# Towards Affinoid Duflo’s Theorem I: Twisted differential operators

Ioan Stanciu

##  1 Introduction

This paper is the first of a series of papers that aims to answer Question A
from [2] regarding the classification of primitive ideals in the affinoid
enveloping algebra of a semisimple Lie algebra defined over a discrete
valuation ring.

######  Theorem 1.1.

Let $\mathfrak{g}$ be a semisimple Lie algebra defined over a field $K$ of
characteristic $0$. Then any primitive ideal in $U(\mathfrak{g})$ with
$K$-rational infinitesimal central character is the annihilator of the simple
quotient of some Verma module. In case $K=\mathbb{C}$, the theorem gives a
classification of all primitive ideals.

Assume for now that our ground field is $\mathbb{C}$ and let $\mathfrak{g}$ be
a semisimple $\mathbb{C}$-Lie algebra. For a complex variety $X$, we will
denote $\mathcal{D}_{X}$ the sheaf of differential operators on $X$. Let $G$
be the semisimple connected affine algebraic group associated with
$\mathfrak{g}$ and fix $B$ a Borel subgroup. Let $X=G/B$ be the flag variety
associated to $\mathfrak{g}$. In [6, Proposition 3.6], the authors prove an
equivalence of categories between $G$-equivariant coherent
$\mathcal{D}_{X\times X}$-modules and $B$-equivariant coherent
$\mathcal{D}_{X}$-modules. Further, the Beilinson-Bernstein theorem [3]
establishes an equivalence of categories between coherent
$\mathcal{D}_{X}$-modules and finitely generated $U(\mathfrak{g})$-modules
with trivial central character. Combining these two results, one obtains a
geometric proof of Duflo’s theorem for ideals with _trivial central
character_. In the next article, we remove some restrictions from [6,
Proposition 3.6]: we prove that the results hold over a general commutative
Noetherian ring and that the equivalence holds between coherent modules over
certain _homogeneous sheaves of twisted differential operators_ , which can be
regarded as _equivariant_ twisted differential operators.

There is a well established theory of twisted differential operators and
homogeneous twisted differential operators over a _complex_ variety introduced
in [4] and treated in more detail in [9]. The authors also explore the
connections between twisted differential operators(tdo’s) and Picard
algebroids and the following question is answered:

######  Question 1.2.

Let $f:Y\to X$ be a map of smooth complex varieties and let $\mathcal{D}$ be a
tdo on $X$. How should we define the pullback of $\mathcal{D}$, call it
$f^{\bullet}\mathcal{D}$, such that $f^{\bullet}\mathcal{D}$ is a tdo on $Y$?

The solution proposed in [4] was to define $f^{\bullet}\mathcal{D}$ using the
$\operatorname{Dif\text{\kern-2.0pt}\,}$ functor introduced by Grothendieck:

$f^{\bullet}\mathcal{D}:=\operatorname{Dif\text{\kern-2.0pt}\,}_{f^{-1}\mathcal{D}}(f^{*}\mathcal{D},f^{*}\mathcal{D})$

is the sheaf of differential operators from $f^{*}\mathcal{D}$ to itself that
commute with the right $f^{-1}\mathcal{D}$-action. In particular when
$\mathcal{D}=\mathcal{D}_{X}$, we obtain
$f^{\bullet}\mathcal{D}_{X}=\mathcal{D}_{Y}$.

Now, let $R$ be a commutative base ring and $X$ be an $R$-scheme that is
smooth, separated and locally of finite type. In order to build a good theory
of twisted differential operators over a commutative ring, there are two basic
questions we need to answer:

Question What constitutes a good definition of a tdo on $X$?

Question Given $f:Y\to X$ a map of smooth, separated and locally of finite
type $R$-schemes and $\mathcal{D}$ a tdo on $X$, how should we define the
pullback of $\mathcal{D}$ such that it is also a tdo on $Y$?

There are two possible candidates of sheaves of differential operators that
one can define over $X$: $\mathcal{D}_{X}$-the sheaf of crystalline
differential operators and $\mathscr{D}_{X}$-the sheaf of Grothendieck’s
differential operators.

One of the key properties satisfied by twisted differential operators over
complex varieties is that they come equipped with a filtration such that the
associated graded is isomorphic with the symmetric algebra of the tangent
sheaf over the ring of functions. The sheaf of Grothendieck’s differential
operators $\mathscr{D}_{X}$ has a natural filtration given by the order of
differential operators, but the associated grading ring does not satisfy the
desired property. Therefore, we choose to work with the sheaf of crystalline
differential operators.

Attempting working with the classical definition of the pullback of tdo’s we
immediately encounter a problem. Assume that $Y=\mathbb{A}^{1}$ is the affine
line over $R$, let $X=\operatorname{Spec\text{\kern-2.0pt}\,}R$ be the base of
$Y$ and $f:Y\to X$ be the natural projection. Let $\mathcal{D}_{X}\cong R$ be
the sheaf of crystalline differential operators on $X$. Then using the
classical definition we obtain

$f^{\bullet}\mathcal{D}\cong\mathscr{D}_{\mathbb{A}^{1}},$

which is the sheaf of _Grothendieck’s differential operators_. In particular,
if we work with the classical definition we obtain that the pullback of a
twisted differential operator does not satisfy the desired property.

To resolve the problem with the definition, we will explore the correspondence
between twisted differential operators (shortened tdo’s frow now on) and
_Picard algebroids_ and define the pullback of tdo’s by first defining the
pullback of Picard algebroids.

Specifically, we will be able to go from tdo’s to Picard algebroids and vice-
versa using two functors $\operatorname{Lie\text{\kern-2.0pt}\,}$ and
$\mathscr{T}$. For a Picard algebroid $\mathcal{L}$ on $X$, we can consider
the pullback $f^{\\#}\mathcal{L}$ as a Lie algebroid on $Y$ and for a general
tdo on $X$ we define

$f^{\\#}\mathcal{D}:=\mathscr{T}(f^{\\#}(\operatorname{Lie\text{\kern-2.0pt}\,}(\mathcal{D}))).$

In the case when $R=\mathbb{C}$, our definition coincides with the definition
found in [4] and [9].

Statement of the main results of the article

Let $G$ be a smooth affine algebraic group of finite type over
$\operatorname{Spec\text{\kern-2.0pt}\,}R$ and let $f:Y\to X$ be a locally
trivial $G$-torsor. Then for a tdo $\mathcal{D}$ equipped with a suitable
$G$-action (we call this a $G$-htdo), we may define its descent
$f_{\\#}\mathcal{D}^{G}$, which is a tdo on $X$.

######  Proposition 1.3 (Corollary 10.12).

The functors $f_{\\#}(-)^{G}$ and $f^{\\#}(-)$ induce quasi-inverse
equivalences between $G$-htdo’s on $Y$ and tdo’s on $X$.

For a $G$-htdo $\mathcal{D}$ we may define the
$\operatorname{Coh\text{\kern-2.0pt}\,}(\mathcal{D},G)$ the category of
$G$-equivariant coherent $\mathcal{D}$-modules. Further, let $\mathcal{A}$ be
a tdo on $X$ and $\mathcal{M}$ a $\mathcal{A}$-module. Then we may endow the
$\mathcal{O}$-module pullback $f^{*}\mathcal{M}$ with the structure of a
$f^{\\#}\mathcal{A}$-module and we call this module $f^{\\#}\mathcal{M}$.

######  Theorem 1.4 (Theorem 11.8).

Assume that $R$ is a Noetherian ring. Let $G$ be a smooth affine algebraic
group of finite type. Let $X,Y$ be smooth separated and locally of finite type
$R$-schemes and let $f:Y\to X$ be a locally trivial $G$-torsor. Further, let
$\mathcal{D}$ be a sheaf of $G$-homogeneous twisted differential operators on
$Y$. The functors:

$\begin{split}&f_{*}(-)^{G}:\operatorname{Coh\text{\kern-2.0pt}\,}(\mathcal{D},G)\to\operatorname{Coh\text{\kern-2.0pt}\,}(f_{\\#}\mathcal{D}^{G})\\\
&f^{\\#}(-):\operatorname{Coh\text{\kern-2.0pt}\,}(f_{\\#}\mathcal{D}^{G})\to\operatorname{Coh\text{\kern-2.0pt}\,}(\mathcal{D},G).\end{split}$
(1)

are quasi-inverse equivalences of categories between coherent $G$-equivariant
$\mathcal{D}$-modules and coherent $(f_{\\#}\mathcal{D})^{G}$-modules.

In fact, we generalise the proposition and the theorem in two directions.
First, for $r\in R$ a regular element, we define the notion of $r$-deformed
Picard algebroids and $r$-deformed $G$-htdo’s. Then we may prove that the
descent of an $r$-deformed $G$-htdo is an $r$-deformed tdo and similarly the
theorem holds for coherent modules over $r$-deformed $G$-htdo’s.

Secondly, for another smooth affine algebraic group of finite type $B$ acting
on $X$ and $Y$, we prove that for a $G\times B$-htdo on $Y$, its descent under
a $B$-equivariant $G$-torsor is a $B$-htdo. A similar result also holds for
the corresponding coherent $G\times B$-modules. This generality will be needed
in future applications: in [12] we use this framework to give a geometric
proof of 1.1 and in [13] we prove an affinoid version of the same theorem.

Structure of the paper

In Sections 2 and 3, we review the theory of equivariant $\mathcal{O}$-modules
and equivariant descent for a locally trivial torsor. Next, we define in
Section 4 the sheaf of crystalline differential operators and the notion of
$r$-deformed twisted differential operators on a smooth, separated and locally
of finite type $R$-scheme. In the next two sections, we establish
correspondences between $r$-deformed Picard algebroids/equivariant Picard
algebroids and $r$-deformed twisted differential operators/homogeneous twisted
differential operators. We then define the pullback of $r$-deformed Picard
algebroids and $r$-deformed tdo’s in Section 7. In the next two sections, we
explore the connections between modules over $r$-deformed Picard algebroids
and $r$-deformed twisted differential operators.

Finally, in Section 10, we prove equivariant descent for $r$-deformed
homogeneous twisted differential operators (Proposition 1.3) and in Section
11, we prove equivariant descent for modules over $r$-deformed homogeneous
twisted differential operators (Theorem 1.4).

Conventions

Throughout this document, $R$ will denote a commutative ring of arbitrary
characteristic and all the schemes will be $R$-schemes. For a map $f:Y\to X$
of $R$-schemes, we will denote $f^{*}$ the pullback in the category of
$\mathcal{O}$-modules and $f_{*}$ the pushforward sheaf. Unadorned tensor
products will be assumed to be taken over $R$. An element $r\in R$ is called
regular if it is not a zero divisor.

##  2 Equivariant $\mathcal{O}$-modules

Let $G$ be an affine algebraic group scheme acting on a scheme $X$; denote the
action by $\sigma_{X}:G\times X\to X$. Furthermore, we denote $p_{X}:G\times
X\to X$ and $p_{2X}:G\times G\times X\to X$ the projections onto the $X$
factor, $p_{23X}:G\times G\times X\to G\times X$ the projection onto the
second and third factor and $m:G\times G\to G$ the multiplication of the group
$G$.

######  Definition 2.1.

Let $G$ an algebraic group scheme acting on a scheme $X$. A $G$-equivariant
$\mathcal{O}_{X}$-module is a pair $(\mathcal{M},\alpha)$, where $\mathcal{M}$
is a quasi-coherent $\mathcal{O}_{X}$-module and
$\alpha:\sigma_{X}^{*}\mathcal{M}\to p_{X}^{*}\mathcal{M}$ is an isomorphism
of $\mathcal{O}_{G\times X}$-modules such that the diagram

${(1_{G}\times\sigma_{X})^{*}p_{X}^{*}\mathcal{M}}$${p_{2X}^{*}\mathcal{M}}$${(1_{G}\times\sigma_{X})^{*}\sigma_{X}^{*}\mathcal{M}}$${(m\times
1_{X})^{*}\sigma_{X}^{*}\mathcal{M}}$$\scriptstyle{p_{23X}^{*}\alpha}$$\scriptstyle{(1_{G}\times\sigma_{X})^{*}\alpha}$$\scriptstyle{id}$$\scriptstyle{(m\times
1_{X})^{*}\alpha}$

of $\mathcal{O}_{G\times G\times X}$-modules commutes (the cocycle condition)
and the pullback

$(e\times 1_{X})^{*}\alpha:\mathcal{M}\to\mathcal{M}$

is the identity map.

We prove a crucial lemma that will be used in the future; it is stated on the
stack project, but the proof is omitted.

######  Lemma 2.2.

[11, 03LG]

Let $G$ be an affine algebraic group acting on schemes $X$ and $Y$ and let
$f:Y\to X$ be a $G$-equivariant morphism. Then the pullback functor $f^{*}$
given by

$(\mathcal{M},\alpha)\mapsto(f^{*}\mathcal{M},(1_{G}\times f)^{*}\alpha)$

defines a functor from $G$-equivariant $\mathcal{O}_{X}$-modules to
$G$-equivariant $\mathcal{O}_{Y}$-modules.

###### Proof.

Let $\mathcal{M}$ be a $G$-equivariant $\mathcal{O}_{X}$-module. Since
$\alpha:\sigma_{X}^{*}\mathcal{M}\to p_{X}^{*}\mathcal{M}$ is an isomorphism,
we get that $(1_{G}\times f)^{*}\alpha:(1_{G}\times
f)^{*}\sigma_{X}^{*}\mathcal{M}\to(1_{G}\times f)^{*}p_{X}^{*}\mathcal{M}$ is
also an isomorphism. We have

$\begin{split}&\sigma_{X}\circ(1_{G}\times
f)(g,y)=gf(y)=f(gy)=f\circ\sigma_{Y}(g,y),\text{ so }(1_{G}\times
f)^{*}\sigma_{X}^{*}\mathcal{M}=\sigma_{Y}^{*}f^{*}\mathcal{M}.\\\
&p_{X}\circ(1_{G}\times f)(g,y)=f(y)=f\circ p_{Y},\text{ so }(1_{G}\times
f)^{*}p_{X}^{*}\mathcal{M}=p_{Y}^{*}f^{*}\mathcal{M}.\end{split}$ (2)

Thus, $(1_{G}\times f)^{*}\alpha:\sigma_{Y}^{*}(f^{*}\mathcal{M})\to
p_{Y}^{*}(f^{*}\mathcal{M})\text{ is an isomorphism}.$

Next, we need to prove that the morphism $(1_{G}\times f)^{*}\alpha$ satisfies
the cocycle condition; that is we need to show that the diagram

${(1_{G}\times\sigma_{Y})^{*}p_{Y}^{*}f^{*}\mathcal{M}}$${p_{2Y}^{*}f^{*}\mathcal{M}}$${(1_{G}\times\sigma_{Y})^{*}\sigma_{Y}^{*}f^{*}\mathcal{M}}$${(m\times
1_{Y})^{*}\sigma_{Y}^{*}f^{*}\mathcal{M}}$$\scriptstyle{p_{23Y}^{*}(1_{G}\times
f)^{*}\alpha}$$\scriptstyle{(1_{G}\times\sigma_{Y})^{*}(1_{G}\times
f)^{*}\alpha}$$\scriptstyle{\operatorname{id\text{\kern-2.0pt}\,}}$$\scriptstyle{(m\times
1_{Y})^{*}(1_{G}\times f)^{*}\alpha}$ (3)

of $\mathcal{O}_{G\times G\times Y}$-modules commutes given that the diagram

${(1_{G}\times\sigma_{X})^{*}p_{X}^{*}\mathcal{M}}$${p_{2X}^{*}\mathcal{M}}$${(1_{G}\times\sigma_{X})^{*}\sigma_{X}^{*}\mathcal{M}}$${(m\times
1_{X})^{*}\sigma_{X}^{*}\mathcal{M}}$$\scriptstyle{p_{23X}^{*}\alpha}$$\scriptstyle{(1_{G}\times\sigma_{X})^{*}\alpha}$$\scriptstyle{\operatorname{id\text{\kern-2.0pt}\,}}$$\scriptstyle{(m\times
1_{X})^{*}\alpha}$ (4)

of $\mathcal{O}_{G\times G\times X}$-modules commutes.

We shall prove that the diagram 3 is the pullback of the diagram 4 under the
morphism $(1_{G}\times 1_{G}\times f)$. We have that

$\begin{split}&f\circ
p_{Y}\circ(1_{G}\times\sigma_{Y})(g_{1},g_{2},y)=f(g_{2}y)=p_{X}\circ(1_{G}\times\sigma_{X})\circ(1_{G}\times
1_{G}\times f)(g_{1},g_{2},y).\\\ &f\circ
p_{2Y}(g_{1},g_{2},y)=y=p_{2X}\circ(1_{G}\times 1_{G}\times
f)(g_{1},g_{2},y).\\\
&f\circ\sigma_{Y}\circ(1_{G}\times\sigma_{Y})(g_{1},g_{2},y)=f(g_{1}g_{2}y)=\sigma_{X}\circ(1_{G}\times\sigma_{X})\circ(1_{G}\times
1_{G}\times f)(g_{1},g_{2},y).\\\ &f\circ\sigma_{Y}\circ(m\times
1_{Y})(g_{1},g_{2},y)=g_{1}g_{2}f(y)=\sigma_{X}\circ(m\times
1_{X})\circ(1_{G}\times 1_{G}\times f)(g_{1},g_{2},y).\end{split}$ (5)

Therefore,

$\begin{split}((1_{G}\times\sigma_{Y})^{*}p_{Y}^{*})f^{*}\mathcal{M}&=(1_{G}\times
1_{G}\times f)^{*}((1_{G}\times\sigma_{X})^{*}p_{X}^{*}\mathcal{M}).\\\
p_{2Y}^{*}(f^{*}\mathcal{M})&=(1_{G}\times 1_{G}\times
f)^{*}(p_{2X}^{*}\mathcal{M}).\\\
((1_{G}\times\sigma_{Y})^{*}\sigma^{*})f^{*}\mathcal{M}&=(1_{G}\times
1_{G}\times f)^{*}((1_{G}\times\sigma_{X})^{*}\sigma_{X}^{*}\mathcal{M}).\\\
((m\times 1_{Y})^{*}\sigma^{*})f^{*}\mathcal{M}&=(1_{G}\times 1_{G}\times
f)^{*}((m\times 1_{X})^{*}\sigma_{X}^{*}\mathcal{M}).\end{split}$ (6)

Similarly, we get

$\begin{split}&(1_{G}\times f)\circ
p_{23Y}(g_{1},g_{2},y)=(g_{2},f(y))=p_{23X}\circ(1_{G}\times 1_{G}\times
f)(g_{1},g_{2},y).\\\ &(1_{G}\times
f)\circ(1_{G}\times\sigma_{Y})(g_{1},g_{2},y)=(g_{1},f(g_{2}y))=(1_{G}\times\sigma_{X})\circ(1_{G}\times
1_{G}\times f)(g_{1},g_{2},y).\\\ &(1_{G}\times f)\circ(m\times
1_{Y})(g_{1},g_{2},y)=(g_{1}g_{2},f(y)=(m\times 1_{X})\circ(1_{G}\times
1_{G}\times f)(g_{1},g_{2},y).\end{split}$ (7)

Thus,

$\begin{split}p_{23Y}^{*}(1_{G}\times f)^{*}\alpha&=(1_{G}\times 1_{G}\times
f)^{*}p_{23X}^{*}\alpha.\\\ (1_{G}\times\sigma)^{*}(1_{G}\times
f)^{*}\alpha&=(1_{G}\times 1_{G}\times
f)^{*}(1_{G}\times\sigma_{X})^{*}\alpha.\\\ (m\times 1_{X})^{*}(1_{G}\times
f)^{*}\alpha&=(1_{G}\times 1_{G}\times f)^{*}(m\times
1_{X})^{*}\alpha.\end{split}$ (8)

Combining equations (6) and (8), we get that the diagram (3) is indeed the
pullback of the diagram (4) under the morphism $1_{G}\times 1_{G}\times f$, so
$(1_{G}\times f)^{*}\alpha$ satisfies the cocycle condition.

Finally, we need to prove that the map $(e\times 1_{Y})^{*}(1_{G}\times
f)^{*}\alpha:f^{*}\mathcal{M}\to f^{*}\mathcal{M}$ is the identity map using
$(e\times 1_{X})^{*}\alpha:\mathcal{M}\to\mathcal{M}$ is the identity map.

We have that $(1_{G}\times f)\circ(e\times 1_{Y})(g,y)=(e,f(y))=(e\times
1_{X})(1_{G}\times f)$, so

$(e\times 1_{Y})^{*}(1_{G}\times f)^{*}\alpha=(1_{G}\times f)^{*}(e\times
1_{X})^{*}\alpha=(1_{G}\times
f)^{*}(\operatorname{id\text{\kern-2.0pt}\,})=\operatorname{id\text{\kern-2.0pt}\,},$

since the identity map is preserved by any functor. ∎

######  Definition 2.3.

Let $G$ an affine algebraic group acting on a scheme $X$ via $\sigma_{X}$. We
define the category of $G$-equivariant quasi-coherent
$\mathcal{O}_{X}$-modules. Objects are given by $G$-equivariant
$\mathcal{O}_{X}$-modules.

A morphism of $G$-equivariant $\mathcal{O}_{X}$-modules
$(\mathcal{M},\alpha_{M})$ and $(\mathcal{N},\alpha_{N})$ is a map
$\phi\in\operatorname{Hom\text{\kern-3.0pt}\,}_{\mathcal{O}_{X}}(\mathcal{M},\mathcal{N})$
such that the following diagram commutes:

${\sigma_{X}^{*}\mathcal{M}}$${p_{X}^{*}\mathcal{M}}$${\sigma_{X}^{*}\mathcal{N}}$${p_{X}^{*}\mathcal{N}.}$$\scriptstyle{\sigma_{X}^{*}\phi}$$\scriptstyle{\alpha_{M}}$$\scriptstyle{p_{X}^{*}\phi}$$\scriptstyle{\alpha_{N}}$

We call such a morphism $G$-equivariant and denote the category of
$G$-equivariant $\mathcal{O}_{X}$-modules together with $G$-equivariant
morphisms by $\operatorname{QCoh\text{\kern-2.0pt}\,}(\mathcal{O}_{X},G)$.

######  Proposition 2.4.

Let $G$ an affine algebraic group acting on a scheme $X$ via $\sigma_{X}$.
Then the category $\operatorname{QCoh\text{\kern-2.0pt}\,}(\mathcal{O}_{X},G)$
is Abelian.

###### Proof.

By construction, we have that
$\operatorname{QCoh\text{\kern-2.0pt}\,}(\mathcal{O}_{X},G)$ is additive. Let
$(\mathcal{M},\alpha_{M})$ and
$(\mathcal{N},\alpha_{N})\in\operatorname{QCoh\text{\kern-2.0pt}\,}(\mathcal{O}_{X},G)$
and let
$\phi\in\operatorname{Hom\text{\kern-3.0pt}\,}_{\operatorname{QCoh\text{\kern-2.0pt}\,}(\mathcal{O}_{X},G)}(\mathcal{M},\mathcal{N})$.
Consider the exact sequence:

$0\to\ker(\phi)\to\mathcal{M}\to\mathcal{N}\to\operatorname{coker\text{\kern-2.0pt}\,}(\phi)\to
0.$

We aim to prove that $\ker(\phi)$ and
$\operatorname{coker\text{\kern-2.0pt}\,}(\phi)$ are in
$\operatorname{QCoh\text{\kern-2.0pt}\,}(\mathcal{O}_{X},G)$. Since
$\sigma_{X}$ and $p_{X}$ are smooth morphisms, the pullback functors
$\sigma_{X}^{*}$ and $p_{X}^{*}$ are exact, so we get two short exact
sequences:

$\begin{split}&0\to\sigma_{X}^{*}\ker(\phi)\to\sigma_{X}^{*}\mathcal{M}\to\sigma_{X}^{*}\mathcal{N}\to\sigma_{X}^{*}\operatorname{coker\text{\kern-2.0pt}\,}(\phi)\to
0,\\\ &0\to p_{X}^{*}\ker(\phi)\to p_{X}^{*}\mathcal{M}\to
p_{X}^{*}\mathcal{N}\to
p_{X}^{*}\operatorname{coker\text{\kern-2.0pt}\,}(\phi)\to 0.\end{split}$ (9)

Consider now the diagram:

${0}$${0}$${\sigma_{X}^{*}\ker(\phi)}$${\sigma_{X}^{*}\mathcal{M}}$${\sigma_{X}^{*}\mathcal{N}}$${0}$${0}$${p_{X}^{*}\ker(\phi)}$${p_{X}^{*}\mathcal{M}}$${p_{X}^{*}\mathcal{N}.}$$\scriptstyle{\alpha_{\mathcal{M}}}$$\scriptstyle{\alpha_{\mathcal{N}}}$

By construction, we have that $\alpha_{\mathcal{M}}$ and
$\alpha_{\mathcal{N}}$ are isomorphisms, so by the Five Lemma we obtain an
isomorphism $\beta:\sigma_{X}^{*}\ker(\phi)\to p_{X}^{*}\ker(\phi)$.
Furthermore, since $\alpha_{\mathcal{M}}$ and $\alpha_{\mathcal{N}}$ satisfy
the cocycle condition, so does $\beta$. Thus, we have proven that
$\ker(\phi)\in\operatorname{QCoh\text{\kern-2.0pt}\,}(\mathcal{O}_{X},G)$. A
similar argument applying the Five Lemma shows that
$\operatorname{coker\text{\kern-2.0pt}\,}(\phi)\in\operatorname{QCoh\text{\kern-2.0pt}\,}(\mathcal{O}_{X},G)$.
Finally, by construction we have that any monorphism and epimorphism in
$\operatorname{QCoh\text{\kern-2.0pt}\,}(\mathcal{O}_{X},G)$ is normal, so
$\operatorname{QCoh\text{\kern-2.0pt}\,}(\mathcal{O}_{X},G)$ is indeed Abelian
category. ∎

From now on, when we use the notion of morphism of $G$-equivariant
$\mathcal{O}_{X}$-modules, we always view it as a morphism in the category
$\operatorname{QCoh\text{\kern-2.0pt}\,}(\mathcal{O}_{X},G)$.

A reformulation of equivariance

We wish to reformulate the notion of an equivariant $\mathcal{O}$-module.
Until the end of the section, we fix $X$ a scheme defined over $R$ acted on by
an affine algebraic group $G$. For any $R$-algebra $A$, we define
$X_{A}:=\operatorname{Spec\text{\kern-2.0pt}\,}A\times_{\operatorname{Spec\text{\kern-2.0pt}\,}R}X$.
We start with a very simple observation: viewing $\mathcal{O}_{X}$ as a left
$\mathcal{O}_{X}$-module,
$(\mathcal{O}_{X},\operatorname{id\text{\kern-2.0pt}\,})$ is a $G$-equivariant
$\mathcal{O}_{X}$-module. We may reformulate this following ideas in [5]: for
each $R$-algebra $A$ inducing a map
$s:\operatorname{Spec\text{\kern-2.0pt}\,}A\to\operatorname{Spec\text{\kern-2.0pt}\,}R$
and for each geometric point
$i_{g}:\operatorname{Spec\text{\kern-2.0pt}\,}A\to G$ which induces an
automorphism $g:X_{A}\to X_{A}$ there exists an isomorphism

$q_{g}:s^{*}\mathcal{O}\to(g^{-1})^{*}s^{*}\mathcal{O},\text{ satisfying }$

$q_{e}=\operatorname{id\text{\kern-2.0pt}\,}\text{ and
}q_{gh}=(g^{-1})^{*}(q_{h})q_{g}$ (10)

in such a way that $(q_{g})$’s are compatible with base change. Let
$r_{g}=g^{*}\circ q_{g}$. For each $U\subset X_{A}$ affine open, $r_{g}$
induces a map $\mathcal{O}_{X_{A}}(U)\to\mathcal{O}_{X_{A}}(g^{-1}U)$. The
equation (10) translates as $r_{e}=\operatorname{id\text{\kern-2.0pt}\,}$ and
$r_{gh}=r_{h}r_{g}$. Furthermore, the $\mathcal{O}$-module compatibility
requires that for any $f_{1},f_{2}\in\mathcal{O}_{X_{A}}(U)$, we have
$r_{g}(f_{1}f_{2})=r_{g}(f_{1})r_{g}(f_{2})$.

We define $r_{g}$ via $r_{g}(f)(x)=f(g^{-1}x)$ for all $R$-algebras $A$,
$U\subset X_{A}$ affine open, $x\in U$, $f\in\mathcal{O}_{X_{A}}(U)$,
$g:X_{A}\to X_{A}$ and it is easy to see that $r_{g}$’s make $\mathcal{O}_{X}$
a $G$-equivariant $\mathcal{O}_{X}$-module according to equation (10). We may
now make an abuse of notation: for each
$i_{g}:\operatorname{Spec\text{\kern-2.0pt}\,}A\to G$ and each
$f\in\mathcal{O}_{X_{A}}(U)$, we denote $g.f=r_{g^{-1}}(f)$ and we translate
the equivariance structure as

$e.f_{1}=f,\quad g.(h.f_{1})=(gh).f_{1},\quad
g.(f_{1}f_{2})=(g.f_{1})(g.f_{2})\text{ for all }g,h\in
G,f_{1},f_{2}\in\mathcal{O}_{X}.$

######  Lemma 2.5.

A $\mathcal{O}_{X}$-module $\mathcal{M}$ is $G$-equivariant if and only if for
each $R$-algebra $A$, for each
$s:\operatorname{Spec\text{\kern-2.0pt}\,}A\to\operatorname{Spec\text{\kern-2.0pt}\,}R$
and for each geometric point
$i_{g}:\operatorname{Spec\text{\kern-2.0pt}\,}A\to G$ which induces an
automorphism $g:X_{A}\to X_{A}$ there exists an isomorphism of
$\mathcal{O}_{A}$-modules

$q_{g}:s^{*}\mathcal{M}\to(g^{-1})^{*}s^{*}\mathcal{M}$

satisfying

$q_{e}=\operatorname{id\text{\kern-2.0pt}\,}\text{ and
}q_{gh}=(g^{-1})^{*}(q_{h})q_{g}$ (11)

in such a way that $(q_{g})$’s are compatible with base change.

###### Proof.

The proof repeats the argument in [5, Proposition 2.2/Proposition 1.3.1]
working over a commutative ring rather than a field and using the structure
sheaf $\mathcal{O}$ instead of the sheaf of differential operators
$\mathcal{D}.$ ∎

Again, by setting $s_{g}=g^{*}\circ q_{g}$, we may reformulate equation (11)
as: for each $R$-algebra $A$ and for each
$i_{g}:\operatorname{Spec\text{\kern-2.0pt}\,}A\to G$, we have an isomorphism
of $\mathcal{O}$-modules $s_{g}:\mathcal{M}_{X_{A}}\to\mathcal{M}_{X_{A}}$
such that for each $U\subset X_{A}$ affine open:

$\begin{split}&s_{e}=\operatorname{id\text{\kern-2.0pt}\,},\\\
&s_{gh}=s_{h}s_{g},\\\ &s_{g}^{\prime}s\text{ are compatible with base
change},\\\ &r_{g}(f.m)=r_{g}(f).s_{g}(m)\text{, for all
}f\in\mathcal{O}_{X_{A}}(U),m\in\mathcal{M}(U).\end{split}$ (12)

Again, we make an abuse of notation: for each
$i_{g}:\operatorname{Spec\text{\kern-2.0pt}\,}A\to G$ and each
$m\in\mathcal{M}_{A}(U)$, we denote $g.m=s_{g^{-1}}(m)$ and we translate the
equivariance structure as:

$\begin{split}&e.m=m,\\\ &gh.m=g.(h.m),\\\ &g.(f.m)=(g.f).(g.m),\end{split}$
(13)

for all $g,h\in G$, $m\in\mathcal{M}$, $f\in\mathcal{O}_{X}$.

Using the definition above, we reformulate the notion of $G$-equivariance of a
morphism of $G$-equivariant $\mathcal{O}_{X}$-modules,
$\phi:\mathcal{M}\to\mathcal{N}$ as:

$g.\phi(m)=\phi(g.m)\text{ for all }g\in G,m\in\mathcal{M}$ (14)

##  3 Equivariant descent for $\mathcal{O}$-modules

######  Definition 3.1.

[1, Section 4.3]

Let $G$ be a smooth affine algebraic group of finite type, $Y$ a scheme
equipped with an action $G\times Y\to Y$ and lastly, let $X$ be a scheme. We
say that a morphism $\xi:Y\to X$ is a $G$-torsor if $\xi$ is faithfully flat
and locally of finite type, the action of $G$ respects $\xi$ and the map

$G\times Y\to Y\times_{X}Y,\qquad(g,y)\mapsto(gy,y)$

is an isomorphism.

An open subscheme $U$ of $X$ is said to trivialise the torsor $\xi$ if there
is a $G$-invariant isomorphism

$G\times U\to\xi^{-1}(U),$

where $G$ acts on $G\times U$ by left multiplication on the first factor.

Finally, let $\mathcal{S}_{X}$ be the set of affine open subschemes $U\subset
X$ such that $U$ trivialises $\xi$ and $\mathcal{O}(U)$ is a finitely
generated $R$-algebra. We say that $\xi$ is a locally trivial torsor if it can
be covered by opens in $\mathcal{S}_{X}$.

######  Definition 3.2 (definition-proposition).

Let $\xi:Y\to X$ be a locally trivial $G$-torsor and let
$(\mathcal{M},\alpha_{M})$ be a quasi-coherent $G$-equivariant
$\mathcal{O}_{Y}$-module. Then the presheaf $(\xi_{*}\mathcal{M})^{G}$
acquires the structure of a quasi-coherent $\mathcal{O}_{X}$-module.
Furthermore, if we are given
$\psi:(\mathcal{M},\alpha_{M})\to(\mathcal{N},\alpha_{N})$ a map of
$G$-equivariant $\mathcal{O}_{Y}$-modules there is a canonical induced map
$(\xi_{*})^{G}\psi:(\xi_{*}\mathcal{M})^{G}\to(\xi_{*}\mathcal{N})^{G}$.

###### Proof.

The question is local, so we may assume that $X$ is affine, $\xi:Y\to X$ is
$p_{X}:G\times X\to X$ and $G$ acts on $G\times X$ via left multiplication on
the first factor. Since $G$ and $X$ are affine, the category of
$G$-equivariant $\mathcal{O}_{G\times X}$-modules is equivalent to the
category of $(\mathcal{O}(G\times X),G)$-modules by the same arguments as in
the proof of [5, Proposition 1.4.1]. Modules in this category are modules
equipped with compatible actions of the ring $\mathcal{O}(G\times X)$ and of
the group $G$.

Let $\mathcal{M}$ be a $G$-equivariant $\mathcal{O}_{G\times X}$-module and
let $M=\Gamma(G\times X,\mathcal{M})$ be its global sections. Since $M$ is a
$(\mathcal{O}(G\times X),G)$-module it acquires a comodule structure
$\rho_{M}:M\to\mathcal{O}(G)\underset{R}{\otimes}{}M$. Furthermore, let
$\rho_{G}:\mathcal{O}(G\times
X)\to\mathcal{O}(G)\underset{R}{\otimes}{}\mathcal{O}(G\times X)$ denote the
comodule structure on $\mathcal{O}(G\times X)$ induced from the $G$-action on
$G\times X$ by left multiplication on the first factor.

As $X$ is affine, to prove the first statement, it is enough to prove that the
Abelian group $(p_{*}\mathcal{M})^{G}(X)=M^{G}$ has the structure of an
$\mathcal{O}(X)$-module.

Let $f\in\mathcal{O}(X)$ and define $\phi\in\mathcal{O}(G\times X)$ by
$\phi(h,x):=f(x)\text{ for all }h\in G,x\in X,$ and for any $m\in M^{G}$
define $f.m:=\phi.m$

We need to prove that $f.m\in M^{G}$. By construction, it is clear that
$\rho_{G}(\phi)=1\otimes\phi$. Since $M$ is a $(\mathcal{O}(G\times
X),G)$-module, we have

$\rho_{M}(\phi.m)=\rho_{G}(\phi)\rho_{M}(m)=(1\otimes\phi)(1\otimes
m)=1\otimes\phi.m,$

so $\phi.m\in M^{G}$, thus the action of $f$ is well-defined.

For the second statement, notice that if
$\varphi:(M,\alpha_{M})\to(N,\alpha_{N})$ is a morphism of
$(\mathcal{O}(G\times X),G)$-modules, it is $G$-equivariant, so $\varphi$
restricts to a map $M^{G}\to N^{G}$.

In general if $\xi:\mathcal{M}\to\mathcal{N}$ is a map of $G$-equivariant
$\mathcal{O}_{Y}$ modules and we let $U$ be affine open in $X$, we have
canonical maps
$(\xi_{*})^{G}\psi:(\xi_{*}\mathcal{M})^{G}(U)\to(\xi_{*}\mathcal{N})^{G}(U)$
compatible with restrictions given by restricting the map $\psi$. Therefore,
glueing together the local morphisms we get a map of sheaves. ∎

In particular, we have proven that if $\xi:Y\to X$ is a locally trivial
$G$-torsor, we obtain a functor $\xi_{*}^{G}$ from $G$-equivariant
$\mathcal{O}_{Y}$-modules to $\mathcal{O}_{X}$-modules. We would like to prove
that this is an equivalence of categories.

Recall that for an $R$-Hopf algebra $H$, a Hopf module $M$ is a left
$H$-module, together with a comodule map $\rho:M\to H\underset{R}{\otimes}{}M$
such that $\rho$ is a map of $H$-modules; here we view $H$ as a module over
itself via left multiplication.

######  Theorem 3.3.

[7, Theorem 4.13] Let $H$ be a Hopf algebra over a commutative ring $R$ and
$M$ a Hopf module. Denote $M^{\operatorname{co\text{\kern-3.0pt}\,}H}$ the
coinvariants of $M$. Then the map

$\mu:H\underset{R}{\otimes}{}M^{\operatorname{co\text{\kern-3.0pt}\,}H}\to
M,\quad\mu(h\otimes m)=h.m$

is an isomorphism.

We may now prove the main result of this section.

######  Proposition 3.4.

[Equivariant descent for $\mathcal{O}$-modules] Let $G$ be a smooth affine
algebraic group of finite type and let $\xi:Y\to X$ be a locally trivial
$G$-torsor. Then the functors $\xi_{*}(-)^{G}$ and $\xi^{*}(-)$ induce quasi-
inverse equivalences of categories between $G$-equivariant quasi-coherent
$\mathcal{O}_{Y}$-modules and quasi-coherent $\mathcal{O}_{X}$-modules.

###### Proof.

For $\mathcal{M}\in\operatorname{QCoh\text{\kern-2.0pt}\,}(\mathcal{O}_{Y},G)$
and $\mathcal{N}\in\operatorname{QCoh\text{\kern-2.0pt}\,}(\mathcal{O}_{X})$
we obtain by functoriality maps
$\mathcal{M}\to\xi^{*}(\xi_{*}\mathcal{M})^{G}$ and
$(\xi_{*}(\xi^{*}\mathcal{N}))^{G}\to\mathcal{N}$, respectively. Thus we only
need to prove the statement locally. We may then assume that $Y=G\times X$,
$p:G\times X\to X$ is the projection onto the second factor and $G$ acts on
$G\times X$ via left multiplication on the first factor.

We start by constructing a natural isomorphism
$\eta:\operatorname{id\text{\kern-2.0pt}\,}\to(p_{*})^{G}p^{*}$. Let
$\mathcal{M}$ be a $\mathcal{O}_{X}$-module. We aim to define a map
$\eta_{M}:\mathcal{M}\to p_{*}^{G}p^{*}\mathcal{M}$. For any open affine
$U\subset X$, we have

$(p_{*}^{G}p^{*}\mathcal{M})(U)=(\mathcal{O}(G)\underset{R}{\otimes}{}\mathcal{M}(U))^{G}.$

Let
$\eta_{\mathcal{M}U}:\mathcal{M}(U)\to(\mathcal{O}(G)\underset{R}{\otimes}{}\mathcal{M}(U))^{G}$
be defined by $m\mapsto 1\otimes m$.

Since $G$ is flat over $R$, $\mathcal{O}(G)$ is a flat $R$-module, so a direct
limit of free $R$-modules. Write $\mathcal{O}_{G}=\lim M_{i}$, where $M_{i}$
is a free $R$-module. We have that $(M_{i}\otimes_{R}\mathcal{M}_{U})^{G}\cong
M_{i}$ and since rational cohomology commutes with direct limits by [8, Lemma
I.4.17], we obtain that $\eta_{\mathcal{M}U}$ is indeed an isomorphism.

Let $V\subset U$ be open affine and let
$\operatorname{res\text{\kern-1.0pt}\,}_{UV}:\mathcal{M}(U)\to\mathcal{M}(V)$
be the restriction map. It is easy to see that the following diagram is
commutative:

${\mathcal{M}(U)}$${(\mathcal{O}(G)\underset{R}{\otimes}{}\mathcal{M}(U))^{G}}$${\mathcal{M}(V)}$${(\mathcal{O}(G)\underset{R}{\otimes}{}\mathcal{M}(V))^{G}.\par\par}$$\scriptstyle{\operatorname{res\text{\kern-1.0pt}\,}_{UV}}$$\scriptstyle{\eta_{\mathcal{M}U}}$$\scriptstyle{\operatorname{id\text{\kern-2.0pt}\,}\underset{R}{\otimes}{}\operatorname{res\text{\kern-1.0pt}\,}_{UV}}$$\scriptstyle{\eta_{\mathcal{M}V}}$

Thus, $\eta$ is a map of sheaves. Next, we prove that the isomorphism $\eta$
is natural. Let $\varphi:\mathcal{M}\to\mathcal{N}$ be a morphism of
$\mathcal{O}_{X}$-modules. It is enough to show that the following diagram is
commutative:

${\mathcal{M}}$${(p_{*})^{G}p^{*}\mathcal{M}}$${\mathcal{N}}$${(p_{*})^{G}p^{*}\mathcal{N}.\par}$$\scriptstyle{\varphi}$$\scriptstyle{\eta_{M}}$$\scriptstyle{(p_{*})^{G}p^{*}\varphi}$$\scriptstyle{\eta_{N}}$

We can work locally. Let $U\subset X$ be affine open and let
$m\in\mathcal{M}(U)$. Then $\eta_{N}(\varphi(m))=1\otimes\varphi(m)$. On the
other hand we have $p^{*}\varphi:p^{*}\mathcal{M}(G\times U)\to
p^{*}\mathcal{N}(G\times U)$ defined by $p^{*}\varphi(F\otimes
m)=F\otimes\varphi(m)$ for any $F\in\mathcal{O}(G),m\in\mathcal{M}(U)$.

Thus, we have that
$(p_{*})^{G}p^{*}\varphi:(p_{*})^{G}p^{*}\mathcal{M}(U)\to(p_{*})^{G}p^{*}\mathcal{M}(U)$
is defined by $(p_{*})^{G}p^{*}\varphi(1\otimes m)=1\otimes\varphi(m)$, for
all $m\in\mathcal{M}(U)$. In particular, we get that

$(p_{*})^{G}p^{*}\varphi(\eta_{M}(m))=(p_{*})^{G}p^{*}\varphi(1\otimes
m)=1\otimes\varphi(m)=\eta_{N}(\varphi(m)),$

which shows that the diagram is indeed commutative.

Now, let $(\mathcal{M},\alpha)$ be a $G$-equivariant $\mathcal{O}_{G\times
X}$-module. By construction $(p_{*})^{G}\mathcal{M}$ is a subsheaf of
$p_{*}\mathcal{M}$ and since there is a canonical sheaf map
$p^{*}(p_{*})\mathcal{M}\to\mathcal{M}$, we get by functoriality that there is
a map $\nu_{\mathcal{M}}:p^{*}(p_{*})^{G}\mathcal{M}\to\mathcal{M}$.

Let $U\subset X$ open affine. Then we have the induced map

$\nu_{\mathcal{M}_{G\times
U}}:\mathcal{O}(G)\underset{R}{\otimes}{}\mathcal{M}(G\times
U)^{G}=p^{*}(p_{*})^{G}\mathcal{M}(G\times U)\to\mathcal{M}(G\times U)$

given by $\nu_{\mathcal{M}_{G\times U}}(f\otimes m)=f.m$. We aim to prove that
this map is an isomorphism. Since $\mathcal{M}$ is $G$-equivariant the
isomorphism $\alpha$ induces an automorphism of $\mathcal{O}_{G\times G\times
X}$-modules on $p^{*}\mathcal{M}$, so in particular we obtain an automorphism
on $\mathcal{O}(G)\otimes\mathcal{M}(G\times U)$ of $\mathcal{O}(G\times
G\times U)$-modules. This induces a Hopf module structure on
$\mathcal{M}(G\times U)$ for the Hopf algebra $\mathcal{O}(G)$. Thus, by
Theorem 3.3 we get that $\nu_{\mathcal{M}_{G\times U}}$ is indeed an
isomorphism.

Let $\\{U_{i}\\}_{i\in I}$ be an affine open cover of $X$ and let $\\{G\times
U_{i}\\}_{i\in I}$ be the corresponding affine cover of $G\times X$. As
$\nu_{\mathcal{M}_{G\times U_{i}}}$ is an isomorphism, the sheaves
$p^{*}(p_{*})^{G}\mathcal{M}$ and $\mathcal{M}$ agree on an affine open cover
and there is a sheaf map between the two, $\nu$ is an isomorphism.

To finish the proof, notice that by construction, we have
$\nu:p^{*}(p_{*})^{G}\to\operatorname{id\text{\kern-2.0pt}\,}$ is a natural
isomorphism. This concludes the proof of the proposition. ∎

We will also consider a slightly more general setting. Let $Y$ be a variety
acted on by two smooth affine algebraic groups of finite type, $G$ and $B$.
Let us denote the two actions by . and $\star$.

######  Observation 3.5.

Let $\mathcal{M}$ be a $G\times B$-equivariant $\mathcal{O}_{Y}$-module. Then
$\mathcal{M}^{G}$ is a $B$-equivariant submodule of $\mathcal{M}$. Let
$\phi:\mathcal{M}\to\mathcal{M}^{\prime}$ be a $G\times B$ equivariant map of
$G\times B$-equivariant modules. Then $\phi$ restricts to a $B$-equivariant
map $\phi:\mathcal{M}^{G}\to\mathcal{M}^{\prime G}$.

###### Proof.

We view the equivariance structure via the equations (13). We have that for
$g\in G,b\in B$ and $m\in\mathcal{M}$ that

$g.(b\star m)=b\star(g.m).$

If $m\in\mathcal{M}^{G}$, then $g.m=m$, so by the equation above $g.(b\star
m)=b\star m$, so $b\star m\in\mathcal{M}^{G}$. Thus, the $B$-equivariance on
$\mathcal{M}$ induces $B$-equivariance on $\mathcal{M}^{G}$.

Similarly, using the equivariance of morphisms in equation (14), we have since
$\phi$ is particular $G$-equivariant that for $m\in\mathcal{M}^{G}$,
$g.\phi(m)=\phi(g.m)=\phi(m)$, so $\phi$ restricts to a map
$\mathcal{M}^{G}\to\mathcal{M}^{\prime G}$.

Finally, since $\phi$ is in particular $B$-equivariant, we have that for
$m\in\mathcal{M}^{G}$

$b\star\phi(m)=\phi(b\star m),$

concluding the proof. ∎

######  Lemma 3.6.

Let $G$ and $B$ be smooth affine algebraic groups of finite type acting on
$R$-schemes $X$ and $Y$ such that the action of $B$ and $G$ on $Y$ commute.
Let $\xi:Y\to X$ be a locally trivial $G$-torsor that is $B$-equivariant.

  * •

Let $\mathcal{M}$ be a $G\times B$-equivariant $\mathcal{O}_{Y}$-module. Then
$(\xi_{*}\mathcal{M})^{G}$ is a $B$-equivariant $\mathcal{O}_{X}$-module.

  * •

Let $\mathcal{N}$ be a $B$-equivariant $\mathcal{O}_{X}$-module. Then
$\xi^{*}\mathcal{N}$ is a $G\times B$-equivariant $\mathcal{O}_{Y}$-module.

###### Proof.

Since $\xi$ is $B$-equivariant, the $B$-action on
$\mathcal{O}_{X}\cong(\xi_{*}\mathcal{O}_{Y})^{G}$ is induced from the
$B$-action on $\mathcal{O}_{Y}$. Further, using the observation above we may
define a $B$-action on $(\xi_{*}\mathcal{M})^{G}$ which is compatible with the
$B$-action on $\mathcal{O}_{Y}$ since $\mathcal{M}$ is $B$-equivariant, so the
first claim is proven.

For the second claim, we let $G$ act on $\mathcal{N}$ via $g.n=n$ for all
$g\in G$ and $n\in\mathcal{N}$, so that $\mathcal{N}$ is $G\times
B$-equivariant. The claim follows from Lemma 2.2. ∎

######  Corollary 3.7.

Let $G$ and $B$ be smooth affine algebraic groups of finite type acting on $Y$
and $X$ such that the action of $B$ and $G$ on $Y$ commute. Let $\xi:Y\to X$
be a locally trivial $G$-torsor that is $B$-equivariant. The functors
$\xi_{*}(-)^{G}$ and $\xi^{*}(-)$ induce quasi-inverse equivalences of
categories between $G\times B$-equivariant quasi-coherent
$\mathcal{O}_{Y}$-modules and quasi-coherent $B$-equivariant
$\mathcal{O}_{X}$-modules.

###### Proof.

This follows from Proposition 3.4, Lemma 3.6 and Observation 3.5. ∎

##  4 Deformed twisted differential operators

######  Definition 4.1.

We call an $R$-scheme $X$ that is smooth, separated and locally of finite type
an _$R$ -variety_.

We write $\mathcal{T}_{X}$ for the sheaf of sections of the tangent bundle
$TX$.

######  Definition 4.2.

[1, Definition 4.2]

Let $X$ be an $R$-variety. The sheaf of crystalline differential operators is
defined to be the enveloping algebra $\mathcal{D}_{X}$ of the Lie algebroid
$\mathcal{T}_{X}$.

We can view $\mathcal{D}_{X}$ as a sheaf of ring generated by
$\mathcal{O}_{X}$ and $\mathcal{T}_{X}$ modulo the relations:

  * •

$f\partial=f\cdot\partial$;

  * •

$\partial f-f\partial=\partial(f)$;

  * •

$\partial\partial^{\prime}-\partial^{\prime}\partial=[\partial,\partial^{\prime}],$

for all $f\in\mathcal{O}_{X}$ and
$\partial,\partial^{\prime}\in\mathcal{T}_{X}$. The sheaf $\mathcal{D}_{X}$
comes equipped with a natural PBW filtration:

$0\subset F_{0}(\mathcal{D}_{X})\subset F_{1}(\mathcal{D}_{X})\subset\ldots$

consisting of coherent $\mathcal{O}_{X}$-modules such that

$F_{0}(\mathcal{D}_{X})=\mathcal{O}_{X},\quad
F_{1}(\mathcal{D}_{X})=\mathcal{O}_{X}\oplus\mathcal{T}_{X},\quad
F_{m}(\mathcal{D}_{X})=F_{1}(\mathcal{D}_{X})\cdot
F_{m-1}(\mathcal{D}_{X})\text{ for }m>1.$

Since $X$ is smooth, the tangent sheaf $\mathcal{T}_{X}$ is locally free and
the associated graded sheaf of algebras of $\mathcal{D}_{X}$ is isomorphic to
the symmetric algebra of $\mathcal{T}_{X}$:

$\operatorname{gr\text{\kern-2.0pt}\,}(\mathcal{D}_{X})=\bigoplus_{m=1}^{\infty}\frac{F_{m}(\mathcal{D}_{X})}{F_{m-1}(\mathcal{D}_{X})}\cong\operatorname{Sym\text{\kern-2.0pt}\,}_{\mathcal{O}_{X}}\mathcal{T}_{X}.$
(15)

If $q:T^{*}X\to X$ is the cotangent bundle of $X$ defined by the locally free
sheaf $\mathcal{T}_{X}$, then we can also identify
$\operatorname{gr\text{\kern-2.0pt}\,}(\mathcal{D}_{X})$ with
$q_{*}\mathcal{O}_{T^{*}X}$.

Let $X$ be an $R$-variety and let
$U=\operatorname{Spec\text{\kern-2.0pt}\,}(A)\subset X$ be open affine.
Further, we consider $\mathcal{M}$ a sheaf of $\mathcal{O}_{X}$-bimodules
quasi-coherent with respect to the left action. We define a filtration on
$M=\mathcal{M}(U)$ given by $F_{\bullet}M$:

  * •

$F_{-1}(M)=0,$

  * •

$F_{n}(M)=\\{m\in
M|\operatorname{ad\text{\kern-2.0pt}\,}(a_{0})\operatorname{ad\text{\kern-2.0pt}\,}(a_{1})...\operatorname{ad\text{\kern-2.0pt}\,}(a_{n})(m)=0\text{,
for any }a_{0},a_{1},\ldots a_{n}\in A\\}$, for $n\geq 0$.

We say that $M$ is differential if $M=\cup_{n\in\mathbb{N}}F_{n}(M)$ and we
call $\mathcal{M}$ a differential $\mathcal{O}_{X}$-bimodule if there is an
affine open cover $(U_{i})_{i\in I}$ such that $\mathcal{M}(U_{i})$ is a
differential bimodule for all $i\in I$.

Let $\mathcal{M},\mathcal{N}$ be two quasi-coherent $\mathcal{O}_{X}$-modules.
Then for any affine open $U$ in $X$, the set
$\operatorname{Hom\text{\kern-3.0pt}\,}_{R}(\mathcal{M}(U),\mathcal{N}(U))$
has the structure of a $\mathcal{O}_{X}(U)$-bimodule. Let
$\mathcal{F}\in\operatorname{Hom\text{\kern-3.0pt}\,}_{R}(\mathcal{M},\mathcal{N}$);
we say that $\mathcal{F}$ is a differential operator of degree $\leq n$ if for
any affine open $U$, $\mathcal{F}(U)\in
F_{n}(\operatorname{Hom\text{\kern-3.0pt}\,}_{R}(\mathcal{M}(U),\mathcal{N}(U)).$
We denote
$\operatorname{Dif\text{\kern-2.0pt}\,}^{n}(\mathcal{M},\mathcal{N})$ the
subsheaf of differential operators of degree $\leq n$ and
$\operatorname{Dif\text{\kern-2.0pt}\,}(\mathcal{M},\mathcal{N})=\cup_{n\in\mathbb{N}}\operatorname{Dif\text{\kern-2.0pt}\,}^{n}(\mathcal{M},\mathcal{N})$
the subsheaf of differential operators of finite degree.

We may construct differential $\mathcal{O}_{X}$-modules using the following
proposition:

######  Proposition 4.3.

Let $\mathcal{M}$ be a coherent $\mathcal{O}_{X}$-module and let $\mathcal{N}$
be a $\mathcal{O}_{X}$-module. Then
$\operatorname{Dif\text{\kern-2.0pt}\,}(\mathcal{M},\mathcal{N})$ is
differential $\mathcal{O}_{X}$-bimodule.

###### Proof.

The proof follows by repeating the argument in [14, Proposition 2.1.3]. ∎

######  Definition 4.4.

Let $\mathcal{A}$ be a $\mathcal{O}_{X}$-algebra. We say that $\mathcal{A}$ is
a differential algebra if $\mathcal{A}$ is a flat $R$-module and
multiplication makes $\mathcal{A}$ a differential $\mathcal{O}_{X}$-bimodule.
The filtration $F_{\bullet}(\mathcal{A})$ becomes a ring filtration and with
respect to this filtration
$\operatorname{gr\text{\kern-2.0pt}\,}^{F}(\mathcal{A})$ is commutative.

######  Definition 4.5.

Let $r\in R$ be a regular element. An algebra of $r$-deformed twisted
differential operators(tdo) is an $\mathcal{O}_{X}$-differential algebra
$\mathcal{D}$ such that:

  1. i)

The natural map $\mathcal{O}_{X}\to F_{0}(\mathcal{D})$ is an isomorphism.

  2. ii)

The morphism
$\operatorname{gr\text{\kern-2.0pt}\,}_{1}^{F}\mathcal{D}\to\mathcal{T}_{X}=\operatorname{Der\text{\kern-3.0pt}\,}_{R}(\mathcal{O}_{X},\mathcal{O}_{X})$
defined by $\psi\mapsto\operatorname{ad\text{\kern-2.0pt}\,}_{\psi}$ for
$\psi\in F_{1}(\mathcal{D})$ induces an isomorphism
$\operatorname{gr\text{\kern-2.0pt}\,}_{1}^{F}\mathcal{D}\cong
r\mathcal{T}_{X}$.

  3. iii)

The morphism of $\mathcal{O}_{X}$-algebras
$\operatorname{Sym\text{\kern-2.0pt}\,}_{\mathcal{O}_{X}}(\operatorname{gr\text{\kern-2.0pt}\,}_{1}^{F}\mathcal{D})\to\operatorname{gr\text{\kern-2.0pt}\,}^{F}\mathcal{D}$
is an isomorphism.

A morphism of tdo’s is a morphism of $\mathcal{O}_{X}$ algebras compatible
with the $\mathcal{O}_{X}$-bimodule structure and the maps in $i)\to iii)$.

We should make some remarks about this definition: when $r=1$ we call
$\mathcal{D}$ a sheaf of twisted differential operators. Classically, working
with twisted differential operators over a complex variety the condition
$iii)$ is implied by $i)$ and $ii)$. This is no longer true in our case.
Further, the sheaf of _Grothendieck_ ’s differential operators does not
satisfy condition $iii)$ for a general ring $R$. This is the main reason why
we develop the theory of twisted differential operators using the connection
with Lie algebroids rather than using the classical
$\operatorname{Dif\text{\kern-2.0pt}\,}$ definition.

######  Lemma 4.6.

Assume that $X$ is locally Noetherian $R$-variety and let $\mathcal{D}$ be an
$r$-deformed tdo on $X$. Then $\mathcal{D}$ is locally Noetherian.

###### Proof.

We have by conditions $ii)$ and $iii)$ that
$\operatorname{gr\text{\kern-2.0pt}\,}^{F}\mathcal{D}\cong\operatorname{Sym\text{\kern-2.0pt}\,}_{\mathcal{O}_{X}}(r\mathcal{T}_{X})$
and because $r$ is regular
$\operatorname{Sym\text{\kern-2.0pt}\,}_{\mathcal{O}_{X}}(r\mathcal{T}_{X})\cong\operatorname{Sym\text{\kern-2.0pt}\,}_{\mathcal{O}_{X}}(\mathcal{T}_{X})$.
Since $\mathcal{T}_{X}$ is a free $\mathcal{O}_{X}$-module and $X$ is locally
Noetherian, we obtain that
$\operatorname{Sym\text{\kern-2.0pt}\,}_{\mathcal{O}_{X}}(\mathcal{T}_{X})$ is
locally Noetherian. Therefore, we have
$\operatorname{gr\text{\kern-2.0pt}\,}^{F}\mathcal{D}$ is locally Noetherian,
which implies the same for $\mathcal{D}$. ∎

##  5 Connections between deformed Lie algebroids
and deformed tdo’s

Throughout this section, we let $X$ denote an $R$-variety and $r\in R$ a
regular element.

######  Definition 5.1.

A Lie algebroid $\mathcal{L}$ on $X$ is a quasi-coherent
$\mathcal{O}_{X}$-module equipped with a morphism of $\mathcal{O}_{X}$-modules
$\rho:\mathcal{L}\to\mathcal{T}_{X}$ (the anchor map) and an $R$-linear
pairing $[\bullet,\bullet]:\mathcal{L}\to\mathcal{T}_{X}$ such that:

  * •

$[\bullet,\bullet]$ defines the structure of a Lie algebra on $\mathcal{L}$
and $\rho$ is a morphism of Lie algebras.

  * •

$[l_{1},fl_{2}]=f[l_{1},l_{2}]+\rho(l_{1})(f)l_{2}$ for
$l_{i}\in\mathcal{L},f\in\mathcal{O}_{X}$.

A morphism of Lie algebroids is a morphism of $\mathcal{O}_{X}$-modules
compatible with the anchor maps and bracket.

In particular, locally we obtain that for any $U\subset X$ affine open that
$\mathcal{L}(U)$ is an $(R,\mathcal{O}_{X}(U))$-Lie Rinehart algebra, see [10]
for definition and basic properties of Lie Rinehart algebras. We may think of
$\mathcal{L}$ as a sheaf of $(R,\mathcal{O}_{X})$-Lie Rinehart algebras; we
will use this local description soon.

######  Definition 5.2.

The universal enveloping algebra of $\mathcal{L}$, denoted $U(\mathcal{L})$,
is the sheaf of $R$-algebras generated by $\mathcal{O}_{X}$ and $\mathcal{L}$
modulo the relations:

  * •

$i:\mathcal{O}_{X}\to U(\mathcal{L})$ is a morphism of $R$-algebras,

  * •

$j:\mathcal{L}\to U(\mathcal{L})$ is a morphism of Lie algebras,

  * •

$j(fl)=i(f)j(l)$ and $[j(l),i(f)]=i(\rho(l)(f))$.

Locally, $U(\mathcal{L})$ is just the enveloping algebra of the corresponding
$(R,\mathcal{O}_{X}(U))$-algebra.

We want to establish a correspondence between Lie algebroids and $r$-deformed
tdo’s on an $R$\- variety $X$. For an $\mathcal{O}_{X}$-differential algebra
$\mathcal{D}$ we define
$\operatorname{Lie\text{\kern-2.0pt}\,}(\mathcal{D}):=F_{1}(\mathcal{D})$; one
may prove that when $\mathcal{D}$ is a tdo,
$\operatorname{Lie\text{\kern-2.0pt}\,}(\mathcal{D})$ is a Lie algebroid, see
[4, 1.2.5]; unfortunately not all Lie algebroids induce tdo’s, so we need a
more specific notion.

######  Definition 5.3.

We call a Lie algebroid $\mathcal{L}$ an $r$-deformed Picard algebroid if
there exists a short exact sequence of Lie algebras and
$\mathcal{O}_{X}$-modules:

$0\to\mathcal{O}_{X}\to\mathcal{L}\to r\mathcal{T}_{X}\to 0.$

One should notice that the Lie algebra structure imposed on $\mathcal{O}_{X}$
is the trivial one. We should also denote $1_{\mathcal{L}}$ the image of
$1\in\mathcal{O}_{X}$ under the inclusion map.

A morphism of $r$-deformed Picard algebroids is a morphism of Lie algebroids
compatible with the maps in the short exact sequence defining $r$-deformed
Picard structure.

######  Proposition 5.4.

Let $\mathcal{L}$ be an $r$-deformed Picard algebroid on $X$. Then the sheaf
of rings $\mathcal{D}:=U(\mathcal{L})/U(\mathcal{L})(i(1)-j(1))$ is an
$r$-deformed tdo with
$\operatorname{Lie\text{\kern-2.0pt}\,}(\mathcal{D})=\mathcal{L}$.

###### Proof.

The question is local so we may assume that $X$ is affine;
$A=\mathcal{O}_{X}(X)$, $T=\mathcal{T}_{X}(X)$, $L=\mathcal{L}(X)$ and let
$i:A\to L$ denote the injection induced by the short exact sequence defining
$\mathcal{L}$.

Consider the enveloping algebra $U(L)$ of the $(R,A)$-Lie algebra $L$. We can
think of it as being generated by $A$ and the universal enveloping algebra of
the Lie algebra $L$ subject to the relations: $fl=f\cdot l$ (the module
action), $fl-lf=\rho(l)(f)$ for $f\in A$ and $l\in L$,
$l_{1}l_{2}-l_{2}l_{1}=[l_{1},l_{2}]$ for $l_{1},l_{2}\in L$. The natural
filtration on $U(L)$ is given by:

  * •

$F_{0}(U(L))=A,$

  * •

For $n\geq 1$, $F_{n}(U(L))=A+l_{1}l_{2}\ldots l_{m}$, where $m\leq n$ and
$l_{1},l_{2},\ldots l_{m}\in L$.

Since $fl-lf=\rho(l)(f)$ for $f\in A$ and $l\in L$, it easy to see that $U(L)$
becomes a differential $A$ algebra with respect to this filtration.

Let $I=\langle i(1)-1_{L}\rangle$ be the central two sided ideal and
$D:=\mathcal{D}(X)=U(L)/IU(L)$. We give $D$ the quotient natural quotient
filtration induced from $U(L)$. By construction, we have that $D$ is also a
differential $A$-algebra since $U(L)$ is. Furthermore, we have $F_{0}(D)\cong
F_{0}(U(L))\cong A$.

Let $g:F_{1}(U(L))\to L$ given by $g(a+l)=i(a)+l$ for $a\in A,l\in L$. Then it
is clear that $g$ is surjective and $\ker(g)=a-i(a)=F_{1}(I)$. Thus, we obtain
$F_{1}(D)\cong L$, so $\operatorname{Lie\text{\kern-2.0pt}\,}(D)\cong L$.

Since $L$ is an $r$-deformed Picard algebroid, we obtain immediately that
$\operatorname{gr\text{\kern-2.0pt}\,}_{1}D\cong L/A\cong rT$. Finally,
because $X$ is an $R$-variety, $T$ is projective as an $A$-module. Therefore,
$L$ is also projective as $A$-module, so we have by [10, Theorem 3.1] that
$\operatorname{gr\text{\kern-2.0pt}\,}D\cong\operatorname{Sym\text{\kern-2.0pt}\,}_{A}(\operatorname{gr\text{\kern-2.0pt}\,}_{1}D).$
Thus, we have proven all the required properties to make $D$ an $r$-deformed
tdo on $X$. ∎

Using the Lemma above we make the following definition:

######  Definition 5.5.

Let $X$ be an $R$-variety. Define a functor $\mathscr{T}:$ category of
$r$-deformed Picard algebroids on $X\to$ category of $r$-deformed twisted
differential operators on $X$ by

$\mathscr{T}(\mathcal{L}):=U(\mathcal{L})/U(\mathcal{L})(i(1)-j(1)).$

######  Lemma 5.6.

Let $\mathcal{D}$ be an $r$-deformed tdo on $X$. The sheaf
$\operatorname{Lie\text{\kern-2.0pt}\,}(\mathcal{D}):=F_{1}(\mathcal{D})$ is
an $r$-deformed Picard algebroid and furthermore

$\mathscr{T}(\operatorname{Lie\text{\kern-2.0pt}\,}\mathcal{D})=U(\operatorname{Lie\text{\kern-2.0pt}\,}(\mathcal{D}))/U(\operatorname{Lie\text{\kern-2.0pt}\,}(\mathcal{D}))(i(1)-j(1))\cong\mathcal{D}.$

###### Proof.

Let $\mathcal{L}:=\operatorname{Lie\text{\kern-2.0pt}\,}(\mathcal{D})$. Since
$\mathcal{D}$ is a differential algebra, $\mathcal{L}$ is a Lie algebroid by
[4, 1.2.5], with the anchor map $\rho:\mathcal{L}\to\mathcal{T}_{X}$ induced
by axiom $ii)$ of Definition 4.5. Further by axioms $i)$ and $ii)$ of 4.5 we
observe that $\ker(\rho)=\mathcal{O}_{X}$ and
$\operatorname{im\text{\kern-2.0pt}\,}(\rho)=r\mathcal{T}_{X}$, so
$\mathcal{L}$ is indeed an $r$-deformed Picard algebroid.

Let $\mathcal{A}:=U(\mathcal{L})/U(\mathcal{L})(i(1)-j(1))$. By Proposition
5.4, $\mathcal{A}$ is an $r$-deformed tdo. Further by construction we have
that there is morphism of filtered algebras $\mathcal{A}\to\mathcal{D}$ and
$\operatorname{gr\text{\kern-2.0pt}\,}_{1}(\mathcal{A})\cong\operatorname{gr\text{\kern-2.0pt}\,}_{1}(\mathcal{D})$.
Since $\mathcal{A}$ and $\mathcal{D}$ are $r$-deformed tdo’s, we have by axiom
$iii)$ of Definition 4.5 that
$\operatorname{gr\text{\kern-2.0pt}\,}(\mathcal{A})\cong\operatorname{Sym\text{\kern-2.0pt}\,}_{\mathcal{O}_{X}}(\operatorname{gr\text{\kern-2.0pt}\,}_{1}\mathcal{A})$
and
$\operatorname{gr\text{\kern-2.0pt}\,}(\mathcal{D})\cong\operatorname{Sym\text{\kern-2.0pt}\,}_{\mathcal{O}_{X}}(\operatorname{gr\text{\kern-2.0pt}\,}_{1}\mathcal{D})$.
Therefore, we get
$\operatorname{gr\text{\kern-2.0pt}\,}(\mathcal{A})\cong\operatorname{gr\text{\kern-2.0pt}\,}(\mathcal{D})$,so
$\mathcal{A}\cong\mathcal{D}$ since there is a filtered morphism between them.

∎

######  Corollary 5.7.

Let $X$ be an $R$-variety. The functors $\mathscr{T}$ and
$\operatorname{Lie\text{\kern-2.0pt}\,}$ induce quasi-inverse equivalences of
categories between the category of $r$-deformed Picard algebroids on $X$ and
the category of $r$-deformed tdo’s on $X$.

###### Proof.

This follows from Proposition 5.4 and Lemma 5.6. ∎

##  6 Equivariant deformed Picard algebroids and deformed homogeneous twisted
differential operators

Throughout this section, we fix $X$ an $R$-variety, $G$ a smooth affine
algebraic group of finite type acting on $X$ and $r\in R$ a regular element.
Recall from 13 that for a $G$-equivariant $\mathcal{O}_{X}$-module
$\mathcal{M}$ we denoted by abuse of notation by $\cdot$ the group action
giving the equivariance.

######  Definition 6.1.

Let $(\mathcal{L},\rho)$ be a Lie algebroid. We say that $\mathcal{L}$ is
$r$-deformed $G$-equivariant if $\mathcal{L}$ is a $G$-equivariant as a
$\mathcal{O}_{X}$-module and it is equipped with a Lie algebra morphism
$i_{\mathfrak{g}}:r\mathfrak{g}\to\mathcal{L}$ such that:

  1. i)

$g\cdot[x,y]=[g\cdot x,g\cdot y]$, for $g\in G$, $x,y\in\mathcal{L}$.

  2. ii)

$g.\rho(l)(f)=\rho(g.l)(g.f)$, for $g\in
G,l\in\mathcal{L},f\in\mathcal{O}_{X}.$ This is equivalent to $\rho$ being
$G$-equivariant.

  3. iii)

$i_{\mathfrak{g}}(g.\psi)=g.i_{\mathfrak{g}}(\psi)$, for $g\in G$ and $\psi\in
r\mathfrak{g}$. Here $G$ acts on $r\mathfrak{g}\subset\mathfrak{g}$ via the
Adjoint action.

Similarly, we may define the notion of equivariant differential algebra.

######  Definition 6.2.

Let $\mathcal{D}$ be a differential $\mathcal{O}_{X}$-algebra. We call
$\mathcal{D}$ an $r$-deformed _$G$ -equivariant_ differential algebra if it is
$G$-equivariant as a left $\mathcal{O}_{X}$-module and it is equipped with a
Lie algebra map $i_{\mathfrak{g}}:r\mathfrak{g}\to\mathcal{D}$ such that:

  1. 1.

$g.1=1$ and $g.(d_{1}d_{2})=(g.d_{1})(g.d_{2})$, for $g\in G$ and
$d_{1},d_{2}\in\mathcal{D}.$

  2. 2.

$g.(fd)=(g.f)(g.d)$, for $f\in\mathcal{O}_{X}$ and $d\in\mathcal{D}$.

  3. 3.

$i_{\mathfrak{g}}(g.\psi)=g.i_{\mathfrak{g}}(\psi)$, for $g\in G$ and $\psi\in
r\mathfrak{g}$.

######  Lemma 6.3.

Let $(\mathcal{L},\rho,i_{\mathfrak{g}})$ be an $r$-deformed $G$-equivariant
Lie algebroid. Then $U(\mathcal{L})$ is an $r$-deformed $G$-equivariant
differential algebra.

###### Proof.

Since $\mathcal{L}$ is quasi-coherent as a $\mathcal{O}_{X}$-module, so is
$U(\mathcal{L})$. We define $G$-action on $U(\mathcal{L})$ by defining
$g.(l_{1}l_{2}\ldots g.j)=(g.l_{1})(g.l_{2})\ldots(g.l_{j})$ for
$l_{1},l_{2},\ldots l_{j}\in\mathcal{L}$ and $g.(fl_{1}l_{2}\ldots
l_{j})=(g.f)(g.l_{1}l_{2}\ldots l_{j})$ for $f\in\mathcal{O}_{X}$ and
$l_{1},l_{2},\ldots l_{j}\in\mathcal{L}$. We have

$\begin{split}g.(l_{1}l_{2}-l_{2}l_{1})&=(g.l_{1})(g.l_{2})-(g.l_{2})(g.l_{1})\\\
&=[g.l_{1},g.l_{2}]\\\ &=g.[l_{1},l_{2}]\end{split}$ (16)

and

$\begin{split}g.(fl-lf)&=(g.f)(g.l)-(g.l)(g.f)\\\ &=[g.f,g.l]\\\
&=\rho(g.l)(g.f)\\\ &=g.\rho(l)(f).\end{split}$ (17)

Since $U(\mathcal{L})$ is generated by $\mathcal{O}_{X}$ and the enveloping
algebra of $\mathcal{L}$, subject to the relations $(fl-lf)=\rho(l)(f)$ and
$fl=f.l$, it follows from the equations and definition that $U(\mathcal{L})$
is $G$-equivariant as $\mathcal{O}_{X}$-module and furthermore, axioms $i)$
and $ii)$ of Definition 6.2 are satisfied. Further it easy to check that
$G$-action preserves the filtration on $U(\mathcal{L})$.

The morphism $i_{\mathfrak{g}}:r\mathfrak{g}\to\mathcal{L}$ can be extended to
a morphism $i_{\mathfrak{g}}:r\mathfrak{g}\to U(\mathcal{L})$ via the natural
map $\mathcal{L}\to U(\mathcal{L})$ and it is clear by construction that under
$G$-action and the map $i_{\mathfrak{g}}$ defined above that $U(\mathcal{L})$
becomes an $r$-deformed $G$-equivariant differential algebra. ∎

As we are interested in deformed Picard algebroids, we define the notion of an
$r$-deformed $G$-equivariant Picard algebroid. The $G$ action on $\mathcal{L}$
induces by differentiation a
$\mathfrak{g}:=\operatorname{Lie\text{\kern-2.0pt}\,}(G)$ action via a map
$\beta_{\mathcal{M}}:\mathfrak{g}\to\operatorname{End\text{\kern-3.0pt}\,}(\mathcal{L})$.
We also let $\eta:\mathfrak{g}\to\mathcal{T}_{X}$ denote the infinitesimal
action of $\mathfrak{g}$ on $X$.

######  Definition 6.4.

Let $\mathcal{L}$ be an $r$-deformed Picard algebroid. We say that
$\mathcal{L}$ is an $r$-deformed $G$-equivariant Picard algebroid if
$\mathcal{L}$ is an $r$-deformed $G$-equivariant algebroid, in the short exact
sequence

$0\to\mathcal{O}_{X}\to\mathcal{L}\to r\mathcal{T}_{X}\to 0$

all the morphisms are $G$-equivariant and

  * •

The derivative of the $G$-action induces a $\mathfrak{g}$ action and thus a
$r\mathfrak{g}$ action on $\mathcal{L}$. This must coincide with the action
$l\mapsto[i_{\mathfrak{g}}(\psi),l]$ for $\psi\in r\mathfrak{g}$ and
$l\in\mathcal{L}$.

  * •

$\eta_{|r\mathfrak{g}}=\rho\circ i_{\mathfrak{g}}$.

A morphism of $r$-deformed $G$-equivariant Picard algebroids is a morphism of
$r$-deformed Picard algebroids compatible with the equivariance structures.

######  Definition 6.5.

Let $\mathcal{D}$ be a $\mathcal{O}_{X}$-algebra. We say that $\mathcal{D}$ is
a sheaf of $r$-deformed $G$-homogeneous twisted differential operators
($r$-deformed $G$-htdo) if $(\mathcal{D},i_{\mathfrak{g}})$ is an $r$-deformed
$G$-equivariant differential $\mathcal{O}_{X}$-algebra and a sheaf of
$r$-deformed twisted differential operators, and furthermore:

  * •

The image of $i_{\mathfrak{g}}$ lies in $F_{1}\mathcal{D}$.

  * •

The derivative of the $G$-action induces a $\mathfrak{g}$ action and thus a
$r\mathfrak{g}$-action on $\mathcal{D}$. This must coincide with the action
$d\mapsto[i_{\mathfrak{g}}(\psi),d]$ for $\psi\in r\mathfrak{g}$ and
$d\in\mathcal{D}$.

  * •

$\eta_{|r\mathfrak{g}}=\rho\circ i_{\mathfrak{g}}$, where
$\rho:F_{1}\mathcal{D}=\operatorname{Lie\text{\kern-2.0pt}\,}(\mathcal{D})\to\mathcal{T}_{X}$
is the natural anchor map.

A morphism of $r$-deformed $G$-htdo’s is a morphism of $r$-deformed tdo’s
compatible with the equivariance structures.

One should notice that since $g.1=1$, the morphism $\mathcal{O}_{X}\to
F_{0}(\mathcal{D})$ is automatically $G$-equivariant.

######  Lemma 6.6.

Let $(\mathcal{L},\rho,i_{\mathfrak{g}})$ be an $r$-deformed $G$-equivariant
Picard algebroid. Then $\mathscr{T}(\mathcal{L})$ is an $r$-deformed $G$-htdo.

###### Proof.

We have by Lemma 6.3 that $U(\mathcal{L})$ is a $G$-equivariant differential
algebra. Now, since $\mathcal{L}$ is $G$-equivariant, the action of $G$
stabilises the ideal generated by $i(1)-j(1)$, so the $G$ action descends on
$U(\mathcal{L})/U(\mathcal{L})(i(1)-j(1))$. Similarly, composing the map
$i_{\mathfrak{g}}:r\mathfrak{g}\to U(\mathcal{L})$ with the natural
projection, we obtain a map $r\mathfrak{g}\to\mathscr{T}(\mathcal{L}).$
Further, by Proposition 5.4, $U(\mathcal{L})/U(\mathcal{L})(i(1)-j(1))$ is an
$r$-deformed tdo. Finally, the axioms of Definition 6.4 imply that
$\mathscr{T}(\mathcal{L})$ is an $r$-deformed $G$-htdo. ∎

######  Lemma 6.7.

Let $\mathcal{D}$ be an $r$-deformed $G$-htdo. Then
$\mathcal{L}:=F_{1}(\mathcal{D})$ is an $r$-deformed $G$-equivariant Picard
algebroid.

###### Proof.

We have by Lemma 5.6 that $\mathcal{L}$ is an $r$-deformed Picard algebroid
and the axioms for $\mathcal{D}$ imply that $\mathcal{L}$ is a
$G$-equivariant. Since $\mathcal{D}$ is a $G$-equivariant differential algebra
the map $\mathcal{L}\to\mathcal{T}_{X}$ is $G$-equivariant. Lastly, the
morphism $\mathcal{O}_{X}\to F_{0}(\mathcal{D})$ is $G$-equivariant and the
other axioms in definition Definition 6.4 follow from the corresponding axioms
in 6.5. ∎

######  Corollary 6.8.

Let $X$ be an $R$-variety and $r\in R$ a regular element. The maps
$\mathscr{T}$ and $\operatorname{Lie\text{\kern-2.0pt}\,}$ induce quasi-
inverse equivalences of categories between the category of $r$-deformed
$G$-equivariant Picard algebroids on $X$ and the category of $r$-deformed
$G$-htdo’s on $X$.

###### Proof.

This follows by combining Lemmas 6.6 and 6.7 and Corollary 5.7. ∎

##  7 Pullback of deformed Picard algebroids

Throughout this section, we fix $f:Y\to X$ a morphism of $R$-varieties and
$r\in R$ a regular element. The map $f$ induces a morphism
$f^{*}\Omega_{X}^{1}\to\Omega_{Y}^{1}$ and by dualising we obtain
$\alpha:\mathcal{T}_{Y}\to f^{*}{\mathcal{T}_{X}}$. Here $\Omega_{X}$ and
$\Omega_{Y}$ denote the sheaf of differential $1$-forms.

######  Definition 7.1.

Let $(\mathcal{L},\rho_{X})$ be an $r$-deformed Picard algebroid on $X$ and
let $\beta=f^{*}(\rho_{X})$. Then we let

$f^{\\#}\mathcal{L}:=r\mathcal{T}_{Y}\times_{rf^{*}\mathcal{T}_{X}}f^{*}\mathcal{L}=\\{(d,l)|d\in
r\mathcal{T}_{Y},l\in f^{*}\mathcal{L},\alpha(d)=\beta(l)\\}.$

We give $f^{\\#}\mathcal{L}$ the structure of a Lie algebroid by setting
$\rho_{Y}(d,l)=d$ and the Lie bracket be induced by

$[(\psi,f\otimes P),(\eta,g\otimes
Q)]:=([\psi,\eta],fg\otimes[P,Q]+\psi(g)\otimes P-\eta(f)\otimes Q),$

for $\psi,\eta\in r\mathcal{T}_{Y},f,g\in\mathcal{O}_{Y},P,Q\in
f^{-1}\mathcal{L}.$ We call $f^{\\#}\mathcal{L}$ the pullback of
$\mathcal{L}$.

######  Lemma 7.2.

Let $(\mathcal{L},\rho_{X})$ be an $r$-deformed Picard algebroid. Then
$(f^{\\#}\mathcal{L},\rho_{Y})$ is an $r$-deformed Picard algebroid.

###### Proof.

Since $\mathcal{L}$ is an $r$-deformed Picard algebroid, it fits into a short
exact sequence:

$0\to\mathcal{O}_{X}\to\mathcal{L}\to r\mathcal{T}_{X}\to 0.$

By assumption, $X$ is a smooth variety, thus $\mathcal{T}_{X}$ is a free
$\mathcal{O}_{X}$-module, so in particular is flat. Since
$r\mathcal{T}_{X}\cong\mathcal{T}_{X}$ as $\mathcal{O}_{X}$-modules, by
pulling back along $f$ we obtain a short exact sequence:

$0\to\mathcal{O}_{Y}\to f^{*}\mathcal{L}\to rf^{*}\mathcal{T}_{X}\to 0.$ (18)

Considering the pullback diagram

${f^{\\#}\mathcal{L}}$${f^{*}\mathcal{L}}$${r\mathcal{T}_{Y}}$${rf^{*}\mathcal{T}_{X},}$

we obtain
$\ker(\rho_{Y})=\ker(f^{\\#}\mathcal{L}\to\mathcal{T}_{Y})\cong\ker(f^{*}\mathcal{L}\to
rf^{*}\mathcal{T}_{X})\cong\mathcal{O}_{Y}.$ Finally, since
$\operatorname{im\text{\kern-2.0pt}\,}(f^{*}\mathcal{L}\to
rf^{*}\mathcal{T}_{X})=rf^{*}\mathcal{T}_{X}$, we obtain that
$\operatorname{im\text{\kern-2.0pt}\,}(\rho_{Y})=r\mathcal{T}_{Y}$. ∎

We would like to describe how the pullback interacts with composition of
morphisms. In general, one would like to prove that for $u:Z\to Y,f:Y\to X$
maps of $R$-varieties and $\mathcal{L}$ an $r$-deformed Picard algebroid on
$X$, we have $u^{\\#}f^{\\#}\mathcal{L}\cong(f\circ u)^{\\#}\mathcal{L}$.
Unfortunately, this is not always true. In the following, we give sufficient
conditions.

######  Lemma 7.3.

Let $u:Z\to Y,f:Y\to X$ be maps of $R$-varieties and $\mathcal{L}$ be an
$r$-deformed Picard algebroid on $X$. Assume that $u$ is flat or $f$ is etale.
Then

$u^{\\#}f^{\\#}\mathcal{L}\cong(f\circ u)^{\\#}\mathcal{L}.$

###### Proof.

First assume $u$ is flat. Then:

$\begin{split}u^{\\#}f^{\\#}\mathcal{L}&=u^{\\#}(r\mathcal{T}_{Y}\times_{rf^{*}\mathcal{T}_{X}}f^{*}\mathcal{L})\\\
&=r\mathcal{T}_{Z}\times_{ru^{*}\mathcal{T}_{Y}}u^{*}(r\mathcal{T}_{Y}\times_{rf^{*}\mathcal{T}_{X}}f^{*}\mathcal{L})\\\
&=r\mathcal{T}_{Z}\times_{ru^{*}\mathcal{T}_{Y}}{ru^{*}\mathcal{T}_{Y}}\times_{ru^{*}f^{*}\mathcal{T}_{X}}u^{*}f^{*}\mathcal{L}\text{
($u$ is flat, so commutes with limits)}\\\ &\cong
r\mathcal{T}_{Z}\times_{ru^{*}f^{*}\mathcal{T}_{X}}u^{*}f^{*}\mathcal{L}\\\
&\cong(f\circ u)^{\\#}\mathcal{L}.\end{split}$ (19)

Now suppose $f$ is etale, thus $f^{*}\mathcal{T}_{X}\cong\mathcal{T}_{Y}$.
Then:

$\begin{split}u^{\\#}f^{\\#}\mathcal{L}&=u^{\\#}(r\mathcal{T}_{Y}\times_{rf^{*}\mathcal{T}_{X}}f^{*}\mathcal{L})\\\
&\cong u^{\\#}f^{*}\mathcal{L}\\\ &\cong
r\mathcal{T}_{Z}\times_{ru^{*}\mathcal{T}_{Y}}u^{*}f^{*}\mathcal{L}\\\ &\cong
r\mathcal{T}_{Z}\times_{ru^{*}f^{*}\mathcal{T}_{X}}u^{*}f^{*}\mathcal{L}\\\
&\cong(f\circ u)^{\\#}\mathcal{L}.\qed\end{split}$ (20)

######  Lemma 7.4.

Let $(\mathcal{L},\rho_{X},i_{\mathfrak{g}})$ be an $r$-deformed
$G$-equivariant Picard algebroid on $X$ and assume further $f:Y\to X$ is a
$G$-equivariant. Then $(f^{\\#}\mathcal{L},\rho_{Y})$ is an $r$-deformed
$G$-equivariant Picard algebroid.

###### Proof.

By the Lemma above, $f^{\\#}\mathcal{L}$ is an $r$-deformed Picard algebroid.
Since $f:Y\to X$ is $G$-equivariant, we obtain by Lemma 2.2 that
$f^{*}\mathcal{L}$ is a $G$-equivariant $\mathcal{O}_{Y}$-module and that the
maps $\alpha$ and $\beta$ are $G$-equivariant. We define a $G$ action on
$f^{\\#}\mathcal{L}$ via

$g.(d,l)=(g.d,g.l)\text{ for }d\in r\mathcal{T}_{Y},l\in f^{*}\mathcal{L}.$

We need to check that this action is well-defined and $\rho_{Y}$ is
$G$-equivariant. The second statement is easy, we have
$g.\rho_{Y}(a,b)=\rho_{Y}(g.(a,b))$. For the other statement, let $d\in
r\mathcal{T}_{Y},l\in f^{*}\mathcal{L}$ such that $\alpha(d)=\beta(l)$. Then,
we have

$\alpha(g.d)=g.\alpha(d)=g.\beta(l)=\beta(g.l).$

So, we are left to prove that $G$-action interacts correctly with the Lie
bracket. Let $\psi,\eta\in r\mathcal{T}_{Y},f,h\in\mathcal{O}_{Y},P,Q\in
f^{-1}\mathcal{L}.$ Then

$\begin{split}g.[(\psi,f\otimes P),(\eta,h\otimes
Q)]&=g.([\psi,\eta],fh\otimes[P,Q]+\psi(h)\otimes P-\eta(f)\otimes Q)\\\
&=(g.[\psi,\eta],g.fh\otimes g.[P,Q]\\\ &+g.\psi(h)\otimes
g.P-g.\eta(f)\otimes g.Q)\\\ &=([g.\psi,g.\eta],(g.f)(g.h)\otimes[g.P,g.Q]\\\
&+\psi(g.h)\otimes g.P-\eta(g.f)\otimes g.Q)\\\ &=[g.(\psi,f\otimes
P),g.(\eta,h\otimes Q)].\end{split}$ (21)

Since $G$ acts on $Y$ we obtain the infinitesimal map
$\eta:\mathfrak{g}\to\mathcal{T}_{Y}$. The map
$i_{\mathfrak{g}}:r\mathfrak{g}\to\mathcal{L}$ can be extended to a map
$i_{\mathfrak{g}}:\mathcal{O}_{X}\otimes r\mathfrak{g}\to\mathcal{L}$ and by
pulling back we obtain a map $i_{\mathfrak{g}}^{*}:\mathcal{O}_{Y}\otimes
r\mathfrak{g}\to f^{*}\mathcal{L}$. We let $i:r\mathfrak{g}\to
f^{*}\mathcal{L}$ be the restriction of $i_{\mathfrak{g}}^{*}$ to
$r\mathfrak{g}$; by construction, we have that the $r\mathfrak{g}$ action
induced by $i$ coincides with the one induced by the $G$ action. Therefore we
obtain a map $(\eta_{|r\mathfrak{g}},i):r\mathfrak{g}\to f^{\\#}\mathcal{L}$
and it follows by the construction that the $r\mathfrak{g}$-action induced by
the derivative of the $G$-action coincides with the $r\mathfrak{g}$-action
induced by $(\eta_{|r\mathfrak{g}},i)$ and further that
$\eta_{|r\mathfrak{g}}=\rho_{Y}\circ(\eta_{|r\mathfrak{g}},i)$. Thus
$(f^{\\#}\mathcal{L},\rho_{Y},(\eta_{|r\mathfrak{g}},i))$ is an $r$-deformed
$G$-equivariant Picard algebroid. ∎

######  Definition 7.5.

Let $\mathcal{D}$ be an $r$-deformed tdo/$G$-htdo on $X$ and let $f:Y\to X$ be
a morphism/$G$-equivariant morphism of $R$-varieties. We call

$f^{\\#}\mathcal{D}:=\mathscr{T}(f^{\\#}\operatorname{Lie\text{\kern-2.0pt}\,}(\mathcal{D}))$

the pullback of $\mathcal{D}$ along $f$.

######  Corollary 7.6.

Let the notation as above. Then $f^{\\#}\mathcal{D}$ is well defined and
furthermore it is an $r$-deformed tdo/htdo.

###### Proof.

This follows from Lemmas 7.2, 7.4 and Corollaries 5.7, 6.8. ∎

######  Corollary 7.7.

Let $u:Z\to Y,f:Y\to X$ be maps of $R$-varieties and $\mathcal{D}$ be an
$r$-deformed tdo on $X$. Assume that $u$ is flat or $f$ is etale. Then

$u^{\\#}f^{\\#}\mathcal{D}\cong(f\circ u)^{\\#}\mathcal{D}.$

###### Proof.

This follows from Lemma 7.3 and Corollary 5.7. ∎

$(e\times 1_{X})^{\\#}\alpha:\mathcal{D}\to\mathcal{D}$

Let us explain how our condition for an $r$-deformed $G$-htdo fits into a
diagram satisfying the cocycle condition. Denote the $G$-action by
$\sigma_{X}:G\times X\to X$. Furthermore, we denote $p_{X}:G\times X\to X$ and
$p_{2X}:G\times G\times X\to X$ the projections on the $X$ factor,
$p_{23X}:G\times G\times X\to G\times X$ the projection onto the second and
third factor and $m:G\times G\to G$ the multiplication of the group $G$.

######  Lemma 7.8.

Let $\mathcal{D}$ be an $r$-deformed $G$-htdo on $X$. Then there exists
$\alpha:\sigma_{X}^{\\#}\mathcal{D}\to p_{X}^{\\#}\mathcal{D}$ an isomorphism
of $\mathcal{O}_{G\times X}$-algebras such that the diagram:

${(1_{G}\times\sigma_{X})^{\\#}p_{X}^{\\#}\mathcal{D}}$${p_{2X}^{\\#}\mathcal{D}}$${(1_{G}\times\sigma_{X})^{\\#}\sigma_{X}^{\\#}\mathcal{D}}$${(m\times
1_{X})^{\\#}\sigma_{X}^{\\#}\mathcal{D}}$$\scriptstyle{p_{23X}^{\\#}\alpha}$$\scriptstyle{(1_{G}\times\sigma_{X})^{\\#}\alpha}$$\scriptstyle{id}$$\scriptstyle{(m\times
1_{X})^{\\#}\alpha}$ (22)

of $\mathcal{O}_{G\times G\times X}$-algebras commutes ( the cocycle
condition) and the pullback

$(e\times 1_{X})^{\\#}\alpha:\mathcal{D}\to\mathcal{D}$

is the identity map. We note that this is the same condition as in [14,
Section 5.2].

###### Proof.

Let $\mathcal{L}=\operatorname{Lie\text{\kern-2.0pt}\,}(D)$; we have by Lemma
6.7 that $\mathcal{L}$ is a $G$-equivariant $r$-deformed Picard algebroid. In
particular, we obtain an isomorphism $\beta:\sigma_{X}^{*}\mathcal{L}\to
p_{X}^{*}\mathcal{L}$, which can be extended to an isomorphism of $r$-deformed
Picard algebroids $\sigma_{X}^{\\#}\mathcal{L}\to p_{X}^{\\#}\mathcal{L}$ and
thus to an isomorphsim of $\mathcal{O}_{G\times X}$-algebras
$\alpha:\sigma_{X}^{\\#}\mathcal{D}=\mathscr{T}(\sigma_{X}^{\\#}\mathcal{L})\to\mathscr{T}(p_{X}^{\\#}\mathcal{L})=p_{X}^{\\#}\mathcal{D}$.
Further, since the map $\beta$ satisfies the cocycle condition, so does
$\alpha$. ∎

##  8 Representations of Lie algebroids

Throughout this section we fix $X$ an $R$-variety, $r\in R$ a regular element
and $(\mathcal{L},\rho_{X})$ a Lie algebroid on $X$.

######  Definition 8.1.

Let $\mathcal{M}$ be a quasi-coherent $\mathcal{O}_{X}$-module. We say that
$\mathcal{M}$ is a _$\mathcal{L}$ -module_ if $\mathcal{M}$ is a sheaf of
modules over the sheaf of Lie algebras $\mathcal{L}$ and for all
$f\in\mathcal{O}_{X}$, $l\in\mathcal{L}$, $m\in\mathcal{M}$, we have

$\begin{split}f.(l.m)&=l.(f.m)-\rho_{X}(l)(f).m,\\\
(f.l).m&=f.(l.m).\end{split}$ (23)

We define a morphism of $\mathcal{L}$-modules to be a morphism of
$\mathcal{O}_{X}$-modules compatible with the $\mathcal{L}$-action.

######  Definition 8.2.

Assume that $(\mathcal{L},\rho_{X})$ is an $r$-deformed Picard algebroid and
let $f:Y\to X$ be a map of $R$-varieties and $\mathcal{M}$ a
$\mathcal{L}$-module. Then we define the pullback of $\mathcal{M}$ along $f$,
via $f^{\\#}\mathcal{M}=f^{*}\mathcal{M}$ as an $\mathcal{O}_{Y}$-module and

$(\psi,P\otimes l).(Q\otimes m):=\psi(Q)\otimes m+PQ\otimes l.m,$

for $\psi\in r\mathcal{T}_{Y}$, $P,Q\in\mathcal{O}_{Y},l\in
f^{-1}\mathcal{L},m\in f^{-1}\mathcal{M}$.

######  Lemma 8.3.

The action defined above makes $f^{\\#}\mathcal{M}$ a
$f^{\\#}\mathcal{L}$-module.

###### Proof.

First, we check the bracket action. We have for $\psi,\eta\in
r\mathcal{T}_{Y}$, $P,Q,R\in\mathcal{O}_{Y}$, $a,b\in f^{-1}\mathcal{L}$ and
$m\in f^{-1}\mathcal{M}$ that

$\begin{split}(\psi,P\otimes a).((\eta,R\otimes b).Q\otimes m)&=(\psi,P\otimes
a).(\eta(Q)\otimes m+RQ\otimes b.m)\\\ &=\psi(\eta(Q))\otimes
m+P\eta(Q)\otimes a.m\\\ &+\psi(RQ)\otimes b.m+PQR\otimes a.(b.m)\end{split}$
(24)

and

$\begin{split}(\eta,R\otimes b).((\psi,P\otimes a).Q\otimes
m)&=\eta(\psi(Q))\otimes m+R\psi(Q)\otimes b.m\\\ &+\eta(PQ)\otimes
a.m+PQR\otimes b.(a.m).\end{split}$ (25)

Thus, combining the equations above, we obtain

$\begin{split}(\psi,P\otimes a).((\eta,R\otimes b).Q\otimes m)&-(\eta,R\otimes
b).((\psi,P\otimes a).Q\otimes m)\\\ &=\psi(\eta(Q))-\eta(\psi(Q))\otimes m\\\
&+PQR\otimes a.(b.m)-b.(a.m)\\\ &+P\eta(Q)-\eta(PQ)\otimes
a.m+\psi(RQ)-R\psi(Q)\otimes b.m\\\ &=[\psi,\eta](Q)\otimes
m+PQR\otimes[a,b].m\\\ &-Q\eta(P)\otimes a.m+Q\psi(R)\otimes b.m\\\
&=[(\psi,P\otimes a),(\eta,R\otimes b)].(Q\otimes m).\end{split}$ (26)

To check the first axiom, we have

$\begin{split}R.((\psi,P\otimes l).Q\otimes m)&=R.(\psi(Q)\otimes m+PQ\otimes
l.m)\\\ &=R\psi(Q)\otimes m+PQR\otimes l.m\\\ &=\psi(RQ)\otimes
m-Q\psi(R)\otimes m+PQR\otimes l.m\\\ &=(\psi,P\otimes l).R.(Q\otimes
m)-\rho_{Y}(\psi,P\otimes l)(R).(Q\otimes m).\end{split}$ (27)

Finally, we have

$\begin{split}R.((\psi,P\otimes l).Q\otimes m)&=R.(\psi(Q)\otimes m+PQ\otimes
l.m)\\\ &=R\psi(Q)\otimes m+PQR\otimes l.m\\\ &=(R.(\psi,P\otimes l)).Q\otimes
m.\qed\end{split}$ (28)

We now define equivariant representations. Let $G$ be a smooth affine
algebraic group of finite type acting on an $R$-variety $X$.

######  Definition 8.4.

Let $(\mathcal{L},\rho,i_{\mathfrak{g}})$ be an $r$-deformed $G$-equivariant
Lie algebroid on $X$. We say that $\mathcal{M}$ is a $G$-equivariant
$\mathcal{L}$-module if:

  1. i)

$\mathcal{M}$ is a $G$-equivariant $\mathcal{O}_{X}$-module and $\mathcal{M}$
is a $\mathcal{L}$-module.

  2. ii)

$g.(l.m)=(g.l).(g.m)$ for any $g\in G$, $l\in\mathcal{L}$ and
$m\in\mathcal{M}$.

  3. iii)

The $r\mathfrak{g}$ action induced by restricting the $\mathfrak{g}$ action
induced from the derivative of the $G$-action on $\mathcal{M}$ coincides with
the $r\mathfrak{g}$-action induced from $i_{\mathfrak{g}}$.

A morphism of $G$-equivariant $\mathcal{L}$-modules is a morphism of
$G$-equivariant $\mathcal{O}_{X}$-modules compatible with the
$\mathcal{L}$-action.

######  Lemma 8.5.

Let $\mathcal{L}$ be an $r$-deformed $G$-equivariant Lie algebroid on $X$ and
$\mathcal{M}$ a $G$-equivariant $\mathcal{L}$-module. Further, let $Y$ be a
variety and $f:Y\to X$ a $G$-equivariant morphism. Then $f^{\\#}\mathcal{M}$
is a $G$-equivariant $f^{\\#}\mathcal{L}$-module.

###### Proof.

We have the $G$ action on $f^{\\#}\mathcal{M}$ induced by the action on the
simple tensors $g.(Q\otimes m)=g.Q\otimes g.m$ for $g\in G$,
$Q\in\mathcal{O}_{Y}$, $m\in f^{-1}\mathcal{M}$. Since
$f^{\\#}\mathcal{M}=f^{*}\mathcal{M}$ as a $\mathcal{O}_{Y}$-module, the first
axiom of Definition 8.4 follows from Lemma 2.2.

Next, we have for $g\in G$, $\psi\in\mathcal{T}_{Y},Q\in\mathcal{O}_{Y},l\in
f^{-1}\mathcal{L},m\in f^{-1}\mathcal{M}$:

$\begin{split}g.((\psi,P\otimes l).Q\otimes m)&=g.(\psi(Q)\otimes m+PQ\otimes
l.m)\\\ &=g.\psi(Q)\otimes g.m+g.(PQ)\otimes g.(l.m)\\\ &=(g.\psi)(g.Q)\otimes
g.m+g.Pg.Q\otimes(g.l).(g.m)\\\ &=(g.\psi,g.(P\otimes l).(g.Q\otimes g.m)\\\
&=(g.(\psi,P\otimes l)).(g.(Q\otimes m)).\end{split}$ (29)

Finally, the third axiom follows easily from the definition of $G$-action on
$f^{\\#}\mathcal{L}$ and on $f^{\\#}\mathcal{M}$. ∎

We should remark that one could define a more general notion of equivariance
over an $r$-deformed $G$-equivariant Lie algebroid $(\mathcal{L},\rho,i_{g})$.
Let $L$ be a closed subgroup of $G$, then one may relax condition $i)$ to
$L$-equivariance, impose condition $ii)$ to hold just for $g\in L$ and change
condition $iii)$ to: the
$r\mathfrak{l}=\operatorname{Lie\text{\kern-2.0pt}\,}(L)$ action induced by
derivative of the $L$-action on $\mathcal{M}$ coincides with the
$r\mathfrak{l}$-action induced by the restriction of $i_{\mathfrak{g}}$ to
$r\mathfrak{l}$.

##  9 Modules over twisted differential operators

We keep the notation from the previous section. As our main interest is in
modules over deformed htdo’s, we need a definition of a representation of a
deformed Picard algebroid. Recall that $(\mathcal{L},\rho)$ is an $r$-deformed
Picard algebroid if $\mathcal{L}$ fits into the following short exact sequence
$0\to\mathcal{O}_{X}\to\mathcal{L}\to r\mathcal{T}_{X}\to 0$, Thus, for a
$\mathcal{L}$-module, we get two actions of the structure sheaf: one since
$\mathcal{M}$ is an $\mathcal{O}_{X}$-module by assumption and one induced by
the short exact sequence.

We say that $\mathcal{M}$ is a _Picard module_ if the two actions defined
above coincide.

######  Lemma 9.1.

Let $(\mathcal{L},\rho)$ be an $r$-deformed Picard algebroid and let
$\mathcal{M}$ be a Picard $\mathcal{L}$-module. Then $\mathcal{M}$ is a module
over $\mathcal{D}:=\mathscr{T}(\mathcal{L})$.

###### Proof.

Since $\mathcal{M}$ is a $\mathcal{L}$-module, we obtain that $\mathcal{M}$ is
also a $U(\mathcal{L})$-module in a similar fashion as Lie algebra
representations correspond to enveloping algebra modules. Further, the
condition that $\mathcal{M}$ is a Picard is exactly the condition that allows
us to descend to a $\mathcal{D}$-module. ∎

As a corollary of the proof, we obtain immediately:

######  Corollary 9.2.

Let $(\mathcal{L},\rho)$ be an $r$-deformed Picard algebroid on $X$ and
$\mathcal{D}=\mathscr{T}(\mathcal{L})$. Then the identity map provides a one-
to-one correspondence between Picard $\mathcal{L}$-modules and
$\mathcal{D}$-modules.

For an $r$-deformed Picard algebroid $\mathcal{L}$ we will denote
$\operatorname{Mod\text{\kern-2.0pt}\,}(\mathcal{L})$ the category of Picard
$\mathcal{L}$-modules. Similarly, for an $r$-deformed tdo $\mathcal{D}$ we
denote $\operatorname{Mod\text{\kern-2.0pt}\,}(\mathcal{D})$ the category of
quasi-coherent $\mathcal{D}$-modules and
$\operatorname{Coh\text{\kern-2.0pt}\,}(\mathcal{D})$ its full subcategory
consisting of coherent modules.

Equivariant representations

We now move to the equivariant setting. Recall that $G$ is an algebraic group
acting on $X$ with Lie algebra $\mathfrak{g}$.

######  Definition 9.3.

Let $(\mathcal{D},i_{\mathfrak{g}})$ be an $r$-deformed $G$-htdo. We say that
a $\mathcal{D}$-module $\mathcal{M}$ is a $G$-equivariant module over
$\mathcal{D}$ if:

  1. i)

$\mathcal{M}$ is a $G$-equivariant $\mathcal{O}_{X}$-module.

  2. ii)

$g.(d.m)=(g.d).(g.m)$, for all $g\in G,d\in\mathcal{D},m\in\mathcal{M}$.

  3. iii)

The $r\mathfrak{g}$-action induced by the derivative of the $G$-action on
$\mathcal{M}$ coincides with the $r\mathfrak{g}$-action induced by
$i_{\mathfrak{g}}$.

######  Lemma 9.4.

Let $(\mathcal{L},\rho,i_{\mathfrak{g}})$ be an $r$-deformed $G$-equivariant
Picard algebroid and $\mathcal{M}$ be a $G$-equivariant Picard
$\mathcal{L}$-module. Further, let $\mathcal{D}=\mathscr{T}(\mathcal{L})$ the
$r$-deformed $G$-htdo corresponding to $\mathcal{L}$. Then $\mathcal{M}$ is a
$G$-equivariant $\mathcal{D}$-module.

###### Proof.

We have by Corollary 9.2 that $\mathcal{M}$ is a $\mathcal{D}$-module, so we
only have to prove equivariance. Axioms $i)$ and $iii)$ follow from the
corresponding axioms in Definition 8.4, while axiom $ii)$ follows by an easy
induction argument by using the definition of the $G$-action on
$U(\mathcal{L})$. We only check the first step: we have that for $g\in G$,
$l_{1},l_{2}\in\mathcal{L}$ and $m\in\mathcal{M}$ that

$\begin{split}g.(l_{1}l_{2}.m)&=g.(l_{1}.(l_{2}.m)\\\
&=(g.l_{1}).(g.(l_{2}.m)\\\ &=[(g.l_{1})(g.l_{2})].(g.m)\\\
&=(g.l_{1}l_{2}).(g.m).\qed\end{split}$ (30)

Similarly to the non-equivariant case, we obtain

######  Corollary 9.5.

Let $X$ be an $R$-variety and $(\mathcal{L},\rho,i_{\mathfrak{g}})$ an
$r$-deformed $G$-equivariant Picard algebroid and let
$\mathcal{D}=\mathscr{T}(\mathcal{L})$ be the corresponding $r$-deformed
$G$-htdo. The identity map provides a one-to-one correspondence between
$G$-equivariant Picard $\mathcal{L}$-modules and $G$-equivariant
$\mathcal{D}$-modules.

For an $r$-deformed $G$-equivariant Picard algebroid $\mathcal{L}$ we will
denote $\operatorname{Mod\text{\kern-2.0pt}\,}(\mathcal{L},G)$ the category of
$G$-equivariant Picard $\mathcal{L}$-modules. Similarly, for a $G$-htdo
$\mathcal{D}$ we denote
$\operatorname{Mod\text{\kern-2.0pt}\,}(\mathcal{D},G)$ the category of quasi-
coherent $G$-equivariant $\mathcal{D}$-modules and
$\operatorname{Coh\text{\kern-2.0pt}\,}(\mathcal{D},G)$ its full subcategory
consisting of coherent modules. A similar argument to the one in Proposition
2.4 proves that these categories are Abelian.

Pullback of modules over twisted differential operators

We conclude the section by defining of the pullback of a module over a sheaf
of twisted differential operators.

######  Definition 9.6.

Let $f:Y\to X$ be a morphism of $R$-varieties, $\mathcal{D}$ an $r$-deformed
tdo on $X$ and $\mathcal{M}$ a $\mathcal{D}$-module. We define the pullback of
$\mathcal{M}$ under the map $f$ to be $f^{\\#}\mathcal{M}$.

We should remark that the sheaf $f^{\\#}\mathcal{M}$ has the structure of a
$f^{\\#}\mathcal{D}$-module by Corollary 5.7, Lemma 8.3 and Corollary 9.2.

######  Lemma 9.7.

Let $\mathcal{D}$ be an $r$-deformed $G$-htdo, $\mathcal{M}$ a $G$-equivariant
$\mathcal{D}$-module and $f:Y\to X$ a $G$-equivariant morphism of
$R$-varieties. Then $f^{\\#}\mathcal{M}$ is a $G$-equivariant
$f^{\\#}\mathcal{D}$-module.

###### Proof.

We have by the definition above that $f^{\\#}\mathcal{M}$ is a
$f^{\\#}\mathcal{D}$ module and by Corollary 9.2, $f^{\\#}\mathcal{D}$ is an
$r$-deformed $G$-htdo, so the statement makes sense. The claim now follows by
combining Lemma 8.5 and Corollary 9.5. ∎

As for the $r$-deformed $G$-htdo’s, we may prove that a $G$-equivariant module
over an $r$-deformed $G$-htdo module satisfies a cocycle condition. Denote the
$G$-action on $X$ by $\sigma_{X}:G\times X\to X$. Further, we denote
$p_{X}:G\times X\to X$ and $p_{2X}:G\times G\times X\to X$ the projections on
the $X$ factor, $p_{23X}:G\times G\times X\to G\times X$ the projection onto
the second and third factor and $m:G\times G\to G$ the multiplication of the
group $G$. Let $\mathcal{D}$ be an $r$-deformed $G$-htdo and $\mathcal{M}$ a
$G$-equivariant $\mathcal{D}$-module. Then there exists
$\alpha:\sigma_{X}^{\\#}\mathcal{M}\to p_{X}^{\\#}\mathcal{M}$ an isomorphism
of $p_{X}^{\\#}\mathcal{D}$-modules such that the diagram 22 commutes, where
we replace $\mathcal{D}$ by $\mathcal{M}$. Again, we note that the condition
is similar to the one in [14, Section 5.2.9].

##  10 Equivariant descent for Lie algebroids and homogeneous twisted
differential operators

Throughout this section, we let $X,Y$ be $R$-varieties, $G$ a smooth affine
algebraic group of finite type acting _freely_ on $Y$. Further, we fix $f:Y\to
X$ a locally trivial $G$-torsor and $(\mathcal{L},\rho,i_{\mathfrak{g}})$ an
$r$-deformed $G$-equivariant Picard algebroid on $Y$. Let
$\alpha:\mathfrak{g}\to\mathcal{T}_{Y}$ be the derivative of the $G$-action on
$Y$. Recall that by Definition 6.4, $\alpha=\rho\circ i_{\mathfrak{g}}$ as
maps from $r\mathfrak{g}$ to $\mathcal{T}_{X}$. Throughout this section, we
will use without further specifying that all the $R$-modules appearing have no
$r$-torsion.

Let $\widetilde{\mathcal{T}}_{X}:=(f_{*}\mathcal{T}_{Y})^{G}$ and let
$\sigma:\widetilde{\mathcal{T}}_{X}\to\mathcal{T}_{X}$ denote the anchor map.
Further, we denote
$\widetilde{\mathfrak{g}_{X}}:=(f_{*}\mathcal{O}_{Y}\otimes\mathfrak{g})^{G}$,
where $G$ acts on $\mathfrak{g}$ via the Adjoint action. We let
$\tilde{\alpha}$ the induced map
$\widetilde{\mathfrak{g}_{X}}\to\widetilde{\mathcal{T}}_{X}$. Since
$f_{*}\mathcal{O}_{Y}\otimes\mathfrak{g}$ has no $r$-torsion, it is easy to
see that $r\widetilde{\mathfrak{g}_{X}}\cong(f_{*}\mathcal{O}_{Y}\otimes
r\mathfrak{g})^{G}$.

######  Lemma 10.1.

The maps $\tilde{\alpha}$ and $\sigma$ induce a short exact sequence

$0\to\widetilde{\mathfrak{g}_{X}}\xrightarrow{\tilde{\alpha}}\widetilde{\mathcal{T}}_{X}\xrightarrow{\sigma}\mathcal{T}_{X}\to
0.$

###### Proof.

The question is local, so we may assume that $X$ is affine $Y=G\times X$, $G$
acts on $Y$ via left multiplication on the first factor and $f$ is the
projection on the second factor. In that case, we have

$\begin{split}\widetilde{\mathfrak{g}_{X}}(X):&=(f_{*}\mathcal{O}_{Y}\otimes\mathfrak{g})^{G}(X)\\\
&=(\mathcal{O}_{Y}(Y)\otimes\mathfrak{g})^{G}\\\
&\cong(\mathcal{O}_{X}(X)\otimes\mathcal{O}_{G}(G)\otimes\mathfrak{g})^{G}\\\
&\cong\mathcal{O}_{X}(X)\otimes(\mathcal{O}_{G}(G)\otimes\mathfrak{g})^{G}\\\
&\cong\mathcal{O}_{X}(X)\otimes(\mathcal{T}(G))^{G}.\end{split}$ (31)

Further, we have by the proof of [1, Lemma 4.4] that
$\widetilde{\mathcal{T}}_{X}(X)\cong\mathcal{O}_{X}(X)\otimes\mathcal{T}(G)^{G}\oplus\mathcal{T}_{X}(X)$,
so the conclusion follows. ∎

By abuse of notation we denote $i_{\mathfrak{g}}$ the map
$i_{\mathfrak{g}}:\mathcal{O}_{Y}\otimes r\mathfrak{g}\to\mathcal{L}$ and
$\operatorname{\tilde{\textit{i}_{\mathfrak{g}}}}:r\widetilde{\mathfrak{g}_{X}}\to(f_{*}\mathcal{L})^{G}$
the induced map.

######  Definition 10.2.

Let $f_{\\#}\mathcal{L}^{G}$ be the $\mathcal{O}_{X}$-module
$(f_{*}\mathcal{L})^{G}/\operatorname{\tilde{\textit{i}_{\mathfrak{g}}}}(r\widetilde{\mathfrak{g}_{X}})$.
We call this the descent of $\mathcal{L}$.

######  Lemma 10.3.

The $\mathcal{O}_{X}$-module $f_{\\#}\mathcal{L}^{G}$ has the structure of an
$r$-deformed Picard algebroid.

###### Proof.

The bracket structure on $\widetilde{\mathcal{L}}:=(f_{*}\mathcal{L})^{G}$ is
induced from the bracket structure on $\mathcal{L}$; furthermore this descends
to a bracket structure on $f_{\\#}\mathcal{L}^{G}$ by setting
$[a+\operatorname{\tilde{\textit{i}_{\mathfrak{g}}}}(r\widetilde{\mathfrak{g}_{X}}),b+\operatorname{\tilde{\textit{i}_{\mathfrak{g}}}}(r\widetilde{\mathfrak{g}_{X}})]=[a,b]+\operatorname{\tilde{\textit{i}_{\mathfrak{g}}}}(r\widetilde{\mathfrak{g}_{X}})$
for $a,b\in(f_{*}\mathcal{L})^{G}$. Since the image of $i_{\mathfrak{g}}$ is
an ideal in $\mathcal{L}$, we obtain that the image of
$\operatorname{\tilde{\textit{i}_{\mathfrak{g}}}}$ is an ideal in
$(f_{*}\mathcal{L})^{G}$, thus the bracket is well defined. Therefore, we are
left to construct an anchor map.

Consider the short exact sequence
$0\to\mathcal{O}_{Y}\to\mathcal{L}\xrightarrow{\rho}r\mathcal{T}_{Y}\to 0$.
Applying Proposition 3.4, we obtain a short exact sequence

$0\to\mathcal{O}_{X}\to\widetilde{\mathcal{L}}\xrightarrow{\tilde{\rho}}r\widetilde{\mathcal{T}}_{X}\to
0.$

By construction, we have
$\tilde{\alpha}=\tilde{\rho}\circ\operatorname{\tilde{\textit{i}_{\mathfrak{g}}}}$,
so the map

$\bar{\widetilde{\rho}}:\widetilde{\mathcal{L}}/\operatorname{\tilde{\textit{i}_{\mathfrak{g}}}}(r\widetilde{\mathfrak{g}_{X}})\to\widetilde{\mathcal{T}}_{X}/\tilde{\alpha}(r\widetilde{\mathfrak{g}_{X}})$

is well defined. Furthermore, since there is no $r$-torsion, we have by Lemma
10.1 an induced isomorphism
$\bar{\sigma}:r\widetilde{\mathcal{T}}_{X}/r\tilde{\alpha}(\widetilde{\mathfrak{g}_{X}})\to
r\mathcal{T}_{X}$. We define the anchor map on
$\widetilde{\mathcal{L}}/\operatorname{\tilde{\textit{i}_{\mathfrak{g}}}}(r\widetilde{\mathfrak{g}_{X}})$
to be $\rho_{X}:=\bar{\sigma}\circ\bar{\widetilde{\rho}}$ and it is clear that
together with the bracket
$\widetilde{\mathcal{L}}/\operatorname{\tilde{\textit{i}_{\mathfrak{g}}}}(r\widetilde{\mathfrak{g}_{X}})$
becomes a Lie algebroid, so it remains to show that it is an $r$-deformed
Picard algebroid. It is easy to see that
$\operatorname{im\text{\kern-2.0pt}\,}\rho_{X}=r\mathcal{T}_{X}$, so it is
enough to prove that
$\ker(\rho_{X})\cong\ker\widetilde{\rho}\cong\mathcal{O}_{X}$.

Since $\bar{\sigma}$ is injective we have that

$\ker{\rho_{X}}=\ker(\bar{\tilde{\rho}})=(\ker\tilde{\rho}+\operatorname{\tilde{\textit{i}_{\mathfrak{g}}}}(r\widetilde{\mathfrak{g}_{X}}))/\operatorname{\tilde{\textit{i}_{\mathfrak{g}}}}(r\widetilde{\mathfrak{g}_{X}})\cong\ker\tilde{\rho}/\ker\tilde{\rho}\cap\operatorname{\tilde{\textit{i}_{\mathfrak{g}}}}(r\widetilde{\mathfrak{g}_{X}})$

by the second isomorphism theorem. Thus, it is enough to prove
$\ker\tilde{\rho}\cap\operatorname{\tilde{\textit{i}_{\mathfrak{g}}}}(r\widetilde{\mathfrak{g}_{X}})=0$
and since
$\tilde{\alpha}=\tilde{\rho}\circ\operatorname{\tilde{\textit{i}_{\mathfrak{g}}}}$,
this reduces to proving $\ker\tilde{\alpha}=0$. By assumptions, the action of
$G$ on $Y$ is free; thus the map $\alpha:\mathfrak{g}\to\mathcal{T}_{Y}$ is
injective and since $\mathcal{O}_{Y}$ is a faithfully flat $R$-module, the
induced map $\alpha:\mathcal{O}_{Y}\otimes r\mathfrak{g}\to r\mathcal{T}_{Y}$
injective and thus so is $\tilde{\alpha}$. ∎

######  Lemma 10.4.

Let $\mathcal{L}$ as before. Then
$f^{\\#}(f_{\\#}\mathcal{L}^{G})\cong\mathcal{L}$.

###### Proof.

Recall that by abuse of notation, we denote
$i_{\mathfrak{g}}:\mathcal{O}_{Y}\otimes r\mathfrak{g}\to\mathcal{L}$ and
$\alpha:\mathcal{O}_{Y}\otimes r\mathfrak{g}\to\mathcal{T}_{Y}$ the induced
maps; we still have $\alpha=\rho\circ i_{\mathfrak{g}}$. Let
$\widetilde{\mathcal{L}}:=(f_{*}\mathcal{L})^{G}$ so that there is a short
exact sequence

$0\to(f_{*}\mathcal{O}_{Y}\otimes
r\mathfrak{g})^{G}\to\widetilde{\mathcal{L}}\to f_{\\#}\mathcal{L}^{G}\to 0$

Pulling back under the torsor $f$ and applying Proposition 3.4, we obtain a
short exact sequence

$0\to\mathcal{O}_{Y}\otimes
r\mathfrak{g}\xrightarrow{i_{\mathfrak{g}}}\mathcal{L}\to
f^{*}(f_{\\#}\mathcal{L}^{G})\to 0,$

so,
$f^{*}(f_{\\#}\mathcal{L}^{G})\cong\mathcal{L}/i_{\mathfrak{g}}(\mathcal{O}_{Y}\otimes
r\mathfrak{g}).$

Similarly, by pulling back under $f$ the short exact sequence in Lemma 10.1
and taking into account there is no $r$-torsion we obtain a short exact
sequence

$0\to\mathcal{O}_{Y}\otimes
r\mathfrak{g}\xrightarrow{\alpha}r\mathcal{T}_{Y}\to rf^{*}\mathcal{T}_{X}\to
0,$ (32)

so $rf^{*}\mathcal{T}_{X}\cong r\mathcal{T}_{Y}/\alpha(\mathcal{O}_{Y}\otimes
r\mathfrak{g}).$

Therefore, we obtain

$f^{\\#}(f_{\\#}\mathcal{L}^{G})\cong
f^{*}(f_{\\#}\mathcal{L}^{G})\times_{rf^{*}\mathcal{T}_{X}}r\mathcal{T}_{Y}\cong\mathcal{L}/i_{\mathfrak{g}}(\mathcal{O}_{Y}\otimes
r\mathfrak{g})\times_{r\mathcal{T}_{Y}/\alpha(\mathcal{O}_{Y}\otimes
r\mathfrak{g})}r\mathcal{T}_{Y}.$

Define $\varphi:\mathcal{L}\to f^{\\#}(f_{\\#}\mathcal{L}^{G})$ by
$\varphi(l)=(l+i_{\mathfrak{g}}(\mathcal{O}_{Y}\otimes
r\mathfrak{g}),\rho(l))$ for any $l\in\mathcal{L}$.

To see that $\varphi$ is injective, we use that $\alpha$ is an injective map,
thus so is $i_{\mathfrak{g}}$, therefore
$i_{\mathfrak{g}}(\mathcal{O}_{Y}\otimes r\mathfrak{g})\cap\ker(\rho)=0$.

Finally, let $(l+i_{\mathfrak{g}}(\mathcal{O}_{Y}\otimes
r\mathfrak{g}),x)\in\mathcal{L}/i_{\mathfrak{g}}(\mathcal{O}_{Y}\otimes
r\mathfrak{g})\times_{r\mathcal{T}_{Y}/\alpha(\mathcal{O}_{Y}\otimes
r\mathfrak{g})}r\mathcal{T}_{Y},$ so that
$\rho(l)+\alpha(\mathcal{O}_{Y}\otimes
r\mathfrak{g})=x+\alpha(\mathcal{O}_{Y}\otimes r\mathfrak{g})$, thus
$x=\rho(l)+i$ for some $i\in\alpha(\mathcal{O}_{Y}\otimes r\mathfrak{g})$.
Since $\alpha$ is injective there exist a unique $j\in\mathcal{O}_{Y}\otimes
r\mathfrak{g}$, so that $\alpha(j)=i$. Thus
$\varphi(l+i_{\mathfrak{g}}(j))=(l+i_{\mathfrak{g}}(\mathcal{O}_{Y}\otimes
r\mathfrak{g}),x),$ so $\varphi$ is surjective. ∎

######  Lemma 10.5.

Let $\mathcal{P}$ be an $r$-deformed Picard algebroid on $X$. Then
$f_{\\#}(f^{\\#}\mathcal{P})^{G}\cong\mathcal{P}$.

###### Proof.

We may view $\mathcal{P}$ as an $r$-deformed $G$-equivariant Picard algebroid
by letting $g.p=p$ for all $g\in G,p\in P$ and the map
$r\mathfrak{g}\to\mathcal{P}$ to be the $0$ map. Then by Lemma 7.4,
$f^{\\#}\mathcal{P}$ is an $r$-deformed $G$-equivariant Picard algebroid. The
induced map $i_{\mathfrak{g}}:r\mathfrak{g}\to
f^{*}\mathcal{P}\times_{rf^{*}\mathcal{T}_{X}}r\mathcal{T}_{Y}$ is defined as
$i_{\mathfrak{g}}(\psi)=(0,\alpha(\psi))$.

Using Proposition 3.4 for $\mathcal{O}$-modules together with the fact that
$(f_{*})^{G}$ commutes with limits we get:

$\begin{split}(f_{*}f^{\\#}\mathcal{P})^{G}&=(f_{*}(f^{*}\mathcal{P}\times_{rf^{*}\mathcal{T}_{X}}r\mathcal{T}_{Y}))^{G}\\\
&\cong(f_{*}f^{*}\mathcal{P})^{G}\times_{r(f_{*}f^{*}\mathcal{T}_{X})^{G}}(rf_{*}\mathcal{T}_{Y})^{G}\\\
&\cong\mathcal{P}\times_{r\mathcal{T}_{X}}r\widetilde{\mathcal{T}}_{X}.\end{split}$
(33)

Therefore, we obtain:

$\begin{split}f_{\\#}(f^{\\#}\mathcal{P})^{G}&\cong(f_{*}f^{\\#}\mathcal{P})^{G}/i_{\mathfrak{g}}(r\widetilde{\mathfrak{g}_{X}})\\\
&\cong(\mathcal{P}\times_{r\mathcal{T}_{X}}r\widetilde{\mathcal{T}}_{X})/i_{\mathfrak{g}}(r\widetilde{\mathfrak{g}_{X}})\\\
&\cong\mathcal{P}\times_{r\mathcal{T}_{X}}r\mathcal{T}_{X}\text{ (by Lemma
\ref{d1sesliealgeqdes})}.\\\ &\cong\mathcal{P}.\qed\end{split}$ (34)

######  Corollary 10.6.

The functors $f_{\\#}(-)^{G}$ and $f^{\\#}(-)$ induce quasi-inverse
equivalences of categories between the category of $r$-deformed
$G$-equivariant Picard algebroids on $Y$ and the category of $r$-deformed
Picard algebroids on $X$.

###### Proof.

The proof follows from Lemmas 10.4 and 10.5. ∎

We consider again a more general setting: let $B$ another smooth affine
algebraic group acting on $Y$ and $X$ such that $f:Y\to X$ is $B$-equivariant.
Recall that we are assuming that $f$ is a locally trivial $G$-torsor.

######  Lemma 10.7.

With the assumption as above we have:

  * •

let $(\mathcal{L},\rho,i_{\mathfrak{g}\times\mathfrak{b}})$ be an $r$-deformed
$G\times B$-equivariant Picard algebroid on $Y$. Then $f_{\\#}\mathcal{L}^{G}$
may be given the structure of an $r$-deformed $B$-equivariant Picard algebroid
on X.

  * •

let $(\mathcal{K},\rho_{X},j_{\mathfrak{b}})$ be an $r$-deformed
$B$-equivariant Picard algebroid on $X$. Then $f^{\\#}\mathcal{K}$ may be
given the structure of an $r$-deformed $G\times B$-equivariant Picard
algebroid on $Y$.

###### Proof.

For the first claim, we start by proving that
$\widetilde{\mathcal{L}}=(f_{*}\mathcal{L})^{G}$ is a $B$-equivariant Lie
algebroid. First, we get a $B$-action on $\widetilde{\mathcal{L}}$ from Lemma
3.6, so that $\widetilde{\mathcal{L}}$ becomes a $B$-equivariant
$\mathcal{O}_{X}$-module. Further, since $\mathcal{L}$ is $B$-equivariant,
axiom $i)$ of definition 6.1 is also satisfied.

Next, since the actions of $G$ and $B$ on $Y$ commute the anchor map
$\sigma:\widetilde{\mathcal{T}}_{X}\to\mathcal{T}_{X}$ is $B$-equivariant.
Composing with the $B$-equivariant map
$\tilde{\rho}:\widetilde{\mathcal{L}}\to\widetilde{\mathcal{T}}_{X}$ we obtain
a $B$-equivariant map
$\sigma\circ\tilde{\rho}:\widetilde{\mathcal{L}}\to\mathcal{T}_{X}$, so axiom
$ii)$ of 6.1 is satisfied.

We let $i_{\mathfrak{g}}$ and $i_{\mathfrak{b}}$ denote the restriction of
$i_{\mathfrak{g}\times\mathfrak{b}}$ to $r\mathfrak{g}\cong
r\mathfrak{g}\times 0\subset r(\mathfrak{g}\times\mathfrak{b})$ and
$r\mathfrak{b}\cong 0\times r\mathfrak{b}\subset
r(\mathfrak{g}\times\mathfrak{b})$, respectively. Let
$\beta:\mathfrak{b}:\to\mathcal{T}_{Y}$,
$\gamma:\mathfrak{b}\to\mathcal{T}_{X}$ denote the infinitesimal action $B$ on
$Y$ and $X$; since $\mathcal{L}$ is in particular $B$-equivariant, we have
$\beta=\rho\circ i_{\mathfrak{b}}$. By descending we obtain maps
$\tilde{\beta}:r\mathfrak{b}\to\widetilde{\mathcal{T}}_{X}$ and
$\tilde{i_{\mathfrak{b}}}:r\mathfrak{b}\to\widetilde{\mathcal{L}}$ such that
$\tilde{\beta}=\tilde{\rho}\circ\tilde{i_{\mathfrak{b}}}$. Since the actions
of $G$ and $B$ commute, we also get $\gamma=\sigma\circ\tilde{\beta}$ so
$\gamma=\sigma\circ\tilde{\beta}$. Therefore combining this two we get:

$\gamma=(\sigma\circ\tilde{\rho})\circ\tilde{i_{\mathfrak{b}}}.$

Therefore we get, since $\mathcal{L}$ is $B$-equivariant, that the Lie
algebroid
$(\widetilde{\mathcal{L}},\sigma\circ\tilde{\rho},\tilde{i_{\mathfrak{b}}})$
is also $B$-equivariant. By the proof of Lemma 10.3, to show that
$f_{\\#}\mathcal{L}^{G}$ is $B$-equivariant it suffices to prove that
$i_{\mathfrak{g}}(r\widetilde{\mathfrak{g}_{X}})$ is $B$-equivariant as an
$\mathcal{O}_{X}$-module. Further, by using Lemma 3.6 this is equivalent to
proving $i_{\mathfrak{g}}(\mathcal{O}_{Y}\otimes r\mathfrak{g})$ is a
$B$-equivariant $\mathcal{O}_{Y}$-module. This is true since
$i_{\mathfrak{g}\times\mathfrak{b}}$ is in particular $B$-equivariant, so
$i_{\mathfrak{g}}$ sends a $B$-equivariant module to a $B$-equivariant module.

Now, we prove the second claim. We may endow $\mathcal{K}$ with a trivial
$G$-action $g.k=k$ for all $k\in\mathcal{K}$ and with the zero map
$j_{\mathfrak{g}}\to\mathcal{K}$ so that
$(\mathcal{K},\rho_{X},j_{\mathfrak{g}}\times j_{\mathfrak{b}})$ becomes
$G\times B$-equivariant. The claim follows from Lemma 7.4. ∎

######  Corollary 10.8.

The functors $f_{\\#}(-)^{G}$ and $f^{\\#}(-)$ induce quasi-inverse
equivalences of categories between the category of $r$-deformed $G\times
B$-equivariant Picard algebroids on $Y$ and the category of $r$-deformed
$B$-equivariant Picard algebroids on $X$.

###### Proof.

This follows from Corollary 10.6 and Lemma 10.7. ∎

Descent of twisted differential operators

We keep the notations from the start of the section. Further, we let
$\mathcal{D}$ be a sheaf of $r$-deformed $G$-homogeneous twisted differential
operators on $Y$.

######  Definition 10.9.

We define the descent of $\mathcal{D}$ under the torsor $f:Y\to X$ to be the
sheaf

$f_{\\#}\mathcal{D}^{G}:=\mathscr{T}(f_{\\#}(\operatorname{Lie\text{\kern-2.0pt}\,}\mathcal{D})^{G}).$

######  Lemma 10.10.

Let the notations be as above. Then:

  1. i)

$f_{\\#}\mathcal{D}^{G}$ is an $r$-deformed tdo on $X$.

  2. ii)

$f^{\\#}(f_{\\#}\mathcal{D}^{G})\cong\mathcal{D}$.

###### Proof.

The first claim follows from Lemma 10.3 and Corollary 5.7. The second claim
follows from Lemma 10.4 and Corollary 6.8. ∎

######  Lemma 10.11.

Let $\mathcal{A}$ be an $r$-deformed tdo on $X$. Then
$(f_{\\#}f^{\\#}\mathcal{A})^{G}\cong\mathcal{A}$.

###### Proof.

We view $\mathcal{A}$ as an $r$-deformed $G$-equivariant htdo on $X$ with the
trivial $G$-action, so that
$\operatorname{Lie\text{\kern-2.0pt}\,}(\mathcal{A})$ is an $r$-deformed
$G$-equivariant Picard algebroid with the trivial $G$-action. The claim
follows from Lemma 10.5 and Corollary 5.7. ∎

Similarly to the deformed Picard algebroids case we obtain:

######  Corollary 10.12.

The functors $f_{\\#}(-)^{G}$ and $f^{\\#}(-)$ induce quasi-inverse
equivalences between the category of $r$-deformed $G$-equivariant htdo’s on
$Y$ and the category of $r$-deformed tdo’s on $X$.

###### Proof.

This follows from the Lemmas 10.10 and 10.11. ∎

######  Corollary 10.13.

Assume that $f$ is also $B$-equivariant. The functors $f_{\\#}(-)^{G}$ and
$f^{\\#}(-)$ induce quasi-inverse equivalences between the category
$r$-deformed $G\times B$-equivariant htdo’s on $Y$ and the category
$r$-deformed $B$-equivariant htdo’s on $X$.

###### Proof.

This follows from the previous corollary, along with Lemma 10.7 and Corollary
6.8. ∎

##  11 Equivariant descent for equivariant Picard algebroids and htdo’s
modules

We keep the notations from the previous section: recall that $G$ is a smooth
affine algebraic group of finite type, $X$ and $Y$ are $R$-varieties, and
$f:Y\to X$ is a locally trivial $G$-torsor. Further, we assume that
$(\mathcal{L},\rho,i_{\mathfrak{g}}$ is an $r$-deformed $G$-equivariant Picard
algebroid on $Y$.

######  Lemma 11.1.

Let $\mathcal{M}$ be a $G$-equivariant Picard $\mathcal{L}$-module. Then
$(f_{*}\mathcal{M})^{G}$ is a Picard $f_{\\#}\mathcal{L}^{G}$-module. We call
$(f_{*}\mathcal{M})^{G}$ the _descent_ of $\mathcal{M}$.

###### Proof.

Let $\widetilde{\mathcal{L}}=(f_{*}\mathcal{L})^{G}$. Then
$(f_{*}\mathcal{M})^{G}$ is a Picard $\widetilde{\mathcal{L}}$-module, so it
remains to prove that the action of
$i_{\mathfrak{g}}(r\widetilde{\mathfrak{g}_{X}})$ kills
$(f_{*}\mathcal{M})^{G}$.

Since the $G$ action on $(f_{*}\mathcal{M})^{G}$ is constant, by
differentiating this action we obtain a trivial $\mathfrak{g}$ action, so a
trivial $r\mathfrak{g}$ action. But by our assumption on $\mathcal{M}$ this
coincides with the $r\mathfrak{g}$ action induced from
$i_{\mathfrak{g}}:r\mathfrak{g}\to\mathcal{L}$; the conclusion follows. ∎

######  Proposition 11.2.

Let $\mathcal{L}$ be an $r$-deformed $G$-equivariant Picard algebroid on $Y$.
The functors

$\begin{split}&f_{*}(-)^{G}:\operatorname{Mod\text{\kern-2.0pt}\,}(\mathcal{L},G)\to\operatorname{Mod\text{\kern-2.0pt}\,}(f_{\\#}\mathcal{L}^{G}),\\\
&f^{\\#}(-):\operatorname{Mod\text{\kern-2.0pt}\,}(f_{\\#}\mathcal{L}^{G})\to\operatorname{Mod\text{\kern-2.0pt}\,}(\mathcal{L},G),\end{split}$
(35)

are quasi-inverse equivalences of categories.

###### Proof.

This follows from the Lemma above, Lemma 10.4 and Proposition 3.4. ∎

######  Corollary 11.3.

Let $\mathcal{D}$ be an $r$-deformed $G$-htdo on $Y$. The functors

$\begin{split}&f_{*}(-)^{G}:\operatorname{Mod\text{\kern-2.0pt}\,}(\mathcal{D},G)\to\operatorname{Mod\text{\kern-2.0pt}\,}(f_{\\#}\mathcal{D}^{G}),\\\
&f^{\\#}(-):\operatorname{Mod\text{\kern-2.0pt}\,}(f_{\\#}\mathcal{D}^{G})\to\operatorname{Mod\text{\kern-2.0pt}\,}(\mathcal{D},G),\end{split}$
(36)

are quasi-inverse equivalences of categories.

###### Proof.

This follows from the Proposition above and Corollaries 9.2 and 9.5. ∎

Let $B$ is another smooth affine algebraic group acting on $Y$ such that the
actions of $G$ and $B$ commute and $f:Y\to X$ is $B$-equivariant. Let
$\mathcal{D}$ be an $r$-deformed $G\times B$-htdo. Similar to the
$\mathcal{O}$-module case (Corollary 3.7), we obtain using Corollary 11.3:

######  Corollary 11.4.

Let $\mathcal{D}$ be an $r$-deformed $G\times B$-htdo on $Y$ and assume $f$ is
$B$-equivariant. The functors

$\begin{split}&f_{*}(-)^{G}:\operatorname{Mod\text{\kern-2.0pt}\,}(\mathcal{D},G\times
B)\to\operatorname{Mod\text{\kern-2.0pt}\,}(f_{\\#}\mathcal{D}^{G},B),\\\
&f^{\\#}(-):\operatorname{Mod\text{\kern-2.0pt}\,}(f_{\\#}\mathcal{D}^{G},B)\to\operatorname{Mod\text{\kern-2.0pt}\,}(\mathcal{D},G\times
B),\end{split}$ (37)

are quasi-inverse equivalences of categories.

To prove the main theorem of the paper, we need two easy results. Recall that
for an $R$-variety $Z$ we denote $\mathcal{D}_{Z}$ the sheaf of crystalline
differential operators on $Z$. For $r\in R$ regular element, we call
$\mathcal{D}_{Z,r}=\mathscr{T}(\mathcal{O}_{Z}\oplus r\mathcal{T}_{Z})$ the
sheaf of $r$-deformed differential operators and
$\mathcal{D}(Z)_{r}=\Gamma(Z,\mathcal{D}_{Z,r})$. It is clear by construction
that $\mathcal{D}_{Z,r}$ is an $r$-deformed tdo. We also make the following
assumption:

######  Assumption 11.5.

The base ring $R$ is Noetherian.

######  Lemma 11.6.

Let $\mathcal{M}$ be a $\mathcal{D}_{X,r}$-module. Then $f^{*}\mathcal{M}$ is
a coherent $\mathcal{D}_{Y,r}$-module if and only $\mathcal{M}$ is a coherent
$\mathcal{D}_{X,r}$-module.

###### Proof.

Since coherence is a local problem, we may assume that $X$ is affine,
$Y=G\times X$, $f$ is the projection onto the second factor and $G$ acts on
$Y$ via left multiplication on the first factor. Let $M$ be a
$\mathcal{D}(X)_{r}$ module, so $f^{*}M\cong\mathcal{O}(G)\otimes M$.

We may view $D(Y)_{r}$ as an $r$-deformation of the ring $D(Y)$. Since
$D(Y)\cong D(G)\otimes D(X)$ and deformations commutes with tensor product, we
obtain that the ring
$\mathcal{D}(Y)_{r}\cong\mathcal{D}(G)_{r}\otimes\mathcal{D}(X)_{r}$ acts
naturally on $f^{*}M$. Further, since our base ring $R$ is Noetherian we have
by Lemma 4.6 that $D(Y)_{r}$ and $D(X)_{r}$ are Noetherian rings, so coherence
is equivalent to local finite generation. Thus it is enough to prove the
following claim:

Claim. $M$ is a finitely generated $\mathcal{D}(X)_{r}$ module if and only if
$\mathcal{O}(G)\otimes M$ is a finitely generated
$\mathcal{D}(G)_{r}\otimes\mathcal{D}(X)_{r}$ module.

The direct implication is trivial. For the reverse implication, we will prove
the contrapositive: if $M$ is not Noetherian, neither is
$\mathcal{O}(G)\otimes M$. Let $M_{1}\subsetneq M_{2}\subsetneq
M_{3}\ldots\subsetneq$ be an infinite ascending of $M$-submodules. Since $G$
is a smooth group, $\mathcal{O}(G)$ is a faithfully flat $R$-module, thus
tensoring with $\mathcal{O}(G)$ preserves injections. In particular, we obtain
an infinite chain

$\mathcal{O}(G)\otimes M_{1}\subsetneq\mathcal{O}(G)\otimes
M_{2}\subsetneq\mathcal{O}(G)\otimes M_{3}\ldots\subsetneq$

of submodules of $\mathcal{O}(G)\otimes M$. Therefore, the claim is proven. ∎

######  Corollary 11.7.

Let $\mathcal{D}$ be an $r$-deformed $G$-htdo on $Y$ and let $\mathcal{M}$ be
a $(f_{\\#}\mathcal{D})^{G}$-module. Then $f^{\\#}\mathcal{M}$ is a coherent
$\mathcal{D}$-module if and only if $\mathcal{M}$ is coherent.

###### Proof.

The claim is local, thus we may assume that $\mathcal{D}=\mathcal{D}_{Y,r}$ so
that $(f_{\\#}\mathcal{D})^{G}=\mathcal{D}_{X,r}$. The claim follows by the
Lemma above and Lemma 10.10. ∎

We may now prove the main theorem.

######  Theorem 11.8.

Assume the base ring $R$ is Noetherian. Let $G$ be a smooth affine algebraic
group of finite type. Let $X,Y$ be $R$-varieties and let $f:Y\to X$ be a
locally trivial $G$-torsor. Further, let $\mathcal{D}$ be a sheaf of
$r$-deformed $G$-homogeneous twisted differential operators on $Y$. The
functors:

$\begin{split}&f_{*}(-)^{G}:\operatorname{Coh\text{\kern-2.0pt}\,}(\mathcal{D},G)\to\operatorname{Coh\text{\kern-2.0pt}\,}(f_{\\#}\mathcal{D}^{G}),\\\
&f^{\\#}(-):\operatorname{Coh\text{\kern-2.0pt}\,}(f_{\\#}\mathcal{D}^{G})\to\operatorname{Coh\text{\kern-2.0pt}\,}(\mathcal{D},G),\end{split}$
(38)

are quasi-inverse equivalences of categories between coherent $G$-equivariant
$\mathcal{D}$-modules and coherent $(f_{\\#}\mathcal{D})^{G}$-modules.

###### Proof.

This follows from Corollaries 11.3 and 11.7. ∎

As a corollary, we obtain using Corollary 11.4:

######  Corollary 11.9.

Let the assumptions be as above. Further, assume that $B$ is a smooth affine
algebraic group of finite type such that $f:Y\to X$ is $B$-equivariant.
Further assume that $\mathcal{D}$ is an $r$-deformed $G\times B$-htdo on $Y$.
The functors:

$\begin{split}&f_{*}(-)^{G}:\operatorname{Coh\text{\kern-2.0pt}\,}(\mathcal{D},G\times
B)\to\operatorname{Coh\text{\kern-2.0pt}\,}(f_{\\#}\mathcal{D}^{G},B),\\\
&f^{\\#}(-):\operatorname{Coh\text{\kern-2.0pt}\,}(f_{\\#}\mathcal{D}^{G},B)\to\operatorname{Coh\text{\kern-2.0pt}\,}(\mathcal{D},G\times
B),\end{split}$ (39)

are quasi-inverse equivalences of categories.

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