A}^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}\coloneqq F_{A}^{int}\circ
W_{A/S}$. We will call $F_{A}^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}$ and
$F_{A}^{int}$ the reduced and intermediate Frobenii of $A$ correspondingly.

In the case when both $A$ and $B$ are reduced, $F_{A}^{int}$ is an isomorphism
and so $F_{A/B}=F_{A}^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}$. However, we will be
primarily interested in the non-reduced setting, where $A^{{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}}$ will turn out to be more relevant than $A^{(1)}$.

######  Remark 2.7.

Let us try to motivate consideration of the reduced Frobenius as compared to
the relative one. The main reason is that when $A$ is pseudo-smooth,
$F_{A}^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}$ still enjoys some properties that the
relative Frobenius $F_{A/B}$ has if we assume that $A$ is actually smooth. For
example, we will see further that under the pseudo-smoothness assumption (see
Remark 2.39) $F_{A}^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}$ is a finite faithfully
flat map, which is no longer true for the relative Frobenius $F_{A/B}$ in this
generality. This is illustrated well by the example of $A\coloneqq
B[x]/x^{p}$, where one has $A^{(1)}\simeq B[x]/x^{p}$ and where the relative
Frobenius $F_{A/B}\colon B[x]/x^{p}\rightarrow B[x]/x^{p}$ is the unique
$B$-algebra map that sends $x$ to 0. This map is clearly not flat. On the
other hand, $A^{{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}}\subset A$ is identified with
$B\subset B[x]/x^{p}$ and the reduced Frobenius $F_{A}^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}\colon A^{{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}}\rightarrow A$ is given
by the above embedding which is indeed finite and faithfully flat.

######  Remark 2.8.

In [BK], for a pseudo-smooth $A$ it is claimed that $A^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}$ can be identified with the tensor product
$A^{p}\otimes_{B^{p}}B$ via the natural map $a^{p}\otimes b\mapsto a^{p}b$.
Unfortunately, this is not exactly true. Put $B\coloneqq\mathbb{F}_{p}[x]$ and
$A=B[t]/(t^{p}-x)$. One checks easily that $\Omega^{1}_{A/B}\simeq A\cdot dt$,
and so $A$ is pseudo-smooth. On the other hand
$B^{p}\simeq\mathbb{F}_{p}[x^{p}]$, $A^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}\simeq
A^{p}\simeq B$, and so the natural map $A^{p}\otimes_{B^{p}}B\rightarrow
A^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}$ is identified with multiplication map
$\mathbb{F}_{p}[x]\otimes_{\mathbb{F}_{p}[x^{p}]}\mathbb{F}_{p}[x]\rightarrow\mathbb{F}_{p}[x]$
which is not an isomorphism.

######  Remark 2.9.

Let $X=\operatorname{{Spec}}A$. By [Stacks, Tag0BR8], $F_{X}^{{\\{\\!\\!\\!\
p\ \\!\\!\\!\\}}}\colon X\rightarrow X^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}\coloneqq\operatorname{{Spec}}A^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}$
is a universal homeomorphism. In particular, it induces an isomorphism of the
underlying topological spaces. In fact all three schemes $X\rightarrow
X^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}\hookrightarrow
X^{(1)}\coloneqq\operatorname{{Spec}}A^{(1)}$ have the same underlying
topological space and the difference between them is only seen on the level of
sheaves of functions.

######  Construction 2.10 (Reduced Frobenius: schemes).

Given a base scheme $S$ and an $S$-scheme $X$, the construction $A\rightarrow
A^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}$ globalizes, producing an $S$-scheme
$X^{{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}}$. This is a scheme which has the same
underlying topological space as the original $X$ and its Frobenius twist
$X^{(1)}\coloneqq X\times_{S,F_{S}}S$, but with the sheaf of functions given
by $\mathcal{O}_{X}^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}\coloneqq\mathcal{O}_{X}^{p}\cdot\mathcal{O}_{S}\subset\mathcal{O}_{X}$.
One has maps $\mathcal{O}_{X}^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}\hookrightarrow\mathcal{O}_{X}$ and
$\mathcal{O}_{X^{(1)}}\twoheadrightarrow\mathcal{O}_{X}^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}$ (given locally by $F_{A}^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}$ and
$F_{A}^{int}$) which induce a map $F_{X}^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}\colon
X\rightarrow X^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}$ and a closed embedding
$F_{X}^{int}\colon X^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}\hookrightarrow X^{(1)}$.
This way one can also view $X^{{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}}$ as the
schematic image of the relative Frobenius $F_{X/S}\colon X\rightarrow X^{(1)}$
inside the Frobenius twist $X^{(1)}\coloneqq X\times_{S,F_{S}}S$. We will call
$F_{X}^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}$ and $F_{X}^{int}$ the reduced and
intermediate Frobenii of $X$; both are morphisms of $S$-schemes. We can also
define a reduced twist map $W_{X}^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}\colon
X^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}\rightarrow X$ (by globalizing
$W_{A}^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}$); this map is not a morphism of
$S$-schemes, instead it is $F_{S}$-linear. One has relations $F_{X}^{int}\circ
F_{X}^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}=F_{X/S}$ and $W_{X/S}\circ
F_{X}^{int}=W_{X}^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}$.

######  Example 2.11.

Assume $S=\operatorname{{Spec}}B$.

  1. (1)

Put $P\coloneqq B[x_{1},\ldots,x_{d}]$ with
$\operatorname{{Spec}}P\simeq\mathbb{A}^{d}_{S}$. Then $P^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}\coloneqq P^{p}\cdot B\simeq B[x_{1}^{p},\ldots,x_{d}^{p}]\subset
P$. On the other hand the Frobenius twist $P^{(1)}\coloneqq
P\otimes_{B,F_{B}}B\simeq B[y_{1},\ldots,y_{d}]$ (with $y_{i}\coloneqq
x_{i}\otimes 1$), and the relative Frobenius $F_{P/B}\colon P^{(1)}\rightarrow
P$ is the unique $B$-algebra map that sends $y_{i}$ to $x_{i}^{p}$. We see
that in this case the intermediate Frobenius
$F_{\mathbb{A}^{d}_{S}}^{int}\colon(\mathbb{A}^{d}_{S})^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}\rightarrow(\mathbb{A}^{d}_{S})^{(1)}$ is an isomorphism.

  2. (2)

Consider the unique surjection $P\twoheadrightarrow B$ of $B$-algebras sending
every $x_{i}$ to 0. It corresponds to the embedding
$\\{0\\}_{S}\hookrightarrow\mathbb{A}^{d}_{S}$ of 0. Put

$Q\coloneqq B\otimes_{P,F^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}_{P}}P\simeq
P/(x_{1}^{p},\ldots,x_{d}^{p}).$

$X\coloneqq\operatorname{{Spec}}Q\hookrightarrow\mathbb{A}^{d}_{S}$ is the
Frobenius neighborhood of 0: namely,
$X\simeq\\{0\\}_{S}\times_{\mathbb{A}^{d}_{S},F^{{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}}_{\mathbb{A}^{d}}}\mathbb{A}^{d}_{S}$. We have $Q^{\\{\\!\\!\\!\
p\ \\!\\!\\!\\}}\coloneqq Q^{p}\cdot B\simeq B\subset Q$. However,
$Q^{(1)}\coloneqq Q\otimes_{B,F_{B}}B\simeq
B[y_{1},\ldots,y_{d}]/(y_{1}^{p},\ldots,y_{d}^{p})$, and the intermidiate
Frobenius $F_{Q}^{int}\colon Q^{(1)}\rightarrow Q^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}\simeq B$ sends $y_{i}$ to $x_{i}^{p}=0$.

######  Remark 2.12.

Since $F_{X}^{{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}}\colon X\rightarrow
X^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}$ is a universal homeomorphism, say by
[Stacks, Tag 0BTY] it induces an equivalence of small étale sites
$(X^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}})_{\text{et}}\simeq X_{\text{et}}$, by

$(U\rightarrow X^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}})\in(X^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}})_{\text{et}}\ \ \mapsto\ \ (X\times_{X^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}}U\rightarrow X)\in X_{\text{et}}.$

###  2.2. Restricted Lie algebras and Jacobson’s formula

Let $A$ be an associative ring such that $p\cdot A=0$. For $a\in A$ denote by
$\operatorname{{ad}}(a)\colon A\rightarrow A$ a map that sends
$b\mapsto[a,b]=ab-ba$.

If $A$ is commutative, given two elements $a,b\in A$ we have
$(a+b)^{p}=a^{p}+b^{p}$. This formula generalizes to the general non-
commutative case as follows

######  Lemma 2.13 (Jacobson, [GD, Chapter II, §7, Proposition 3.2]).

Let $a,b\in A$. Then

$(a+b)^{p}=a^{p}+b^{p}+\sum_{i=1}^{p-1}L_{i}(a,b)$

where $L_{i}(a,b)$ is a Lie polynomial described as $i^{-1}$ times the
coefficient of $t^{i-1}$ in the expression
$\operatorname{{ad}}(a+tb)^{p-1}(b)\in A[t]$.

######  Remark 2.14.

In particular, $L_{1}(a,b)=\operatorname{{ad}}(a)^{p-1}(b)$.

######  Remark 2.15.

For an element $a\in A$ denote by $\operatorname{{ad}}(a)\colon A\rightarrow
A$ the commutator map $x\mapsto[a,x]$. Note that
$\operatorname{{ad}}(a)^{p}=\operatorname{{ad}}(a^{p})$. Indeed,
$\operatorname{{ad}}(a)=\ell(a)-r(a)$ where $\ell(a),r(a)\colon A\rightarrow
A$ are left and right multiplications by $a$ correspondingly. Since $\ell(a)$
and $r(a)$ commute we have

$\operatorname{{ad}}(a)^{p}=(\ell(a)-r(a))^{p}=\ell(a)^{p}-r(a)^{p}=\ell(a^{p})-r(a^{p})=\operatorname{{ad}}(a^{p}).$

We now remind the definition of a restricted Lie algebra.

######  Definition 2.16.

A restricted Lie algebra over a ring $B$ is given by a $B$-module $L$ with a
$B$-linear Lie bracket $[-,-]$ and a restricted $p$-power operation $-^{[p]}$
that are compatible in the following way:

  1. (1)

$(bx)^{[p]}=b^{p}x^{[p]}$ for any $b\in B$ and $x\in L$;

  2. (2)

$[x^{[p]},y]=\operatorname{{ad}}(x)^{p}(y)$ for any $x,y\in L$;

  3. (3)

$(x+y)^{[p]}=x^{[p]}+y^{[p]}+\sum_{i=1}^{p-1}L_{i}(x,y)$, where $L_{i}(x,y)$
are the Lie polynomials from Lemma 2.13.

The definition of restricted Lie algebra is mainly motivated by the following
example:

######  Example 2.17.

Let $A$ be an associative $B$-algebra. Then by Lemma 2.13 and Remark 2.15, $A$
endowed with the Lie bracket given by commutator $[-,-]$ and the restricted
structure $a^{[p]}\coloneqq a^{p}$ is a restricted Lie algebra.

Another natural source of examples of restricted Lie algebras is differential
geometry in char $p$ (see Remark 2.19 below). In particular, Lie algebra of
any group scheme $G$ is endowed with a natural restricted Lie algebra
structure.

###  2.3. Crystalline differential operators

We essentially follow [BMR, Section 2], except we work in a relative and
pseudo-smooth setting.

Let $X\rightarrow S$ be a pseudo-smooth $S$-scheme. Let
$\mathcal{T}_{X}\coloneqq\mathcal{T}_{X/S}$ be the sheaf of
$\mathcal{O}_{S}$-linear differentiations of $\mathcal{O}_{X}$. We will
ocasionally call (local) sections of $\mathcal{T}_{X}$ vector fields.

Denote $\Omega^{1}_{X}\coloneqq\Omega^{1}_{X/S}$. One has
$\mathcal{T}_{X}\simeq(\Omega^{1}_{X})^{\vee}$, and, since $X$ is pseudo-
smooth, we get that $\mathcal{T}_{X}$ is a locally free
$\mathcal{O}_{X}$-module of finite rank.

The sheaf $\mathcal{T}_{X}$ has a natural ($\mathcal{O}_{S}$-linear) Lie-
bracket given by commutator999Namely, the commutator of two differentiations
is again a differentiation. which endows it with the structure of a Lie
algebroid over $X$. Let $\mathcal{D}_{X}$ be the universal enveloping algebra
of $\mathcal{T}_{X}$. Explicitly, $\mathcal{D}_{X}$ is generated over
$\mathcal{O}_{S}$ by $\mathcal{O}_{X}$ and $\mathcal{T}_{X}$ with the
relations given by $f_{1}\cdot f_{2}-f_{2}\cdot f_{1}=0$ for
$f_{i}\in\mathcal{O}_{X}$, $v\cdot f-f\cdot v=v(f)\in\mathcal{O}_{X}$ for
$v\in\mathcal{T}_{X}$ and $f\in\mathcal{O}_{X}$, and $v_{1}\cdot
v_{2}-v_{2}\cdot v_{1}=[v_{1},v_{2}]\in\mathcal{T}_{X}$ for
$v_{i}\in\mathcal{T}_{X}$. A data of $\mathcal{O}_{X}$-quasicoherent
$\mathcal{D}_{X}$-module is equivalent to what is usually called a “D-module”
on $X$: namely, a quasicoherent sheaf $\mathcal{E}$ endowed with a flat
connection
$\nabla\colon\mathcal{E}\rightarrow\mathcal{E}\otimes\Omega^{1}_{X}$.

######  Remark 2.18.

The algebra $\mathcal{D}_{X}$ comes with a natural increasing “PBW-filtration”
which is identified with the filtration by order of the differential operator.
Namely, $\mathcal{D}_{X,\leq n}\subset\mathcal{D}_{X}$ is the
$\mathcal{O}_{X}$-submodule locally generated by the image of the
multiplication map
$\underbrace{\mathcal{T}_{X}\otimes_{\mathcal{O}_{S}}\ldots\otimes_{\mathcal{O}_{S}}\mathcal{T}_{X}}_{n}\rightarrow\mathcal{D}_{X}$.
Note that since $[\mathcal{T}_{X},\mathcal{T}_{X}]\subset\mathcal{T}_{X}$ we
have $[\mathcal{D}_{X,\leq i},\mathcal{D}_{X,\leq
j}]\subset\mathcal{D}_{X,\leq i+j-1}$. By a version of PBW-theorem for Lie
algebroids ref one has a natural isomorphism

$\operatorname{{gr}}_{*}\mathcal{D}_{X}\simeq\operatorname{{Sym}}^{*}_{\mathcal{O}_{X}}\mathcal{T}_{X}.$

Consequently, if we have a trivialization
$\mathcal{T}_{X}\simeq\mathcal{O}_{X}^{d}$ given by a frame of vector fields
$\partial_{1},\ldots,\partial_{d}$ we can decompose

$\mathcal{D}_{X}\simeq\oplus_{\alpha\in\mathbb{N}^{d}}\mathcal{O}_{X}\cdot\partial^{\alpha}$

as a left $\mathcal{O}_{X}$-module. Here, for $\alpha\in\mathbb{N}^{d}$,
$\partial^{\alpha}$ denotes the monomial
$\partial_{1}^{\alpha_{1}}\ldots\partial_{d}^{\alpha_{d}}$. In terms of this
decomposition $\mathcal{D}_{X,\leq
n}\simeq\oplus_{\alpha\in\mathbb{N}^{d}}\mathcal{O}_{X}\cdot\partial^{\alpha}$
with

$|\alpha|=\alpha_{1}+\ldots+\alpha_{d}\leq n.$

######  Remark 2.19 (Restricted Lie algebra structure on $\mathcal{T}_{X}$).

In char $p$, $\mathcal{T}_{X}$ is naturally a sheaf of restricted Lie algebras
over $\mathcal{O}_{S}$. Namely, note that the $p$-th iterate of an
$\mathcal{O}_{S}$-linear derivation $v\in\mathcal{T}_{X}$ again defines a
$\mathcal{O}_{S}$-linear derivation of $\mathcal{O}_{X}$, which we will denote
$v^{[p]}\in\mathcal{T}_{X}$. Since $v^{[p]}$ is equal to the image of $v^{p}$
in the $\mathcal{O}_{S}$-linear endomorphisms
$\mathcal{E}nd_{\mathcal{O}_{S}}(\mathcal{O}_{X})$ and the Lie bracket on
$\mathcal{T}_{X}$ is induced by the commutator in
$\mathcal{E}nd_{\mathcal{O}_{S}}(\mathcal{O}_{X})$, one sees from Example 2.17
that the relations in Definition 2.16 are satisfied.

The algebra $\mathcal{D}_{X}$ naturally acts on $\mathcal{O}_{X}$ by
$\mathcal{O}_{S}$-linear endomorphisms: namely, $\mathcal{O}_{X}$ acts by
multiplication, while $\mathcal{T}_{X}$ acts by the corresponding
$\mathcal{O}_{S}$-linear derivations. Note that any $\mathcal{O}_{S}$-linear
derivation acts by zero on all functions in $\mathcal{O}_{X}^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}=\mathcal{O}_{X}^{p}\cdot\mathcal{O}_{S}\subset\mathcal{O}_{X}$.
It follows that $\mathcal{O}_{X}^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}\subset\mathcal{O}_{X}$ lies in the center $Z(\mathcal{D}_{X})$,
and so one can consider $\mathcal{D}_{X}$ as a sheaf of
$\mathcal{O}_{X}^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}$-algebras. By adjunction,
from the identification $(F_{X}^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}})^{-1}\mathcal{O}_{X^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}}\simeq\mathcal{O}_{X}^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}$, we get a
map $\mathcal{O}_{X^{\\!{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}}}\rightarrow
Z(F_{X*}^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}\mathcal{D}_{X})$ and this way we can
consider $F_{X*}^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}\mathcal{D}_{X}$ as a sheaf of
algebras over $X^{{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}}$.

######  Remark 2.20.

By construction, the map $F_{X}^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}\colon
X\rightarrow X^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}$ is affine. Consequently, one
has a monoidal equivalence of categories of quasi-coherent sheaves on $X$ and
quasi-coherent sheaves of $F_{X*}^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}\mathcal{O}_{X}$-modules on $X^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}$.
This then gives an equivalence of categories of quasi-coherent
$\mathcal{D}_{X}$-modules on $X$ and quasi-coherent $F_{X*}^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}\mathcal{D}_{X}$-modules on $X^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}$.

######  Remark 2.21.

Note that by base change, $\Omega^{1}_{X^{(1)}}\simeq
W_{X/S}^{*}\Omega^{1}_{X}\coloneqq\Omega^{1}_{X}\otimes_{\mathcal{O}_{X}}\mathcal{O}_{X^{(1)}}$.
In particular, if $X$ is pseudo-smooth, then so is $X^{(1)}$. We warn that
this is not necessarily true for $X^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}$ (see
Remark 2.33).

Let us also define the appropriate variants of twisted tangent and cotangent
bundles.

######  Construction 2.22 (Twisted tangent and cotangent bundles).

For the usual Frobenius twist, one can consider
$\Omega^{1}_{X^{(1)}}\simeq\mathcal{T}_{X}\otimes_{\mathcal{O}_{X}}\mathcal{O}_{X^{(1)}}$
and $\mathcal{T}_{X^{(1)}}\simeq
W_{X/S}^{*}\mathcal{T}_{X}\coloneqq\mathcal{T}_{X}\otimes_{\mathcal{O}_{X}}\mathcal{O}_{X^{(1)}}$.
Analogously, we define the (reduced Frobenius) twisted tangent and cotangent
bundles $\mathcal{T}_{X}^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}\coloneqq
W_{X}^{{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}*}\mathcal{T}_{X}\coloneqq\mathcal{T}_{X}\otimes_{\mathcal{O}_{X}}\mathcal{O}_{X}^{{\\{\\!\\!\\!\
p\ \\!\\!\\!\\}}}$ and $\Omega_{X}^{1,{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}}\coloneqq\Omega_{X}^{1}\otimes_{\mathcal{O}_{X}}\mathcal{O}_{X}^{{\\{\\!\\!\\!\
p\ \\!\\!\\!\\}}}$ (here the map
$\mathcal{O}_{X}\rightarrow\mathcal{O}_{X}^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}$
sends $f\mapsto f^{p}$). One has $\Omega_{X}^{1,{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}}\simeq(\mathcal{T}_{X}^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}})^{\vee}$.
We warn that it is no longer necessarily true that
$\mathcal{T}_{X}^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}$ (resp.
$\Omega_{X}^{1,{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}}$) is isomorphic to
$\mathcal{T}_{X^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}}$ (resp.
$\Omega^{1}_{X^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}}$), e.g. since
$X^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}$ can happen to be not smooth. Nevertheless
there is still some natural compatibility (see Corollary 2.24).

For a vector bundle $\mathcal{E}$ on a scheme $X$ let
$\mathrm{Tot}_{X}(\mathcal{E})\coloneqq\operatorname{{Spec}}_{X}\operatorname{{Sym}}^{*}_{\mathcal{O}_{X}}\mathcal{E}^{\vee}$
be the total space of $\mathcal{E}$. In the case $\mathcal{E}$ is
$\mathcal{T}_{X}$ or $\Omega^{1}_{X}$ we denote it by $TX$ and $T^{*}\\!X$
correspondingly. We denote by $\mathcal{E}^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}\coloneqq W_{X}^{{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}*}\mathcal{E}\simeq\mathcal{E}\otimes_{\mathcal{O}_{X}}\mathcal{O}_{X}^{\\{\\!\\!\\!\
p\ \\!\\!\\!\\}}$ the pull-back of $\mathcal{E}$ to $X^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}$.

######  Lemma 2.23.

One has a natural isomorphism

$\mathrm{Tot}_{X^{\\!{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}}}(\mathcal{E}^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}})=\left(\mathrm{Tot}_{X}(\mathcal{E})\right)^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}.$

###### Proof.

Note that $\mathrm{Tot}_{X^{\\!{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}}}(\mathcal{E}^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}})\simeq\mathrm{Tot}_{X}(\mathcal{E})\times_{X}X^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}$ under the map $W_{X}^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}\colon
X^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}\rightarrow X$. Indeed,
$\operatorname{{Sym}}_{\mathcal{O}_{X}^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}}^{*}(\mathcal{E}^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}})\simeq(\operatorname{{Sym}}_{\mathcal{O}_{X}}^{*}\mathcal{E})\otimes_{\mathcal{O}_{X}}\mathcal{O}_{X}^{\\{\\!\\!\\!\
p\ \\!\\!\\!\\}}$ and the isomorphism is obtained by taking
$\operatorname{{Spec}}_{X^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}}$ of both sides. By
taking $W_{\mathrm{Tot}_{X}(\mathcal{E})}^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}\colon\left(\mathrm{Tot}_{X}(\mathcal{E})\right)^{\\{\\!\\!\\!\
p\ \\!\\!\\!\\}}\rightarrow\mathrm{Tot}_{X}(\mathcal{E})$ and the projection
$\left(\mathrm{Tot}_{X}(\mathcal{E})\right)^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}\rightarrow X^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}$ we obtain a map

$\left(\mathrm{Tot}_{X}(\mathcal{E})\right)^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}\rightarrow\mathrm{Tot}_{X}(\mathcal{E})\times_{X}X^{\\{\\!\\!\\!\
p\ \\!\\!\\!\\}}\simeq\mathrm{Tot}_{X^{\\!{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}}}(\mathcal{E}^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}).$

To check that it is an isomorphism we can work locally on $X$ and $S$. So we
can assume $X=\operatorname{{Spec}}A$, $Y=\operatorname{{Spec}}B$ and
$\mathcal{E}$ is trivial. Then we need to check that the natural map

$A[x_{1},\ldots,x_{n}]\otimes_{A}A^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}\rightarrow(A[x_{1},\ldots,x_{n}])^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}$

given by $f\otimes s\mapsto f^{p}s$ is an isomorphism. But it is clear that
the right hand side is the polynomial ring on $x_{1}^{p},\ldots,x_{n}^{p}$
over $A^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}$ and so this map is indeed an
isomorphism. ∎

Plugging $\mathcal{E}$ to be $\mathcal{T}_{X}$ or $\Omega^{1}_{X}$ we get

######  Corollary 2.24.

One has isomorphisms

$\mathrm{Tot}_{X^{\\!{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}}}(\mathcal{T}_{X}^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}})\simeq(TX)^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}\text{ and
}\mathrm{Tot}_{X^{\\!{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}}}(\Omega_{X}^{1,{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}})\simeq(T^{*}\\!X)^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}.$

###  2.4. Differential operators on smooth schemes

For this section we assume that $X$ is a smooth scheme over $S$. We start with
identifying the reduced Frobenius in this setting.

Assume that $X=\mathbb{A}^{d}_{S}$ is the affine space over $S$ of dimension
$d$. Then locally over $S$ we have $S=\operatorname{{Spec}}B$ and
$X=\operatorname{{Spec}}B[x_{1},\ldots,x_{d}]$. Let $P\coloneqq
B[x_{1},\ldots,x_{d}]$. We have $P^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}=B[x_{1}^{p},\ldots,x_{d}^{p}]\subset P$ and it is clear that $P$
is a free $P^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}$-module of rank $p^{d}$. In
particular, we see that $F_{X}^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}\colon
X\rightarrow X^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}$ is a finite faithfully flat
map.

Moreover, following Example 2.11(1), $F_{P}^{int}$ induces an isomorphism
$P^{(1)}\xrightarrow{\sim}P^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}$. This shows that
$F_{X}^{int}\colon X^{(1)}\xrightarrow{\sim}X^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}$
is an isomorphism, and that under this identification $F_{X}^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}=F_{X/S}$.

In general smooth setting, Zariski locally $X$ has an étale map $q\colon
X\rightarrow\mathbb{A}^{d}_{S}$ for some $d$. Since $q$ is étale, the relative
Frobenius $F_{X/\mathbb{A}^{d}_{S}}$ is an isomorphism, and so one has a
fibered square

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Since $q$ is flat, the pull-back
$F_{X/S}^{-1}(\mathcal{O}_{X^{(1)}})\rightarrow\mathcal{O}_{X}$ is locally
injective, and so $\mathcal{O}_{X}^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}\simeq
F_{X/S}^{-1}(\mathcal{O}_{X^{(1)}})$. This shows that $F_{X}^{int}\colon
X^{(1)}\xrightarrow{\sim}X^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}$ is an isomorphism
and $F_{X}^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}=F_{X/S}$, as in the case of affine
space. From the latter we also get that $F_{X}^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}$ is a finite faithfully flat map of rank $p^{\dim_{S}X}$.

Given the above natural identification
$X^{(1)}\xrightarrow{\sim}X^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}$, it follows that
in the smooth case the vector bundle $\mathcal{T}_{X}^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}$ can be identified with $\mathcal{T}_{X^{(1)}}$ and,
consequently, $(T^{*}X)^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}$ can also be
identified with $T^{*}\\!X^{(1)}$.

######  Construction 2.25 ($p$-curvature map).

Let $X$ be a smooth $S$-scheme. Consider the map
$\theta\coloneqq\mathcal{T}_{X}\rightarrow\mathcal{D}_{X}$ of sheaves on $X$
given by the following formula

$v\mapsto\theta(v)\coloneqq v^{p}-v^{[p]},$

where $-^{[p]}$ is the restricted power (see Remark 2.19). We claim that

  * •

$\theta$ is additive: $\theta(v_{1}+v_{2})-(\theta(v_{1})+\theta(v_{2}))=0$;

  * •

$\theta$ is Frobeinus-linear: $\theta(fv)-f^{p}\cdot\theta(v)=0$;

  * •

$\theta(v)$ lies in the center $Z(\mathcal{D}_{X})$ for any
$v\in\mathcal{T}_{X}$: $[\theta(v),f]=[\theta(v),w]=0$ for any
$f\in\mathcal{O}_{X}$ and $w\in\mathcal{T}_{X}$.

Indeed, note that $\theta(v)\in\mathcal{D}_{X,\leq p}$ and that its image in
$\operatorname{{gr}}_{p}\mathcal{D}_{X}\simeq\operatorname{{Sym}}^{p}_{\mathcal{O}_{X}}(\mathcal{T}_{X})$
is $v^{p}$. It is easy101010For all cases except $[\theta(v),w]$ this follows
from the fact that the associated graded of $\mathcal{D}_{X}$ is commutative.
For $[\theta(v),w]$ one can use that $[-,w]$ induces a derivation of
$\operatorname{{gr}}_{*}\mathcal{D}_{X}$ by sending
$x\in\operatorname{{gr}}_{i}\mathcal{D}_{X}$ to the class of
$[\tilde{x},w]\in\mathcal{D}_{X,\leq i}$ in
$\operatorname{{gr}}_{i}\mathcal{D}_{X}$, and that any derivation is 0 on the
$p$-th power $v^{p}$. to see that all expressions above have 0 image in
$\operatorname{{gr}}_{p}\mathcal{D}_{X}$, and so belong to
$\mathcal{D}_{X,\leq p-1}$. Thus, by Lemma 2.26 below, they are zero if and
only of their images in $\mathcal{E}nd_{\mathcal{O}_{S}}(\mathcal{O}_{X})$
are. However, $\theta(v)$ acts by 0 on any function $f$ in $\mathcal{O}_{X}$
since by definition $v^{[p]}$ acts as $v^{p}$, and so the image of all of the
expressions above in $\mathcal{E}nd_{\mathcal{O}_{S}}(\mathcal{O}_{X})$ is
identically 0.

This way we obtain a map

$\theta^{\prime}\colon\mathcal{T}_{X}\otimes_{\mathcal{O}_{X}}\mathcal{O}_{X}^{p}\rightarrow
Z(\mathcal{D}_{X})$

of $\mathcal{O}_{X}^{p}$-modules. Since $Z(\mathcal{D}_{X})$ is in fact an
$\mathcal{O}_{X}^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}\simeq\mathcal{O}_{X}^{p}\cdot\mathcal{O}_{S}$-algebra we can
extend this to a map
$\mathcal{T}_{X}\otimes_{\mathcal{O}_{X}}\mathcal{O}_{X}^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}\rightarrow Z(\mathcal{D}_{X})$ of
$\mathcal{O}_{X}^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}$, or in other words a map of
sheaves on $X^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}$

(2.1) $\theta\colon\mathcal{T}_{X}^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}\rightarrow
Z(F_{X*}^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}\mathcal{D}_{X}).$

which we call the “$p$-curvature map”.

######  Lemma 2.26.

Let $X$ be smooth over $S$. Then the natural $\mathcal{O}_{S}$-linear action
of $\mathcal{D}_{X}$ on $\mathcal{O}_{X}$ induces an embedding
$\mathcal{D}_{X,\leq
p-1}\hookrightarrow\mathcal{E}nd_{\mathcal{O}_{S}}(\mathcal{O}_{X})$.

###### Proof.

The statement is Zariski-local, so we can assume that we have an étale map
$f\colon X\rightarrow\mathbb{A}^{n}_{S}$ given by functions
$f_{1},\ldots,f_{n}$. Let $z_{1},\ldots,z_{n}$ be the coordinates on
$\mathbb{A}^{n}$. We have an isomorphism $\mathcal{T}_{X}\simeq
f^{*}\mathcal{T}_{\mathbb{A}^{n}_{S}}$ and can consider the trivialization of
$\mathcal{T}_{X}$ given by $\partial_{i}\coloneqq f^{-1}(\partial_{z_{i}})$.
We have $f_{i}=f^{*}(z_{i})$ and this way $\partial_{i}(f_{j})=\delta_{ij}$.
$\mathcal{T}_{X}$ is trivial with basis $\partial_{i}$, $i=1,\ldots,n$ where
$n=\operatorname{{rk}}(\Omega^{1}_{X})$. Let $D\in\mathcal{D}_{X,\leq p-1}$;
by Remark 2.18 it can be written as a sum

$D=\sum_{\alpha\in\mathbb{N}^{d}}g_{\alpha}\cdot\partial^{\alpha}$

where $g_{\alpha}\in\mathcal{O}_{X}$ and
$\alpha=(\alpha_{1},\ldots,\alpha_{n})$ is such that $|\alpha|\leq p-1$. Take
a maximal $\alpha$ with respect to the lexicographic order on $\mathbb{N}^{d}$
such that $g_{\alpha}\neq 0$; then it is easy to check that
$D(f_{1}^{\alpha_{1}}\ldots
f_{n}^{\alpha_{n}})=\alpha_{1}!\ldots\alpha_{n}!\cdot g_{\alpha}$, which is
non-zero since all $\alpha_{i}<p$. ∎

Recall the twisted tangent and cotangent bundles (Construction 2.22
$\mathcal{T}_{X}^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}$, $\Omega^{1,{\\{\\!\\!\\!\
p\ \\!\\!\\!\\}}}_{X}$). By Corollary 2.24 we have

$(T^{*}X)^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}\simeq\operatorname{{Spec}}_{X^{\\!{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}}}(\operatorname{{Sym}}^{*}_{\mathcal{O}_{X}^{\\!{\\{\\!\\!\\!\
p\ \\!\\!\\!\\}}}}\mathcal{T}_{X}^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}})$

be the total space of $\Omega^{1,{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}}_{X}$. The
natural projection $(T^{*}X)^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}\rightarrow
X^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}$ is affine and so one has an equivalence of
categories of quasi-coherent sheaves on $(T^{*}\\!X)^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}$ and quasi-coherent
$\operatorname{{Sym}}^{*}_{\mathcal{O}_{X}^{\\!{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}}}\mathcal{T}_{X{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}}$-modules on
$X^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}$. The map $\theta$ above extends to a map
of sheaves of $\mathcal{O}_{X}^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}$-algebras

$\theta\colon\operatorname{{Sym}}^{*}_{\mathcal{O}_{X}^{\\!{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}}}\\!\mathcal{T}_{X}^{{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}}\rightarrow
Z(F_{X*}^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}\mathcal{D}_{X}),$

and this way $F_{X*}^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}\mathcal{D}_{X}$ defines a
sheaf of algebras over $(T^{*}\\!X)^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}$.

Then, following [BMR] (or rather [OV, Section 2.1] in the relative setting) we
have the following structural result for $F_{X*}^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}\mathcal{D}_{X}$.

######  Proposition 2.27.

Let $X$ be a smooth $S$-scheme. Then

  1. (1)

The $p$-curvature map (defined in the end of Section 2.3)

$\theta\colon\operatorname{{Sym}}^{*}_{\mathcal{O}_{X}^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}}\\!\mathcal{T}_{X^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}}\rightarrow
Z(F_{X*}^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}\mathcal{D}_{X})$

is an isomorphism.

  2. (2)

The sheaf of algebras over $(T^{*}\\!X)^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}$
defined by $F_{X*}^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}\mathcal{D}_{X}$ is an
Azumaya algebra of rank $2p^{\dim_{S}X}$.

######  Example 2.28.

Assume that $S=\operatorname{{Spec}}B$ is affine and that
$X\coloneqq\mathbb{A}^{d}_{S}$ with coordinates $z_{1},\ldots,z_{d}$. In this
case global sections of $\mathcal{D}_{X}$ are given by the Weyl algebra

$W_{d}(B)\coloneqq B\langle
z_{1},\ldots,z_{d},\partial_{z_{1}},\ldots,\partial_{z_{d}}\rangle/([z_{i},z_{j}]=[\partial_{z_{i}},\partial_{z_{j}}]=0,[\partial_{z_{j}},z_{i}]=\delta_{ij}).$

The $p$-th power $\partial_{z_{i}}^{p}$ acts by zero on $\mathcal{O}_{X}$, so
$\partial_{z_{i}}^{[p]}=0$ and
$\theta(\partial_{x_{i}})=\partial_{z_{i}}^{p}$. Then proposition 2.27 tells
us that the center $Z(\mathcal{D}_{X})$ is isomorphic to the polynomial ring
$B[z_{i}^{p},\partial_{z_{i}}^{p}]\subset\mathcal{D}_{X}$ and that
$\mathcal{D}_{X}$ is an Azumaya algebra over it.

We finish with some remarks that will be useful in the next section:

######  Remark 2.29.

Consider the homomorphism
$\operatorname{{Sym}}^{*}_{\mathcal{O}_{X}^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}}\\!\mathcal{T}_{X^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}}\twoheadrightarrow\mathcal{O}_{X}^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}$ sending $\mathcal{T}_{X}^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}$ to 0.
It corresponds to the embedding of the zero section $i_{0}\colon
X^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}\hookrightarrow(T^{*}\\!X)^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}$. Then the pull-back $\mathcal{D}_{X,0}\coloneqq
i_{0}^{*}(F_{X*}^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}\mathcal{D}_{X})$ is an
Azumaya algebra on $X^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}$ which is canonically
split. Indeed, $F_{X*}^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}\mathcal{O}_{X}$ is a
vector bundle of rank $p^{\dim_{S}X}$ and $\mathcal{T}_{X}^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}\subset Z(F_{X*}^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}\mathcal{D}_{X})$
acts on it by 0 (since $v^{[p]}$ by definition acts as $v^{p}$): so the action
of $F_{X*}^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}\mathcal{D}_{X}$ on
$F_{X*}^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}\mathcal{O}_{X}$ factors through
$\mathcal{D}_{X,0}$. Comparing the ranks we get that $F_{X*}^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}\mathcal{O}_{X}$ is a splitting bundle for $\mathcal{D}_{X,0}$,
and so

$\mathcal{D}_{X,0}\simeq\mathcal{E}nd_{\mathcal{O}_{X}^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}}(F_{X*}^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}\mathcal{O}_{X}).$

######  Remark 2.30 (Cartier’s equivalence).

From the above remark one gets an equivalence of categories of quasi-coherent
sheaves of $\mathcal{D}_{X,0}$-modules and quasi-coherent sheaves on
$X^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}$. The corresponding pair of qausi-inverse
functors is given by $-\otimes_{\mathcal{O}_{X}^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}}\\!F_{X*}^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}\mathcal{O}_{X}$ and
$\mathcal{H}om_{\mathcal{D}_{X,0}}(F_{X*}^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}\mathcal{O}_{X},-)$. Under this equivalence the
$\mathcal{D}_{X,0}$-module $F_{X*}^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}\mathcal{O}_{X}$ corresponds to $\mathcal{O}_{X}^{\\{\\!\\!\\!\
p\ \\!\\!\\!\\}}$.

######  Remark 2.31.

Let $\mathcal{I}\subset\mathcal{O}_{X}$ be a quasicoherent sheaf of ideals
that is invariant under the action of $\mathcal{D}_{X}$. Note that
$F_{X}^{{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}}$ is affine, so the pushforward
$F_{X*}^{{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}}$ is exact, and for any quasicoherent
sheaf $\mathcal{F}$ on $X$ the map $(F_{X}^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}})^{-1}(\mathcal{F}_{X*}^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}\mathcal{F})\rightarrow\mathcal{F}$ is an equivalence. The
pushforward $\mathcal{F}_{X*}^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}(\mathcal{I})$
then gives a sub-$\mathcal{D}_{X,0}$-module of
$\mathcal{F}_{X*}^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}\mathcal{O}_{X}$ which by the
equivalence in Remark 2.30 is necessarily of the form
$\mathcal{J}\otimes_{\mathcal{O}_{X}^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}}\mathcal{F}_{X*}^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}\mathcal{O}_{X}\subset\mathcal{F}_{X*}^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}\mathcal{O}_{X}$ for some ideal
$\mathcal{J}\subset\mathcal{O}_{X}^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}$. Under the
equivalence of categories between quasicoherent modules over
$\mathcal{F}_{X*}^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}\mathcal{O}_{X}$ and quasi-
coherent sheaves on $X$ the module
$\mathcal{J}\otimes_{\mathcal{O}_{X}^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}}\mathcal{F}_{X*}^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}\mathcal{O}_{X}$
exactly corresponds to $(F_{X}^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}})^{*}\mathcal{J}$, and so we get that $\mathcal{I}\simeq
F_{X}^{{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}*}(\mathcal{J})$ for some ideal
$\mathcal{J}\subset\mathcal{O}_{X}^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}$.

###  2.5. Local structure of a pseudo-smooth scheme

The goal of this section is to give a local description of a general pseudo-
smooth scheme which will allow us to make a reduction to smooth case. We start
with the following lemma:

######  Lemma 2.32.

Let $Y$ be a smooth $S$-scheme, and let $Z\subset Y^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}$ be a subscheme. Then the preimage $(F_{Y}^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}})^{-1}(Z)=Z\times_{Y^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}}Y\subset Y$
is a pseudo-smooth scheme.

###### Proof.

Let $\mathcal{J}\subset\mathcal{O}_{Y}^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}$ be the
sheaf of ideals defining $Z$. Then $\mathcal{I}\coloneqq F_{Y}^{{\\{\\!\\!\\!\
p\ \\!\\!\\!\\}}*}(\mathcal{J})\subset\mathcal{O}_{X}$ is the sheaf of ideals
defining $X\coloneqq(F_{Y}^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}})^{-1}(Z)$. Note
that $\mathcal{I}$ is locally generated by elements in
$\mathcal{O}_{Y}^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}\simeq(F_{Y}^{{\\{\\!\\!\\!\
p\ \\!\\!\\!\\}}})^{-1}\mathcal{O}_{Y{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}}$. Let
$i\colon X\rightarrow Y$ be the corresponding embedding. We have an exact
sequence

$\ldots\rightarrow
i^{-1}(\mathcal{I}/\mathcal{I}^{2})\xrightarrow{d}i^{*}\Omega^{1}_{Y}\twoheadrightarrow\Omega^{1}_{X}\rightarrow
0$

of sheaves on $X$ where the left map is induced by the map that sends
$[f]\in\mathcal{I}/\mathcal{I}^{2}$ to $df\in\Omega^{1}_{Y}$. However, for any
$f\in\mathcal{O}_{Y}^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}$ we have $df=0$. Since
$\mathcal{I}$ is locally generated by elements in
$\mathcal{O}_{Y}^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}$ we get that the map $d$
above is equal to 0 and so $\Omega^{1}_{X}\simeq i^{*}\Omega^{1}_{Y}$. ∎

######  Remark 2.33.

We claim that in the above case the reduced Frobenius twist $X^{\\{\\!\\!\\!\
p\ \\!\\!\\!\\}}$ is given by $Z$. Indeed, $\mathcal{O}_{X}$ (or rather
$i_{*}\mathcal{O}_{X}$) is described as the reduction
$\mathcal{O}_{Y}\otimes_{\mathcal{O}_{Y}^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}}\mathcal{O}_{Y}^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}\\!/\mathcal{J}$.
Indeed, $\mathcal{O}_{X}^{{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}}$ is the subalgebra
in $\mathcal{O}_{X}$ generated by $\mathcal{O}_{X}^{p}$ and $\mathcal{O}_{S}$,
which is easily seen to be isomorphic to $\mathcal{O}_{Y}^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}\\!/\mathcal{J}\simeq\mathcal{O}_{Z}$. Moreover, this way the
correponding embedding $i^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}\colon
X^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}\hookrightarrow Y^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}$ is identified with the original $Z\hookrightarrow
Y^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}$. Consequently, we also see that the
commutative diagram

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F_{X}^{\\{\\!\\!\\!\
p\
\\!\\!\\!\\}}}$$\scriptstyle{i}$$\textstyle{Z\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i^{\\{\\!\\!\\!\
p\
\\!\\!\\!\\}}}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F_{Y}^{\\{\\!\\!\\!\
p\ \\!\\!\\!\\}}}$$\textstyle{Y^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}}$

is in fact a fiber square. In particular, $F_{X}^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}$ is a finite locally free map of degree $p^{\dim_{S}Y}$.

Note that $Z$ here was an arbitrary closed scheme of $Y$; in particular, it
could be very singular.

######  Remark 2.34.

Recall that in the course of proof of Lemma 2.32 we showed that
$i^{*}\Omega^{1}_{Y}\simeq\Omega^{1}_{X}$ via the natural map. It then
formally follows that $i^{*}\Omega^{j}_{Y}\simeq\Omega^{j}_{X}$ for all $j\geq
0$. By considering the commutative diagram

$\textstyle{X^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{W_{X}^{\\{\\!\\!\\!\
p\ \\!\\!\\!\\}}}$$\scriptstyle{i^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\textstyle{Y^{\\{\\!\\!\\!\
p\
\\!\\!\\!\\}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{W_{Y}^{\\{\\!\\!\\!\
p\ \\!\\!\\!\\}}}$$\textstyle{Y}$

it also follows that the natural map $i^{{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}*}\Omega^{j,{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}}_{Y}\rightarrow\Omega^{j,{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}}_{X}$
induced by the identification $i^{{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}*}\circ
W_{Y}^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}=W_{X}^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}\circ i$ is an isomorphism for any $j\geq 0$.

In fact, all pseudo-smooth schemes Zariski-locally look like Frobenius
neighborhoods in smooth schemes:

######  Proposition 2.35.

Let $X$ be a pseudo-smooth $S$-scheme. Then Zariski-locally it is of the form
$(F_{Y}^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}})^{-1}(Z)=Z\times_{Y^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}}Y\subset Y$ with $Y$ smooth over $S$ and $Z\hookrightarrow Y$ is
a closed subscheme (see Lemma 2.32).

###### Proof.

Let $x\in X$ be a point. We will show that there is a neighborhood of $x$ of
the form above.

Since $X$ is locally of finite type, passing to an open we can assume that
there is an embedding $i\colon X\hookrightarrow\mathbb{A}^{n}$; the natural
map $i^{*}\colon i^{*}\Omega^{1}_{\mathbb{A}^{n}}\rightarrow\Omega^{1}_{X}$ is
a surjection. Let $\mathcal{I}\subset\mathcal{O}_{\mathbb{A}^{n}}$ be the
ideal defining $X$. As before, we have an exact sequence

$\ldots\rightarrow
i^{-1}(\mathcal{I}/\mathcal{I}^{2})\xrightarrow{d}i^{*}\Omega^{1}_{\mathbb{A}^{n}}\xrightarrow{i^{*}}\Omega^{1}_{X}\rightarrow
0.$

Since $X$ is pseudo-smooth, the kernel of $i^{*}$ is a vector bundle. Let
$d\coloneqq\operatorname{{rk}}H^{0}(X,\Omega^{1}_{X})$. After passing to a
neighborhood $U$ of $x$ in $\mathbb{A}^{n}$ we can assume that
$\operatorname{{Ker}}(i^{*})$ is a trivial vector bundle with basis given by
differentials $df_{1},\ldots,df_{n-d}$ of $(n-d)$ functions
$f_{1},\ldots,f_{n-d}\in\mathcal{I}$. Consider the map
$s\colon\mathcal{O}_{U}^{\oplus n-d}\rightarrow\Omega^{1}_{U}$ defined by
$(n-d)$ sections $df_{1},\ldots,df_{n-d}$. The support of its kernel is a
closed subset which doesn’t intersect $X$ (and in particular doesn’t contain
$x$) thus considering the complement we can assume that $s$ is an embedding.
Moreover, the dimension of the stalk of the cokernel is an upper-
semicontinuous function bounded below by $n-(n-d)=d$. Since it is equal to $d$
on $X$ we see that it is also equal to $d$ on some neighborhood, and so
restriction of the cokernel to this neighborhood is a vector bundle of rank
$d$. Shrinking the open further we can assume that $\Omega^{1}_{U}$ is trivial
with basis given by differentials
$df_{1},\ldots,df_{n-d},dg_{1},\ldots,dg_{d}$ (with $f_{i}$’s being the
functions we considered before, and $g_{i}$’s some other functions).

Now, let $i_{Y}\colon Y\hookrightarrow U$ be the subscheme defined by
$\mathcal{I}_{Y}\coloneqq(f_{1},\ldots,f_{n-d})\subset\mathcal{I}$. Then we
claim that the map $q\colon Y\rightarrow\mathbb{A}^{d}$ given by
$g_{1},\ldots,g_{d}$ is étale. Indeed, one can realize $Y$ as a subscheme in
$U\times\mathbb{A}^{d}\subset\mathbb{A}^{n}\times\mathbb{A}^{d}$ cut out by
equations $f_{1}=\ldots=f_{n-d}=t_{1}-g_{1}=\ldots=t_{d}-g_{d}=0$ where
$t_{i}$’s are coordinates on $\mathbb{A}^{d}$. Since
$df_{1},\ldots,df_{n-d},dg_{1},\ldots,dg_{d}$ are linearly independent over
$\mathcal{O}_{U}$ fiberwise, the Jacobian of this system of equations
(relative to $\mathbb{A}^{d}$) is a unit, and so projection to
$\mathbb{A}^{d}$ is étale.

This way we get that $Y$ is in fact smooth over $S$. Let
$\mathcal{I}^{\prime}\subset\mathcal{O}_{Y}$ be the ideal defining
$i_{X}\colon X\hookrightarrow Y$. We have an exact sequence

$\ldots\rightarrow i^{-1}(\mathcal{I}^{\prime}/\mathcal{I}^{\prime
2})\xrightarrow{d}i^{*}_{X}\Omega^{1}_{Y}\xrightarrow{i_{X}^{*}}\Omega^{1}_{X}\rightarrow
0.$

However, note that the map $i_{X}^{*}$ is now an isomorphism. Indeed,
$i_{X}^{*}\Omega^{1}_{Y}$ is exactly the quotient of $i^{*}\Omega^{1}_{U}$ by
$df_{i}$, which are spanning the kernel of $i^{*}_{X}$. Thus, we get that the
map $d$ is 0, or, in other words, that
$d(\mathcal{I}^{\prime})\subset\mathcal{I}^{\prime}$. This is equivalent to
$\mathcal{I}^{\prime}$ being invariant under the action of $\mathcal{D}_{Y}$,
from which it follows by Remark 2.31 that $\mathcal{I}^{\prime}\simeq
F_{Y}^{{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}*}\mathcal{J}$ for some
$\mathcal{J}\subset\mathcal{O}_{Y^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}}$. Then,
putting $Z\subset\mathcal{Y}^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}$ to be the
subscheme defined by $\mathcal{J}$ we exactly get that
$X\simeq(F_{Y}^{{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}})^{-1}(Z)$. ∎

Proposition 2.35 allows to extend the discussion in Section 2.4 to the pseudo-
smooth setting. We start with an extension of Lemma 2.26:

######  Corollary 2.36.

Let $X$ be a pseudo-smooth scheme over $S$. Then the natural map

$\mathcal{D}_{X,\leq
p-1}\rightarrow\mathcal{E}nd_{\mathcal{O}_{S}}(\mathcal{O}_{X})$

induced by the action of $\mathcal{D}_{X}$ on $\mathcal{O}_{X}$ is an
embedding.

###### Proof.

The statement is local, so we can assume that $X=(F_{Y}^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}})^{-1}(Z)\hookrightarrow Y$ for a closed subscheme
$Z\hookrightarrow Y^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}$ in smooth scheme. We can
also assume that $Y$ has an étale map $f\colon Y\rightarrow\mathbb{A}^{n}$.
Let $\partial_{i}$ and $f_{i}$ be as in the proof of Lemma 2.26. Let $i\colon
X\rightarrow Y$ be the embedding. Recall (see proof of Lemma 2.32) that
$i^{*}\Omega^{1}_{Y}\xrightarrow{\sim}\Omega^{1}_{X}$, and, consequently,
$\mathcal{T}_{X}\simeq i^{*}\mathcal{T}_{Y}$. Let
$\partial_{i}^{\prime}\coloneqq i^{-1}(\partial_{i})\in\mathcal{T}_{X}$ and
let $f_{i}^{\prime}\coloneqq i^{*}(f_{i})\in\mathcal{O}_{X}$. We have
$\partial_{i}^{\prime}(f_{j}^{\prime})=\delta_{ij}$. The same argument as in
Lemma 2.26 then shows that for any $D\in\mathcal{D}_{X,\leq p-1}$ there is a
function $g\in\mathcal{O}_{X}$ such that $D(g)\neq 0$. ∎

######  Construction 2.37 ($p$-curvature map in the pseudo-smooth setting).

As in the smooth case, consider the map
$\theta\coloneqq\mathcal{T}_{X}\rightarrow\mathcal{D}_{X}$ of sheaves on $X$
given by the following formula

$v\mapsto\theta(v)\coloneqq v^{p}-v^{[p]}.$

Using Corollary 2.36 in place of Lemma 2.26 as in Construction 2.25 one shows
that

  * •

$\theta$ is additive: $\theta(v_{1}+v_{2})-(\theta(v_{1})+\theta(v_{2}))=0$;

  * •

$\theta$ is Frobeinus-linear: $\theta(fv)-f^{p}\cdot\theta(v)=0$;

  * •

$\theta(v)$ lies in the center $Z(\mathcal{D}_{X})$ for any
$v\in\mathcal{T}_{X}$: $[\theta(v),f]=[\theta(v),w]=0$ for any
$f\in\mathcal{O}_{X}$ and $w\in\mathcal{T}_{X}$.

This way we obtain a map

$\theta^{\prime}\colon\mathcal{T}_{X}\otimes_{\mathcal{O}_{X}}\mathcal{O}_{X}^{p}\rightarrow
Z(\mathcal{D}_{X})$

of $\mathcal{O}_{X}^{p}$-modules. Since $Z(\mathcal{D}_{X})$ is in fact an
$\mathcal{O}_{X}^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}\simeq\mathcal{O}_{X}^{p}\cdot\mathcal{O}_{S}$-algebra we can
extend this to a map
$\mathcal{T}_{X}\otimes_{\mathcal{O}_{X}}\mathcal{O}_{X}^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}\rightarrow Z(\mathcal{D}_{X})$ of
$\mathcal{O}_{X}^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}$, or, equivalently, a map of
sheaves on $X^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}$

(2.2) $\theta\colon\mathcal{T}_{X}^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}\rightarrow
Z(F_{X*}^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}\mathcal{D}_{X}).$

which we still call the “$p$-curvature map”.

######  Corollary 2.38.

Let $X$ be a pseudo-smooth scheme over $S$. As before, consider
$(T^{*}\\!X)^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}\simeq\operatorname{{Spec}}_{X^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}}(\operatorname{{Sym}}^{*}_{\mathcal{O}_{X}^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}}\\!\mathcal{T}_{X}^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}})$.

  1. (1)

The map $\theta\colon\mathcal{T}_{X}^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}\rightarrow Z(F_{X*}^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}\mathcal{D}_{X})$ induces an isomorphism

$\theta\colon\operatorname{{Sym}}^{*}_{\mathcal{O}_{X}^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}}\\!\mathcal{T}_{X}^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}\xrightarrow{\sim}Z(F_{X*}^{{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}}\mathcal{D}_{X}).$

  2. (2)

$F_{X*}^{{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}}\mathcal{D}_{X}$, considered as a
sheaf of algebras over $(T^{*}\\!X)^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}$, is an
Azumaya algebra of rank111111A priori this is a locally constant function on
$X$. $p^{2\operatorname{{rk}}\Omega^{1}_{X}}$.

###### Proof.

Both statements are local (e.g. in Zariski topology) and so we can assume
$X\simeq(F_{Y}^{{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}})^{-1}(Z)$ where $Y$ is smooth
over $S$ and $Z\hookrightarrow Y$ is a closed subscheme. In this case,
$\mathcal{D}_{X}$ is a central reduction of $\mathcal{D}_{Y}$ via an embedding
$T^{*}Y^{(1)}|_{Z}\hookrightarrow T^{*}Y^{(1)}\simeq(T^{*}Y)^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}$. More precisely, let $i\colon X\hookrightarrow Y$ be the
embedding. Following the proof of Lemma 2.32 in this situation we have
isomorphisms $\Omega^{1}_{X}\simeq i^{*}\Omega^{1}_{Y}$, and consequently,
$\mathcal{T}_{X}\simeq i^{*}\mathcal{T}_{Y}$. By naturality, the latter is an
isomorphism of Lie algebroids, and induces an isomorphism
$\mathcal{D}_{X}\simeq i^{*}\mathcal{D}_{Y}$. Also

$\operatorname{{Sym}}^{*}_{\mathcal{O}_{X}^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}}\mathcal{T}_{X}^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}\simeq\operatorname{{Sym}}^{*}_{\mathcal{O}_{Y}^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}}\mathcal{T}_{Y}^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}\otimes_{\mathcal{O}_{Y}^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}}\mathcal{O}_{Z}$

and so $(T^{*}\\!X)^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}$ can be identified with
the restriction of $(T^{*}Y)^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}$ to
$Z\hookrightarrow Y^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}$. Moreover, if we denote
by $i_{Z}\colon Z\hookrightarrow Y^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}$ the
corresponding embedding, applying $F_{X*}$ we get

$F_{X*}^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}\mathcal{D}_{X}\simeq
F_{X*}^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}(i^{*}\mathcal{D}_{Y})\simeq
i_{Z}^{*}F_{Y*}^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}\mathcal{D}_{Y}\simeq
F_{Y*}^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}\mathcal{D}_{Y}\otimes_{\mathcal{O}_{Y}^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}}\mathcal{O}_{Z}\simeq F_{Y*}^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}\mathcal{D}_{Y}\otimes_{\mathcal{O}_{(T^{*}Y)^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}}}\mathcal{O}_{(T^{*}\\!X)^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}},$

where in the last isomorphism we considered $F_{Y*}^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}\mathcal{D}_{Y}$ as an Azumaya algebra over
$(T^{*}Y)^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}$. Thus we get that
$F_{X*}^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}\mathcal{D}_{X}$ as a sheaf of algebras
over $(T^{*}\\!X)^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}$ is a restriction of an
Azumaya algebra, and as such is an Azumaya algebra itself. Its center then
also is necessarily isomorphic to $\mathcal{O}_{(T^{*}\\!X)^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}}\simeq\operatorname{{Sym}}^{*}_{\mathcal{O}_{X}^{\\{\\!\\!\\!\
p\ \\!\\!\\!\\}}}\mathcal{T}_{X}^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}$. ∎

######  Remark 2.39.

From the proof of Proposition 2.35 it also follows that $F_{X}^{\\{\\!\\!\\!\
p\ \\!\\!\\!\\}}\colon X\rightarrow X^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}$ is a
finite faithfully flat map of degree $p^{\operatorname{{rk}}\Omega^{1}_{X}}$.
Indeed, it is locally obtained as a base change of relative Frobenius for a
smooth scheme $Y$ of (relative) dimension $\operatorname{{rk}}\Omega^{1}_{X}$
(see Remark 2.33).

###  2.6. De Rham complex and Cartier operation

Let $X$ be a pseudo-smooth scheme over $S$. Put
$r\coloneqq\operatorname{{rk}}\Omega^{1}_{X}$; this is a locally constant
function on $X$. Consider the de Rham complex

$\mathrm{dR}_{X}\coloneqq\left[0\rightarrow\mathcal{O}_{X}\rightarrow\Omega^{1}_{X}\rightarrow\ldots\rightarrow\Omega^{r}_{X}\rightarrow
0\right]{:}$

this is a complex of sheaves of $\mathcal{O}_{S}$-modules on $X$ with
differential given by the (relative) de Rham differential. Since the relative
de Rham differential is $\mathcal{O}_{X}^{p}\cdot B$-linear, the pushforward
$F_{X*}^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}\mathrm{dR}_{X}$ defines a complex of
coherent sheaves on $X^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}$.

Let
$\mathcal{H}_{\mathrm{dR},X}^{*}\coloneqq\mathcal{H}^{*}(F_{X*}^{\\{\\!\\!\\!\
p\ \\!\\!\\!\\}}\mathrm{dR}_{X})$ denote the sheaf cohomology algebra of
$F_{X*}^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}\mathrm{dR}_{X}$. Recall the notation
$\Omega^{1,{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}}_{X}\coloneqq W_{X}^{{\\{\\!\\!\\!\
p\ \\!\\!\\!\\}}*}\Omega^{1}_{X}$. One can define the inverse Cartier morphism

$C^{-1}\colon\wedge^{*}_{\mathcal{O}_{X}^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}}\\!\\!(\Omega^{1,{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}}_{X})\rightarrow\mathcal{H}_{\mathrm{dR},X}^{*}$

as follows. Namely in degree 0 it is given by $(F_{X}^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}})^{-1}\colon\mathcal{O}_{X^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}}\rightarrow F_{X*}\mathcal{O}_{X}$, while in degree 1 locally
one sends $(W_{X}^{{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}})^{-1}(df)\in\Omega^{1,{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}}_{X}$ to
$[f^{p-1}df]\in\mathcal{H}_{\mathrm{dR},X}^{1}$. By a standard computation the
latter gives a well-defined additive map to $\mathcal{H}^{1}_{\mathrm{dR},X}$,
which is $F_{X}^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}$-linear. For higher degrees
$C^{-1}$ uniquely extends by multiplicativity.

The classical result of Cartier states that $C^{-1}$ is an equivalence if $X$
over $S$ is smooth. This remains true in the pseudo-smooth setting as well.

######  Proposition 2.40.

Let $X$ be a pseudo-smooth $S$-scheme. Then the inverse Cartier map is an
isomorphism.

###### Proof.

We deduce the statement from the smooth case. The statement is Zariski local,
so we can assume $X=(F_{Y}^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}})^{-1}(Z)$ is as in
Lemma 2.32 (so $Y$ is smooth and $Z\hookrightarrow Y^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}$). In this case $X^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}\simeq Z$. Let
$i\colon X\hookrightarrow Y$ and $i^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}\colon
Z\hookrightarrow\mathcal{Y}^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}$. By Remarks 2.33,
2.34 and base change, $F_{X*}^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}\mathrm{dR}_{X}$
can be naturally identified with $i^{{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}*}F_{Y*}^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}\mathrm{dR}_{Y}$. By
Cartier isomorphism in the smooth case, the cohomology sheaves
$\mathcal{H}_{\mathrm{dR},Y}^{*}$ are isomorphic to
$\wedge^{*}_{\mathcal{O}_{Y}^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}}\Omega^{1,{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}}_{Y}$; in particular,
they are vector bundles and thus are flat over $\mathcal{O}_{Y}^{\\{\\!\\!\\!\
p\ \\!\\!\\!\\}}$. From this and Remark 2.34 we get that

$\mathcal{H}_{\mathrm{dR},X}^{*}\simeq\mathcal{H}^{*}(i^{{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}*}F_{Y*}^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}\mathrm{dR}_{Y})\simeq
i^{{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}*}\mathcal{H}_{\mathrm{dR},Y}^{*}\simeq
i^{{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}*}(\wedge^{*}_{\mathcal{O}_{Y}^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}}\\!\Omega^{1,{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}}_{Y})\simeq\wedge^{*}_{\mathcal{O}_{X^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}}}\Omega^{1,{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}}_{X}.\qed$

Let $C$ be the inverse to $C^{-1}$. In each degree $i$, $C$ gives an
isomorphism $\mathcal{H}^{i}_{\mathrm{dR},X}\simeq\Omega^{i,{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}}_{X}$ which we can precompose with the projection
$F_{X*}^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}\Omega^{i}_{X,cl}\twoheadrightarrow\mathcal{H}^{i}_{\mathrm{dR},X}$
from the closed $i$-forms. We will call the resulting map
$\mathsf{C}^{i}\colon F_{X*}^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}\Omega^{i}_{X,cl}\twoheadrightarrow\Omega^{i,{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}}_{X}$ the Cartier operation. For $i=1$ one gets a short exact
sequence

(2.3) $0\rightarrow F_{X*}^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}\mathcal{O}_{X}/\mathcal{O}_{X}^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}\xrightarrow{d}F_{X*}^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}\Omega^{1}_{X,cl}\xrightarrow{\mathsf{C}}\Omega^{1,{\\{\\!\\!\\!\
p\ \\!\\!\\!\\}}}_{X}\rightarrow 0,$

of sheaves on $X^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}$ where the term on the left
is identified with the exact 1-forms via $d$.

######  Remark 2.41.

The map $\mathsf{C}\colon F_{X*}^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}\Omega^{1}_{X,cl}\rightarrow\Omega^{1,{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}}_{X}$ dually induces a map $\mathcal{T}_{X}^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}\otimes F_{X*}^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}\Omega^{1}_{X,cl}\rightarrow\mathcal{O}_{X}^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}$ which lifts the pairing map $\mathcal{T}_{X}^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}\otimes\Omega^{1,{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}}_{X}\rightarrow\mathcal{O}_{X}^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}$.
Substituting a local section $\xi\in\mathcal{T}_{X}^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}$ we obtain the corresponding “contraction” maps

$\widetilde{\iota_{\xi}}\colon F_{X*}^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}\Omega^{1}_{X,cl}\rightarrow\Omega^{1,{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}}_{X}\ \ \text{ and }\ \
\iota_{\xi}\colon\Omega^{1,{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}}_{X}\rightarrow\mathcal{O}_{X}^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}.$

Following [BK], we have the following explicit formula for
$\widetilde{\iota_{\xi}}$. Namely, given a closed 1-form
$\omega\in\Omega^{1}_{X,cl}$ and a vector field $\xi\in\mathcal{T}_{X}$
(giving a section $\xi^{{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}}\coloneqq(W_{X}^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}})^{-1}(\xi)\in\mathcal{T}_{X}^{{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}}$)
one has the following formula

(2.4) $\widetilde{\iota_{\xi^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}}}(\mathsf{C}(\omega))=\langle\xi^{[p]},\omega\rangle-\xi^{p-1}(\langle\xi,\omega\rangle)$

for the corresponding contraction.

###  2.7. Line bundles with connection and Milne’s exact sequence

Let $(\mathcal{E},\nabla)$, with
$\nabla\colon\mathcal{E}\rightarrow\mathcal{E}\otimes\Omega^{1}_{X}$ be a
quasi-coherent sheaf with flat connection on a pseudo-smooth $S$-scheme $X$.
The connection $\nabla$ induces a map
$\mathcal{D}_{X}\rightarrow\mathcal{E}nd_{\mathcal{O}_{S}}(\mathcal{E})$,
whose restriction to $Z(\mathcal{D}_{X})$ factors through the subalgebra of
$\mathcal{O}_{X}$-linear endomorphisms
$\mathcal{E}nd_{\mathcal{O}_{X}}(\mathcal{E})^{\nabla^{\prime}=0}\subset\mathcal{E}nd_{\mathcal{O}_{S}}(\mathcal{E})$
that are flat with respect to the natural connection121212Here the connection
$\nabla^{\prime}$ on $\mathcal{E}nd_{\mathcal{O}_{X}}(\mathcal{E})$ is given
by $(\nabla^{\prime}(\psi))(s)\coloneqq\nabla(\psi(s))-\psi(\nabla(s)$ for
$s\in\mathcal{E}$ and $\psi\in\mathcal{E}nd_{\mathcal{O}_{X}}(\mathcal{E})$.
Thus an endomorphism $\psi\in\mathcal{E}nd_{\mathcal{O}_{X}}(\mathcal{E})$ is
flat with respect to $\nabla^{\prime}$ if and only if it commutes with the
action of $\mathcal{T}_{X}$ induced by $\nabla$. on
$\mathcal{E}nd_{\mathcal{O}_{X}}(\mathcal{E})$. Indeed, central elements
commute both with multiplication on functions and action by vector fields.

######  Construction 2.42 ($p$-curvature map for line bundles).

Let $\mathcal{E}\coloneqq\mathcal{L}$ in $(\mathcal{E},\nabla)$ be a line
bundle. In this case, the endomorphism sheaf
$(\mathcal{E}nd_{\mathcal{O}_{X}}(\mathcal{L}),\nabla^{\prime})$ with its
natural connection is given by $(\mathcal{O}_{X},d)$. In particular,
$\mathcal{E}nd_{\mathcal{O}_{X}}(\mathcal{L})^{\nabla^{\prime}=0}\subset\mathcal{O}_{X}$
is given by $\ker d$, which is identified with $\mathcal{O}_{X}^{\\{\\!\\!\\!\
p\ \\!\\!\\!\\}}\subset\mathcal{O}_{X}$ by Proposition 2.40. Using the
$p$-curvature map (2.2) we obtain a map of sheaves
$\mathcal{T}_{X}^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}\rightarrow\mathcal{O}_{X}^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}$, or,
in other words, a section of $\Omega^{1,{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}}_{X}$,
which we will denote
$\mathsf{c}_{p}(\mathcal{L},\nabla)\in\Omega^{1,{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}}_{X}$. It is not hard to see from the definition (and the way
$\mathcal{T}_{X}^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}\subset
Z(F_{X*}^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}\mathcal{D}_{X})$ acts on the tensor
product), that

$\mathsf{c}_{p}((\mathcal{L}_{1},\nabla_{1})\otimes(\mathcal{L}_{2},\nabla_{2}))=\mathsf{c}_{p}(\mathcal{L}_{1},\nabla_{1})+\mathsf{c}_{p}(\mathcal{L}_{2},\nabla_{2}).$

If we are given a trivialization $\mathcal{L}\simeq\mathcal{O}_{X}$, $\nabla$
can be written in the form $d+\alpha$ for some closed 1-form
$\alpha\in\Omega^{1}_{X,cl}$. Restricting to line bundles of this form we get
a map of sheaves

$\mathsf{c}_{p}\colon F_{X*}^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}\Omega^{1}_{X,cl}\rightarrow\Omega^{1,{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}}_{X},\ \ \ \
\mathsf{c}_{p}\colon\alpha\mapsto\mathsf{c}_{p}(\mathcal{O}_{X},d+\alpha).$

Since
$(\mathcal{O}_{X},d+\alpha_{1})\otimes(\mathcal{O}_{X},d+\alpha_{2})\simeq(\mathcal{O}_{X},d+\alpha_{1}+\alpha_{2})$
the above map is a map of sheaves of abelian groups.

######  Remark 2.43.

One has an explicit formula for the map $\mathsf{c}_{p}$ in terms of the
Cartier operation. Namely, given $\alpha\in\Omega^{1}_{X,cl}$ let’s compute
the pairing $\langle\xi^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}},\mathsf{c}_{p}(\alpha)\rangle$ for any $\xi\in\mathcal{T}_{X}$,
$\xi^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}\in(W_{X}^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}})^{-1}(\xi)\in\mathcal{T}_{X}^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}$:
this will define $\mathsf{c}_{p}(\alpha)$ uniquely. By definition, we need to
compute the action of $\theta(\xi)=\xi^{p}-\xi^{[p]}$ on line bundle
$(\mathcal{O}_{X},d+\alpha)$. This is given by

$(\xi+\langle\xi,\alpha\rangle)^{p}-\xi^{[p]}-\langle\xi^{[p]},\alpha\rangle=\langle\xi,\alpha\rangle^{p}-\langle\xi^{\\{\\!\\!\\!\
p\
\\!\\!\\!\\}},\mathsf{C}(\alpha)\rangle\in\mathcal{E}nd_{\mathcal{O}_{S}}(\mathcal{O}_{X})$

where for the equality we used that $\xi^{p}-\xi^{[p]}$ acts by 0 on
$\mathcal{O}_{X}$, formula (2.4) and also Jacobson’s formula. Note that
$\langle\xi,\alpha\rangle^{p}=(W_{X}^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}})^{-1}(\langle\xi,\alpha\rangle)=\langle\xi^{{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}},\alpha^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}\rangle$, where
$\alpha^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}\coloneqq(W_{X}^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}})^{-1}\alpha$. This gives

$\mathsf{c}_{p}(\alpha)=\alpha^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}-\mathsf{C}(\alpha).$

We now consider a non-commutative counterpart of the above $p$-curvature map.

######  Construction 2.44 (Central reductions of $\mathcal{D}_{X}$ associated
to 1-forms).

Let $\alpha\in H^{0}(X^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}},\Omega^{1,{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}}_{X})$ be a global
section. Let $i_{\alpha}\colon X^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}\hookrightarrow(T^{*}\\!X)^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}$ be
the graph of $\alpha$. Under the identification $(T^{*}\\!X)^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}\simeq\operatorname{{Spec}}_{X}\operatorname{{Sym}}^{*}_{\mathcal{O}_{X}^{\\{\\!\\!\\!\
p\ \\!\\!\\!\\}}}\mathcal{T}_{X}^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}$, the image
$\Gamma_{\alpha}=\operatorname{{Im}}(i_{\alpha})\subset(T^{*}\\!X)^{\\{\\!\\!\\!\
p\ \\!\\!\\!\\}}$ of $i_{\alpha}$ is defined by the sheaf of ideals generated
by linear expressions

$\mathcal{I}_{\alpha}\coloneqq(\xi-\langle\xi,\alpha\rangle)_{\xi\in\mathcal{T}_{X}^{\\{\\!\\!\\!\
p\
\\!\\!\\!\\}}}\subset\operatorname{{Sym}}^{*}_{\mathcal{O}_{X}^{\\{\\!\\!\\!\
p\ \\!\\!\\!\\}}}\mathcal{T}_{X}^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}.$

We denote by $\mathcal{D}_{X,\alpha}\coloneqq
i_{\alpha}^{*}(F_{X*}\mathcal{D}_{X})$ the corresponding pull-back of
$F_{X*}^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}\mathcal{D}_{X}$ considered as a sheaf
of algebras over $(T^{*}\\!X)^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}$. For any
$\alpha$ the sheaf $\mathcal{D}_{X,\alpha}$ defines an Azumaya algebra over
$X^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}$ of rank
$p^{2\operatorname{{rk}}\Omega^{1}_{X}}$.

######  Construction 2.45 (Line bundles with connection vs splittings.).

Suppose we are given a line bundle with flat connection $(\mathcal{L},\nabla)$
and put $\alpha\coloneqq\mathsf{c}_{p}(\mathcal{L},\nabla)$. Then by
definition of $\mathsf{c}_{p}$ we have that
$\xi\in\mathcal{T}_{X}^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}$ acts on
$F_{X*}^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}\mathcal{L}$ by
$\langle\xi,\alpha\rangle\in\mathcal{O}_{X}^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}$.
It follows that the action of $F_{X*}^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}\mathcal{D}_{X}$ on $F_{X*}^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}\mathcal{L}$ induced by $\nabla$ factors through
$\mathcal{D}_{X,\alpha}$. Moreover, by Remark 2.39 $F_{X*}^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}\mathcal{L}$ is a vector bundle of rank
$p^{\operatorname{{rk}}\Omega^{1}_{X}}$ on $X^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}$
and thus is a splitting bundle for $\mathcal{D}_{X,\alpha}$: namely, $\nabla$
induces an isomorphism

$\nabla\colon\mathcal{D}_{X,\alpha}\xrightarrow{\sim}\mathcal{E}nd_{\mathcal{O}_{X}^{\\{\\!\\!\\!\
p\ \\!\\!\\!\\}}}(F_{X*}^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}\mathcal{L}).$

Vice versa, let $\mathcal{E}$ be a splitting bundle
$\mathcal{D}_{X,\alpha}$-module on $X^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}$. Then
$\mathcal{E}$ is a vector bundle of rank
$p^{\operatorname{{rk}}\Omega^{1}_{X}}$. However $\mathcal{E}$ is also a
quasi-coherent $F_{X*}\mathcal{O}_{X}$-module (with the action via the
composition $F_{X*}^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}\mathcal{O}_{X}\rightarrow
F_{X*}^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}\mathcal{D}_{X}\twoheadrightarrow\mathcal{D}_{X,\alpha})$ and so
comes as a pushforward $(F_{X*}^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}})\mathcal{F}$
of some quasi-coherent sheaf $\mathcal{F}$ on $X$. Since $F_{X}^{\\{\\!\\!\\!\
p\ \\!\\!\\!\\}}$ is a finite locally free map of degree
$p^{\operatorname{{rk}}\Omega^{1}_{X}}$ the only option is that
$\mathcal{F}\simeq(F_{X*}^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}})^{-1}(\mathcal{E})$
is a line bundle. The action of $\mathcal{D}_{X}\simeq(F_{X*}^{\\{\\!\\!\\!\
p\ \\!\\!\\!\\}})^{-1}(F_{X*}^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}\mathcal{D}_{X})$
on $\mathcal{F}$ induces a natural flat connection, whose $p$-curvature is
given by $\alpha$. Moreover, isomorphisms between two line bundles
$(\mathcal{L}_{1},\nabla_{1})$ and $(\mathcal{L}_{2},\nabla_{2})$ with the
same $p$-curvature $\alpha$ exactly correspond to isomorphisms between the
corresponding $\mathcal{D}_{X,\alpha}$-modules $F_{X*}^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}\mathcal{L}_{1}$ and $F_{X*}^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}\mathcal{L}_{2}$. In particular, any two such line bundles a
étale (and, in fact even Zariski) locally isomorphic.

This gives an equivalence of two groupoids: line bundles with flat connection
on $X$ with $p$-curvature $\alpha$, and splitting bundles131313Here an object
is given by a vector bundle $\mathcal{E}$ on $X^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}$ together with an isomorphism
$\gamma\colon\mathcal{D}_{X,\alpha}\simeq\mathcal{E}nd_{\mathcal{O}_{X}^{\\{\\!\\!\\!\
p\ \\!\\!\\!\\}}}(\mathcal{E})$, and morphisms are isomorphisms of vector
bundles that preserve this data. for $\mathcal{D}_{X,\alpha}$ on
$X^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}$ correspondingly.

Consider the map
$d\log\colon\mathcal{O}_{X}^{\times}\rightarrow\Omega^{1}_{X,cl}$ of étale
sheaves on $\mathcal{O}_{X}$ sending $f$ to $df/f$. It induces a map of étale
sheaves $F_{X*}^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}\mathcal{O}_{X}^{\times}\rightarrow F_{X*}^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}\Omega^{1}_{X,cl}$ on $X^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}$.

######  Proposition 2.46.

There is a short exact sequence of étale sheaves on $X^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}$

$0\rightarrow F_{X*}^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}\mathcal{O}_{X}^{\times}/(\mathcal{O}_{X}^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}})^{\times}\xrightarrow{d\log}F_{X*}^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}\Omega^{1}_{X,cl}\xrightarrow{\mathsf{c}_{p}}\Omega^{1,{\\{\\!\\!\\!\
p\ \\!\\!\\!\\}}}_{X}\rightarrow 0,$

where $\mathsf{c}_{p}$ is the map from Construction 2.42.

###### Proof.

The idea is to use the equivalence of groupoids described in Construction
2.45. First of all, the kernel of $d\log\colon F_{X*}^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}\mathcal{O}_{X}^{\times}\rightarrow F_{X*}^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}\Omega^{1}_{X,cl}$ is given by the intersection of
$F_{X*}^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}\mathcal{O}_{X}^{\times}$ and $\ker
d\subset F_{X*}^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}\mathcal{O}_{X}$. The latter is
isomorphic to $\mathcal{O}_{X}^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}$ by Proposition
2.40, so the intersection is exactly $(\mathcal{O}_{X}^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}})^{\times}$. This shows exactness from the left. To show that
$\mathsf{c}_{p}$ is surjective, for any local section
$\alpha\in\Omega^{1,{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}}_{X}$ it is enough to show
that étale locally on $X$ there is a line bundle with $p$-curvature given by
(the corresponding pull-back of) $\alpha$. By the above this is equivalent to
finding a splitting bundle for the Azumaya algebra $\mathcal{D}_{X,\alpha}$,
which is always possible étale locally. It remains to show the exactness in
the middle. Let $\alpha_{1},\alpha_{2}$ be two closed forms on an étale open
$U\rightarrow X$ such that
$\mathsf{c}_{p}(\alpha_{1})=\mathsf{c}_{p}(\alpha_{2})$. Then by the
discussion in Construction 2.45 line bundles with connection
$(\mathcal{O}_{U},d+\alpha_{1})$ and $(\mathcal{O}_{U},d+\alpha_{2})$ are
locally isomorphic, which means141414Indeed, if $(\mathcal{O}_{U},\nabla)$ and
$(\mathcal{O}_{U},\nabla^{\prime})$ are isomorphic, the isomorphism is given
by multiplication by $f$ and then $\nabla^{\prime}=\nabla+d\log f$. that
(after some cover) $\alpha_{1}=\alpha_{2}+d\log f$ for some function $f$. ∎

###  2.8. Restricted Poisson structures

We recall and try to motivate the definition of restricted Poisson structure
due to [BK].

Recall that a Poisson bracket on a commutative ring $A$ is a Lie bracket
$\\{-,-\\}$ on $A$ that is also a derivation in each variable. If the
corresponding derivations are $B$-linear for some algebra $B$ with a
homomorphism $B\rightarrow A$ then we will say that the Poisson bracket is
over $B$ (or $B$-linear).

######  Construction 2.47 (Free Poisson algebra).

Let $V$ be a $B$-module. Let $T(V)$ be the tensor algebra over $B$ and $L(V)$
the free Lie algebra over $B$ on $V$ correspondingly. Then $T(V)$ is the
universal envelopping algebra of $L(V)$. Viewed this way, $T(V)$ acquires a
PBW-filtration $F_{*}^{PBW}T(V)$, for which
$\operatorname{{gr}}_{*}^{PBW}T(V)\simeq\operatorname{{Sym}}^{*}_{B}V$. One
can also consider the corresponding Rees algebra151515In [BK] $Q(V)$ is called
the algebra of quantized polynomials in $V$, hence the notation.
$Q(V)\coloneqq\oplus_{i}F_{i}^{PBW}T(V)\cdot h^{i}$ corresponding to
$F_{*}^{PBW}T(V)$; this is an $h$-torsion free $B[h]$-algebra such that
$Q(V)[h^{-1}]\simeq T(V)[h,h^{-1}]$ and
$Q(V)/h\simeq\operatorname{{gr}}_{*}^{PBW}T(V)$. Since
$[F_{i}^{PBW}T(V),F_{j}^{PBW}T(V)]\subset F_{i+j-1}^{PBW}T(V)$ one has
$[Q(V),Q(V)]\subset h\cdot Q(V)$ and there is a natural Poisson bracket on
$\operatorname{{gr}}_{*}^{PBW}T(V)$ defined by

$h\cdot\\{f,g\\}\coloneqq[\tilde{f},\tilde{g}]\mod h^{2}$

where $\tilde{f}$ and $\tilde{g}$ are some lifts of
$f,g\in\operatorname{{gr}}_{*}^{PBW}T(V)\simeq Q(V)/h$ to $Q(V)$.

The resulting Poisson bracket on
$\operatorname{{gr}}_{*}^{PBW}T(V)\simeq\operatorname{{Sym}}^{*}_{B}L(V)$ is
identified with the Kostant-Kirillov bracket, namely the unique bracket that
on linear functions $z,w\in L(v)\subset\operatorname{{Sym}}^{*}_{B}L(V)$ is
given by $\\{z,w\\}\coloneqq[z,w]\in L(V)$. We call
$\operatorname{{Sym}}^{*}_{B}L(V)$ with the above bracket the free Poisson
algebra on $V$ and denote it $\mathrm{Pois}(V)$.

This is justified by the universal property: namely, any $B$-linear map
$f\colon V\rightarrow A$ to a Poisson algebra $A$ extends to a Poisson algebra
homomorphism $\mathrm{Pois}(V)\rightarrow A$. Indeed, $f$ extends to a Lie
algebra map $f^{\prime}\colon L(V)\rightarrow A$ with respect to the Poisson
bracket on $A$, and then we can extend $f^{\prime}$ to an algebra map
$\operatorname{{Sym}}^{*}_{B}L(V)\rightarrow A$. Since the Poisson bracket on
$\operatorname{{Sym}}^{*}_{B}L(V)$ is uniquely defined by its restriction to
$L(V)$ we get that this map is automatically Poisson. In particular, for any
Poisson algebra $A$ there is a natural Poisson algebra map
$\mathrm{Pois}(A)\rightarrow A$.

######  Remark 2.48.

The construction $V\mapsto\mathrm{Pois}(V)$ defines a monad on $B$-modules.
Indeed, we have a natural map $V\rightarrow\mathrm{Pois}(V)$ and a natural map
$\mathrm{Pois}(\mathrm{Pois}(V))\rightarrow\mathrm{Pois}(V)$ induced by the
Poisson structure on $\mathrm{Pois}(V)$ that endow $\mathrm{Pois}(-)$ with a
structure of a monad. Then, using the morphism $\mathrm{Pois}(A)\rightarrow A$
one can define Poisson $B$-algebras simply as modules over $\mathrm{Pois}(-)$.

The idea of a restricted Poisson structure on $A$ is that together with a
Poisson bracket one should include a restricted $p$-power operation
$-^{[p]}\colon A\rightarrow A$ that would turn $(A,\\{-,-\\})$ in a restricted
Lie algebra. However, the natural question is how $-^{[p]}$ should interact
with the mulptiplication on $A$. This is what we will try to explain in the
rest of this section. First, motivated by Construction 2.47 we can build the
free restricted Poisson algebra:

######  Construction 2.49 (Free restricted Poisson algebra: part I).

Let $V$ be a $B$-module and consider the free restricted Lie algebra
$L_{\mathsf{rest}}(V)$ on $V$. The underlying $B$-module of
$L_{\mathsf{rest}}(V)$ has a natural filtration $L_{\mathsf{rest}}(V)^{\leq*}$
with $L_{\mathsf{rest}}(V)^{\leq-1}=0$, $L_{\mathsf{rest}}(V)^{\leq 0}\simeq
L(V)$, and $L_{\mathsf{rest}}(V)^{\leq i+1}$ is inductively defined as
$L_{\mathsf{rest}}(V)^{\leq i}+(L_{\mathsf{rest}}(V)^{\leq i})^{[p]}$, where
by the Jacobson’s formula the latter expression is a well-defined $B$-module.
One has $\operatorname{{gr}}_{*}(L_{\mathsf{rest}}(V))\simeq\oplus_{i\geq
0}L(V)^{(i)}$, where $L(V)^{(i)}\coloneqq L(V)\otimes_{B,F_{B}^{i}}B$ is the
$i$’th Frobenius twist.

We can consider the corresponding universal enveloping algebra and its Rees
algebra
$Q_{\mathsf{rest}}(V)\coloneqq\oplus_{i}F_{i}^{PBW}U(L_{\mathsf{rest}}(V))$
with respect to the PBW-filtration. One can then define the free restricted
Poisson algebra $\mathrm{Pois}_{\mathsf{rest}}(V)$ as
$Q_{\mathsf{rest}}(V)/h\simeq\operatorname{{Sym}}L_{\mathsf{rest}}(V)$. As in
Construction 2.47 this is naturally a Poisson algebra via the Lie bracket on
$L_{\mathsf{rest}}(V)$. However, we also need to endow
$\operatorname{{Sym}}L_{\mathsf{rest}}(V)$ with a natural restricted
structure. For this we need a little more preparation.

Let $L$ be a restricted Lie algebra over $B$ (e.g. $L_{\mathsf{rest}}(V)$),
and consider its universal enveloping algebra $U(L)$. Let us denote by
$U_{h}(L)\coloneqq\oplus_{n}F_{n}^{PBW}U(L)\cdot h^{n}$ its Rees algebra with
respect to the PBW-filtration. $L$ embeds in $U_{h}(L)$ as $h\cdot L\subset
h\cdot F_{1}^{PBW}U(L)\subset U_{h}(L)$ and generates $U_{h}(L)$ over $B[h]$.

Consider the map $\theta\colon L\rightarrow U_{h}(L)$ which sends $x\in L$ to
$x^{p}-h^{p-1}x^{[p]}$. Note that it is $F_{B}$-linear.

######  Lemma 2.50.

$\theta$ is additive and its image lands in the center $Z(U_{h}(L))\subset
U_{h}(L)$.

###### Proof.

We will denote by $[-,-]_{L}$ the Lie bracket in $L$ to distinguish it from
the one in $U_{h}(L)$. The additivity follows from the Jacobson’s formula (see
Lemma 2.13 and part(2) of Definition 2.16). For the statement about the
center, it is enough to show that $[\theta(x),y]=0$ for $y\in L$. Note that
for $x,y\in L$ we have $[x,y]=h\cdot[x,y]_{L}$. Thus for $y\in L$ we have
$[x^{p},y]=\operatorname{{ad}}(x)^{p}(y)=h^{p}\operatorname{{ad}}_{L}(x)^{p}(y)$.
On the other hand, $[x^{[p]},y]=h\operatorname{{ad}}_{L}(x)^{p}(y)$. This
shows that $[\theta(x),y]=0$. ∎

The map from 2.50 gives a map of $B$-algebras

$\theta\colon\operatorname{{Sym}}L^{(1)}\rightarrow Z(U_{h}(L)).$

Let $-^{(1)}\colon L\rightarrow L^{(1)}$ be the natural map sending $x$ to
$x\otimes 1$. Consider also the map $p\colon
Q_{\mathsf{rest}}(V)\twoheadrightarrow\mathrm{Pois}_{\mathsf{rest}}(V)\simeq
Q_{\mathsf{rest}}(V)/h$ given by reduction modulo $h$. Note that given
$x\in\mathrm{Pois}_{\mathsf{rest}}(V)$ and any lift $\widetilde{x}\in
Q_{\mathsf{rest}}(V)$ the element $(\widetilde{x})^{p}\in
Q_{\mathsf{rest}}(V)$ is well-defined modulo $h^{p}$. Indeed, by Jacobson’s
formula

$(\widetilde{x}+hy)^{p}=\widetilde{x}^{p}+\sum_{i=1}^{p-1}{L_{i}}(\widetilde{x},hy)\pmod{h^{p}},$

but each ${L_{i}}(\widetilde{x},hy)$ is Lie polynomial of total Lie
degree161616Meaning that it is expressed as a sum of terms each involving a
composition of $p-1$ commutators. $p-1$ and (see the explicit description of
$L_{i}$ in Lemma 2.13)
$L_{i}(\widetilde{x},hy)=h^{i}L_{i}(\widetilde{x},{y})$. Thus
$(\widetilde{x}+hy)^{p}=\widetilde{x}^{p}\mod{h^{p}}.$

Then one can reconstruct the restricted structure on $L$ in terms of $\theta$,
namely

$h^{p-1}x^{[p]}=\tilde{x}^{p}-\theta(x^{(1)})\pmod{h^{p}}.$

Under the identification $h^{p-1}Q_{\mathsf{rest}}(V)/h^{p}\simeq
Q_{\mathsf{rest}}(V)/h$ this defines $x^{[p]}$ uniquely. The idea now is to
define the restricted structure on the whole of
$\operatorname{{Sym}}L_{\mathsf{rest}}(V)$ in a similar manner using the map
$\theta$.

######  Lemma 2.51.

  1. (1)

For any $x\in\mathrm{Pois}_{\mathsf{rest}}(V)$ and any lift $\widetilde{x}\in
Q_{\mathsf{rest}}(V)$ the element $\widetilde{x}^{p}-\theta(x^{(1)})$ is
divisible by $h^{p-1}$.

  2. (2)

The map
$-^{[p]}\colon\mathrm{Pois}_{\mathsf{rest}}(V)\rightarrow\mathrm{Pois}_{\mathsf{rest}}(V)$
defined by

$h^{p-1}x^{[p]}=\tilde{x}^{p}-\theta(x^{(1)})\pmod{h^{p}}$

endows
$\mathrm{Pois}_{\mathsf{rest}}(V)\simeq\operatorname{{Sym}}L_{\mathsf{rest}}(V)$
with a restricted Lie algebra structure that agrees with the one on
$L_{\mathsf{rest}}(V)$.

###### Proof.

Consider the $B[h]/h^{p-1}$-algebra given by $Q_{\mathsf{rest}}(V)/h^{p-1}$.
By the discussion above concerning the Jacobson’s formula,
$x\mapsto\tilde{x}^{p}$ produces a well-defined map
$\mathrm{Pois}_{\mathsf{rest}}(V)\simeq Q_{\mathsf{rest}}(V)/h\rightarrow
Q_{\mathsf{rest}}(V)/h^{p-1}$. By [BK, Lemma 1.3] this is in fact an algebra
homomorphism. Moreover, it agrees with $\theta$ on $L_{\mathsf{rest}}$ and
since
$\mathrm{Pois}_{\mathsf{rest}}(V)\coloneqq\operatorname{{Sym}}L_{\mathsf{rest}}(V)$
it agrees with $\theta$ on the whole $\mathrm{Pois}_{\mathsf{rest}}(V)$. We
get that $\widetilde{x}^{p}-\theta(x^{(1)})=0$ modulo $h^{p-1}$ for any
$x\in\mathrm{Pois}_{\mathsf{rest}}(V)$.

For (2) we need to check the defining properties of restricted Lie algebra
(Definition 2.16). First of all, $-^{[p]}$ is Frobenius-linear since
$x\rightarrow\tilde{x}^{p}$ and $\theta$ are. Note that since
$\\{x,y\\}=h[\tilde{x},\tilde{y}]\mod h^{2}$ one has

$\operatorname{{ad}}_{[-,-]}(\tilde{x}^{p})(\tilde{y})=\operatorname{{ad}}_{[-,-]}(\tilde{x})^{p}(\tilde{y})=h^{p}\cdot\operatorname{{ad}}_{\\{-,-\\}}(x)^{p}(y)\mod
h^{p+1}.$

Since $\theta(x^{(1)})$ is central applying $[-,\tilde{y}]$ to the expression
for $x^{[p]}$ we get that

$h^{p}\operatorname{{ad}}_{\\{-,-\\}}(x^{[p]})(y)=h^{p}\operatorname{{ad}}_{\\{-,-\\}}(x)^{p}(y)\mod
h^{p+1}.$

Finally, the formula for $(x+y)^{[p]}$ follows from the Jacobson formula.
Namely, $\theta$ is a homomorphism, so we have
$\theta(x^{(1)}+y^{(1)})=\theta(x^{(1)})+\theta(y^{(1)})$, and

$h^{p-1}\cdot((x+y)^{[p]}-x^{[p]}-y^{[p]})=(\tilde{x}+\tilde{y})^{p}-\tilde{x}^{p}-\tilde{y}^{p}\mod
h^{p}$

where the right hand side is given by $h^{p-1}\sum_{i}L_{i}(x,y)$ (2.13, here
since each $L_{i}$ is a Lie polynomial of degree $p-1$ we have
$L_{i}(\tilde{x},\tilde{y})=h^{p-1}L_{i}(x,y)$). ∎

######  Construction 2.52 (Free restricted Poisson algebra: part II).

Continuing Construction 2.49, via Lemma 2.51 we can now endow
$\mathrm{Pois}_{\mathsf{rest}}(V)$ not only with a Poisson bracket, but also a
$p$-power operation $-^{[p]}$ such that
$(\mathrm{Pois}_{\mathsf{rest}}(V),\\{-,-\\},-^{[p]})$ becomes a restricted
Lie algebra. This allows to consider
$\mathrm{Pois}_{\mathsf{rest}}(\mathrm{Pois}_{\mathsf{rest}}(V))$ and a
homomorphism of algebras

$\mathrm{Pois}_{\mathsf{rest}}(\mathrm{Pois}_{\mathsf{rest}}(V))\rightarrow\mathrm{Pois}_{\mathsf{rest}}(V)$

defined (via the identification
$\mathrm{Pois}_{\mathsf{rest}}(\mathrm{Pois}_{\mathsf{rest}}(V))\simeq\operatorname{{Sym}}L_{\mathsf{rest}}(\mathrm{Pois}_{\mathsf{rest}}(V))$)
by sending $L_{\mathsf{rest}}(\mathrm{Pois}_{\mathsf{rest}}(V))$ to
$\mathrm{Pois}_{\mathsf{rest}}(V)$ using the structure of restricted Lie
algebra, and then extending by multiplicativity. We also have an embedding
$V\subset L_{\mathsf{rest}}(V)\subset\mathrm{Pois}_{\mathsf{rest}}(V)$ which
with the map above endows $\mathrm{Pois}_{\mathsf{rest}}$ with a structure of
a monad on the category of $B$-modules.

######  Definition 2.53 (Restricted Poisson algebra).

A restricted Poisson algebra $A$ over $B$ is a module over the monad given by
$V\mapsto\mathrm{Pois}_{\mathsf{rest}}(V)$.

######  Remark 2.54.

We note that there are a lot of structures underlying the restricted Poisson
structure. Let $A$ be a restricted Poisson $B$-algebra.

  * •

There is a natural map of monads
$\mathrm{Pois}(-)\rightarrow\mathrm{Pois}_{\mathsf{rest}}(-)$ induced by the
embedding
$\operatorname{{Sym}}^{*}(L(V))\rightarrow\operatorname{{Sym}}^{*}(L_{\mathsf{rest}}(V))$,
which in turn is induced by $L(V)\rightarrow L_{\mathsf{rest}}(V)$ for each
$B$-module $V$. Thus by Remark 2.48, $A$ has a natural Poisson algebra
structure.

  * •

There is a natural map of monads
$L_{\mathsf{rest}}(V)\rightarrow\mathrm{Pois}_{\mathsf{rest}}(-)$ induced by
the embedding
$L_{\mathsf{rest}}(V)\rightarrow\operatorname{{Sym}}^{*}(L_{\mathsf{rest}}(V))$
for each $B$-module $V$. This gives a restricted Lie algebra structure on $A$.

  * •

There is a natural diagram of monads

$\textstyle{L(-)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{Pois}(-)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{L_{\mathsf{rest}}(-)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{Pois}_{\mathsf{rest}}(-)}$

given by the above maps and embeddings
$L(V)\rightarrow\operatorname{{Sym}}^{*}(L(V))$ and $L(V)\rightarrow
L_{\mathsf{rest}}(V)$ for each $B$-module $V$ correspondingly, which shows
that the Lie algebra structure on $A$ given by the above Poisson structure and
restricted Lie structure on $A$ agree.

Finally, let us describe more explicitly how $(-)^{[p]}$ in
$\mathrm{Pois}_{\mathsf{rest}}(V)$ interects with multiplication.

######  Construction 2.55 (Polynomial $P(x,y)$ and formula for $(xy)^{[p]}$.).

Consider $V\simeq Bx\oplus By$ a free $B$-module of rank 2 with basis given by
$x$ and $y$. Consider the (non-restricted) algebras $Q(V)$ and
$\mathrm{Pois}(V)$. Then by [BK, Equation 1.3] (which in turn follows from
[BK, Lemma 1.3]) the expression $(xy)^{p}-x^{p}y^{p}$ is divisible by
$h^{p-1}$. One can the define a Poisson polynomial
$P(x,y)\in\mathrm{Pois}(x,y)\coloneqq\mathrm{Pois}(V)$ by

$h^{p-1}P(x,y)=(xy)^{p}-x^{p}y^{p}\pmod{h^{p}Q(V)}.$

Then, for the restricted structures in $Q_{\mathsf{rest}}(V)$ we have an
equality

$\displaystyle h^{p-1}(xy)^{[p]}\coloneqq(xy)^{p}$
$\displaystyle-\theta(xy)=x^{p}y^{p}+h^{p-1}P(x,y)-(x^{p}-h^{p-1}x^{[p]})(y^{p}-h^{p-1}y^{[p]})=$
$\displaystyle=h^{p-1}(x^{p}y^{[p]}+x^{[p]}y+P(x,y))\pmod{h^{p}},$

which shows that
$(xy)^{[p]}=x^{p}y^{[p]}+x^{[p]}y+P(x,y)\in\mathrm{Pois}_{\mathsf{rest}}(V)$.
Note that for any $V$ this formula uniquely defines the restricted structure
on
$\mathrm{Pois}_{\mathsf{rest}}(V)\simeq\operatorname{{Sym}}^{*}(L_{\mathsf{rest}}(V))$
by induction on the degree.

This way for any restricted Poisson algebra $A$ and any two elements $x,y\in
A$ we have

(2.5) $(xy)^{[p]}=x^{p}y^{[p]}+x^{[p]}y+P(x,y),$

where the value of $P(x,y)\in A$ is computed via the map
$\mathrm{Pois}(A)\rightarrow A$. In fact, since the above formula defines the
restricted structure on a free restricted Poisson algebra uniquely, one can
give an alternative definition of a restricted Poisson algebra as a Poisson
$B$-algebra $A$ with a restricted $p$-power operation $-^{[p]}$ such that
$(A,\\{-,-\\},-^{[p]})$ is restricted Lie algebra and that the relation (2.5)
is satisfied for any $x,y\in A$. This is how it is defined in [BK] (namely,
see [BK, Definiton 1.8]).

Equation 2.5 is hard to check in practice in the original form essentially
because the definition of $P(x,y)$ is complicated (and not very explicit).
But, at least when $p>2$ there is an equivalent condition that is easier to
check:

######  Remark 2.56.

Let $p>2$. Let $x\in\mathrm{Pois}(V)$ be some element. Then, for any lift
$\widetilde{x}\in Q(V)$ we have
$(\widetilde{x}^{2})^{p}=\widetilde{x}^{p}\cdot\widetilde{x}^{p}$ and so
$P(x,x)=0$. From this and (2.5) we get that

(2.6) $(x^{2})^{[p]}=2x^{[p]}x^{p}.$

Note that $xy=\frac{1}{4}((x+y)^{2}-(x-y)^{2})$. Thus, applying the restricted
power, we get

$(xy)^{[p]}=\tfrac{1}{4}((x+y)^{2}-(x-y)^{2})^{[p]}$

which, using Jacobson’s formula, gives an expression for $P(x,y)$ in terms of
$-^{[p]}$ applied to squares and some Poisson polynomial in $x$, $y$,
$x^{[p]}$ and $y^{[p]}$. This shows that checking Equation (2.5) reduces to
checking that $-^{[p]}$ together with the Poisson bracket give a restricted
structure in the Lie algebra structure and that Equation (2.5) holds for all
$x$.

In particular, $(A,\\{-,-\\},-^{[p]})$ defines a restricted Poisson algebra
2.5 if and only if

  * •

$(A,\\{-,-\\})$ is a Poisson algebra;

  * •

$(A,\\{-,-\\},-^{[p]})$ defines a restricted Lie algebra;

  * •

$(x^{2})^{[p]}=2x^{[p]}x^{p}$ for any $x\in A$.

This simplification of [BK, Definiton 1.8] was also previously observed in
[BYZ].

###  2.9. Restricted symplectic geometry

######  Definition 2.57.

A symplectic scheme $(X,\omega)$ over $S$ is a pseudo-smooth $S$-scheme $X$
with a non-degenerate globally defined closed 2-form $\omega\in
H^{0}(X,\Omega^{2}_{X})$.

Since $\omega$ is non-degenerate it induces an identification
$\mathcal{T}_{X}\xrightarrow{\sim}\Omega^{1}_{X}$ via
$\xi\mapsto\iota_{\xi}\omega$, where $\iota_{\xi}$ is the contraction with
$\xi$. This way $\omega$ also defines a section $\omega^{-1}$ of
$\wedge^{2}_{\mathcal{O}_{X}}\mathcal{T}_{X}$, which then gives a Poisson
bracket on $\mathcal{O}_{X}$: $\\{f,g\\}\coloneqq\langle\omega^{-1},df\wedge
dg\rangle$. The property that $d\omega=0$ translates to the Jacobi identity
for $\\{-,-\\}$: in particular $(\mathcal{O}_{X},\\{-,-\\})$ defines a sheaf
of Lie algebras over $\mathcal{O}_{S}$.

Having a Poisson bracket $\\{-,-\\}$, for any $f\in\mathcal{O}_{X}$ the
bracket $\\{f,-\\}$ defines a derivation
$H_{f}\colon\mathcal{O}_{X}\rightarrow\mathcal{O}_{X}$. This gives a map of
sheaves $H_{-}\colon\mathcal{O}_{X}\rightarrow\mathcal{T}_{X}$ locally sending
$f\mapsto H_{f}$.

The following definition is classical:

######  Definition 2.58.

A derivation $\xi\in H^{0}(X,\Omega^{1})$ is called Hamiltonian if $\xi=H_{f}$
for some $f\in H^{0}(X,\mathcal{O}_{X})$.

######  Remark 2.59.

Tracing through the definitions one can see that the equation $\xi=H_{f}$ is
in fact equivalent to $\iota_{\xi}\omega=df$. Since $\omega$ is non-degenerate
one sees from this that Hamiltonian vector fields locally generate
$\mathcal{T}_{X}$ over $\mathcal{O}_{X}$.

One of the main ideas in [BK] is that in char $p$ there is a natural
enhancement of the usual Poisson structure called restricted Poisson structure
(see Definition 2.53). Namely, together with the Poisson bracket $\\{-,-\\}$
on $X$ they ask for a restricted $p$-th power operation
$-^{[p]}\colon\mathcal{O}_{X}\rightarrow\mathcal{O}_{X}$ that turns
$(\mathcal{O}_{X},\\{-,-\\})$ into a restricted Lie algebra compatibly with
such structure on $\mathcal{T}_{X}$ (meaning that $H_{f^{[p]}}=H_{f}^{[p]}$).
It should also interact with multiplication in a particular way, namely for
any $f,g\in\mathcal{O}_{X}$ we should have

(2.7) $(fg)^{[p]}=f^{p}g^{[p]}+f^{[p]}g^{p}+P(f,g),$

where $P(x,y)$ is a certain Poisson polynomial (see Remark 2.55). By Remark
2.56, if $p>2$, this is also equivalent to the simpler relation

$(f^{2})^{[p]}=2f^{[p]}f^{p}$

for any $f\in\mathcal{O}_{X}$.

In [BK, Theorem 1.11] Bezrukavnikov and Kaledin show that given a symplectic
$S$-scheme $(X,\omega)$ Hamiltonian vector fields are closed under the
restricted $p$-th power $-^{[p]}$ if and only if171717Here $C\colon
F_{X*}^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}\Omega^{2}_{X,cl}\rightarrow\Omega^{2,{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}}_{X}$ is the Cartier operator and the condition $C(\omega)=0$ is
equivalent to $\omega$ being exact locally in Zariski topology. $C(\omega)=0$.
Under this assumption, a consequence of [BK, Theorem 1.12] is that restricted
structures extending the Poisson bracket given by $\omega$ are in bijection
with classes $[\eta]\in H^{0}(X,\Omega^{1}_{X}/d(\mathcal{O}_{X}))$ such that
$d[\eta]=\omega$. Note that such classes $\eta$ form a torsor on
$X^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}$ over the vector bundle
$F_{X*}^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}\Omega^{1}_{X,cl}/d(F_{X*}^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}\mathcal{O}_{X})\simeq\Omega^{1,{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}}_{X}$.

######  Definition 2.60.

A restricted symplectic $S$-scheme $(X,[\eta])$ is a pair of a pseudo-smooth
$S$-scheme $X$ and a section $[\eta]\in
H^{0}(X,\Omega^{1}_{X}/d(\mathcal{O}_{X}))$ such that $\omega\coloneqq
d\eta\in H^{0}(X,\Omega^{2}_{X})$ is non-degenerate.

######  Remark 2.61.

Given $[\eta]$, one reconstructs the restricted structure $-^{[p]}$ via an
explicit formula $f^{[p]}=H_{f}^{p-1}(\langle H_{f},\eta\rangle)-\langle
H_{f}^{[p]},\eta\rangle$. Also, any two restricted structures
$-^{[p]_{1}},-^{[p]_{2}}$ differ by a Frobenius-differentiation of
$\mathcal{O}_{X}$: this can be seen directly from Equation 2.7.

The following remark will be important.

######  Remark 2.62 (Doing differential geometry on $X$ as a scheme over
$X^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}$ and $S$ is the same).

Let $X$ be an $S$-scheme and let $X\rightarrow X^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}$ be the reduced Frobenius. Then we claim that providing $X$ with
the structure of a restricted symplectic scheme over $S$ is the samle as
providing it with the same structure over the reduced twist $X^{\\{\\!\\!\\!\
p\ \\!\\!\\!\\}}$. Indeed, first of all we have a commutative diagram

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X^{\\{\\!\\!\\!\
p\
\\!\\!\\!\\}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S}$

which gives an exact sequence

$F_{X}^{{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}*}\Omega^{1}_{X^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}}\xrightarrow{F_{X}^{{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}*}}\Omega^{1}_{X}\rightarrow\Omega^{1}_{X/X^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}}\rightarrow 0.$

Since the pull-back map $F_{X}^{{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}*}\Omega^{1}_{X^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}}\rightarrow\Omega^{1}_{X}$ is 0 we get that
$\Omega^{1}_{X/X^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}}\simeq\Omega^{1}_{X}$ and so
$X$ is pseudo-smooth over $X^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}$. We also have
isomorphisms $\Omega^{i}_{X/X^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}}\simeq\Omega^{i}_{X}$ and, more generally of the de Rham complex
$\mathrm{dR}_{X/X^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}}\simeq\mathrm{dR}_{X}$.
Lastly, essentially by the definition of reduced twist, $X^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}$ for $X$ considered as an $X^{\\{\\!\\!\\!\ p\
\\!\\!\\!\\}}$-scheme and as $S$-scheme coincide (because
$\mathcal{O}_{X}^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}$ is defined as the image of
relative Frobenius). Thus symplectic and, more generally, restricted
symplectic structures on $X$ as an $X^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}$-scheme
or an $S$-scheme are the same thing.

###  2.10. The group scheme $G_{0}$ and Darboux lemma

Another insight of Bezrukavnikov and Kaledin is that restricted Poisson
structures on $X$ can be interpreted in terms of “formal geometry” and related
torsors. To recall it let us introduce one more piece of notation.

######  Construction 2.63 (Algebra $A_{0}$).

Fix a number $d\in\mathbb{N}$ and let $S=\operatorname{{Spec}}\mathbb{F}_{p}$.
Let $V$ be a vector space scheme of dimension $d$ over $\mathbb{F}_{p}$ (so
$V\simeq\operatorname{{Spec}}\mathbb{F}_{p}[x_{1},\ldots,x_{d}]$) and let
$W=V\oplus V^{\vee}$ be the total space of the cotangent bundle to $V$: thus
$W$ is a symplectic vector space of dimension $2d$. Let $y_{i}$ be the dual
coordinates to $x_{i}$; one has
$W\simeq\operatorname{{Spec}}\mathbb{F}_{p}[x_{1},\ldots,x_{d},y_{1},\ldots,y_{d}]$.

Consider the ring of functions on the Frobenius neighborhood
$F_{W}^{-1}(\\{0\\})$ of $0$ in $W$ (see Example 2.11). Explicitly,
put181818This algebra is denoted $A$ in [BK, Section 3].

$A_{0}\coloneqq\mathbb{F}_{p}[x_{i},y_{j}]_{1\leq i,j\leq
d}/x_{i}^{p}=y_{j}^{p}=0.$

One has $\Omega^{1}_{A_{0}}\simeq(\oplus_{i=1}^{d}A_{0}\cdot
dx_{i})\oplus(\oplus_{i=1}^{d}A_{0}\cdot dy_{i})$ (namely, since
$dx_{i}^{p}=dy_{j}^{p}=0$ the map
$\Omega^{1}_{W}\otimes_{\mathbb{F}_{p}[x_{i},y_{j}]}A_{0}\rightarrow\Omega^{1}_{A_{0}}$
is an isomorphism). In particular, $A_{0}$ is pseudo-smooth over
$\mathbb{F}_{p}$. $\operatorname{{Spec}}A_{0}$ endowed with a natural
restricted symplectic structure given by the 1-form

$\eta_{\mathsf{can}}=\sum_{i=1}^{d}y_{i}dx_{i}\in\Omega^{1}_{A_{0}}.$

Indeed, the 2-form $\omega\coloneqq d\eta_{\mathsf{can}}=\sum_{i}dy_{i}\wedge
dx_{i}\in\Omega^{2}_{A_{0}}$ is easily seen to be non-degenerate.

Similarly, for any $\mathbb{F}_{p}$-algebra $R$ we can consider an $R$-algebra

$A_{0}(R)\coloneqq A_{0}\otimes_{\mathbb{F}_{p}}R.$

We have
$\Omega^{1}_{A_{0}(R)/R}\simeq\Omega^{1}_{A_{0}}\otimes_{\mathbb{F}_{p}}R$ and
via this identification the pair
$(\operatorname{{Spec}}A_{0}(R),[\eta_{\mathsf{can}}\otimes 1])$ defines a
restricted symplectic $R$-scheme (see Definition 2.60). Further we will denote
$\eta_{\mathsf{can}}\otimes 1$ by $\eta_{\mathsf{can}}$ for convenience.

######  Construction 2.64 (Group scheme $G_{0}$).

Consider the functor

$A_{0}\colon R\mapsto A_{0}(R)$

on $\mathbb{F}_{p}$-algebras. Let $\underline{\mathrm{Aut}}(A_{0})$ be the
functor

$\underline{\mathrm{Aut}}(A_{0})\colon
R\mapsto\mathrm{Aut}_{R}(A_{0}(R))\simeq\mathrm{Iso}_{R}(\operatorname{{Spec}}A_{0}(R),\operatorname{{Spec}}A_{0}(R))$

that sends an $\mathbb{F}_{p}$-algebra $R$ to the group of $R$-linear algebra
automorphisms. Since $A_{0}(R)$ is a free module over $R$ one sees that
$\underline{\mathrm{Aut}}(A_{0})$ with represented by the affine group
subscheme of $\operatorname{{GL}}(A_{0})$ given by those linear automorphisms
that preserve multiplication on $A_{0}$. Finally, let
$G_{0}\subset\underline{\mathrm{Aut}}(A_{0})$ be the functor

$R\mapsto\mathrm{Aut}_{R}^{\mathsf{rest}}((\operatorname{{Spec}}A_{0}(R),[\eta_{\mathsf{can}}]))$

that sends $R$ to the group of $R$-algebra automorphisms of $A_{0}(R)$ that
preserve the restricted symplectic structure given by $\eta_{\mathsf{can}}$.
It can be seen to be represented by a group subscheme of
$\underline{\mathrm{Aut}}(A_{0})$: namely, one adds equations to
$\phi\in\underline{\mathrm{Aut}}(A_{0})$ saying that
$\mathsf{C}(\phi^{*}\eta_{\mathsf{can}}-\eta_{\mathsf{can}})=0$ (which is
equivalent to $[\eta_{\mathsf{can}}]\in\Omega^{1}_{A_{0}(R)}/d(A_{0}(R))$
being preserved under $\phi$).

######  Remark 2.65.

The group scheme $H\coloneqq\underline{\mathrm{Aut}}(A_{0})$ is not
particularly nice, for example it is not reduced. Indeed, let
$I_{0}=(x_{1},\ldots,x_{d},y_{1},\ldots,y_{d})\subset A_{0}$ be the maximal
ideal. Note that for a reduced $R$ any $\phi\in\mathrm{Aut}_{R}(A_{0}(R))$
preserves $I_{0}(R)\coloneqq I_{0}\otimes R$: indeed $I_{0}(R)$ in this case
coincides with nilradical of $A_{0}(R)$. If $H$ were reduced this would also
be true for the $\mathcal{O}(H)$-point of $H$ given by the identity map
$H\xrightarrow{\mathrm{id}}H$. Since any $R$-point of $H$ factors through this
one we would get that any $\phi\in\mathrm{Aut}_{R}(A_{0}(R))$ even for non-
reduced $R$ also preserve $I_{0}(R)$. However, this is not true: whenever
$\alpha_{p}(R)\neq 0$ one can consider an automorphism sending $x_{i}$ to
$x_{i}+\varepsilon$ for some non-zero $\varepsilon\in\alpha_{p}(R)$: this
gives a well defined automorphism of $A_{0}(R)$ that doesn’t preserve
$I_{0}(R)$. Similarly, one sees that $G_{0}$ is also not reduced: namely
$\eta_{\mathsf{can}}$ is also preserved by the above automorphism.

######  Construction 2.66 (Torsor of “Frobenius-frames”).

Let $X$ be a pseudo-smooth scheme over a base scheme $S$ such that
$\operatorname{{rk}}H^{0}(X,\Omega^{1}_{X/S})=2d$. Consider the algebra
$A_{0}$ from Construction 2.63 for that particular $d$. Consider the reduced
Frobenius map $F_{X}^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}\colon X\rightarrow
X^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}$ (Construction 2.10).

One can construct a canonical $\underline{\mathrm{Aut}}(A_{0})$-torsor
$\mathcal{M}_{X}$ on $X^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}$ as follows. Namely,
for a map $T\rightarrow X^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}$ one puts

$\mathcal{M}_{X}(T)\coloneqq\mathrm{Iso}_{T}(\operatorname{{Spec}}A_{0}\times
T,X\times_{X^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}}T)$

This is a sheaf in flat topology on the category of schemes over
$X^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}$. Note that $\mathcal{M}_{X}$ has a natural
action of $\underline{\mathrm{Aut}}(A_{0})$. Moreover, flat locally on
$X^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}$ one has an isomorphism

${\mathcal{M}_{X}|}_{U}\simeq\underline{\mathrm{Aut}}(A_{0})\times U.$

Indeed, let $U\rightarrow X$ be a Zariski cover that trivializes
$\Omega^{1}_{X/S}$. Then $U$ is still pseudo-smooth, so $F_{U}^{\\{\\!\\!\\!\
p\ \\!\\!\\!\\}}\colon U\rightarrow U^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}$ is
faithfully flat (as well as the composite map $U\rightarrow U^{\\{\\!\\!\\!\
p\ \\!\\!\\!\\}}\rightarrow X^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}$) and by Lemma
2.67 below

$X\times_{X^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}}U\simeq U\times_{U^{\\{\\!\\!\\!\
p\ \\!\\!\\!\\}}}U\simeq\operatorname{{Spec}}A_{0}\times U.$

This way the restriction ${\mathcal{M}_{X}|}_{U}$ is identified with the
functor

$(T\rightarrow U)\mapsto\mathrm{Iso}_{T}(\operatorname{{Spec}}A_{0}\times
T,\operatorname{{Spec}}A_{0}\times T)$

which is exactly $X\times\underline{\mathrm{Aut}}(A_{0})$. This way we see
that the action of $\underline{\mathrm{Aut}}(A_{0})$ on $\mathcal{M}_{X}$ is
effective and that the latter indeed defines an
$\underline{\mathrm{Aut}}(A_{0})$-torsor over $X^{\prime}$ (in flat topology).
In [BK] it is called the torsor of Frobenius frames on $X$.

######  Lemma 2.67.

Let $X$ be a pseudo-smooth scheme over a base scheme $S$. Then Zariski locally
on $X$ one has a natural isomorphism

$X\times_{X^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}}X\simeq
X\times\operatorname{{Spec}}A_{0}.$

###### Proof.

When $X$ is smooth over $S$, $F_{X}^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}\colon
X\rightarrow X^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}$ is given by the relative
Frobenius $F_{X/S}\colon X\rightarrow X^{(1)}$ and the statement is classical:
Zariski locally one has an étale map $X\rightarrow\mathbb{A}^{2d}_{S}$ and in
the case of affine space the isomorphism is easy to construct explicitly.

For general pseudo-smooth scheme we use Proposition 2.35: namely, $X$ Zariski
locally is identified with the Frobenius neighborhood $(F_{Y}^{\\{\\!\\!\\!\
p\ \\!\\!\\!\\}})^{-1}(Z)$ in a smooth $S$-scheme $Y$ of a closed subscheme
$Z\hookrightarrow Y^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}$. Following Remark 2.33
the reduced Frobenius map $F_{X}^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}\colon
X\rightarrow X^{\\{\\!\\!\\!\ p\ \\!\\!\\!\\}}\simeq Z$ is the pull-back of