# On stacky surfaces and noncommutative surfaces

Eleonore Faber  School of Mathematics, University of Leeds, LS2 9JT Leeds,
United Kingdom<EMAIL_ADDRESS>,  Colin Ingalls  School of Mathematics
and Statistics, Carleton University, Ottawa, ON K1S 5B6, Canada
<EMAIL_ADDRESS>,  Shinnosuke Okawa  Department of Mathematics,
Graduate School of Science, Osaka University, Machikaneyama 1-1, Toyonaka,
Osaka 560-0043, Japan<EMAIL_ADDRESS>and  Matthew Satriano
Department of Pure Mathematics, University of Waterloo, Canada
<EMAIL_ADDRESS>
###### Abstract.

Let $\mathbf{k}$ be an algebraically closed field of characteristic $\geq 7$
or zero. Let $\mathcal{A}$ be a tame order of global dimension $2$ over a
normal surface $X$ over $\mathbf{k}$ such that
$\operatorname{Z}(\mathcal{A})=\mathcal{O}_{X}$ is locally a direct summand of
$\mathcal{A}$. We prove that there is a $\mu_{N}$-gerbe $\mathcal{X}$ over a
smooth tame algebraic stack whose generic stabilizer is trivial, with coarse
space $X$ such that the category of 1-twisted coherent sheaves on
$\mathcal{X}$ is equivalent to the category of coherent sheaves of modules on
$\mathcal{A}$. Moreover, the stack $\mathcal{X}$ is constructed explicitly
through a sequence of root stacks, canonical stacks, and gerbes. This extends
results of Reiten and Van den Bergh to finite characteristic and the global
situation. As applications, in characteristic $0$ we prove that such orders
are geometric noncommutative schemes in the sense of Orlov, and we study
relations with Hochschild cohomology and Connes’ convolution algebra.

Part of the work by EF was carried out at the Isaac Newton Institute in
Cambridge, supported by EPSRC grant EP/R014604/1.

CI was partially supported by a Discovery Grant from the National Science and
Engineering Research Council of Canada

SO was partially supported by JSPS Grants-in-Aid for Scientific Research
(18H01120, 19KK0348, 20H01797, 20H01794, 21H04994).

MS was partially supported by a Discovery Grant from the National Science and
Engineering Research Council of Canada and a Mathematics Faculty Research
Chair.

###### Contents

  1. 1 Introduction
  2. 2 Generalizing Reiten-Van den Bergh to characteristic $p\geq 7$
  3. 3 The root stack $(\mathcal{X}_{r},\mathcal{A}_{r})$
  4. 4 The canonical stack $\left(\mathcal{X}_{c},\mathcal{A}_{c}\right)$
  5. 5 The gerbe $\mathcal{X}_{g}$
  6. 6 Example: reducing to positive characteristic
  7. 7 The Centre
  8. 8 Applications and complements

##  1\. Introduction

Noncommutative algebraic geometry is a burgeoning field connecting
noncommutative algebra, representation theory, derived categories, and
algebraic geometry. Understanding the extent to which noncommutative objects
are determined by commutative ones is of central interest and is a theme which
underlies the noncommutative McKay correspondence. In particular, we are
interested in understanding when given orders $\mathcal{A}$ over schemes $X$,
we can find stacks $\mathcal{X}$ with
$\operatorname{Mod}\mathcal{A}\simeq\operatorname{qcoh}\mathcal{X}.$

This question can be modified by asking for $\operatorname{Mod}\mathcal{A}$ to
be a component of the category $\operatorname{qcoh}\mathcal{X}$, or a semi-
orthogonal component of the derived category
$\mathbf{D}^{\mathrm{b}}\operatorname{coh}\mathcal{X}$. The following two
examples are well known instances of this situation.

######  Example 1.1.

Let $G$ be a finite group and $R$ a commutative ring with $G$-action. Let
$R\ast G$ be the skew group ring and $\left[\operatorname{Spec}R/G\right]$ be
the quotient stack. It follows immediately from the definitions that we have
the following equivalence of categories

$\displaystyle\operatorname{Mod}(R\ast G)\simeq\mbox{($G$-equivariant
$R$-modules)}\simeq\operatorname{qcoh}\left[\operatorname{Spec}R/G\right].$
(1.1)

######  Example 1.2 (cf. Corollary 5.5).

Let $\mathcal{A}$ be an Azumaya algebra of order $N$ on a scheme $X$. Then we
can associate a $\mu_{N}$-gerbe $\mathcal{X}$ to $\mathcal{A}$ such that the
following equivalences of categories hold. Here,
$\operatorname{qcoh}^{(i)}\mathcal{X}$ denotes the category of $i$-twisted
quasi-coherent sheaves on $\mathcal{X}$.

$\displaystyle\operatorname{mod}\mathcal{A}^{\otimes
i}\simeq\operatorname{qcoh}^{(i)}\mathcal{X}$ (1.2)

$\displaystyle\bigoplus_{i=0}^{N-1}\operatorname{mod}\mathcal{A}^{\otimes
i}\simeq\operatorname{mod}\prod_{i=0}^{N-1}\mathcal{A}^{\otimes
i}\simeq\operatorname{qcoh}\mathcal{X}$ (1.3)

Inspired by work of Connes [Con94], Chan and the second author [CI04] showed
that to every stack $\mathcal{X}$ with a choice of finite flat presentation,
there is an associated noncommutative coordinate ring (the convolution
algebra); this ring may be thought of as a noncommutative scheme. Furthermore,
they showed that in the case of noncommutative _curves_ , their construction
is reversible: every noncommutative curve is the noncommutative coordinate
ring of a smooth Deligne–Mumford curve [CI04, Theorem 7.7]. Finding a groupoid
whose convolution algebra is a given algebra has been studied in the setting
of $C^{*}$-algebras, see for example [Kum86, LR19, Bic21].

It is desirable to have a converse to [CI04] beyond the curve case since it
allows one to import (stacky) geometric techniques in the study of
noncommutative rings. It is even further desirable to ensure that in such a
converse construction, the stack is Morita equivalent to the module category
of the noncommutative ring. We say that an additive category $\mathcal{C}$ is
a component of the additive category $\mathcal{D}$ if there is an additive
category $\mathcal{E}$ such that $\mathcal{D}$ is equivalent to the direct sum
$\mathcal{D}\simeq\mathcal{C}\oplus\mathcal{E}$. The following questions are
of interest:

######  Question 1.3.

Let $\mathcal{A}$ be a sheaf of algebras over a scheme $S$.

  * •

When does there exist a stack $\mathcal{X}$ such that
$\operatorname{mod}\mathcal{A}$ is a component of
$\operatorname{qcoh}\mathcal{X}$?

  * •

When does there exist a stack $\mathcal{X}$ with coarse moduli space
$\pi\colon\mathcal{X}\to S$ with a vector bundle $V$ on $\mathcal{X}$ such
that $\mathcal{A}\simeq\pi_{\ast}\mathop{{\mathcal{E}}nd}\nolimits(V)$?

We achieve both of these goals in the case of noncommutative surfaces. Our
result is much more involved than the case of noncommutative curves due to the
presence of singularities.

######  Definition 1.4.

Let $\mathbf{k}$ be an algebraically closed field of characteristic $0$ or
$p\geq 7$. We say the pair $(X,\mathcal{A})$ is a _smooth tame split order of
dimension $2$_ if the following hold. First, $X$ is normal, connected, quasi-
projective surface over $\mathbf{k}$ and $\mathcal{A}$ is a split tame order
on $X$ of $\operatorname{gl.dim}\mathcal{A}=2$; this implies $\mathcal{A}$ is
a reflexive coherent sheaf on $X$. Second,
$\operatorname{Z}(\mathcal{A})=\mathcal{O}_{X}$ and $\mathcal{O}_{X}$ is
locally a direct summand of $\mathcal{A}$, which holds for example if
$\operatorname{char}(\mathbf{k})\nmid\operatorname{rank}\mathcal{A}$.

The theorem below is our main result and is a combination of Theorems 4.4 and
4.5.

######  Theorem 1.5.

Let $\mathbf{k}$ be an algebraically closed field of characteristic $0$ or
$p\geq 7$ and let $(X,\mathcal{A})$ be a smooth, tame split order of dimension
$2$. Then there exists a smooth tame algebraic stack $\mathcal{X}$ with coarse
space $X$ and a Morita equivalence

$\displaystyle\operatorname{mod}\mathcal{A}\simeq\operatorname{coh}^{(1)}\mathcal{X}$
(1.4)

between the category of coherent right $\mathcal{A}$-modules and $1$-twisted
coherent sheaves on $\mathcal{X}$. Furthermore, if
$\mathcal{A}\otimes\mathbf{k}(X)$ is a matrix ring over the quotient field
$\mathbf{k}(X)$ of $X$, then $\mathcal{X}$ is generically a scheme and
$\operatorname{mod}\mathcal{A}\simeq\operatorname{coh}\mathcal{X}$.

The following corollary is an immediate application of Theorem 1.5. To the
best of the authors’ knowledge, this is the only known proof so far and the
problem is open if we allow the global dimension of $\mathcal{A}$ to be
greater than $2$.

######  Corollary 1.6.

Let $\mathcal{A}$ and $X$ be as in Theorem 1.5 and assume $X$ is projective.
Further assume that $\mathcal{X}$ is smooth Deligne–Mumford and generically
tame (this automatically holds in characteristic zero). Then the derived
category $\mathbf{D}^{\mathrm{b}}\operatorname{mod}\mathcal{A}$ is a
_geometric noncommutative scheme_ in the sense of [Orl16, Definition 4.3].
Namely, it has an admissible embedding into the derived category of a smooth
projective variety over $\mathbf{k}$.

Equation (1.4) implies that $\operatorname{mod}\mathcal{A}$, and hence its
derived category, is equivalent to a component of an orthogonal decomposition
of $\mathbf{D}^{\mathrm{b}}\operatorname{coh}\mathcal{X}$. On the other hand,
our hypotheses on $\mathcal{X}$ combined with [Kre09] tell us $\mathcal{X}$ is
a global quotient. Hence $\mathcal{X}$ is a smooth projective tame algebraic
stack (see [BLS16, Definition 2.5] for the definition of projective stacks),
so it is a geometric noncommutative scheme by [BLS16, Theorem 6.4]. Thus we
obtain Corollary 1.6.

As another immediate corollary of Theorem 1.5, we obtain an equivalence of
Hochschild cohomology by, say, [Kel03].

######  Corollary 1.7.

There is an equivalence of Hochschild cohomology

$\displaystyle\operatorname{HH}^{\bullet}(\operatorname{mod}\mathcal{A})\simeq\operatorname{HH}^{\bullet}(\operatorname{coh}^{(1)}\mathcal{X})$
(1.5)

The category $\operatorname{coh}\mathcal{X}$ admits an orthogonal
decomposition by the subcategories of ${\overline{m}}$-twisted sheaves for
${\overline{m}}\in\mathbb{Z}/N\mathbb{Z}$ when $\mathcal{X}$ is a
$\mu_{N}$-gerbe (see Corollary 5.5). In particular we have a product
decomposition of Hochschild cohomology as follows:

$\displaystyle\operatorname{HH}^{\bullet}\left(\mathcal{X}\right)=\prod_{{\overline{m}}\in\mathbb{Z}/N\mathbb{Z}}\operatorname{HH}^{\bullet}\left(\operatorname{coh}^{({\overline{m}})}\mathcal{X}\right)$
(1.6)

The decomposition (1.6) admits the following more concrete description based
on

$\displaystyle\operatorname{HH}^{\bullet}(\mathcal{X})=\operatorname{Hom}_{\mathcal{X}\times\mathcal{X}}^{\bullet}\left(\Delta_{\ast}\mathcal{O}_{\mathcal{X}},\Delta_{\ast}\mathcal{O}_{\mathcal{X}}\right),$
(1.7)

where $\Delta\colon\mathcal{X}\to\mathcal{X}\times\mathcal{X}$ is the diagonal
morphism. Note that $\mathcal{X}\times\mathcal{X}$ is a
$\mu_{N}\times\mu_{N}$-gerbe, and since $\mathcal{O}_{\mathcal{X}}$ is
0-twisted, the decomposition (5.6) of $\Delta_{\ast}\mathcal{O}_{\mathcal{X}}$
corresponding to the character group
$\widehat{\mu_{N}\times\mu_{N}}=\mathbb{Z}/N\mathbb{Z}\times\mathbb{Z}/N\mathbb{Z}$
is concentrated in the anti-diagonal part as follows:

$\displaystyle\Delta_{\ast}\mathcal{O}_{\mathcal{X}}=\bigoplus_{{\overline{m}}\in\mathbb{Z}/N\mathbb{Z}}\left(\Delta_{\ast}\mathcal{O}_{\mathcal{X}}\right)_{({\overline{m}},-{\overline{m}})}$
(1.8)

Thus we obtain the decomposition

$\displaystyle\operatorname{Hom}_{\mathcal{X}\times\mathcal{X}}^{\bullet}\left(\Delta_{\ast}\mathcal{O}_{\mathcal{X}},\Delta_{\ast}\mathcal{O}_{\mathcal{X}}\right)=\prod_{{\overline{m}}\in\mathbb{Z}/N\mathbb{Z}}\operatorname{Hom}_{\mathcal{X}\times\mathcal{X}}^{\bullet}\left(\left(\Delta_{\ast}\mathcal{O}_{\mathcal{X}}\right)_{({\overline{m}},-{\overline{m}})},\left(\Delta_{\ast}\mathcal{O}_{\mathcal{X}}\right)_{({\overline{m}},-{\overline{m}})}\right)$
(1.9)

and the ${\overline{m}}$-th component on the right hand side corresponds to
that of the right hand side of (1.6).

By definition
$\operatorname{coh}^{(1)}\mathcal{X}=\operatorname{coh}^{(\overline{1})}\mathcal{X}$,
so that via (1.5) the Hochschild cohomology of $\mathcal{A}$ is equivalent to
a direct summand of the Hochschild cohomology of $\mathcal{X}$. It is
conceivable that the deformations of $\mathcal{A}$ can be investigated by
means of the Hochschild cohomology of the gerbe $\mathcal{X}$ via (1.5), which
in turn is expected to admit a version of the Hochschild–Kostant–Rosenberg
isomorphism. Unfortunately, however, the HKR isomorphism for stacks is not
fully established in the literature and in particular not for our gerbe
$\mathcal{X}$ (see, say, [ACH19], where the HKR isomorphism is proved for
global quotients of smooth varieties by finite groups in characteristic $0$).

Our construction of $\mathcal{X}$ itself draws an interesting parallel between
noncommutative algebraic geometry and the “bottom-up” construction [GS17] due
to Geraschenko and the fourth author. Given a smooth tame Deligne–Mumford
stack $\mathcal{Y}$ with trivial generic stabilizer, one may consider the
branch divisor $D$ of the coarse space $\pi\colon\mathcal{Y}\to Y$. The
bottom-up construction [GS17, Theorem 1] proves that $D$ completely determines
$\mathcal{Y}$. Moreover, one may reconstruct $\mathcal{Y}$ from $D$ through an
iterated procedure of two simple operations known as root stacks [Cad07,
AGV08] and canonical stacks [Vis89]; specifically, $\mathcal{Y}$ is the
canonical stack over the root stack over the canonical stack of $Y$, where the
root stack is taken along $D$. It is important to note that not every $(Y,D)$
arises as the branch divisor of a smooth Deligne–Mumford stack, see [GS17,
Example 7].

In analogy with the branch divisor of the coarse space map
$\pi\colon\mathcal{Y}\to Y$, one may consider the branch divisor of a
noncommutative surface over its underlying commutative surface, i.e. given a
smooth tame split order $(X,\mathcal{A})$ of dimension $2$, one may consider
the ramification locus $D$ of $\mathcal{A}$. Our proof of Theorem 1.5 shows
that $D$ arises as the branch locus of a smooth tame stack $\mathcal{X}$.
Moreover, our construction of $\mathcal{X}$ follows the bottom-up procedure
described above and hence gives precise information about the geometric
structure of $\mathcal{X}$. Specifically, we construct morphisms of stacks and
orders

${\mathcal{A}_{c}}$${\mathcal{A}_{r}}$${\mathcal{A}}$${\mathcal{X}\coloneqq\mathcal{X}_{g}}$${\mathcal{X}_{c}}$${\mathcal{X}_{r}}$${X}$$\scriptstyle{g}$$\scriptstyle{c}$$\scriptstyle{r}$
(1.10)

which sequentially improve the singularities of $X$ and $\mathcal{A}$. The
morphisms and algebras satisfy the following properties:

  1. (1)

$\mathcal{X}_{r}$ is normal and $\mathcal{A}_{r}$ is a maximal order Azumaya
in codimension one such that

$\displaystyle\operatorname{mod}\mathcal{A}_{r}\simeq\operatorname{mod}\mathcal{A}.$
(1.11)

The map $r$ resolves the singularities of the pair $(X,D)$ in codimension one,
where $D$ is the $\mathbb{Q}$-divisor with standard coefficients determined by
$\mathcal{A}$.

  2. (2)

$\mathcal{X}_{c}$ is smooth and $\mathcal{A}_{c}$ is the reflexive hull of
$c^{\ast}\mathcal{A}_{r}$; it is an Azumaya algebra and satisfies

$\displaystyle\operatorname{mod}\mathcal{A}_{c}\simeq\operatorname{mod}\mathcal{A}_{r}.$
(1.12)

  3. (3)

$g\colon\mathcal{X}_{g}\to\mathcal{X}_{c}$ is the $\mu_{N}$-gerbe
corresponding to $\mathcal{A}_{c}$, where $[\mathcal{A}_{c}]\in
H^{2}_{\text{\'{e}t}}(\mathcal{X}_{c},\mu_{N})$. There is an equivalence of
categories

$\displaystyle\operatorname{coh}^{(1)}\mathcal{X}_{g}\simeq\operatorname{mod}\mathcal{A}_{c}.$
(1.13)

Our constructions described above are both global as well as positive
characteristic generalizations of those by Reiten and Van den Bergh [RV89,
Section 5]; the latter results are established in the case of complete local
rings of surfaces in characteristic zero. In the case considered in [RV89],
the extensions from $X$ to $\mathcal{X}_{r}$ and from $\mathcal{X}_{r}$ to
$\mathcal{X}_{c}$ are both Galois (when interpreted appropriately). In [RV89]
they prove that the composition of extensions is again Galois, which allows
them to directly show the Morita equivalence between $\mathcal{A}$ and
$\mathcal{A}_{c}$. On the contrary, in our situation the extensions are not
necessarily Galois but are quotients by finite linearly reductive group
schemes. In particular, it is not clear whether the composition of the
extensions is also such a quotient. In this paper we use the Galois theory of
Hopf algebras as explained in [Mon93], and instead of trying to combine the
extensions into one, we prove the Morita equivalence in two steps; namely, we
prove the local version of the equivalence between $\mathcal{A}$ and
$\mathcal{A}_{r}$ first in Corollary 2.18 and then (the local version of) the
equivalence between $\mathcal{A}_{r}$ and $\mathcal{A}_{c}$ in Corollary 2.27.

## Acknowledgments

This project was initiated at the American Institute of Mathematics workshop
“Noncommutative surfaces and Artin’s conjecture.” We thank the American
Institute of Mathematics for its warm hospitality. It is a pleasure to also
thank Pieter Belmans, Valery Lunts, Sid Mathur, and Michel Van den Bergh.
Lastly, we thank Daniel Chan for enlightening discussions concerning the
example in Section 6.

##  2\. Generalizing Reiten-Van den Bergh to characteristic $p\geq 7$

Our goal in this section is to generalize some of the results established in
[RV89] from characteristic $0$ to characteristic $p\geq 7$. We begin with some
preliminaries.

First, throughout this paper we consider _right_ modules unless otherwise
stated. For an algebra $\mathcal{A}$ (over a symmetric monoidal category), we
let $\mathcal{A}^{e}\coloneqq\mathcal{A}^{\mathrm{op}}\otimes\mathcal{A}$
denote the enveloping algebra. An $\mathcal{A}$-bimodule is a module over
$\mathcal{A}^{e}$. We let $\operatorname{Z}(\mathcal{A})$ denote the _centre_
of the algebra $\mathcal{A}$.

Next, Azumaya algebras will play an important role in this paper; a standard
reference for this material is [Mil80, Chapter 4]. We recall the following
definition.

######  Definition 2.1.

Let $R$ be a commutative local ring. An _Azumaya algebra_ over $R$ is an
$R$-algebra $A$ which is a free $R$-module of finite rank such that the
following natural homomorphism of $R$-algebras is an isomorphism.

$\displaystyle A\otimes_{R}A^{\mathrm{op}}\to\operatorname{End}_{R}(A);\quad
a\otimes b^{\mathrm{op}}\mapsto\left[x\mapsto axb\right]$ (2.1)

More generally, if $\mathcal{X}$ is an algebraic stack and $\mathcal{A}$ is a
coherent sheaf of $\mathcal{O}_{\mathcal{X}}$-algebras, we say $\mathcal{A}$
is a sheaf of Azumaya algebras if $\mathcal{A}$ is locally free as an
$\mathcal{O}_{\mathcal{X}}$-module and the following natural map, similar to
(2.1), is an isomorphism.

$\displaystyle\mathcal{A}\otimes_{\mathcal{O}_{\mathcal{X}}}\mathcal{A}^{\mathrm{op}}\to\mathop{{\mathcal{E}}nd}\nolimits_{\mathcal{O}_{\mathcal{X}}}\left(\mathcal{A}\right)$
(2.2)

######  Remark 2.2.

The Azumaya condition can checked locally in the following sense. Let
$\mathcal{A}$ be a sheaf of algebras over a stack $\mathcal{X}$, and let
$f\colon U\to\mathcal{X}$ be a flat surjective morphism of stacks; note that
$f$ is not assumed to be locally finitely presented. Then by faithfully flat
descent, $\mathcal{A}$ is a sheaf of Azumaya algebras if and only if the
pullback $f^{\ast}\mathcal{A}$ is.

We consider the following class of noncommutative surfaces, which generalizes
the set-up in Reiten-Van den Bergh [RV89].

######  Situation 2.3 (Local version of Definition 1.4).

  *   * •

$\mathbf{k}$ is an algebraically closed field of characteristic $0$ or $p\geq
7$.

  * •

$R$ is an integrally closed noetherian complete local $\mathbf{k}$-algebra of
dimension $2$ such that $R/\mathfrak{m}=\mathbf{k}$. Let $K\coloneqq Q(R)$ be
the field of fractions.

  * •

$A$ is a central simple algebra over $K$.

  * •

$\Lambda$ is a tame $R$-order in $A$ of global dimension $2$.

  * •

$R$ is a direct summand of $\Lambda$ as an $R$-module. This is the case, for
example, when
$\operatorname{char}(\mathbf{k})\not|\operatorname{rank}\Lambda$.

The notion of tameness appearing in Situation 2.3 is defined as follows.

######  Definition 2.4.

Let $(R,A)$ be as in Situation 2.3. An $R$-order $\Lambda$ in $A$ is _tame_ if
it is reflexive as an $R$-module and $\Lambda\otimes_{R}R_{\mathfrak{p}}$ is a
hereditary $R_{\mathfrak{p}}$-algebra for any prime ideal
$\mathfrak{p}\subseteq R$ of height $1$; i.e., it is an
$R_{\mathfrak{p}}$-order of global dimension $1$.

######  Remark 2.5.

It would be interesting to know if the last assumption of Situation 2.3
follows from the rest when $\operatorname{char}(\mathbf{k})\geq 7$.

######  Definition 2.6.

An $R$-order $\Lambda$ is _of finite representation type_ if there are only
finitely many finitely generated indecomposable reflexive $\Lambda$-modules up
to isomorphisms.

######  Remark 2.7.

A tame order of global dimension $2$ is of finite representation type (see
[RV89, p. 12]). In fact, by [Ram69, Theorem 1.10], any reflexive
$\Lambda$-module is projective. Alternatively, see [IW14, Proposition 2.17].

We next recall the dualizing bimodule following [SV08, pp. 662–663]. Consider
the enveloping algebra
$\Lambda^{e}=\Lambda^{\mathrm{op}}\otimes_{\mathbf{k}}\Lambda$ of $\Lambda$
and the derived category $\mathbf{D}(\Lambda^{e})$ of right
$\Lambda^{e}$-modules.

######  Definition 2.8.

The _rigid dualizing complex_ of $\Lambda$ is a pair consisting of an object
$D_{\Lambda}\in\mathbf{D}(\Lambda^{e})$ and an isomorphism $\chi\colon
D_{\Lambda}\xrightarrow{\sim}\mathop{\mathbb{R}\mathrm{Hom}}\nolimits_{\Lambda^{e}}(\Lambda,D_{\Lambda}\otimes
D_{\Lambda})$ in $\mathbf{D}(\Lambda^{e})$ which is uniquely characterized by
the following properties:

  * •

$D_{\Lambda}$ has finite injective dimension on both sides.

  * •

For each $i$, $\mathcal{H}^{i}(D_{\Lambda})$ is finitely generated both as a
left $\Lambda$-module and as a right $\Lambda$-module.

  * •

Both of the natural maps
$\Lambda\to\mathop{\mathbb{R}\mathrm{Hom}}\nolimits_{\Lambda}(D_{\Lambda},D_{\Lambda})$
and
$\Lambda\to\mathop{\mathbb{R}\mathrm{Hom}}\nolimits_{\Lambda^{\mathrm{op}}}(D_{\Lambda},D_{\Lambda})$
are isomorphisms in $\mathbf{D}(\Lambda^{e})$.

The _dualizing bimodule_ $\omega_{\Lambda}$ is defined as follows.

$\displaystyle\omega_{\Lambda}=\mathcal{H}^{-\dim R}(D_{\Lambda})$ (2.3)

######  Remark 2.9.

  *   * •

Rigid dualizing complexes exist for algebras $\Lambda$ as in Situation 2.3 by
[Yek99, Proposition 5.7]. Hence
$\omega_{\Lambda}=\mathcal{H}^{-2}(D_{\Lambda})$ exists as well.

  * •

We have the following explicit descriptions for $D_{\Lambda}$ and
$\omega_{\Lambda}$. The proof of [SV08, Lemma 2.5] applies to the $\Lambda$
under consideration. Below $D_{R}\simeq\omega_{R}[2]$ denotes the rigid
dualizing complex of $R$ (recall that $R$ as in Situation 2.3 is Cohen-
Macaulay).

$\displaystyle
D_{\Lambda}\simeq\mathop{\mathbb{R}\mathrm{Hom}}\nolimits_{R}(\Lambda,D_{R})$
(2.4)
$\displaystyle\omega_{\Lambda}\simeq\operatorname{Hom}_{R}(\Lambda,\omega_{R})$
(2.5)

  * •

By the uniqueness of the balanced dualizing complexes, both they and dualizing
bimodules commute with localizations to (Zariski/ étale/ formal) neighborhoods
and glue together if they exist locally. In particular the dualizing bimodules
in the global Definition 1.4, which are defined similarly to Definition 2.8,
exist and are given by the following formula.

$\displaystyle\omega_{\mathcal{A}}=\mathop{{\mathcal{H}}om}\nolimits_{\mathcal{O}_{X}}(\mathcal{A},\omega_{X})$
(2.6)

######  Definition 2.10.

In Situation 2.3, let $I\subset A$ be a _divisorial ideal_ (cf. [RV89, p. 1,
p. 63]); i.e., a finitely generated $\Lambda$-submodule of $A$ such that
$IK=A$ which is reflexive as an $R$-module and $I_{\mathfrak{p}}$ is
invertible for any prime ideal $\mathfrak{p}\subset R$ of height $1$. We
associate to $I$ the following $\mathbb{Z}$-graded $A$-algebra, where
$I^{(n)}$ denotes the reflexive hull of $I^{\otimes n}$.

$\displaystyle\Lambda[I]\coloneqq\bigoplus_{n\in\mathbb{Z}}I^{(n)}x^{n}\subseteq
A[x,x^{-1}]$ (2.7)

If there is $n>0$ such that $I^{(n)}=\Lambda a\Lambda$ for some $a\in
K^{\ast}$, set

$\displaystyle{\Lambda[I]}_{n}\coloneqq\Lambda[I]/(1-ax^{n}).$ (2.8)

We may assume that there is no $(b,m)$ such that $m|n,m<n,$ and $a=b^{m}$
([RV89, p. 63]).

Applying the above construction to $I=\omega_{\Lambda}^{-1}$, where the
existence of $a$ and $n>0$ as above is explained in the first paragraph of
[RV89, p. 67], we have

$\displaystyle\Lambda_{1}\coloneqq{\Lambda[\omega_{\Lambda}^{-1}]}_{n}$ (2.9)
$\displaystyle R_{1}\coloneqq\operatorname{Z}(\Lambda_{1})$ (2.10)

######  Remark 2.11.

In [RV89], $\Lambda_{1}$ is associated with the divisorial ideal
$I=D(\Lambda/R)$ (defined in [RV89, p. 63]). As it turns out it is the same as
$\omega_{\Lambda}^{-1}.$ Indeed, since both sides are reflexive, it is enough
to confirm the assertion locally at all height $1$ primes $\mathfrak{p}$ of
$R$ and use the structure theorem for the hereditary order
$\Lambda_{\mathfrak{p}}$. Note that the proof of [SV08, Proposition 2.7] gives
us the formula

$\omega_{\Lambda}\simeq\omega_{R}\otimes\operatorname{rad}\Lambda\mathfrak{p}^{-1}$
(2.11)

in codimension one at $\mathfrak{p}$, where $e$ is the ramification index of
$\Lambda$ at $\mathfrak{p}$, i.e. $e\geq 1$ is the smallest integer such that

$\operatorname{rad}\Lambda^{e}=\mathfrak{p}.$ (2.12)

Note furthermore that $e|\operatorname{rank}\Lambda$.

Although [RV89] discusses the $\mathbb{Z}/n$-graded algebra $\Lambda_{1}$ and
its centre $R_{1}$, we could instead consider the $\mathbb{Z}$-graded versions
as follows.

$\displaystyle\widetilde{\Lambda}\coloneqq\Lambda[\omega_{\Lambda}^{-1}]$
(2.13)
$\displaystyle{\widetilde{R}}\coloneqq\operatorname{Z}(\widetilde{\Lambda})$
(2.14)

In the following lemma, we show that the $\mathbb{Z}$-graded algebra
$\widetilde{\Lambda}$ and the $\mathbb{Z}/n$-graded algebra $\Lambda_{1}$ are
essentially equivalent. Consider the $(\mathbb{Z}/n\times\mathbb{Z})$-algebra
$\Lambda_{1}[\xi,\xi^{-1}]$, where the $\mathbb{Z}$-degree of $\Lambda_{1}$ is
taken to be $0$ and

$\displaystyle\deg\xi=\left(\overline{-1},1\right)\in\mathbb{Z}/n\times\mathbb{Z}.$
(2.15)

######  Lemma 2.12.

There is an isomorphism of $\mathbb{Z}$-graded algebras

$\displaystyle{\left(\Lambda_{1}[\xi,\xi^{-1}]\right)}_{\overline{0}}\simeq\widetilde{\Lambda},$
(2.16)

where the left hand side is the degree $\overline{0}$ part with respect to the
$\mathbb{Z}/n$-grading.

###### Proof.

To avoid notational complication, put

$\displaystyle\Delta\coloneqq\Lambda_{1}=\bigoplus_{{\overline{\imath}}\in\mathbb{Z}/n}\Delta_{{\overline{\imath}}}.$
(2.17)

Then

$\displaystyle{\left(\Delta[\xi,\xi^{-1}]\right)}_{\overline{0}}=\bigoplus_{j\in\mathbb{Z}}\Delta_{{\overline{\jmath}}}\xi^{j}=\bigoplus_{{\overline{\imath}}\in\mathbb{Z}/n}\bigoplus_{\begin{subarray}{c}j\in\mathbb{Z}\\\
\overline{j}={\overline{\imath}}\end{subarray}}\Delta_{{\overline{\imath}}}\xi^{j}.$
(2.18)

Define the isomorphism
$\varphi\colon\left(\Delta[\xi,\xi^{-1}]\right)_{\overline{0}}\to\widetilde{\Lambda}$
as follows. For $0\leq\ell<n$ and $q\in\mathbb{Z}$, put $j=\ell+qn$. Let
$q_{\ell}\colon\widetilde{\Lambda}_{\ell}\xrightarrow{\sim}\Delta_{{\overline{\ell}}}$
be the natural bijection. Then define the degree $j$ component of $\varphi$ as
follows.

$\displaystyle\varphi_{j}\colon\Delta_{{\overline{\jmath}}}\xi^{j}\to\widetilde{\Lambda}_{j};\quad\lambda\xi^{j}\mapsto
q_{\ell}^{-1}(\lambda)a^{q}x^{j}=q_{\ell}^{-1}(\lambda)a^{\lfloor\frac{j}{n}\rfloor}x^{j}$
(2.19)

One can easily confirm that $\varphi$ is a homomorphism of $\mathbb{Z}$-graded
algebras. In order to see that $\varphi$ is an isomorphism, note that

$\displaystyle\widetilde{\Lambda}_{j}=\widetilde{\Lambda}_{\ell}\left(\widetilde{\Lambda}_{n}\right)^{q}=\widetilde{\Lambda}_{\ell}a^{q}.$
(2.20)

∎

######  Corollary 2.13.

$\displaystyle\left(R_{1}[\xi,\xi^{-1}]\right)_{\overline{0}}\simeq{\widetilde{R}}.$
(2.21)

###### Proof.

This follows from Lemma 2.12 and the following computation.

$\displaystyle\operatorname{Z}\left(\left(\Lambda_{1}[\xi,\xi^{-1}]\right)_{\overline{0}}\right)\simeq\left(\operatorname{Z}\left(\Lambda_{1}[\xi,\xi^{-1}]\right)\right)_{\overline{0}}=\left(\operatorname{Z}\left(\Lambda_{1}\right)[\xi,\xi^{-1}]\right)_{\overline{0}}$
(2.22)

∎

######  Corollary 2.14.

${\widetilde{R}}$ is an algebra of finite type over $R$, and
$\widetilde{\Lambda}$ is finitely generated as an ${\widetilde{R}}$-module.

###### Proof.

The first assertion follows from Corollary 2.13. To see this, since $\mu_{n}$
is reductive, it is enough to show that $R_{1}[\xi,\xi^{-1}]$ is of finite
type over $R$. This in turn follows from the fact that $R_{1}$ is a finitely
generated $R$-module, as it is an $R$-submodule of the $R$-module
$\Lambda_{1}$, which is obviously finitely generated.

The second assertion similarly follows from Lemma 2.12. In fact, as
$\Lambda_{1}$ is a coherent $R_{1}$-module, $\Lambda_{1}[\xi,\xi^{-1}]$ is
coherent as an $R_{1}[\xi,\xi^{-1}]$-module, which in turn is a coherent
${\widetilde{R}}$-module by Corollary 2.13. Since $\widetilde{\Lambda}$ is an
${\widetilde{R}}$-submodule of $\Lambda_{1}[\xi,\xi^{-1}]$ by Lemma 2.12, it
is also a coherent ${\widetilde{R}}$-module. ∎

######  Corollary 2.15.

There is an isomorphism of algebraic stacks

$\displaystyle\left[\operatorname{Spec}{\widetilde{R}}/\mathbb{G}_{m}\right]\simeq\left[\operatorname{Spec}R_{1}/\mu_{n}\right].$
(2.23)

###### Proof.

By Corollary 2.13, both sides are isomorphic to the quotient stack

$\displaystyle\left[\operatorname{Spec}R_{1}[\xi,\xi^{-1}]/\left(\mu_{n}\times\mathbb{G}_{m}\right)\right].$
(2.24)

To see this, note that the actions of the subgroups $\mu_{n}$ and
$\mathbb{G}_{m}$ on $\operatorname{Spec}R_{1}[\xi,\xi^{-1}]$ are both free. ∎

The isomorphism of algebraic stacks (2.23) induces an equivalence of
categories of coherent algebras as follows.

$\displaystyle\operatorname{Alg}\left(\operatorname{coh}\left[\operatorname{Spec}{\widetilde{R}}/\mathbb{G}_{m}\right]\right)\simeq\operatorname{Alg}\left(\operatorname{coh}\left[\operatorname{Spec}R_{1}/\mu_{n}\right]\right).$
(2.25)

Note that $\widetilde{\Lambda}$ can be regarded as an object of the left hand
side of (2.25), whereas $\Lambda_{1}$ can be regarded as an object of the
right hand side. They correspond to each other under (2.25). In particular,
the $\mathbb{Z}$-graded algebra $\widetilde{\Lambda}$ and the
$\mathbb{Z}/n$-graded algebra $\Lambda_{1}$ are “Morita equivalent” as
follows.

######  Proposition 2.16.

There is an equivalence of categories

$\displaystyle\alpha\colon\operatorname{mod}^{\mathbb{Z}}\widetilde{\Lambda}\rightleftarrows\operatorname{mod}^{\mathbb{Z}/n}\Lambda_{1}\colon\beta$
(2.26)

###### Proof.

The definitions of $\alpha$ and $\beta$ follow from the fact that
$\widetilde{\Lambda}$ and $\Lambda_{1}$ correspond to each other under the
equivalence of categories (2.25). For the convenience of the reader, we write
them down explicitly.

For $N\in\operatorname{mod}^{\mathbb{Z}}\widetilde{\Lambda}$

$\displaystyle\alpha(N)\coloneqq\left(N\otimes_{\widetilde{\Lambda}}\Lambda_{1}[\xi,\xi^{-1}]\right)_{0}\in\operatorname{mod}^{\mathbb{Z}/n}\Lambda_{1}.$
(2.27)

Conversely, given $M\in\operatorname{mod}^{\mathbb{Z}/n}\Lambda_{1}$,

$\displaystyle\beta(M)\coloneqq\left(M[\xi,\xi^{-1}]\right)_{\overline{0}}\in\operatorname{mod}^{\mathbb{Z}}\left(\Lambda_{1}[\xi,\xi^{-1}]\right)_{\overline{0}}\stackrel{{\scriptstyle\text{\lx@cref{creftype~refnum}{corollary:Rtilde
as invariant
ring}}}}{{\simeq}}\operatorname{mod}^{\mathbb{Z}}\widetilde{\Lambda}.$ (2.28)

∎

######  Proposition 2.17.

$\Lambda_{1}\ast\mathcal{O}_{\mu_{n}}^{\vee}\simeq\operatorname{End}_{\Lambda}(\Lambda_{1}).$
(2.29)

###### Proof.

First note that $\mathcal{O}_{\mu_{n}}=k[t]/(t^{n}-1)$ with comultiplication
$\Delta(t)=t\otimes t$, and $\Lambda_{1}$ is a
$\mathcal{O}_{\mu_{n}}$-comodule. Hence $\Lambda_{1}$ is a
$\mathcal{O}_{\mu_{n}}^{\vee}$-module. The Hopf algebra
$\mathcal{O}_{\mu_{n}}^{\vee}\simeq\mathcal{O}_{\mathbb{Z}/n}$ and so there is
a complete set of orthogonal idempotents $e_{0},\ldots,e_{n-1}$ which give a
basis of $\mathcal{O}_{\mathbb{Z}/n}$ with pairing $e_{i}(t^{j})=\delta_{ij}$.
Let $\Lambda=\sum\lambda_{i}\in\Lambda_{1}$ be an element of $\Lambda_{1}$
with its decomposition into graded components. Then action of
$\mathcal{O}_{\mathbb{Z}/n}$ is described by $e_{i}\lambda=\lambda_{i}$. Let
$d\in\Lambda_{1}$ be homogeneous of degree $j$, then

$e_{i}(d\lambda)=d\lambda_{i-j}=de_{i-j}\lambda$ (2.30)

hence we see that

$\Lambda_{1}\ast\mathcal{O}_{\mu_{n}}^{\vee}=\bigoplus\Lambda_{1}e_{i}$ (2.31)

with relations $e_{i}d=de_{i-j}$ for all $d\in\Lambda_{1}$ of degree $j$ as in
[Mon93, Example 4.1.7]. Hence

$e_{i}\Lambda_{1}e_{j}=\omega_{\Lambda}^{(i-j)}e_{j}$ (2.32)
$e_{i}\Lambda_{1}e_{j}e_{j}\Lambda_{1}e_{\ell}=\omega_{\Lambda}^{(i-j)}\omega_{\Lambda}^{(j-\ell)}e_{\ell}$
(2.33)

and so the multiplication in $\Lambda_{1}\ast\mathcal{O}_{\mu_{n}}^{\vee}$ is
isomorphic to that of

$\operatorname{End}_{\Lambda}(\operatorname*{\mathchoice{\textstyle\bigoplus}{\textstyle\bigoplus}{\scriptstyle\bigoplus}{\scriptscriptstyle\bigoplus}}\omega_{\Lambda}^{(i)})=\bigoplus_{i,j}\operatorname{Hom}_{\Lambda}(\omega_{\Lambda}^{(i)},\omega_{\Lambda}^{(j)})=\bigoplus_{i,j}\omega_{\Lambda}^{(j-i)}.$
(2.34)

∎

######  Corollary 2.18.

There is an equivalence of categories

$\displaystyle\operatorname{mod}^{\mathbb{Z}/n}\Lambda_{1}\simeq\operatorname{mod}\Lambda.$
(2.35)

###### Proof.

The equivalence is obtained as the composition of the following equivalences.

$\displaystyle\operatorname{mod}^{\mathbb{Z}/n}\Lambda_{1}\simeq\operatorname{mod}\Lambda_{1}\ast\mu_{n}\stackrel{{\scriptstyle\text{
\lx@cref{creftype~refnum}{proposition:Lambda1 * mun simeq End Lambda1}
}}}{{\simeq}}\operatorname{mod}\operatorname{End}_{\Lambda}\Lambda_{1}\simeq\operatorname{mod}\Lambda$
(2.36)

The last equivalence follows from the fact that $\Lambda$ has global dimension
$2$. In fact, $\Lambda_{1}$ as a $\Lambda$-module, contains $\Lambda$ as a
direct summand and is a projective module by [Ram69, Theorem 1.10].
Alternatively, see [IW14, Proposition 2.17 (2)$\Rightarrow$(3)]. ∎

######  Corollary 2.19.

$\Lambda_{1}$ has global dimension $2$.

###### Proof.

By Corollary 2.18, since $\Lambda$ has global dimension $2$, the category
$\operatorname{mod}^{\mathbb{Z}/n}\Lambda_{1}$ has global dimension $2$. As
$\operatorname{mod}^{\mathbb{Z}/n}\Lambda_{1}\simeq\operatorname{mod}\Lambda_{1}\ast\mu_{n}$,
the algebra $\Lambda_{1}\ast\mu_{n}$ also has global dimension $2$. As
$\Lambda_{1}$ is a direct summand of this algebra as a bimodule over itself
and $\Lambda_{1}\ast\mu_{n}$ is a free module over $\Lambda_{1}$, it follows
from [MR01, 7.2.8 Theorem (i)] that the global dimension of $\Lambda_{1}$ is
at most $2$. It is clearly at least $2$, hence the conclusion. ∎

Let $R_{1}$ be the centre of $\Lambda_{1}$. It inherits the
$\mathbb{Z}/n$-grading of $\Lambda_{1}$.

Before stating the next result, we recall (and extend) the definition of
linearly reductive quotient singularities.

######  Definition 2.20.

Let $X$ be a normal surface over $\mathbf{k}$. A closed point $x\in
X(\mathbf{k})$ is a _linearly reductive quotient singularity_ if there is a
finite linearly reductive group scheme $H$ acting linearly on $\mathbb{A}^{2}$
such that

$\displaystyle\widehat{\mathcal{O}}_{X,x}\simeq\widehat{\mathcal{O}}_{\mathbb{A}^{2}/H,0}.$
(2.37)

By [Art69, Corollary (2.6)], this is equivalent to the existence of a smooth
affine variety $U$ over $\mathbf{k}$, a finite linearly reductive group scheme
$H$ over $\mathbf{k}$ acting on $U$, and an étale morphism

$\displaystyle U/H\to X$ (2.38)

whose image contains $x\in X$.

More generally, let $X$ be a normal algebraic stack over a field $\mathbf{k}$
with finite inertia and $2$-dimensional coarse space. We say a
$\mathbf{k}$-rational point $x\colon\operatorname{Spec}\mathbf{k}\to X$ is a
linearly reductive quotient singularity if there exist $H\curvearrowright U$
as above and a _smooth_ (not necessarily étale) morphism as in (2.38) whose
image contains $x\in X$.

######  Remark 2.21.

For schemes, linearly reductive quotient singularities are defined in terms of
étale covers. However, for stacky surfaces $\mathcal{X}$ we defined linearly
reductive quotient singularities in terms of the existence of _smooth_ covers
(since étale covers by schemes generally do not exist). We show that these two
definitions are consistent, i.e. if $X$ is a surface and there is a smooth
cover of the form $U/H$ with $U$ smooth and $H$ finite linearly reductive,
then there is an étale cover of this form.

If the characteristic of $\mathbf{k}$ is at least 7, then for schemes of
dimension $2$, $F$-regular singularities are the same as linearly reductive
quotient singularities by [LMM21, Theorem 11.5]. Also, $U$ is $F$-regular
since it is smooth over a perfect field and since direct summands inherit
$F$-regularity, $U/H$ is strongly $F$-regular. Next, since smooth morphisms
étale locally have sections, we may assume we have a smooth surjection $U/H\to
X$ with a section. The existence of such a section implies that
$\mathcal{O}_{X}$ is a direct summand of the structure sheaf of $U/H$, so $X$
is strongly $F$-regular, hence has linearly reductive quotient singularities.

In characteristic $0$, we can similarly argue by replacing “$F$-regular” with
“log terminal” to conclude that, again in dimension 2, being a quotient
singularity is a local property in smooth topology (see, say, [Kol13,
Proposition 2.15]).

Unfortunately it is not clear to the authors if [LMM21, Theorem 11.5]
generalizes to singularities over perfect fields. Actually, this prevents them
from applying the main arguments of this paper to not-necessarily
algebraically closed fields. The issue lies in that the classification of
canonical surface singularities is known only over algebraically closed
fields.

For a general perfect field $\mathbf{k}$, we can still find a finite Galois
extension $\mathbf{k}\subset L$ and present the base change of the original
singularity to $L$ as a quotient by a linearly reductive group scheme (over
$L$). As we illustrate in Example 2.22, in good cases the Galois group is
linearly reductive and we can combine it with (a descent to $\mathbf{k}$ of)
the group scheme, so as to prove that the original singularity over
$\mathbf{k}$ is also a linearly reductive quotient singularity. However, it is
unclear if such an argument always works or not.

######  Example 2.22.

Consider the following singularity.

$\displaystyle\mathbf{k}$ $\displaystyle=\mathbb{F}_{3}$ (2.39) $\displaystyle
R$ $\displaystyle=\mathbf{k}[x,y,z]/(z^{3}-x^{2}-y^{2})$ (2.40)

Note that $R$ is not isomorphic to the $A_{2}$-singularity unless we extend
the base field $\mathbf{k}$. On the other hand, in doing so we can find a
presentation of $R$ as a linearly reductive quotient singularity as follows.

Put

$\displaystyle L=\mathbf{k}[\sqrt{-1}]=\mathbf{k}[i]/(i^{2}+1),$ (2.41)

so that

$\displaystyle R_{L}=R\otimes_{\mathbf{k}}L\simeq
L[x,y,z]/\left(z^{3}-(x+iy)(x-iy)\right).$ (2.42)

It is an $L$-subalgebra of $S=L[u,v]$ by the following map.

$\displaystyle x+iy$ $\displaystyle\mapsto u^{3}$ (2.43) $\displaystyle x-iy$
$\displaystyle\mapsto v^{3}$ (2.44) $\displaystyle z$ $\displaystyle\mapsto
uv$ (2.45)

The $L$-algebra $S$ admits a $\mathbb{Z}/3\mathbb{Z}$-grading given by

$\displaystyle\deg u=\overline{1},\deg v=\overline{-1}.$ (2.46)

It corresponds to a $\mu_{3}$-action on $S$ and the homomorphism above
identifies $R_{L}$ with the invariant subring $S^{\mu_{3}}$.

On the other hand, the $\mathbf{k}$-algebra

$\displaystyle S\simeq\mathbf{k}[u,v,i]/(i^{2}+1)$ (2.47)

admits a $\mu_{6}$-action corresponding to the
$\mathbb{Z}/6\mathbb{Z}$-grading as follows.

$\displaystyle\deg u$ $\displaystyle=\overline{2}$ (2.48) $\displaystyle\deg
v$ $\displaystyle=\overline{-2}$ (2.49) $\displaystyle\deg i$
$\displaystyle=\overline{3}$ (2.50)

We have the following short exact sequence of group schemes over $\mathbf{k}$.

$\displaystyle 1\to\mu_{3}\to\mu_{6}\to\operatorname{Aut}_{\mathbf{k}}(L)\to
1$ (2.51)

Indeed, the Galois group
$\operatorname{Aut}_{\mathbf{k}}(L)\simeq\mathbb{Z}/2\mathbb{Z}$ is generated
by the automorphism

$\displaystyle i\mapsto-i$ (2.52)

of $L$ and it lifts to the action $\mu_{6}\curvearrowright S$ via
$\left(\mu_{6}\right)_{\mathrm{red}}\xrightarrow{\sim}\operatorname{Aut}_{\mathbf{k}}(L)$.
The map $\mu_{3}\to\mu_{6}$ is the dual of the standard quotient map
$\mathbb{Z}/6\mathbb{Z}\twoheadrightarrow\mathbb{Z}/3\mathbb{Z}$. The induced
action $\mu_{3}\curvearrowright S$ coincides with the one above. Hence
$S^{\mu_{6}}\simeq R$ and $R$ is a linearly reductive quotient singularity
over $\mathbf{k}$.

Let us resume the analysis of $\Lambda_{1}$.

######  Proposition 2.23.

There is a finite linearly reductive group scheme $H$ and a linear action on
$S\coloneqq\mathbf{k}[[x,y]]$ which is free on
$\operatorname{Spec}S\setminus\\{0\\}$ such that $R_{1}=S^{H}$.

###### Proof.

By [RV89, Proposition 5.1], $\Lambda_{1}$ is reflexive Azumaya and tame. In
particular, it follows that

$\displaystyle\Lambda_{1}\simeq\left(\Lambda\otimes_{R}R_{1}\right)^{\vee\vee}.$
(2.53)

Since we assume in Situation 2.3 that $R$ is a direct summand of $\Lambda$, it
follows from (2.53) that $R_{1}$ is a direct summand of $\Lambda_{1}$. Hence
it is of finite representation type. In particular, the divisor class group of
$R_{1}$ is finite and hence $R_{1}$ is $\mathbb{Q}$-Gorenstein. Let
$\operatorname{Spec}A\to\operatorname{Spec}R_{1}$ be an index 1 cover of
$\operatorname{Spec}R_{1}$, so that $A$ is $\mathbb{Z}/r$-graded for some
$r>0$ and $R_{1}=A_{\overline{0}}$ ($A$ is nothing but the algebra
$\Lambda_{1}$ as defined in (2.9) for $\Lambda=R_{1}$). It follows that the
divisor class group of $A$ is also finite. Hence by [Lip69, Proposition
(17.3)], it is a rational singularity. Since $A$ is also Gorenstein, it
follows that $A$ is a canonical singularity.

Since we assume that the characteristic is $0$ or $p\geq 7$,
$\operatorname{Spec}A$ is a linearly reductive quotient singularity by [LS14,
Proposition 4.2].

Thus we have confirmed that $R_{1}$ is a direct summand of the ring of formal
power series in $2$-variables, so that it is strongly $F$-regular. Hence
$R_{1}$ is a linearly reductive quotient singularity by [LMM21, Theorem 11.5
(3)$\Rightarrow$(2)]. Let $N\triangleleft H$ be the subgroup scheme generated
by pseudo-reflections, see [Sat12, Definition 1.2]. Replacing $H$ with $H/N$
and $\mathbb{A}^{2}$ with $\mathbb{A}^{2}/N\simeq\mathbb{A}^{2},$ by
Proposition 2.10 of (loc. cit.), we may assume $H$ acts on $\mathbb{A}^{2}$
without pseudo-reflections. Then Theorem 1.9 of (loc. cit.) shows that the
action of $H$ on $\operatorname{Spec}S$ is free outside the origin and
$R_{1}\simeq S^{H}$. ∎

######  Remark 2.24.

Proposition 2.23 is the only proof where we use the assumption that $R$ is a
direct summand.

Take any $H\curvearrowright\operatorname{Spec}S$ as in Proposition 2.23. Put

$\displaystyle\Gamma\coloneqq\Lambda_{1}\overline{\otimes}_{R_{1}}S\stackrel{{\scriptstyle\text{\cite[cite]{[\@@bibref{}{MR978602}{}{},
p.~{}4]}}}}{{\coloneqq}}\left(\Lambda_{1}\otimes_{R_{1}}S\right)^{\ast\ast}.$
(2.54)

The algebra $\Gamma$ inherits the left $H$-action on $\operatorname{Spec}S$;
i.e., it is a right comodule algebra over the Hopf algebra $\mathcal{O}_{H}$
by the coaction map

$\displaystyle\rho\colon\Gamma\to\Gamma\otimes_{\mathbf{k}}\mathcal{O}_{H}.$
(2.55)

Dually, $\Gamma$ is a left module algebra over the dual Hopf algebra
$\mathcal{O}_{H}^{\vee}$ by the following action map, which corresponds to
$\rho$ under the obvious adjunction.

$\displaystyle\mathcal{O}_{H}^{\vee}\otimes_{\mathbf{k}}\Gamma\to\Gamma$
(2.56)

The relationship among the relevant algebras is summarized in the following
commutative diagram.

${\Lambda}$${\Lambda_{1}}$${\Gamma}$${R=R_{0}}$${R_{1}}$${S}$$\scriptstyle{\mu_{n}}$$\scriptstyle{H}$
(2.57)

######  Lemma 2.25.

Let $(R,\Lambda)$ be as in Situation 2.3, so that up to isomorphisms there are
only finitely many indecomposable $\Lambda$-modules which are reflexive
($\iff$ maximal Cohen-Macaulay) $R$-modules. Then $\Lambda$ is Morita
equivalent to the subcategory of $\mathop{\text{ref}}M$ consisting of
indecomposable reflexive modules, regarded as an algebra. In particular, the
Morita equivalence class of $\Lambda$ depends only on the equivalence class of
the category $\mathop{\text{ref}}M$.

###### Proof.

By [Ram69, Theorem 1.10] or [IW14, Proposition 2.17], reflexive
$\Lambda$-modules coincide with projective $\Lambda$-modules. Since the
category of finitely generated $\Lambda$-modules is Krull–Schmidt by, say,
[Rei03, p. 88], any indecomposable projective module is a direct summand of
$\Lambda$. Thus we obtain the Morita equivalence. ∎

######  Proposition 2.26.

There is an isomorphism of algebras as follows.

$\displaystyle\Gamma\ast
H\simeq\operatorname{End}_{\Lambda_{1}}\left(\Gamma\right)$ (2.58)

###### Proof.

By [Mon93, THEOREM 8.3.3 1)$\Rightarrow$2)], it is enough to show that

$\displaystyle\Lambda_{1}=\Gamma^{H}\coloneqq\Gamma^{\mathcal{O}_{H}^{\vee}}=\Gamma^{\mathop{\text{co}}\mathcal{O}_{H}},$
(2.59)

where the last term denotes the subalgebra of coinvariants ([Mon93, Definition
1.7.1 1)]), and that $\Lambda_{1}\to\Gamma$ is a right
$\mathcal{O}_{H}$-Galois extension in the sense of [Mon93, Definition 8.1.1];
i.e., the following map is an isomorphism.

$\displaystyle\beta\colon\Gamma\otimes_{\Lambda_{1}}\Gamma\to\Gamma\otimes_{\mathbf{k}}\mathcal{O}_{H};\quad
a\otimes b\mapsto(a\otimes 1)\rho(b)$ (2.60)

Note that both $\Gamma$ and $\Lambda_{1}$ are reflexive $R_{1}$-modules. So is
$\Gamma^{\mathcal{O}_{H}^{\vee}}$, as it is a direct summand of $\Gamma$ as an
$R_{1}$-module for the reason that $H$ is linearly reductive. Therefore the
equality (2.59) can be checked in codimension one over $R_{1}$.

Similarly, note that $\Gamma$ is a free module over $S$, hence so is the right
hand side of (2.60). On the other hand, since $\Lambda_{1}$ has right global
dimension $2$ by Corollary 2.19 and $\Gamma$ is a reflexive $R_{1}$-module by
construction, it is a projective right $\Lambda_{1}$-module by [Ram69, Theorem
1.10] or [IW14, Proposition 2.17 (2)$\Rightarrow$(3)]. Hence the left hand
side of (2.60) is a projective right $\Gamma$-module. In particular, it is a
free $S$-module. Therefore it is enough to show $\beta$ is an isomorphism in
codimension $1$ over $S$.

As $R_{1}=S^{H}$ and the action $H\curvearrowright\operatorname{Spec}S$ is
free outside the origin, both assertions hold in codimension $1$. To see this,
let $X,Y$ be the punctured spectra of $S$ and $R_{1}$, respectively, so that
the natural map $\pi\colon X\to Y$ is isomorphic to the quotient map
$X\to\left[X/H\right]$. For an $S$-module $M$, put $M^{\circ}\coloneqq M|_{X}$
and so on (similarly for $R_{1}$-modules). By construction, it then follows
that
$\pi^{\ast}\Lambda_{1}^{\circ}=\Lambda_{1}^{\circ}\otimes_{\mathcal{O}_{Y}}\mathcal{O}_{X}\simeq\Gamma^{\circ}$
and $\left(\pi_{\ast}\Gamma^{\circ}\right)^{H}\simeq\Lambda_{1}^{\circ}$,
hence (2.59) holds in codimension $1$ over $\operatorname{Spec}R_{1}$. On the
other hand, it follows from $Y\simeq\left[X/H\right]$ that the natural map

$\displaystyle\mathcal{O}_{X}\otimes_{\mathcal{O}_{Y}}\mathcal{O}_{X}\to\mathcal{O}_{X}\otimes_{\mathbf{k}}\mathcal{O}_{H}$
(2.61)

is an isomorphism. Applying $\Lambda_{1}^{\circ}\otimes_{\mathcal{O}_{Y}}-$ to
this, we see that (2.60) is an isomorphism in codimension $1$ over
$\operatorname{Spec}S$. ∎

######  Corollary 2.27.

There is an equivalence of categories as follows.

$\displaystyle\operatorname{mod}^{H}\Gamma\simeq\operatorname{mod}\Lambda_{1}$
(2.62)

###### Proof.

The proof is parallel to that of Corollary 2.18, where Proposition 2.17 is
replaced with Proposition 2.26. Recall also that $\Lambda_{1}$ has global
dimension $2$ by Corollary 2.19. ∎

##  3\. The root stack $(\mathcal{X}_{r},\mathcal{A}_{r})$

Let $(X,\mathcal{A})$ be as in Definition 1.4. In this section we construct an
algebraic stack $r\colon\mathcal{X}_{r}\to X$ over $X$ and an order
$\mathcal{A}_{r}$ on $\mathcal{X}_{r}$, which is Azumaya in codimension $1$.
Furthermore, this stack is _tame_ in the sense of [AOV08]. Tame stacks are
algebraic stacks whose stabilizers are finite linearly reductive group
schemes, see [AOV08, Theorem 3.2]; note that this includes examples with
$\mu_{p}$-stabilizers in characteristic $p$ and hence tame stacks are more
general than tame Deligne–Mumford stacks (i.e. Deligne–Mumford stacks whose
stabilizers have order prime to the characteristic). Although tame stacks are
Artin stacks in general, they enjoy many of the same properties of tame
Deligne–Mumford stacks, e.g. they have a coarse space whose formation commutes
with _arbitrary_ base change, and the push-forward functor to the coarse space
is exact.

Consider the $\mathbb{Z}$-graded $\mathcal{A}$-algebra
$\widetilde{\mathcal{A}}$ which is defined as follows, where
$\omega_{\mathcal{A}}=\mathop{{\mathcal{H}}om}\nolimits_{X}\left(\mathcal{A},\omega_{X},\right)$
is the dualizing bimodule of $\mathcal{A}$ (see (2.6) for the definition) and

$\displaystyle\omega_{\mathcal{A}}^{-1}\coloneqq\mathop{{\mathcal{H}}om}\nolimits_{\mathcal{A}}\left(\omega_{\mathcal{A}},\mathcal{A}\right)$
(3.1)

is the (reflexive) dual $\mathcal{A}$-bimodule of $\omega_{\mathcal{A}}$. The
symbol ${(-)}^{(i)}$ denotes the reflexive hull of $(-)^{\otimes i}$.

$\displaystyle\widetilde{\mathcal{A}}\coloneqq\mathcal{A}[\omega_{\mathcal{A}}^{\pm}]=\bigoplus_{i\in\mathbb{Z}}\left(\omega_{\mathcal{A}}^{-1}\right)^{(i)}$
(3.2)

The centre $\operatorname{Z}(\widetilde{\mathcal{A}})$ is a
$\mathbb{Z}$-graded $\mathcal{O}_{X}$-algebra and hence

$\displaystyle{\widetilde{X}}\coloneqq\operatorname{Spec}_{X}\operatorname{Z}(\widetilde{\mathcal{A}})$
(3.3)

is equipped with a $\mathbb{G}_{m}$-action. Put

$\displaystyle\mathcal{X}_{r}\coloneqq\left[{\widetilde{X}}/\mathbb{G}_{m}\right].$
(3.4)

Note that the natural morphism

$\displaystyle r\colon\mathcal{X}_{r}\to X$ (3.5)

is the coarse moduli map.

######  Lemma 3.1.

$\operatorname{Z}(\widetilde{\mathcal{A}})$ is of finite type over
$\mathcal{O}_{X}$ and $\widetilde{\mathcal{A}}$ is coherent as an
$\mathcal{O}_{{\widetilde{X}}}$-module.

###### Proof.

Since the ramification index at each prime divides $n$, the rank of
$\mathcal{A}$ by Remark 2.11, we see that at each point $x\in X$ the
completion of $\left(\omega_{\mathcal{A}}^{-1}\right)^{(n)}$ is isomorphic to
the completion
$\widehat{}\mathcal{A}\coloneqq\mathcal{A}\otimes\widehat{\mathcal{O}}_{X,x}$
of $\mathcal{A}$, as $\widehat{}\mathcal{A}$-bimodules. Therefore the
completion of the following sheaf of homomorphisms of $\mathcal{A}$-bimodules

$\displaystyle\mathcal{L}\coloneqq\mathop{{\mathcal{H}}om}\nolimits_{\mathcal{A}^{e}}\left(\mathcal{A},\left(\omega_{\mathcal{A}}^{-1}\right)^{(n)}\right)$
(3.6)

satisfies

$\displaystyle\mathcal{L}\otimes\widehat{\mathcal{O}}_{X,x}\simeq\mathop{{\mathcal{H}}om}\nolimits_{\widehat{}\mathcal{A}^{e}}\left(\widehat{}\mathcal{A},\widehat{}\mathcal{A}\right)\simeq\operatorname{Z}\left(\widehat{}\mathcal{A}\right)=\widehat{\mathcal{O}}_{X,x}.$
(3.7)

Thus we conclude that $\mathcal{L}$ is an invertible sheaf on $X$ and that the
natural map
$\mathcal{L}\otimes_{\mathcal{O}_{X}}\mathcal{A}\to\left(\omega_{\mathcal{A}}^{-1}\right)^{(n)}$
is an isomorphism of $\mathcal{A}$-bimodules.

Take a Zariski open subset $U\subseteq X$ over which $\mathcal{L}$ is trivial,
and fix an isomorphism of $\mathcal{A}|_{U}$-bimodules
$\varphi\colon\mathcal{A}|_{U}\xrightarrow{\sim}\left(\omega_{\mathcal{A}}^{-1}\right)^{(n)}|_{U}$.
Put $t\coloneqq\varphi(1)$. Then

$\displaystyle\mathcal{O}_{U}[t,t^{-1}]\subseteq\operatorname{Z}\left(\left(\widetilde{\mathcal{A}}|_{U}\right)^{(n)}\right),$
(3.8)

where $\left(\widetilde{\mathcal{A}}|_{U}\right)^{(n)}$ is the $n$-th Veronese
subalgebra of $\widetilde{\mathcal{A}}|_{U}$. This implies that the algebra
$\widetilde{\mathcal{A}}|_{U}$ is generated by the $\mathcal{O}_{U}$-coherent
sheaf $\bigoplus_{i=0}^{n-1}\mathcal{A}^{(i)}|_{U}$ as an algebra over
$\mathcal{O}_{U}[t,t^{-1}]$. This implies that $\widetilde{\mathcal{A}}|_{U}$
is finitely generated as a module over $\mathcal{O}_{U}[t,t^{-1}]$ and hence
$\operatorname{Z}\left(\widetilde{\mathcal{A}}|_{U}\right)$ is finitely
generated as an algebra over $\mathcal{O}_{U}$. ∎

Let $\mathcal{A}_{r}$ be the $\mathcal{O}_{\mathcal{X}_{r}}$-algebra
corresponding to $\widetilde{\mathcal{A}}$ under the following equivalence of
monoidal categories.

$\displaystyle\operatorname{coh}\mathcal{X}_{r}\simeq\operatorname{coh}^{\mathbb{G}_{m}}{\widetilde{X}}$
(3.9)

Let $U=\operatorname{Spec}R\to X$ be the standard morphism from the completion
$R$ of the local ring of a closed point of $X$. Put
$\Lambda=\mathcal{A}\otimes_{\mathcal{O}_{X}}R$.

######  Lemma 3.2.

The pair $(R,\Lambda)$ satisfies the assumptions of Situation 2.3.

###### Proof.

By the assumptions of Definition 1.4, $\mathcal{A}$ is of global dimension $2$
and tame. This implies that $\Lambda=\mathcal{A}\otimes_{\mathcal{O}_{X}}R$
has the same properties.

It also follows that $\operatorname{Z}(\Lambda)=R$, since in general the
centre of the completion of an algebra coincides with the completion of the
centre. This follows from the general fact that if $\Lambda$ is an algebra
with $\operatorname{Z}(\Lambda)=R$ and $R\to S$ is a flat morphism of
commutative algebras, then there is a sequence of isomorphisms as follows.

$\displaystyle\operatorname{Z}(\Lambda\otimes_{R}S)\simeq\operatorname{HH}^{0}\left(\Lambda\otimes_{R}S/\mathbf{k}\right)\simeq\operatorname{HH}^{0}\left(\Lambda\otimes_{R}S/S\right)$
(3.10) $\displaystyle\stackrel{{\scriptstyle\text{$R$ is flat over
$S$}}}{{\simeq}}\operatorname{HH}^{0}\left(\Lambda/R\right)\otimes_{R}S\simeq\operatorname{HH}^{0}\left(\Lambda/\mathbf{k}\right)\otimes_{R}S\simeq\operatorname{Z}(\Lambda)\otimes_{R}S.$
(3.11)

∎

The next result relates the local objects introduced in Section 2 to the
global objects which have we just constructed.

######  Proposition 3.3.

$\displaystyle{\widetilde{X}}\times_{X}U$
$\displaystyle\simeq\operatorname{Spec}{\widetilde{R}}\quad\text{($\mathbb{G}_{m}$-equivariant)}$
(3.12) $\displaystyle\mathcal{X}_{r}\times_{X}U$
$\displaystyle\simeq\left[\operatorname{Spec}{\widetilde{R}}/\mathbb{G}_{m}\right]\simeq\left[\operatorname{Spec}R_{1}/\mu_{n}\right]$
(3.13) $\displaystyle\mathcal{A}_{r}|_{U}$
$\displaystyle\simeq\widetilde{\Lambda}$ (3.14)

###### Proof.

For each $i\in\mathbb{Z}$, we have

$\displaystyle\left(\omega_{\mathcal{A}}^{-1}\right)^{(i)}\otimes_{\mathcal{O}_{X}}R$
$\displaystyle\simeq\mathop{{\mathcal{H}}om}\nolimits_{X}\left(\mathop{{\mathcal{H}}om}\nolimits_{X}\left(\left(\omega_{\mathcal{A}}^{-1}\right)^{\otimes
i},\mathcal{O}_{X}\right),\mathcal{O}_{X}\right)\otimes_{\mathcal{O}_{X}}R$
(3.15)
$\displaystyle\simeq\operatorname{Hom}_{R}\left(\operatorname{Hom}_{R}\left(\left(\omega_{\mathcal{A}}^{-1}\otimes_{\mathcal{O}_{X}}R\right)^{\otimes
i},R\right),R\right)\simeq\left(\omega_{\Lambda}^{-1}\right)^{(i)}.$ (3.16)

By [SV08, Lemma 2.5 (3)] or Remark 2.9, the dualizing bimodule localizes and
so
$\omega_{\mathcal{A}}^{-1}\otimes_{\mathcal{O}_{X}}R\simeq\omega_{\Lambda}^{-1}$.
Thus we obtain the following isomorphism of $\mathbb{Z}$-graded $R$-algebras

$\displaystyle\widetilde{\mathcal{A}}\otimes_{\mathcal{O}_{X}}R\simeq\widetilde{\Lambda}$
(3.17)

Hence

$\displaystyle\operatorname{Z}(\widetilde{\mathcal{A}})\otimes_{\mathcal{O}_{X}}R\simeq\operatorname{Z}(\widetilde{\mathcal{A}})\otimes_{\mathcal{O}_{{\widetilde{X}}}}{\widetilde{R}}\stackrel{{\scriptstyle\text{\lx@cref{creftype~refnum}{lemma:Xtilde
is of finite type over
X}}}}{{\simeq}}\operatorname{Z}(\widetilde{\mathcal{A}}\otimes_{\mathcal{O}_{{\widetilde{X}}}}{\widetilde{R}})\simeq\operatorname{Z}\left(\widetilde{\mathcal{A}}\otimes_{\mathcal{O}_{X}}R\right)\simeq\operatorname{Z}(\widetilde{\Lambda})={\widetilde{R}},$
(3.18)

so that we obtain the following $\mathbb{G}_{m}$-equivariant isomorphism.

$\displaystyle{\widetilde{X}}\times_{X}U\simeq\operatorname{Spec}{\widetilde{R}}$
(3.19)

The rest of the assertions immediately follow from this. ∎

######  Proposition 3.4.

The stack $\mathcal{X}_{r}$ is a normal tame algebraic stack with only
linearly reductive quotient singularities.

###### Proof.

We already know that $\mathcal{X}_{r}$ is an algebraic stack. It follows from
(3.12) and [AOV08, Theorem 3.2] that it is a tame algebraic stack. Note that,
although [AOV08, Theorem 3.2 (c)] is stated only for fppf covers, it follows
from [AOV08, Corollary 3.4] and [AOV08, Corollary 3.5] that if $f\colon Y\to
X$ is any surjective morphism of schemes and the base change of a stack over
$X$ by $f$ is tame, then so is the original stack over $X$. Finally, it
follows from (3.12) and the results in Section 2 that $\mathcal{X}_{r}$ is
normal and has only linearly reductive quotient singularities. ∎

##  4\. The canonical stack $\left(\mathcal{X}_{c},\mathcal{A}_{c}\right)$

In Proposition 3.4, we showed $\mathcal{X}_{r}$ is tame and algebraic with
linearly reductive quotient singularities. We therefore have an associated
_canonical stack_

$\displaystyle c\colon\mathcal{X}_{c}\to\mathcal{X}_{r},$ (4.1)

as defined by the fourth author [Sat12], generalizing a previous construction
of Vistoli [Vis89]; see Remark 4.1 below. The stack $\mathcal{X}_{c}$ is a
smooth tame algebraic stack, and $c$ is a relative coarse space map which is
an isomorphism over the smooth locus of $\mathcal{X}_{r}$.

######  Remark 4.1.

For varieties with tame quotient singularities (i.e. étale locally the
quotient by a finite group whose order is prime to the characteristic),
Vistoli constructed a canonical smooth Deligne–Mumford stack, see [Vis89,
Proof of Proposition 2.8 and (2.9)]; for varieties which have linearly
reductive quotient singularities (i.e. étale locally the quotient by a finite
linearly reductive group scheme), the fourth author constructed a canonical
smooth tame algebraic stack, see [Sat12, Theorem 1.10]. Although (loc. cit.)
is only stated for the case where the base is a scheme, the proof works
equally well when the base is a stack. If $M$ is a tame algebraic stack with
only linearly reductive quotient singularities, then by definition, there is a
smooth cover $\\{M_{i}\to M\\}$ with $M_{i}=U_{i}/G_{i}$, $U_{i}$ a smooth
scheme, and $G_{i}$ a finite linearly reductive group scheme. Let
$\mathcal{X}_{i}\to M_{i}$ be the canonical tame stack given by Theorem 1.10
of (loc. cit.). Letting $M_{ij}=M_{i}\times_{M}M_{j}$, Lemma 5.5 of (loc.
cit.) yields an isomorphism between $\mathcal{X}_{i}\times_{M_{i}}M_{ij}$ and
$\mathcal{X}_{j}\times_{M_{i}}M_{ij}$ which is unique up to 2-isomorphism;
this is because canonical stacks commute with smooth base change. Thus, we
obtain our canonical stack $\mathcal{X}\to M$ with the property that
$\mathcal{X}_{i}=\mathcal{X}\times_{M}M_{i}$.

We then put

$\displaystyle\mathcal{A}_{c}\coloneqq\left(c^{\ast}\mathcal{A}_{r}\right)^{\vee\vee}.$
(4.2)

Let $R$ be the completion of a local ring of $X$ and let
$U=\operatorname{Spec}R\to X$ be the induced map. Then $R$ is as in Section 2.
We freely use the notation introduced there. Consider the base change of (4.2)
by $U\to X$:

$\displaystyle
c|_{U}\colon\mathcal{X}_{c}\times_{X}U\to\mathcal{X}_{r}\times_{X}U\simeq\left[\operatorname{Spec}R_{1}/\mu_{n}\right].$
(4.3)

By Proposition 2.23, there is a linearly reductive finite subgroup scheme
$H<\operatorname{GL}(2,\mathbf{k})$ and a linear action
$H\curvearrowright\operatorname{Spec}S$, where $S=\mathbf{k}[[x,y]]$, which is
free outside the origin and $R_{1}\simeq S^{H}$. The following commutative
diagram describes the relationship between global objects and local objects.

${\left[\operatorname{Spec}S/H\right]}$${\operatorname{Spec}R_{1}}$${\left[\operatorname{Spec}R_{1}/\mu_{n}\right]}$${\operatorname{Spec}R}$${\mathcal{X}_{c}}$${\mathcal{X}_{r}}$${X}$${\ulcorner}$$\scriptstyle{/\mu_{n}}$${\ulcorner}$$\scriptstyle{c}$$\scriptstyle{r}$
(4.4)

Since reflexive hulls commute with flat pullback, we see that

$\displaystyle\Gamma\simeq\mathcal{A}_{c}|_{\operatorname{Spec}S}.$ (4.5)

Recall that $\Gamma$ is equipped with an $H$-action. The smash product
$\Gamma\ast H$ is identified with a sheaf of
$\mathcal{O}_{\left[\operatorname{Spec}S/H\right]}$-algebras, which will be
denoted by the same symbol by abuse of notation. We have the following
isomorphism.

$\displaystyle M_{n}(S)\ast H\simeq\Gamma\ast
H\simeq\mathcal{A}_{c}|_{\left[\operatorname{Spec}S/H\right]}$ (4.6)

######  Proposition 4.2.

$\mathcal{A}_{c}$ is a sheaf of Azumaya algebras on the smooth tame algebraic
stack $\mathcal{X}_{c}$.

###### Proof.

This follows from the isomorphism (4.5) and Remark 2.2, since we know that
$\Gamma$ is an Azumaya algebra over $S$ (see Section 2). Note that the
morphism $\operatorname{Spec}S\to\mathcal{X}_{c}$ is flat and jointly
surjective if we vary $\operatorname{Spec}R\to X$ over all points of $X$. ∎

Let $\pi=r\circ c\colon\mathcal{X}_{c}\to X$ be the coarse space morphism.

######  Lemma 4.3.

There is a canonical isomorphism of sheaves of $\mathcal{O}_{X}$-algebras as
follows.

$\displaystyle\alpha\colon$ $\displaystyle\mathcal{A}\to
r_{\ast}\mathcal{A}_{r}$ (4.7) $\displaystyle\beta\colon$
$\displaystyle\mathcal{A}_{r}\to c_{\ast}\mathcal{A}_{c}$ (4.8)

###### Proof.

We have the following canonical isomorphism

$\displaystyle\mathcal{A}=\left(\text{the degree $0$ part of
}\widetilde{\mathcal{A}}\right)\simeq r_{\ast}\mathcal{A}_{r}$ (4.9)

which yields (4.7).

As $\mathcal{A}_{c}=(c^{\ast}\mathcal{A}_{r})^{\vee\vee}$, there is a
canonical morphism $c_{\ast}c^{\ast}\mathcal{A}_{r}\to
c_{\ast}\mathcal{A}_{c}$. The left hand side is isomorphic to
$\mathcal{A}_{r}$ by the projection formula and the fact that
$c_{\ast}\mathcal{O}_{\mathcal{X}_{c}}\simeq\mathcal{O}_{\mathcal{X}_{r}}$.
Thus we obtain the morphism (4.8). Note that $c_{\ast}\mathcal{A}_{c}$ is
torsion free. In fact, otherwise there is a non-trivial subsheaf
$\mathcal{T}\hookrightarrow c_{\ast}\mathcal{A}_{c}$ whose support is strictly
smaller than $\mathcal{X}_{r}$. The corresponding morphism
$c^{\ast}\mathcal{T}\to\mathcal{A}_{c}$ would be non-zero, but this is a
contradiction since $c^{\ast}\mathcal{T}$ is torsion and $\mathcal{A}_{c}$ is
torsion free. Finally, since $c$ is an isomorphism in codimension $1$,
$\mathcal{A}_{r}$ is reflexive, and $c_{\ast}\mathcal{A}_{c}$ is torsion free,
it follows that (4.8) is an isomorphism. ∎

######  Theorem 4.4.

There is an equivalence of $\mathbf{k}$-linear categories

$\displaystyle\operatorname{coh}\mathcal{A}\simeq\operatorname{coh}\mathcal{A}_{r}.$
(4.10)

###### Proof.

From the isomorphism (4.7) we obtain the pair of adjoint functors

$\displaystyle
G\colon\operatorname{coh}\mathcal{A}\rightleftarrows\operatorname{coh}\mathcal{A}_{r}\colon
F$ (4.11)

given by $F(\mathcal{M})=r_{\ast}\mathcal{M}$ and
$G(\mathcal{N})=r^{\ast}\mathcal{N}\otimes_{r^{\ast}\mathcal{A},\alpha^{\prime}}\mathcal{A}_{r}$,
where $\alpha^{\prime}\colon r^{\ast}\mathcal{A}\to\mathcal{A}_{r}$
corresponds to $\alpha$ under the adjoint pair $r^{\ast}\dashv r_{\ast}$.

We first show $G$ is fully faithful; equivalently, we show that the adjunction
unit $\mu\colon\operatorname{id}\to FG$ is a natural isomorphism. For each
$\mathcal{N}\in\operatorname{coh}\mathcal{A}$, the map
$\mu_{\mathcal{N}}\colon\mathcal{N}\to F(G(\mathcal{N}))$ coincides with the
composition of the following morphisms

$\mathcal{N}\xrightarrow[\text{$\alpha$ is
iso}]{\sim}\mathcal{N}\otimes_{\mathcal{A},\alpha}r_{\ast}\mathcal{A}_{r}\mathrel{\mathop{\longrightarrow}^{\sim}}r_{\ast}(r^{\ast}\mathcal{N}\otimes_{r^{\ast}\mathcal{A}}\mathcal{A}_{r})$
(4.12)

so it is enough to confirm that the righthand morphism is an isomorphism.
Since this map is defined globally, we need only prove that its restriction to
each affine open subset of $X$ is an isomorphism. To ease notation, let
$\mathcal{N},\mathcal{A},$ etc. denote their restriction to one of those
affine opens. Then the morphism is obviously an isomorphism for
$\mathcal{N}=\mathcal{A}$. Both sides are right exact functors in
$\mathcal{N}$, and the category of coherent modules over $\mathcal{A}$ is
generated by $\mathcal{A}$ (over such an open affine subset). Thus we are
done.

It remains to show that $G$ is essentially surjective. To achieve this, take
an arbitrary object $\mathcal{M}\in\operatorname{coh}\mathcal{A}_{r}$ and
consider the adjunction counit map

$\displaystyle\eta=\eta_{\mathcal{M}}\colon
GF(\mathcal{M})=r^{\ast}r_{\ast}\mathcal{M}\otimes_{r^{\ast}\mathcal{A}}\mathcal{A}_{r}\to\mathcal{M}.$
(4.13)

It suffices to prove that $\eta$ is an isomorphism. By faithfully flat
descent, we can and will check this locally on $X$. Take a formal neighborhood
$\operatorname{Spec}R\to X$ as before and consider the pullback
$\eta^{\prime}$ of $\eta$ to $\left[\operatorname{Spec}R_{1}/\mu_{n}\right]$.
Since $\operatorname{Spec}R\to X$ is flat, we can rewrite $\eta^{\prime}$ as
follows. To ease notation, put $r^{\prime}\coloneqq
r|_{\left[\operatorname{Spec}R_{1}/\mu_{n}\right]}$,
$\mathcal{A}^{\prime}_{r}\coloneqq\mathcal{A}_{r}|_{\left[\operatorname{Spec}R_{1}/\mu_{n}\right]}$,
and
$\mathcal{M}^{\prime}\coloneqq\mathcal{M}|_{\left[\operatorname{Spec}R_{1}/\mu_{n}\right]}$.

$\displaystyle\eta^{\prime}\colon{r^{\prime}}^{\ast}{r^{\prime}}_{\ast}\mathcal{M}^{\prime}\otimes_{{r^{\prime}}^{\ast}\Lambda}\mathcal{A}^{\prime}_{r}\to\mathcal{M}^{\prime}$
(4.14)

This is a morphism in the category
$\operatorname{mod}\mathcal{A}^{\prime}_{r}$. Let
$M\in\operatorname{mod}^{\mathbb{Z}/n}\Lambda_{1}$ be the object corresponding
to $\mathcal{M}^{\prime}$ under the equivalence of categories

$\displaystyle\operatorname{coh}\mathcal{A}^{\prime}_{r}\simeq\operatorname{mod}^{\mathbb{Z}/n}\Lambda_{1}.$
(4.15)

Then $\eta^{\prime}$ is identified with the following morphism of
$\operatorname{mod}^{\mathbb{Z}/n}\Lambda_{1}$-modules, which is known to be
an isomorphism.

$\displaystyle M_{\overline{0}}\otimes_{\Lambda}\Lambda_{1}\to M$ (4.16)

In fact, this is the adjunction counit map of the equivalence of categories
Corollary 2.18. ∎

######  Theorem 4.5.

There is an equivalence of $\mathbf{k}$-linear categories

$\displaystyle\operatorname{coh}\mathcal{A}_{r}\simeq\operatorname{coh}\mathcal{A}_{c}.$
(4.17)

###### Proof.

The proof is quite analogous to that of Theorem 4.4. From the isomorphism
(4.8) we obtain the pair of adjoint functors

$\displaystyle
G\colon\operatorname{coh}\mathcal{A}_{r}\rightleftarrows\operatorname{coh}\mathcal{A}_{c}\colon
F$ (4.18)

given by $F(\mathcal{M})=c_{\ast}\mathcal{M}$ and
$G(\mathcal{N})=c^{\ast}\mathcal{N}\otimes_{c^{\ast}\mathcal{A}_{r},\beta^{\prime}}\mathcal{A}_{c}$,
where $\beta^{\prime}\colon c^{\ast}\mathcal{A}_{r}\to\mathcal{A}_{c}$
corresponds to $\beta$ under the adjoint pair $c^{\ast}\dashv c_{\ast}$. We
first show $G$ is fully faithful; equivalently, we show that the adjunction
unit $\mu\colon\operatorname{id}\to FG$ is a natural isomorphism. This follows
from that, for each $\mathcal{N}\in\operatorname{coh}\mathcal{A}$, the map
$\mu_{\mathcal{N}}\colon\mathcal{N}\to F(G(\mathcal{N}))$ coincides with the
composition of the following isomorphisms.

$\mathcal{N}\xrightarrow[\text{$\beta$ is
iso}]{\sim}\mathcal{N}\otimes_{\mathcal{A}_{r},\beta}c_{\ast}\mathcal{A}_{c}\mathrel{\mathop{\longrightarrow}^{\sim}}c_{\ast}(c^{\ast}\mathcal{N}\otimes_{c^{\ast}\mathcal{A}_{r}}\mathcal{A}_{c})$
(4.19)

To see the second isomorphism, note that the assertion is obviously correct
for $\mathcal{N}=\mathcal{A}_{r}$ and that both sides are right exact functors
in $\mathcal{N}$, so that the proof is locally reduced to the case
$\mathcal{N}=\mathcal{A}_{r}$. It remains to show that $G$ is essentially
surjective. To achieve this, take an arbitrary object
$\mathcal{M}\in\operatorname{coh}\mathcal{A}_{c}$ and consider the adjunction
counit map

$\displaystyle\eta=\eta_{\mathcal{M}}\colon
GF(\mathcal{M})=c^{\ast}c_{\ast}\mathcal{M}\otimes_{c^{\ast}\mathcal{A}_{r}}\mathcal{A}_{c}\to\mathcal{M}.$
(4.20)

It suffices to prove that $\eta$ is an isomorphism. By faithfully flat
descent, we can and will check this locally on $\mathcal{X}_{r}$. Take a
formal neighborhood $\operatorname{Spec}R\to X$ as before and consider the
pullback $\eta^{\prime}$ of $\eta$ to $\operatorname{Spec}R_{1}$. Since
$\operatorname{Spec}R_{1}\to X$ is flat, we can rewrite $\eta^{\prime}$ as
follows. To ease notation, put $c^{\prime}\coloneqq
c|_{\left[\operatorname{Spec}S/H\right]}$,
$\mathcal{A}^{\prime}_{c}\coloneqq\mathcal{A}_{c}|_{\left[\operatorname{Spec}S/H\right]}$,
and
$\mathcal{M}^{\prime}\coloneqq\mathcal{M}|_{\left[\operatorname{Spec}S/H\right]}$.

$\displaystyle\eta^{\prime}\colon{c^{\prime}}^{\ast}{c^{\prime}}_{\ast}\mathcal{M}^{\prime}\otimes_{{c^{\prime}}^{\ast}\Lambda_{1}}\mathcal{A}^{\prime}_{c}\to\mathcal{M}^{\prime}$
(4.21)

This is a morphism in the category
$\operatorname{coh}\mathcal{A}^{\prime}_{c}$. On the other hand, let
$M\in\operatorname{mod}(\Gamma\ast H)$ be the object corresponding to
$\mathcal{M}^{\prime}$ under the equivalence of categories

$\displaystyle\operatorname{coh}\mathcal{A}^{\prime}_{c}\simeq\operatorname{mod}(\Gamma\ast
H).$ (4.22)

Then $\eta^{\prime}$ is identified with the following morphism of $\Gamma\ast
H$-modules.

$\displaystyle\operatorname{Hom}_{\Gamma\ast
H}\left(\Gamma,M\right)\otimes_{\Lambda_{1}}\Gamma\to M$ (4.23)

In fact, this follows from the commutative diagram below. The functors
$F^{\prime}$ and $G^{\prime}$ in the diagram are obtained as the restriction
of the adjoint pair (4.18) to the formal neighborhood:

$\displaystyle
G^{\prime}\colon\operatorname{mod}\Lambda_{1}\leftrightarrows\operatorname{mod}\mathcal{A}^{\prime}_{c}\colon
F^{\prime}$ (4.24) ${\operatorname{mod}\Gamma\ast
H}$${\operatorname{mod}\Lambda_{1}}$${\operatorname{mod}\mathcal{A}^{\prime}_{c}}$$\scriptstyle{\operatorname{Hom}(\Gamma,-)}$$\scriptstyle{\sim}$$\scriptstyle{G^{\prime}}$$\scriptstyle{-\otimes\Gamma}$$\scriptstyle{F^{\prime}}$
(4.25)

We already know that (4.23) is an isomorphism, as it is the adjunction counit
map of the equivalence of categories Corollary 2.27. ∎

##  5\. The gerbe $\mathcal{X}_{g}$

In this section we associate a $\mu_{N}$-gerbe

$\displaystyle g\colon\mathcal{X}_{g}\to\mathcal{X}_{c}$ (5.1)

to the pair $\left(\mathcal{X}_{c},\mathcal{A}_{c}\right)$ of smooth tame
(Artin) stacky surface and a sheaf of Azumaya algebras; we show that there is
an equivalence of categories

$\displaystyle\operatorname{coh}^{(1)}\mathcal{X}_{g}\simeq\operatorname{mod}\mathcal{A}_{c}$
(5.2)

between the category of _$1$ -twisted coherent sheaves_ on $\mathcal{X}_{g}$
and the category of $\mathcal{A}_{c}$-modules. See [Ols16, Definition 12.3.2]
for $1$-twisted coherent sheaves.

###  5.1. Review of gerbes and twisted sheaves

For the convenience of the reader, we briefly explain some basics of gerbes
and $m$-twisted sheaves mainly after [Lie08]. Let $p\colon
X\to\left(\mathcal{S}ch/\mathbf{k}\right)$ be an algebraic stack over
$\mathbf{k}$, where we equip $\left(\mathcal{S}ch/\mathbf{k}\right)$ with the
fppf topology. Let $f\colon\mathcal{X}\to X$ be a 1-morphism of algebraic
stacks over $\mathbf{k}$. The _relative inertia stack_ $If$ of $f$ is defined
by the following pullback diagram of stacks.

${If}$${\mathcal{X}}$${\mathcal{X}}$${\mathcal{X}\times_{X}\mathcal{X}}$$\scriptstyle{\Delta_{f}}$$\scriptstyle{\Delta_{f}}$
(5.3)

More explicitly, $If$ is equivalent to the category where

  * •

an object is a pair $(x,g)$ consisting of an object $x$ of $\mathcal{X}$ and
an automorphism $g$ of $x$ such that $f(g)=1_{f(x)}$, and

  * •

a morphism from $(x,g)$ to $(y,h)$ is a morphism from $x$ to $y$ which is
compatible with $g$ and $h$ in the obvious manner.

Let $p\colon If\to\mathcal{X}$ be one of the projections. In view of the
explicit description of $If$ given above, the functor $p$ simply forgets the
automorphism $g$. In this sense $If$ is the universal automorphism group of
$\mathcal{X}$ relative to $X$, and actually $If$ comes with the canonical
structure of a relative group algebraic space over $\mathcal{X}$. Recall that
the category $\mathcal{X}$ admits a standard Grothendieck topology where a
collection of morphisms $(x_{i}\to x)_{i\in I}$ is a covering if and only if
the collection of underlying morphisms of schemes is a covering of the site
$\left(\mathcal{S}ch/\mathbf{k}\right)$. We will let $\mathcal{C}$ denote this
site. Then the functor $p\colon If\to\mathcal{X}$ makes $If$ a sheaf over the
site $\mathcal{C}$. Also we can naturally think of a quasi-coherent sheaf $F$
on $\mathcal{X}$ as a sheaf over $\mathcal{C}$, which will be denoted by the
same symbols by abuse of notation. Then there is a natural map

$\displaystyle F\times If\to F$ (5.4)

of sheaves over $\mathcal{C}$ defined as follows. Take an object
$x\in\mathcal{C}$. Then a section $g$ of $If$ over $x$ is an automorphism of
$x$, so we obtain the pullback map $g^{\ast}\colon
F(x)\xrightarrow{\sim}F(x)$. The map

$\displaystyle F(x)\times If(x)\to F(x)$ (5.5)

is simply given by $(f,g)\mapsto g^{\ast}f$, and it is a right action by
construction. We call it the _inertia action_ of the sheaf of groups $If$ on
the sheaf $F$. For a closed subgroup scheme $i\colon
D\hookrightarrow\mathbb{G}_{m}$, a _$D$ -gerbe_ over $X$, or a gerbe over $X$
banded by the sheaf of groups $D_{\mathcal{X}}$, is a stack
$f\colon\mathcal{X}\to X$ over $X$ which is a gerbe in the usual sense (see,
say, [Lie08, Definition 2.2.1.2]) when seen as a stack over the site
canonically associated to $X$ similar to the definition of $\mathcal{C}$
above, and an isomorphism $If\simeq\mathcal{X}\times D$ of group stacks over
$\mathcal{X}$. Through the inertia action, quasi-coherent sheaves on a
$D$-gerbe comes with a right action of $D_{\mathcal{X}}$.

######  Definition 5.1 ($=$[Lie08, Definition 3.1.1]).

A quasi-coherent sheaf $F$ on a $D$-gerbe $\mathcal{X}$ over $X$ is
$\mathcal{X}$-twisted (or $1$-twisted) if the right action of
$D_{\mathcal{X}}$ on $F$ coincides with the left action of $D_{\mathcal{X}}$
on $F$ obtained by pulling back the
$\mathbb{G}_{a,\mathcal{X}}=\mathcal{O}_{\mathcal{X}}$-action on $F$ by the
natural map of group schemes $D_{\mathcal{X}}\stackrel{{\scriptstyle
i}}{{\hookrightarrow}}\mathbb{G}_{m,\mathcal{X}}\to\mathbb{G}_{a,\mathcal{X}}$.

Let $\widehat{D}$ be the character group of $D$. In general, a quasi-coherent
sheaf $F$ on $\mathcal{X}$ admits a decomposition

$\displaystyle F=\bigoplus_{\chi\in\widehat{D}}F_{\chi},$ (5.6)

where $F_{\chi}\subseteq F$ is the subsheaf of sections on which the inertia
action of $D$ is by the character $\chi$.

######  Definition 5.2.

A quasi-coherent sheaf $F$ on a $D$-gerbe $\mathcal{X}$ over $X$ is
$m$-twisted if $F=F_{i^{m}}$, where $i\in\widehat{D}$ is the standard
embedding $i\colon D\hookrightarrow\mathbb{G}_{m}$. The category of
$m$-twisted coherent sheaves on $\mathcal{X}$ will be denoted by
$\operatorname{coh}^{(m)}\mathcal{X}\subseteq\operatorname{coh}\mathcal{X}$.

######  Remark 5.3.

The closed subgroup scheme $D\hookrightarrow\mathbb{G}_{m}$ is either
$\mathbb{G}_{m}$ itself or $\mu_{N}$ for some $N\geq 1$. From now on we assume
that $D=\mu_{N}$, so that $\widehat{D}=\mathbb{Z}/N\mathbb{Z}$. For
$m\in\mathbb{Z}$ we use the standard notation
${\overline{m}}\in\mathbb{Z}/N\mathbb{Z}(=\widehat{D})$ for its residue class,
and put
$\operatorname{coh}^{({\overline{m}})}\mathcal{X}=\operatorname{coh}^{(m)}\mathcal{X}$.

######  Lemma 5.4.

If $F\in\operatorname{coh}^{({\overline{m}})}\mathcal{X}$ for some
${\overline{m}}\neq 0\in\widehat{D}$, then
$f_{\ast}F=0\in\operatorname{coh}X$.

###### Proof.

It is enough to show
$H^{0}(\mathcal{X}\times_{X}V,F|_{\mathcal{X}\times_{X}V})=0$ for each affine
$V\subset X$. The vanishing directly follows from the definition of $H^{0}$
for quasi-coherent sheaves on stacks and the definition of twisted sheaves
given in Definition 5.1. ∎

######  Corollary 5.5.

There is an orthogonal decomposition of abelian categories as follows.

$\displaystyle\operatorname{coh}\mathcal{X}=\bigoplus_{{\overline{m}}\in\widehat{D}}\operatorname{coh}^{({\overline{m}})}\mathcal{X}$
(5.7)

###### Proof.

For $F\in\operatorname{coh}^{({\overline{m}})}\mathcal{X}$ and
$G\in\operatorname{coh}^{({\overline{n}})}\mathcal{X}$ it follows that
$\mathop{{\mathcal{H}}om}\nolimits_{\mathcal{X}}(F,G)\in\operatorname{coh}^{({\overline{n}}-{\overline{m}})}\mathcal{X}$
([Lie08, Lemma 3.1.1.7 (ii)]). Hence $\operatorname{Hom}_{\mathcal{X}}(F,G)=0$
by Lemma 5.4. ∎

###  5.2. $\mathcal{X}_{g}$ and the equivalence of categories

Let $g\colon\mathcal{G}\to\mathcal{X}_{c}$ be the gerbe associated to
$\mathcal{A}_{c}$ in [Gir71, Chapter V.4.4]. See also [Mat21, Theorem 19]. In
order to make this explicit, let
$p\colon\mathcal{X}_{c}\to\left(\mathcal{S}ch/\mathbf{k}\right)$ be the
structure functor of the stack $\mathcal{X}_{c}$. An object of the category
$\mathcal{G}$ is a triple $(x,\mathcal{E},\xi)$, where $x$ is an object of
$\mathcal{X}_{c}$, $\mathcal{E}$ is a locally free sheaf on the scheme $p(x)$,
and
$\xi\colon\mathop{{\mathcal{E}}nd}\nolimits(\mathcal{E})\xrightarrow{\sim}x^{\ast}\mathcal{A}_{c}$
is an isomorphism of sheaves of algebras on $p(x)$. A morphism from
$(x,\mathcal{E},\xi)$ to $(y,\mathcal{F},\eta)$ is a pair of a morphism
$f\colon x\to y$ in $\mathcal{X}_{c}$ and an isomorphism
$\varphi\colon\mathcal{E}\xrightarrow{\sim}p(f)^{\ast}\mathcal{F}$ of locally
free sheaves on $p(x)$ which is compatible with the isomorphisms $\xi,\eta$ in
the obvious sense. The stack $\mathcal{G}$ comes with the obvious functor
$g\colon\mathcal{G}\to\mathcal{X}_{c}$ which sends the triple
$(x,\mathcal{E},\xi)$ to $x$ and the pair $(f,\varphi)$ to $f$. The stack
$\mathcal{G}$ is also equipped with the tautological locally free sheaf
$\mathcal{E}_{g}$ together with the universal isomorphism
$\mathop{{\mathcal{E}}nd}\nolimits(\mathcal{E}_{g})\xrightarrow{\sim}g^{\ast}\mathcal{A}_{c}$.
Moreover $\mathcal{G}$ is a $\mathbb{G}_{m}$-gerbe over $\mathcal{X}_{c}$ via
$g$, where the isomorphism
$\mathbb{G}_{m}\xrightarrow{\sim}\operatorname{Aut}(x,\mathcal{E},\xi)$ is
given by the following map.

$\displaystyle\mathbb{G}_{m}\ni t\mapsto[\mathcal{E}\stackrel{{\scriptstyle
t\cdot}}{{\xrightarrow{\sim}}}\mathcal{E}]\in\operatorname{Aut}(\mathcal{E})$
(5.8)

Therefore $\mathcal{E}_{g}$ is a $1$-twisted sheaf on the gerbe $\mathcal{G}$.
We then obtain functors

$\displaystyle
F\colon\operatorname{coh}^{(1)}\mathcal{G}\leftrightarrows\operatorname{coh}\mathcal{A}_{c}\colon
G$ (5.9)

given by

$F(\mathcal{M})=g_{\ast}(\mathcal{M}\otimes_{\mathcal{O}_{\mathcal{G}}}\mathcal{E}^{\vee}),\quad\quad
G(\mathcal{N})=g^{\ast}\mathcal{N}\otimes_{g^{\ast}\mathcal{A}_{c}}\mathcal{E}.$
(5.10)

Since $\mathcal{N}\to FG(\mathcal{N})$ is an isomorphism, we see $F$ is fully
faithful. Next, by [Lie08, Lemma 3.1.1.7], there is an equivalence of
categories as follows.

$\displaystyle
g_{\ast}\colon\operatorname{coh}^{(0)}\mathcal{G}\xrightarrow{\sim}\operatorname{coh}\mathcal{X}_{c}$
(5.11)

See also [Ols16, Lemma 12.3.3] and its proof. One can easily confirm that the
quasi-inverse is given by $g^{\ast}$, so that for each
$\mathcal{N}\in\operatorname{coh}^{(0)}\mathcal{G}$ the adjunction counit map
$g^{\ast}g_{\ast}\mathcal{N}\to\mathcal{N}$ is an isomorphism. To show $F$ is
essentially surjective, we show $GF(\mathcal{M})\to\mathcal{M}$ is an
isomorphism. We compute
$GF(\mathcal{M})=g^{\ast}g_{\ast}(\mathcal{M}\otimes\mathcal{E}^{\vee})\otimes_{g^{\ast}\mathcal{A}_{c}}\mathcal{E}$.
Since $\mathcal{M}$ and $\mathcal{E}$ are $1$-twisted,
$\mathcal{M}\otimes\mathcal{E}^{\vee}$ is untwisted by [Lie08, Lemma 3.1.1.7],
so
$g^{\ast}g_{\ast}(\mathcal{M}\otimes\mathcal{E}^{\vee})=\mathcal{M}\otimes\mathcal{E}^{\vee}$.
As a result,
$GF(\mathcal{M})=\mathcal{M}\otimes\mathcal{E}^{\vee}\otimes_{g^{\ast}\mathcal{A}_{c}}\mathcal{E}=\mathcal{M}$.
Finally, suppose that the cohomological Brauer class of $\mathcal{A}_{c}$
comes from a class in $H^{2}(\mathcal{X}_{c},\mu_{N})$. Then there is a
$\mu_{N}$-gerbe $g\colon\mathcal{X}_{g}\to\mathcal{X}_{c}$, a natural morphism
$\iota\colon\mathcal{X}_{g}\to\mathcal{G}$, and the following equivalence of
categories (see [Lie08, Lemma 3.1.1.12] and the paragraph before that).

$\displaystyle\iota^{\ast}\colon\operatorname{coh}^{(1)}\mathcal{G}\xrightarrow{\sim}\operatorname{coh}^{(1)}\mathcal{X}_{g}$
(5.12)

Thus we have obtained the desired gerbe $\mathcal{X}_{g}$, which is a smooth
tame (Artin) stack with finite linearly reductive stabilizers. Notice that if
$\mathcal{A}$ is a matrix ring over the quotient field of $X$, then
generically, the Brauer class defined by $\mathcal{A}_{c}$ is trivial, and
hence $\mathcal{X}_{g}\to\mathcal{X}_{c}$ is an isomorphism. In this case
$\operatorname{coh}^{(1)}\mathcal{X}_{g}\simeq\operatorname{coh}\mathcal{X}_{c}\simeq\operatorname{coh}\mathcal{X}_{g}$.
This completes the proof of Theorem 1.5.

##  6\. Example: reducing to positive characteristic

In this section, we give an example illustrating that the reduction to
positive characteristics of a nice noncommutative projective surface in
characteristic $0$ (arising from an AS-regular algebra) may not fulfill the
tameness assumption in Definition 1.4. Consider the following graded algebra

$\displaystyle S=\mathbf{k}\langle x,y,z\rangle/\left(xy-yx-z^{2},xz-zx,yz-
zy\right),$ (6.1)

where $\deg(x)=\deg(y)=\deg(z)=1$. It is a 3-dimensional AS-regular quadratic
algebra. If $\mathbf{k}$ is of characteristic $p>0$, then
$\operatorname{Z}(S)=\mathbf{k}[x^{p},y^{p},z]$. Let $\mathcal{A}$ be the
sheaf of algebras on
$\mathbb{P}^{2}\simeq\mathbb{P}(p,p,1)\simeq\operatorname{Proj}\operatorname{Z}(S)\eqqcolon
X$ associated to $S$, so that

$\displaystyle\operatorname{coh}\mathcal{A}\simeq\operatorname{qgr}S.$ (6.2)

Hence $\operatorname{gl.dim}\mathcal{A}=2$. Moreover, since $S$ is a free
module over $\operatorname{Z}(S)$ of rank $p^{2}$, the structure sheaf
$\mathcal{O}_{X}$ is (locally) a direct summand of $\mathcal{A}$. The
restriction of $\mathcal{A}$ to $(z\neq 0)\subset X$ is the Weyl algebra
$\mathbf{k}\langle x,y\rangle/(xy-yx-1)$, which is known to be an Azumaya
algebra over its centre $\mathbf{k}[x^{p},y^{p}]$, see [Rev73, Théorème 2]. If
we restrict $\mathcal{A}$ to the open set $(x\neq 0)\subset X$, we obtain the
algebra $T=S[y^{-p}]_{0}$ which is generated by
$a=xy^{-1},b=y^{-1}x,c=zy^{-1}$ with the relation $a-b=c^{2}$. Hence we can
eliminate $A$ and we obtain an algebra generated by $b,c$ with relation
$[b,c]=c^{3}$. The centre $Z$ of $T$ is generated by $u=c^{p}$ and
$v=b^{p}+ub^{p-1}$. Let $R$ be the DVR given by localization at the prime
ideal $(u)$, so $R=k[u,v]_{(u)}$. Then $c^{p}\in J(T\otimes_{Z}R)$ and so
$c\in J(T\otimes_{Z}R)$ and we have that $T\otimes_{Z}R/J(T\otimes_{Z}R)\simeq
k(b)$. So $\mathcal{A}\otimes R$ has a unique simple module. If
$\mathcal{A}\otimes R$ were to be hereditary, with a unique simple module, it
must be Azumaya, but then $\mathcal{A}\otimes R/J(\mathcal{A}\otimes R)$ would
be the residue field of $R$, which is not the case. So $\mathcal{A}$ is not
tame. In other words, the restriction of $\mathcal{A}$ to the line
$(z=0)\subset X$ and then to its generic point $\mathfrak{p}$ is isomorphic to
$\mathbf{k}\left(\frac{x}{y}\right)$, where the function field of the line is
$\mathbf{k}\left(\frac{x^{p}}{y^{p}}\right)$ and is unramified along the other
divisors of $X$. Note that $\mathcal{A}$ is not tame. Recall that if
$\mathcal{A}_{\mathfrak{p}}$ is hereditary, then by the structure theorem of
hereditary orders, this field extension must be cyclic, which is not.

######  Remark 6.1.

For $X=\mathbb{P}^{2}$ and $\mathcal{A}$ the sheaf of algebras associated to
$S$ as in (6.1), there does not exist a tame algebraic stack $\mathcal{X}$
with coarse space $X$ for which
$\operatorname{mod}\mathcal{A}\simeq\operatorname{coh}^{(1)}\mathcal{X}$ since
such a stack would have to be smooth, but not smooth in codimension one.

##  7\. The Centre

In this section we prove a special case of the Hochschild-Kostant-Rosenberg
isomorphism in degree $0$ for algebraic stacks with finite stabilizers over
fields. This is presumably well known to experts, but for the lack of a
suitable reference we include a proof. Let $\mathcal{X}$ be an algebraic stack
over a field $\mathbf{k}$. Consider the following Cartesian diagram defining
the inertia stack $I\mathcal{X}$, where $d=e=\Delta_{\mathcal{X}/\mathbf{k}}$.

${I\mathcal{X}}$${\mathcal{X}}$${\mathcal{X}}$${\mathcal{X}\times_{\mathbf{k}}\mathcal{X}}$$\scriptstyle{q}$$\scriptstyle{p}$$\scriptstyle{e}$$\scriptstyle{d}$
(7.1)

######  Proposition 7.1.

Let $\mathcal{X}$ be a smooth algebraic stack with finite stabilizers over a
field $\mathbf{k}$. Then there is an isomorphisms of
$H^{0}\left(\mathcal{X},\mathcal{O}_{\mathcal{X}}\right)$-modules as follows.

$\displaystyle\operatorname{Hom}_{\mathcal{X}\times\mathcal{X}}\left(\Delta_{\mathcal{X}\ast}\mathcal{O}_{\mathcal{X}},\Delta_{\mathcal{X}\ast}\mathcal{O}_{\mathcal{X}}\right)$
$\displaystyle\simeq\bigoplus_{\begin{subarray}{c}W\subset
I\mathcal{X}:\text{connected component}\\\ \dim
W=\dim\mathcal{X}\end{subarray}}\operatorname{Hom}_{\mathcal{X}}\left(p_{\ast}\mathcal{O}_{W},\mathcal{O}_{\mathcal{X}}\right)$
(7.2)

###### Proof.

Note first that

$\displaystyle\operatorname{Hom}_{\mathcal{X}\times\mathcal{X}}\left(d_{\ast}\mathcal{O}_{\mathcal{X}},e_{\ast}\mathcal{O}_{\mathcal{X}}\right)\simeq\operatorname{Hom}_{\mathcal{X}}\left(e^{\ast}d_{\ast}\mathcal{O}_{\mathcal{X}},\mathcal{O}_{\mathcal{X}}\right).$
(7.3)

The _derived_ functor $\mathbb{L}e^{\ast}d_{\ast}$ is represented by the
convolution kernel

$\displaystyle K\coloneqq\Gamma_{d\ast}\mathcal{O}_{\mathcal{X}}\ast
S_{\ast}\left(\Gamma_{e\ast}\mathcal{O}_{\mathcal{X}}\right),$ (7.4)

where $\Gamma_{f}\colon X\to X\times Y$ denotes the graph of a morphism
$f\colon X\to Y$, $\ast$ denotes the convolution of kernels, and
$S\colon\mathcal{X}\times(\mathcal{X}\times\mathcal{X})\xrightarrow{\sim}(\mathcal{X}\times\mathcal{X})\times\mathcal{X}$
exchanges the factors (see, say, [Huy06, Section 5]. We think of the kernels
as right modules). In particular,

$\displaystyle\mathbb{L}e^{\ast}d_{\ast}\mathcal{O}_{\mathcal{X}}\simeq\Phi_{K}(\mathcal{O}_{\mathcal{X}})\simeq\mathbb{R}\mathrm{pr}_{2\ast}K.$
(7.5)

Then there is an isomorphism between $K$ and the structure sheaf of the
derived loop space as in Lemma 7.2 below. It then follows that

$\displaystyle\mathcal{H}^{0}(K)\simeq
d_{\ast}\mathcal{O}_{\mathcal{X}}\otimes_{\mathcal{O}_{\mathcal{X}\times\mathcal{X}}}e_{\ast}\mathcal{O}_{\mathcal{X}}\simeq(d\circ
p)_{\ast}\mathcal{O}_{I\mathcal{X}},$ (7.6)

so that

$\displaystyle
e^{\ast}d_{\ast}\mathcal{O}_{\mathcal{X}}=\mathcal{H}^{0}\left(\mathbb{L}e^{\ast}d_{\ast}\mathcal{O}_{\mathcal{X}}\right)\stackrel{{\scriptstyle\text{\eqref{equation:FM
transform}}}}{{\simeq}}\mathcal{H}^{0}\left(\mathbb{R}\mathrm{pr}_{2\ast}K\right)\simeq\mathrm{pr}_{2\ast}\mathcal{H}^{0}(K)\stackrel{{\scriptstyle\text{\eqref{equation:HKR
in degree 0}}}}{{\simeq}}p_{\ast}\mathcal{O}_{I\mathcal{X}}.$ (7.7)

The second-to-the-last isomorphism follows from that $K$ is supported on the
closed substack $\Delta_{\mathcal{X}}\subset\mathcal{X}\times\mathcal{X}$,
which is finite over $\mathcal{X}$. Thus we see

$\displaystyle\operatorname{Hom}_{\mathcal{X}}\left(e^{\ast}d_{\ast}\mathcal{O}_{\mathcal{X}},\mathcal{O}_{\mathcal{X}}\right)$
$\displaystyle\stackrel{{\scriptstyle\text{\eqref{equation:key
isomorphism}}}}{{\simeq}}\operatorname{Hom}_{\mathcal{X}}(p_{\ast}\mathcal{O}_{I\mathcal{X}},\mathcal{O}_{\mathcal{X}})\simeq\bigoplus_{\begin{subarray}{c}W\subseteq
I\mathcal{X}\\\ \dim
W=\dim\mathcal{X}\end{subarray}}\operatorname{Hom}_{\mathcal{X}}(p|_{W\ast}\mathcal{O}_{W},\mathcal{O}_{\mathcal{X}}),$
(7.8)

where the last isomorphism follows from the smoothness of $\mathcal{X}$, which
implies that $\mathcal{O}_{\mathcal{X}}$ is pure. ∎

######  Lemma 7.2.

There is an isomorphism as follows.

$\displaystyle K\simeq
d_{\ast}\mathcal{O}_{\mathcal{X}}\otimes^{\mathbb{L}}_{\mathcal{O}_{\mathcal{X}\times\mathcal{X}}}e_{\ast}\mathcal{O}_{\mathcal{X}}$
(7.9)

###### Proof.

In this proof, all functors are derived. By definition, $K$ is the following
object of
$\mathbf{D}^{\mathrm{b}}\operatorname{coh}(\mathcal{X}\times\mathcal{X})$; the
isomorphism $S$ disappears due to the obvious symmetry of the object
$\Gamma_{e\ast}\mathcal{O}_{\mathcal{X}}$.

$\displaystyle
p_{14\ast}\left(p_{123}^{\ast}\Gamma_{d\ast}\mathcal{O}_{\mathcal{X}}\otimes
p_{234}^{\ast}\Gamma_{e\ast}\mathcal{O}_{\mathcal{X}}\right)$ (7.10)

We use the following equalities.

$\displaystyle\Gamma_{d}=(d\times 1_{\mathcal{X}})\circ d$ (7.11)
$\displaystyle\Gamma_{e}=(1_{\mathcal{X}}\times e)\circ e$ (7.12)

We also use the following isomorphism of (derived) functors.

$\displaystyle p_{123}^{\ast}(d\times 1_{\mathcal{X}})_{\ast}\simeq(d\times
1_{\mathcal{X}}\times 1_{\mathcal{X}})_{\ast}p_{12}^{\ast}$ (7.13)

This follows from the following Cartesian square and the flat base change
theorem.

${\mathcal{X}\times\mathcal{X}\times\mathcal{X}}$${\mathcal{X}\times\mathcal{X}\times\mathcal{X}\times\mathcal{X}}$${\mathcal{X}\times\mathcal{X}}$${\mathcal{X}\times\mathcal{X}\times\mathcal{X}}$$\scriptstyle{d\times
1_{\mathcal{X}}\times
1_{\mathcal{X}}}$$\scriptstyle{p_{12}}$$\scriptstyle{p_{123}}$$\scriptstyle{d\times
1_{\mathcal{X}}}$ (7.14)

hence

$\displaystyle K\simeq p_{14\ast}\left((d\times 1_{\mathcal{X}}\times
1_{\mathcal{X}})_{\ast}p_{12}^{\ast}d_{\ast}\mathcal{O}_{\mathcal{X}}\otimes
p_{234}^{\ast}\Gamma_{e\ast}\mathcal{O}_{\mathcal{X}}\right)$ (7.15)
$\displaystyle\simeq p_{14\ast}(d\times 1_{\mathcal{X}}\times
1_{\mathcal{X}})_{\ast}\left(p_{12}^{\ast}d_{\ast}\mathcal{O}_{\mathcal{X}}\otimes(d\times
1_{\mathcal{X}}\times
1_{\mathcal{X}})^{\ast}p_{234}^{\ast}\Gamma_{e\ast}\mathcal{O}_{\mathcal{X}}\right)$
(7.16) $\displaystyle\simeq
p_{13\ast}\left(p_{12}^{\ast}d_{\ast}\mathcal{O}_{\mathcal{X}}\otimes{(1\times
e)}_{\ast}e_{\ast}\mathcal{O}_{\mathcal{X}}\right)$ (7.17)
$\displaystyle\simeq p_{13\ast}{(1\times e)}_{\ast}\left({(1\times
e)}^{\ast}p_{12}^{\ast}d_{\ast}\mathcal{O}_{\mathcal{X}}\otimes
e_{\ast}\mathcal{O}_{\mathcal{X}}\right)$ (7.18) $\displaystyle\simeq
d_{\ast}\mathcal{O}_{\mathcal{X}}\otimes e_{\ast}\mathcal{O}_{\mathcal{X}}.$
(7.19)

∎

######  Remark 7.3.

We can further compute the summands of the right-hand-side of (7.6) as
follows, if the stabilizer groups of $\mathcal{X}$ are assumed to be étale.

$\displaystyle\operatorname{Hom}_{\mathcal{X}}\left(p|_{W\ast}\mathcal{O}_{\mathcal{X}},\mathcal{O}_{W}\right)\simeq\operatorname{Hom}_{W}(\mathcal{O}_{W},(p|_{W})^{!}\mathcal{O}_{\mathcal{X}})\simeq\operatorname{End}_{W}(\mathcal{O}_{W})\simeq
H^{0}(W,\mathcal{O}_{W})$ (7.20)

######  Example 7.4.

Though the isomorphism (7.2), the algebra structure on the left hand side is
transferred to the right hand side. Here we describe it explicitly in the case
of classifying stacks.

Let $\mathbf{k}$ be a field and $G$ be a finite group whose order is coprime
to the characteristic of $\mathbf{k}$. By abuse of notation, we let $G$ denote
the constant group scheme $\operatorname{Spec}\left(\mathbf{k}^{G}\right)$
too. Let us consider the Hochschild cohomology of the classifying stack

$\displaystyle X\coloneqq BG=\left[\operatorname{Spec}\mathbf{k}/G\right].$
(7.21)

######  Lemma 7.5.

The $\mathbf{k}$-algebra structure of $\operatorname{HH}^{0}\left(X\right)$
for $X=BG$ coincides with the one obtained from the group stack structure of
$IX$ over $X$.

###### Proof.

The diagram(7.1) is isomorphic to the bottom square of the following tower of
Cartesian squares.

${G\times G}$${G}$${\left[\frac{G}{G}\right]}$${BG}$${BG}$${BG\times
BG}$$\scriptstyle{q}$$\scriptstyle{p}$$\scriptstyle{e}$$\scriptstyle{d}$
(7.22)

In the diagram (7.22) above, the quotient stack is taken with respect to the
left action $G\curvearrowright G$ by conjugation. It also follows from (7.22)
that the diagonal map $d=e$ is flat, so that the bottom square coincides with
the derived fiber product. In particular we obtain the natural isomorphism of
functors as follows. Note that all arrows in (7.22) are finite and flat, so
that both the pushforward and the pullback by any of them is exact already as
functors between abelian categories.

$\displaystyle e^{\ast}\circ d_{\ast}\simeq q_{\ast}\circ p^{\ast}$ (7.23)

Hence

$\displaystyle\operatorname{HH}^{0}\left(BG\right)=\operatorname{Hom}_{BG\times
BG}\left(d_{\ast}\mathcal{O}_{BG},e_{\ast}\mathcal{O}_{BG}\right)\simeq\operatorname{Hom}_{BG}\left(e^{\ast}d_{\ast}\mathcal{O}_{BG},\mathcal{O}_{BG}\right)$
(7.24) $\displaystyle\stackrel{{\scriptstyle\text{\eqref{equation:base change
formula for
BG}}}}{{\xrightarrow{\sim}}}\operatorname{Hom}_{BG}\left(q_{\ast}p^{\ast}\mathcal{O}_{BG},\mathcal{O}_{BG}\right)\simeq\operatorname{Hom}_{BG}\left(q_{\ast}\mathcal{O}_{\left[\frac{G}{G}\right]},\mathcal{O}_{BG}\right).$
(7.25)

On the other hand, by straightforward computations one can confirm that the
multiplication map

$\displaystyle IX\times_{X}IX\to IX$ (7.26)

is isomorphic to the following map, where the source of the map is the
quotient stack with respect to the diagonal conjugacy action:

$\displaystyle\left[\frac{G\times G}{G}\right]$
$\displaystyle\to\left[\frac{G}{G}\right]$ (7.27) $\displaystyle(g,h)$
$\displaystyle\mapsto gh$ (7.28)

The group structure on $IX=\left[\frac{G}{G}\right]$ given by (7.27)
corresponds to the coalgebra structure on
$q_{\ast}\mathcal{O}_{\left[\frac{G}{G}\right]}$, which is turned into the
algebra structure of $\operatorname{HH}^{0}\left(BG\right)$ through the
sequence of isomorphisms (7.24). We prove that thus-obtained algebra structure
of $\operatorname{HH}^{0}\left(BG\right)$ is nothing but the standard one;
i.e., the one coming from the fact that $\operatorname{HH}^{0}\left(BG\right)$
is the endomorphism algebra $\operatorname{End}_{BG\times
BG}\left(d_{\ast}\mathcal{O}_{BG}\right)$. To see this, we use the following
equivalences of symmetric monoidal categories.

$\displaystyle\operatorname{coh}BG\simeq\operatorname{coh}_{G}\operatorname{Spec}\mathbf{k}$
$\displaystyle\simeq\operatorname{rep}_{\mathbf{k}}G\simeq\operatorname{mod}\mathbf{k}G$
(7.29) $\displaystyle\left(V,V\to
V\otimes_{\mathbf{k}}\left(\mathbf{k}^{G}\right)\right)$
$\displaystyle\mapsto\left(V,\mathbf{k}G\otimes V\to V\right)$ (7.30)

This follows from the isomorphism of Hopf algebras as follows.

$\displaystyle\mathbf{k}^{G}=\mathcal{O}_{G}\simeq\left(\mathbf{k}G\right)^{\vee}$
(7.31)

Since the Hochschild cohomology is Morita invariant as algebras, we have the
following isomorphism of $\mathbf{k}$-algebras.

$\displaystyle\operatorname{HH}^{0}\left(BG\right)=\operatorname{HH}^{0}\left(\operatorname{coh}BG\right)\stackrel{{\scriptstyle\text{\eqref{equation:Morita
equivalence for
BG}}}}{{\simeq}}\operatorname{HH}^{0}\left(\operatorname{mod}\mathbf{k}G\right)\simeq\operatorname{Z}\left(\mathbf{k}G\right)$
(7.32)

The last isomorphism is rather easy to confirm. Therefore it remains to
confirm that the $\mathbf{k}$-algebra
$\operatorname{Hom}_{BG}\left(q_{\ast}\mathcal{O}_{\left[\frac{G}{G}\right]},\mathcal{O}_{BG}\right)$,
which appears as the last term of (7.24), where the algebra structure is
defined to be the dual of the coalgebra structure

$\displaystyle q_{\ast}\mathcal{O}_{\left[\frac{G}{G}\right]}\to
q_{\ast}\mathcal{O}_{\left[\frac{G}{G}\right]}\otimes
q_{\ast}\mathcal{O}_{\left[\frac{G}{G}\right]}$ (7.33)

corresponding to the morphism (7.27), matches the algebra structure of
$\operatorname{Z}\left(\mathbf{k}G\right)$ under the isomorphisms (7.24) and
(7.32). As an object of $\operatorname{mod}\mathbf{k}G$,
$q_{\ast}\mathcal{O}_{\left[\frac{G}{G}\right]}$ is nothing but the
$\mathbf{k}$-vector space $\mathbf{k}^{G}$ on which $G$ is acting by
conjugation. Hence

$\displaystyle\operatorname{Hom}_{BG}\left(q_{\ast}\mathcal{O}_{\left[\frac{G}{G}\right]},\mathcal{O}_{BG}\right)\simeq\operatorname{Hom}_{\operatorname{mod}\mathbf{k}G}\left(\mathbf{k}^{G},\mathbf{k}\right)$
(7.34)
$\displaystyle=\operatorname{Hom}_{\operatorname{mod}\mathbf{k}}\left(\mathbf{k}^{G},\mathbf{k}\right)^{G}\stackrel{{\scriptstyle\text{\eqref{equation:OG
= kG
dual}}}}{{\simeq}}\left(\mathbf{k}G\right)^{G}=\operatorname{Z}\left(\mathbf{k}G\right).$
(7.35)

Moreover we have the following commutative diagram in which all vertical
arrows are isomorphisms and the bottom horizontal arrow ${\widetilde{m}}$ is
defined by the commutativity of the diagram.

${\operatorname{Hom}_{BG}\left(q_{\ast}\mathcal{O}_{\left[\frac{G}{G}\right]}\otimes
q_{\ast}\mathcal{O}_{\left[\frac{G}{G}\right]},\mathcal{O}_{BG}\right)}$${\operatorname{Hom}_{BG}\left(q_{\ast}\mathcal{O}_{\left[\frac{G}{G}\right]},\mathcal{O}_{BG}\right)}$${\operatorname{Hom}_{BG}\left(q_{\ast}\mathcal{O}_{\left[\frac{G\times
G}{G}\right]},\mathcal{O}_{BG}\right)}$${\operatorname{Hom}_{\operatorname{mod}\mathbf{k}}\left(\mathbf{k}^{G}\otimes\mathbf{k}^{G},\mathbf{k}\right)^{G}}$${\left(\mathbf{k}G\otimes\mathbf{k}G\right)^{G}}$${\operatorname{Z}\left(\mathbf{k}G\right)}$$\scriptstyle{{\widetilde{m}}}$
(7.36)

By direct computations, one can confirm that the horizontal map
${\widetilde{m}}$ is nothing but the restriction of the multiplication map of
the group ring $\mathbf{k}G$ to the $G$-invariant subspaces. This concludes
the proof, since what we wanted to show is that the composition of
${\widetilde{m}}$ with the following map coincides with the ring structure of
$\operatorname{Z}\left(\mathbf{k}G\right)$.

$\displaystyle\operatorname{Z}\left(\mathbf{k}G\right)\otimes_{\mathbf{k}}\operatorname{Z}\left(\mathbf{k}G\right)=\left(\mathbf{k}G\right)^{G}\otimes_{\mathbf{k}}\left(\mathbf{k}G\right)^{G}\hookrightarrow\left(\mathbf{k}G\otimes\mathbf{k}G\right)^{G}$
(7.37)

∎

##  8\. Applications and complements

In this section we restrict to our base field $\mathbf{k}$ having
characteristic $0$. Our main result, Theorem 1.5, allows us to associate a
stack to a tame order of global dimension $2$. In this section, we first
discuss the converse construction given in [CI04]. This pair of operations is
similar to a Galois connection. The main results of this section are Corollary
8.7 and Corollary 8.8, which describe what happens when these operations are
applied in conjunction to create a closure operation.

We first recall how to associate an order to a finite flat scheme cover
$U\to\mathcal{X}$ of a stack $\mathcal{X}$. The cover induces a groupoid of
schemes:

${R\times_{U}R}$${R}$${U}$$\scriptstyle{c}$$\scriptstyle{i}$$\scriptstyle{s}$$\scriptstyle{t}$$\scriptstyle{u}$
(8.1)

The objects and morphisms in (8.1) are defined as follows.

$\displaystyle R$ $\displaystyle\coloneqq U\times_{\mathcal{X}}U$ (8.2)
$\displaystyle s$ $\displaystyle\coloneqq\mathrm{pr}_{1}$ (8.3) $\displaystyle
t$ $\displaystyle\coloneqq\mathrm{pr}_{2}$ (8.4) $\displaystyle u$
$\displaystyle\coloneqq\Delta_{U/\mathcal{X}}$ (8.5) $\displaystyle i$
$\displaystyle\coloneqq\text{``swap''}$ (8.6) $\displaystyle c$
$\displaystyle\coloneqq\left[R\times_{U}R=U\times_{\mathcal{X}}U\times_{U}U\times_{\mathcal{X}}U\xrightarrow{\sim}U\times_{\mathcal{X}}U\times_{\mathcal{X}}U\xrightarrow[]{\mathrm{pr}_{13}}U\times_{\mathcal{X}}U\right]$
(8.7)

Let $X$ be a scheme. The category of $\mathcal{O}_{X}$-bimodules is the
category of quasi-coherent sheaves on $X\times X$ with monoidal structure
given by convolution (see, e.g., [Van12, p. 6282] for the definition of
convolutions). An _$\mathcal{O}_{X}$ -bimodule algebra_ is an algebra in this
monoidal category. Note that an $\mathcal{O}_{X}$-bimodule algebra has
possibly distinct structures as a left and right $\mathcal{O}_{X}$-modules. An
_$\mathcal{O}_{X}$ -algebra_ is an $\mathcal{O}_{X}$-bimodule algebra in which
$\mathcal{O}_{X}$ is central.

We have the diagram of $\mathcal{O}_{U}$-bimodule algebras as follows, which
is dual to the diagram (8.1).

${\mathcal{O}_{R}\otimes_{\mathcal{O}_{U}}\mathcal{O}_{R}}$${\mathcal{O}_{R}}$${\mathcal{O}_{U}}$$\scriptstyle{i^{\ast}}$$\scriptstyle{c^{\ast}}$$\scriptstyle{u^{\ast}}$$\scriptstyle{s^{\ast}}$$\scriptstyle{t^{\ast}}$
(8.8)

We define a _Hopf algebroid_ over a base scheme $S$ to be a cogroupoid of
quasi-coherent sheaves of $\mathcal{O}_{S}$-algebras which will be abbreviated
as $\mathcal{O}_{R}\leftleftarrows\mathcal{O}_{U}$. We say this Hopf algebroid
is _commutative_ if $\mathcal{O}_{R}$ and $\mathcal{O}_{U}$ are commutative
$\mathcal{O}_{S}$-algebras. The category of
$\mathcal{O}_{R}\leftleftarrows\mathcal{O}_{U}$ comodules is symmetric
monoidal. In fact we have the following equivalence

${\left\\{R\rightrightarrows U\colon\mbox{ a groupoid of schemes affine over
}S\right\\}}$${\\{\mathcal{O}_{R}\leftleftarrows\mathcal{O}_{U}\colon\mbox{a
commutative Hopf algebroid over
}S\\}}$$\scriptstyle{\mathcal{O}}$$\scriptstyle{\mathop{{\mathcal{S}}pec}\nolimits}$
(8.9)

with an equivalence of symmetric monoidal categories as follows.

$\displaystyle\operatorname{qcoh}[U/R]\simeq\mathop{\text{Comod}}\left(\mathcal{O}_{R}\leftleftarrows\mathcal{O}_{U}\right)$
(8.10)

Let $\mathcal{A}\leftleftarrows\mathcal{B}$ be a Hopf algebroid over $S$, and
note that the co-composition morphism defines a co-multiplication
$\mathcal{A}\otimes_{\mathcal{B}}\mathcal{A}\leftarrow\mathcal{A}$. We say
$\mathcal{A}$ is cocommutative if this map is symmetric and $\mathcal{B}$ is
commutative. Given a Hopf algebroid $\mathcal{A}\leftleftarrows\mathcal{B}$ we
define the dual $\mathcal{A}^{\vee}$ to be the sheaf of left module
homomorphisms
$\mathop{{\mathcal{H}}om}\nolimits_{\mathcal{B}-\operatorname{Mod}}\left({}_{\mathcal{B}}\mathcal{A},\mathcal{B}\right)$,
where $\mathcal{A}$ is a left $\mathcal{B}$-module by a choice of the two
structure morphisms. In particular, when $\mathcal{A}=\mathcal{O}_{R}$, we
have that $\mathcal{O}_{R}$ is a bimodule and has $\mathcal{O}_{U}$-module
structures induced by $s^{\ast}$ and $t^{\ast}$. The construction of
$\mathcal{O}_{R}^{\vee}$ is independent of this choice. We call the algebra
$\mathcal{O}_{R}^{\vee}$ the convolution algebra of the groupoid
$R\rightrightarrows U$. We have the following correspondences

${\\{U\to\mathcal{X}\colon\mbox{ a finite flat scheme cover of
}\mathcal{X}\\}}$${\\{R\rightrightarrows U\colon\mbox{a finite flat groupoid
of schemes}\\}}$${\\{\mathcal{O}_{R}\leftleftarrows\mathcal{O}_{U}\colon\mbox{
locally free commutative Hopf algebroid over
}S\\}}$${\\{\mathcal{O}_{R}^{\vee}\leftleftarrows\mathcal{O}_{U}\colon\mbox{
locally free cocommutative Hopf algebroid over
}S\\}}$$\scriptstyle{[U/R]}$$\scriptstyle{\mathcal{O}}$$\scriptstyle{\mathop{{\mathcal{S}}pec}\nolimits}$$\scriptstyle{(-)^{\vee}}$$\scriptstyle{(-)^{\vee}}$
(8.11)

We use the notation $\mathcal{A}\leftleftarrows\mathcal{B}$ for
$\mathcal{O}_{R}^{\vee}\leftleftarrows\mathcal{O}_{U}$. The dual of a Hopf
algebroid is again a Hopf algebroid by the main result of [Sch17], and so
there are structure maps
$\mathcal{O}_{R}^{\vee}\rightrightarrows\mathcal{O}_{U}$ dual to
$\mathcal{O}_{R}\leftleftarrows\mathcal{O}_{U}$, but our shorthand only
focuses on $\mathcal{O}_{R}^{\vee}\leftleftarrows\mathcal{O}_{U}$. In our
situation, these are the same algebra map, induced by the identity map of the
groupoid. Note that the image of $\mathcal{O}_{U}$ is not in the centre of
$\mathcal{O}_{R}^{\vee}$ in general. We have equivalences of monoidal
categories as follows.

$\displaystyle\mathop{\text{Comod}}\mathcal{O}_{R}\simeq\operatorname{Mod}\mathcal{O}_{R}^{\vee}$
(8.12)

Note that $\mathcal{O}_{R}\simeq\mathcal{O}_{U\times_{\mathcal{X}}U}$ in (8.8)
has the structure of a commutative Hopf algebroid. We now apply the dual
functor
${{(-)}}^{\vee}=\mathop{{\mathcal{H}}om}\nolimits_{U}(-,\mathcal{O}_{U})$ to
obtain a cocommutative Hopf algebroid structure on $\mathcal{O}_{R}^{\vee}$.

The comultiplication gives the category
$\operatorname{Mod}\mathcal{O}_{R}^{\vee}$ the structure of a closed symmetric
monoidal category.

######  Theorem 8.1 ([CI04]).

Let $\mathcal{X}$ be a stack with a finite flat scheme cover $p\colon
U\to\mathcal{X}$ which admits a quasi-projective coarse moduli space $S$. Then
the algebra

$\displaystyle\mathcal{A}\coloneqq\operatorname{Hom}_{U}\left(s_{\ast}\mathcal{O}_{U\times_{\mathcal{X}}U},\mathcal{O}_{U}\right),$
(8.13)

which is associated to $p$ via the correspondence (8.11), is a cocommutative
Hopf algebroid over $\mathcal{O}_{U}$. There is an equivalence of symmetric
monoidal categories
$\operatorname{mod}\mathcal{A}\simeq\operatorname{coh}\mathcal{X}$. In
addition, if $U$ and $\mathcal{X}$ are normal and $\mathcal{X}$ is generically
tame, in the sense that the stabilizer of the generic point is linearly
reductive, then the Hopf algebroid $\mathcal{A}$ is an order over $S$
(possibly with centre larger than $\mathcal{O}_{S}$).

###### Proof.

The $\mathcal{O}_{U}$-bimodule $\mathcal{O}_{U\times_{\mathcal{X}}U}$ is
finite over its centralizer. Therefore we can apply [CI04, Proposition 4.4] to
see that there is an equivalence of categories
$\operatorname{mod}\mathcal{A}\simeq\operatorname{coh}\mathcal{X}$. The Hopf
algebroid and symmetric monoidal category structures are not treated in [CI04,
Theorem 3.3], but follow easily from the constructions in that paper. For the
last statement, $\mathcal{A}$ as a sheaf on $S$ is coherent since both $S$ and
the canonical map $U\to S$ are finite. It is also torsion free as $U$ is
normal and hence $\mathcal{O}_{U}$ is torsion free. Lastly, we have that
$\mathcal{A}$ is a semisimple algebra at the generic point $\eta$ since
$\operatorname{mod}\mathcal{A}_{\eta}\simeq\operatorname{coh}\mathcal{X}_{\eta}$
which is semi-simple since $\mathcal{X}$ is generically tame. ∎

The following result of Kresch and Vistoli gives conditions that ensure the
existence of a finite flat scheme cover. Quotient stacks are also
characterized in [Tot04].

######  Corollary 8.2.

Let $\mathcal{X}$ be a Deligne–Mumford stack that is separated and of finite
type over $\mathbf{k}$ with quasi-projective coarse moduli space $S$. Suppose
further that $\mathcal{X}$ is a quotient stack (in particular this holds if
$\mathcal{X}$ is smooth and generically tame). Then there exists a finite flat
scheme cover $p\colon U\to\mathcal{X}$.

###### Proof.

The finite flat scheme cover $p\colon U\to\mathcal{X}$ exists under the
assumptions of the statement (including those in the parentheses) by Kresch
and Vistoli [KV04, Theorems 1, 2, and 3] and Gabber’s Theorem [deJ]. By [CI04,
Theorem 3.3] the map $U\to S$ is finite. ∎

Given the correspondence between stacks $\mathcal{X}$ and orders $\mathcal{A}$
in Theorems 1.5 and 8.1, it is natural to ask how properties of $\mathcal{X}$
and $\mathcal{A}$ relate. Lemma 8.3–Corollary 8.5 describe the correspondence
between stacks and algebras in dimension 0, and will be applied to the generic
point of our scheme $X$. In Corollary 8.6 we describe how properties of the
stack translate to those of the order, the opposite direction of our main
theorem. Lastly, we look at the closure operation given by applying these
operations forward and backward in Corollaries 8.8 and 8.8.

Given a field $K$ and a group scheme $A$ over $K$, the cocycles in the non-
abelian Galois cohomology $H^{2}(K,A)$ are described in [Spr66] and are given
by a Galois extension $L$ of $K$ with Galois group $G$ and maps $f\colon
G\times A\to A$, $g\colon G\times G\to A$ satisfying the following conditions
for all $a\in A$ and $r,s,t\in G$.

  * •

$a\mapsto f_{s}(a)$ is an action of $G$ on $A$.

  * •

The following cocycle conditions are satisfied.

$\displaystyle f_{s}(f_{t}(a))$ $\displaystyle=g_{s,t}f_{st}(a)g_{s,t}^{-1}$
(8.14) $\displaystyle f_{r}(g_{s,t})g_{r,st}$ $\displaystyle=g_{r,s}g_{rs,t}$
(8.15)

######  Lemma 8.3.

Let $\mathcal{X}$ be a smooth and separated Deligne–Mumford stack over a
characteristic $0$ field $K$ such that the structure map
$\mathcal{X}\to\operatorname{Spec}K$ is a coarse moduli space. Then
$\mathcal{X}$ is a banded $A$-gerbe for some finite étale group scheme $A$
over $K$ and is determined by a Galois extension $L$ of $K$ with Galois group
$G$ and a cocycle $\sigma=(f,g)$ in the non-abelian Galois cohomology group
$\mathrm{H}^{2}(K,A)$ as in (8.14). There is a groupoid presenting the stack
$\mathcal{X}$

${A\times G\times A\times G\times\operatorname{Spec}L}$${(A\times
G\times\operatorname{Spec}L)\times_{\operatorname{Spec}L}(A\times
G\times\operatorname{Spec}L)}$${A\times
G\times\operatorname{Spec}L}$${\operatorname{Spec}L}$$\scriptstyle{\simeq}$$\scriptstyle{c}$$\scriptstyle{s}$$\scriptstyle{t}$
(8.16)

with convolution, source, and target maps as follows.

$\displaystyle(k,s,\ell,t,x)$ $\displaystyle\stackrel{{\scriptstyle
c}}{{\mapsto}}(kf_{s}(\ell)g_{s,t},st,x)$ (8.17) $\displaystyle(k,g,x)$
$\displaystyle\stackrel{{\scriptstyle s}}{{\mapsto}}x$ (8.18)
$\displaystyle(k,g,x)$ $\displaystyle\stackrel{{\scriptstyle t}}{{\mapsto}}gx$
(8.19)

###### Proof.

Let $\alpha\colon\mathcal{X}\to{\overline{\mathcal{X}}}$ be the rigidification
of $\mathcal{X}$. By [Ols07, Proposition 2.1], the structure map
$\mathcal{X}\to\operatorname{Spec}K$ factors as
$\mathcal{X}\xrightarrow{\alpha}{\overline{\mathcal{X}}}\xrightarrow{\beta}\operatorname{Spec}K$,
where $\beta$ is a coarse space map which is an isomorphism over a dense open
subspace of $\operatorname{Spec}K$. As a result, $\beta$ is an isomorphism. By
[AOV08, Remark A.2 and Example A.3], the map $\alpha$ is an $A$-gerbe for some
finite étale group scheme $A$ over $K$.

The classification of banded gerbes by cohomology is in Giraud [Gir71] with a
concrete realization over fields as nonabelian Galois cohomology in Springer
[Spr66]. A groupoid presentation of a gerbe constructed by a cocycle is the
same as in the proof of [Moe02, Theorem 3.1], suitably interpreted as Galois
cohomology. ∎

######  Definition 8.4.

Let $K$ be a field, let $L$ be a Galois extension of $K$ with Galois group
$G$. Let $A$ be a finite group scheme over $K$. Further assume that
$|A(L)|=\deg A$. Let $LA$ be the group algebra $LA(L)=(\mathcal{O}_{A}\otimes
L)^{\vee}$ of the constant group $A(L)$ with the coefficient field $L$. Let
$\sigma$ be a cocycle in $H^{2}(K,A)$ as above determined by $(f,g)$ with $f$
given by the action of $G$ on $LA$. We define the _crossed product algebra_
$LA\rtimes_{\sigma}G$ as the $K$-vector space $LA\otimes_{K}KG$ with
multiplication given by

$(a\otimes s)(b\otimes t)=af_{s}(b)g_{s,t}\otimes st$ (8.20)

for $a,b\in LA$ and $s,t\in G$, as in [Mon93, Definition 7.1.1].

######  Corollary 8.5.

With hypotheses as in Lemma 8.3, there is an equivalence of categories

$\operatorname{mod}(LA\rtimes_{\sigma}KG)\simeq\operatorname{coh}\mathcal{X}.$
(8.21)

###### Proof.

The description of the multiplication of the groupoid in (8.17) induces the
multiplication of the convolution algebra in (8.20) and so the result follows
from Theorem 8.1. ∎

######  Corollary 8.6.

Let $R\rightrightarrows U$ be a finite flat groupoid of schemes presenting a
Deligne–Mumford stack $\mathcal{X}=[U/R]$ over a characteristic $0$ field,
with $U$ normal and coarse moduli space $X$. We further assume that
$\mathcal{X}$ is smooth over the generic point $k(X)$. Then
$\mathcal{O}_{R}^{\vee}$ is an order in a semisimple $\mathbf{k}(X)$-algebra
which is Morita equivalent to $\operatorname{Mod}(LA\rtimes_{\sigma}KG)$ as in
Corollary 8.5 with

$\operatorname{mod}(LA\rtimes_{\sigma}KG)\simeq\operatorname{coh}(\mathcal{X}\times_{X}\mathbf{k}(X)).$
(8.22)

Moreover, $\mathcal{O}_{R}^{\vee}$ is a tame order and
$Z(\mathcal{O}_{R}^{\vee})$ is the integral closure of $\mathcal{O}_{X}$ in
$\operatorname{Z}(LA\rtimes_{\sigma}KG)$. If $\mathcal{X}$ is smooth, we can
also identify the centre in the following way. Let $I$ be the union of the
maximal dimensional components of the coarse moduli space of the inertia stack
of $\mathcal{X}$. Then
$\mathcal{O}_{I}^{\vee}\coloneqq\operatorname{Z}(\mathcal{O}_{R}^{\vee})$ has
$\mathcal{O}_{I}^{\vee}\simeq\mathcal{O}_{I}$ as a coherent sheaf over $X$.

###### Proof.

The behaviour at the generic point follows from Corollary 8.5. We can localize
$\operatorname{coh}\mathcal{X}\simeq\operatorname{mod}\mathcal{O}_{R}^{\vee}$
to codimension one to show that $\mathcal{O}_{R}^{\vee}$ is hereditary. Since
$U$ is normal, $\mathcal{O}_{R}^{\vee}$ is reflexive. Since
$\mathcal{O}_{R}^{\vee}$ is tame, its centre is normal, giving the first
description of the centre. The last statement about the centre follows from
Proposition 7.1 and Remark 7.3. ∎

If we start with an order, associate a stack using Theorem 1.5, and then take
its convolution algebra via Theorem 8.1, we obtain an order whose module
category is monoidal. The result is Corollary 8.7 below. Recall from
Definition 5.2 that given a $\mu_{N}$-gerbe $\mathcal{X}$ the category of
$i$-twisted sheaves is $\operatorname{qcoh}\mathcal{X}^{(i)}$.

######  Corollary 8.7.

Let $\mathcal{A}_{1}$ be a tame split order of global dimension 2 over a
surface $X$ as in Definition 1.4 with generic point $\eta$. Then there exists
a $\mu_{N}$-gerbe $\mathcal{X}$ over a smooth tame algebraic stack of
dimension $2$ whose coarse space is $X$ for some $N$ and algebras
$\mathcal{A}_{2},\ldots,\mathcal{A}_{N}$ over $X$ such that

  1. (1)

$\operatorname{mod}\prod\mathcal{A}_{i}\simeq\operatorname{qcoh}\mathcal{X}$.

  2. (2)

$\operatorname{mod}\mathcal{A}_{i}\simeq\operatorname{qcoh}\mathcal{X}^{(i)}$.

  3. (3)

$\mathcal{A}_{i,\eta}$ is Morita equivalent to $\mathcal{A}_{1,\eta}^{\otimes
i}$

  4. (4)

$N$ is the order of $\mathcal{A}_{1,\eta}$ in
$\operatorname{Br}\mathbf{k}(\eta)$.

As a consequence of this result, given $\mathcal{A}_{1}$ as above, we can
associate ${\overline{\mathcal{A}}}\coloneqq\prod_{i}\mathcal{A}_{i}$ which
enjoys the geometric properties of being Morita equivalent to a stack. In
particular, we have the following structures for
$\operatorname{mod}{\overline{\mathcal{A}}}$:

  * •

a symmetric monoidal structure

  * •

tangent and cotangent sheaves

  * •

Picard group

Next, if we start with a stacky surface and finite flat scheme cover, we may
form the convolution algebra by Theorem 8.1, and then construct the associated
stack by Theorem 1.5. The result is as follows. We define
$\operatorname{\mathcal{HH}}^{0}(\mathcal{X})$ to be the sheaf of algebras
over the coarse moduli space given by Hochschild cohomology locally.

######  Corollary 8.8.

Let $\mathcal{X}$ be a smooth tame algebraic stack of dimension at most $2$
which admits a finite flat scheme cover $R\rightrightarrows U\to\mathcal{X}$.
Then the convolution algebra $\mathcal{O}_{R}^{\vee}$ is a tame order over
$\operatorname{\mathcal{HH}}^{0}(\mathcal{X})$ with global dimension equal to
$\dim\mathcal{X}$. Furthermore, there is a stack ${\overline{\mathcal{X}}}$
such that ${\overline{\mathcal{X}}}=\coprod\mathcal{X}_{i}$ where
$\mathcal{X}_{i}$ are $\mu_{n_{i}}$-gerbes over smooth tame stacks with
trivial generic stabilizer over components of
$\mathop{{\mathcal{S}}pec}\nolimits\operatorname{\mathcal{HH}}^{0}(\mathcal{X})$.

Furthermore, $\operatorname{qcoh}\mathcal{X}$ is a component of
$\operatorname{qcoh}{\overline{\mathcal{X}}}$ as monoidal categories. If
$\mathcal{X}$ has trivial generic stabilizer, then
$\mathcal{X}\simeq{\overline{\mathcal{X}}}$.

###### Proof.

For the first statement, we have $\mathcal{X}=[U/R]$ and we may take
$U\to\mathcal{X}$ as our finite cover with convolution algebra
$\mathcal{O}_{R}^{\vee}.$ This is a tame order by Corollary 8.6. We have an
equivalence of categories
$\operatorname{qcoh}\mathcal{X}\simeq\operatorname{mod}\mathcal{O}_{R}^{\vee}$.
So the global dimension of $\mathcal{O}_{R}^{\vee}$ is the dimension of the
coarse moduli space of $\mathcal{X}$. To build the stack
${\overline{\mathcal{X}}}$ we apply our main construction to each component of
$\mathcal{O}_{R}^{\vee}$.

For the last statement, we know that
$\operatorname{qcoh}{\overline{\mathcal{X}}}\simeq\operatorname{qcoh}\mathcal{X}$
as monoidal categories so the reconstruction theorem [Lur04, Remark 5.12] for
stacks applies to show the result. ∎

######  Example 8.9.

Let $\mathcal{X}=BG$ for some linearly reductive group scheme $G$. Then the
scheme cover $\operatorname{Spec}\mathbf{k}\to\mathcal{X}$ gives us the
convolution algebra $\mathbf{k}G$. Since $G$ is linearly reductive we have
$\mathbf{k}G\simeq\prod_{i}D_{i}^{n_{i}\times n_{i}}$ for some division
algebras $D_{i}$ with order $\ell_{i}$ in
$\operatorname{Br}\operatorname{Z}\left(D_{i}\right)$. On the other hand we
have
$\prod_{\operatorname{Irrep}G}\mathbf{k}\simeq\operatorname{Z}\left(\prod_{i}D_{i}\right)\simeq\operatorname{Z}\left(\mathbf{k}G\right)\simeq\operatorname{HH}^{0}(\mathcal{X})$,
where, as we saw in Lemma 7.5, the ring structure of the last term is obtained
as the $\mathbf{k}$-dual of the comultiplication of
$\mathcal{O}_{I\mathcal{X}}$ corresponding to the group structure of
$I\mathcal{X}$ over $\mathcal{X}$. We attach the trivial
$\mu_{\ell_{i}}$-gerbe $\mathcal{X}_{i}\coloneqq B\mu_{\ell_{i}}$ over the
corresponding component of
$\operatorname{Spec}\operatorname{HH}^{0}(\mathcal{X})$ to each $D_{i}$ with
$\operatorname{mod}D_{i}\simeq\mathcal{X}_{i}^{(1)}$. Then
${\overline{\mathcal{X}}}=\coprod_{i}\mathcal{X}_{i}$. Note that
${\overline{\mathcal{X}}}$ has only abelian stabilizer subgroups whereas $G$
may be non-abelian.

## References

  * [ACH19] Dima Arinkin, Andrei Căldăraru and Márton Hablicsek  “Formality of derived intersections and the orbifold HKR isomorphism”  In _J. Algebra_ 540, 2019, pp. 100–120  DOI: 10.1016/j.jalgebra.2019.08.002
  * [AGV08] Dan Abramovich, Tom Graber and Angelo Vistoli  “Gromov-Witten theory of Deligne-Mumford stacks”  In _Amer. J. Math._ 130.5, 2008, pp. 1337–1398  DOI: 10.1353/ajm.0.0017
  * [AOV08] Dan Abramovich, Martin Olsson and Angelo Vistoli  “Tame stacks in positive characteristic”  In _Ann. Inst. Fourier (Grenoble)_ 58.4, 2008, pp. 1057–1091  URL: http://aif.cedram.org/item?id=AIF_2008__58_4_1057_0
  * [Art69] M. Artin  “Algebraic approximation of structures over complete local rings”  In _Inst. Hautes Études Sci. Publ. Math._ , 1969, pp. 23–58  URL: https://mathscinet.ams.org/mathscinet-getitem?mr=268188
  * [Bic21] Tristan Bice  “An algebraic approach to the Weyl groupoid”  In _J. Algebra_ 568, 2021, pp. 193–240  DOI: 10.1016/j.jalgebra.2020.10.010
  * [BLS16] Daniel Bergh, Valery A. Lunts and Olaf M. Schnürer  “Geometricity for derived categories of algebraic stacks”  In _Selecta Math. (N.S.)_ 22.4, 2016, pp. 2535–2568  DOI: 10.1007/s00029-016-0280-8
  * [Cad07] Charles Cadman  “Using stacks to impose tangency conditions on curves”  In _Amer. J. Math._ 129.2, 2007, pp. 405–427  DOI: 10.1353/ajm.2007.0007
  * [CI04] Daniel Chan and Colin Ingalls  “Non-commutative coordinate rings and stacks”  In _Proc. London Math. Soc. (3)_ 88.1, 2004, pp. 63–88  DOI: 10.1112/S0024611503014278
  * [Con94] Alain Connes  “Noncommutative geometry”  Academic Press, Inc., San Diego, CA, 1994, pp. xiv+661
  * [deJ] Johan deJong  “A result of Gabber”  URL: https://www.math.columbia.edu/~dejong/papers/2-gabber.pdf
  * [Gir71] Jean Giraud  “Cohomologie non abélienne”, Die Grundlehren der mathematischen Wissenschaften, Band 179  Springer-Verlag, Berlin-New York, 1971, pp. ix+467
  * [GS17] Anton Geraschenko and Matthew Satriano  “A “bottom up” characterization of smooth Deligne-Mumford stacks”  In _Int. Math. Res. Not. IMRN_ , 2017, pp. 6469–6483  DOI: 10.1093/imrn/rnw201
  * [Huy06] D. Huybrechts  “Fourier-Mukai transforms in algebraic geometry”, Oxford Mathematical Monographs  Oxford: The Clarendon Press Oxford University Press, 2006, pp. viii+307  DOI: 10.1093/acprof:oso/9780199296866.001.0001
  * [IW14] Osamu Iyama and Michael Wemyss  “Maximal modifications and Auslander-Reiten duality for non-isolated singularities”  In _Invent. Math._ 197.3, 2014, pp. 521–586  DOI: 10.1007/s00222-013-0491-y
  * [Kel03] Bernhard Keller  “Derived invariance of higher structures on the Hochschild complex” available at https://webusers.imj-prg.fr/~bernhard.keller/publ/dih.pdf, 2003
  * [Kol13] János Kollár  “Singularities of the minimal model program” With a collaboration of Sándor Kovács 200, Cambridge Tracts in Mathematics  Cambridge University Press, Cambridge, 2013, pp. x+370  DOI: 10.1017/CBO9781139547895
  * [Kre09] Andrew Kresch  “On the geometry of Deligne-Mumford stacks”  In _Algebraic geometry—Seattle 2005. Part 1_ 80, Proc. Sympos. Pure Math.  Amer. Math. Soc., Providence, RI, 2009, pp. 259–271  DOI: 10.1090/pspum/080.1/2483938
  * [Kum86] Alexander Kumjian  “On $C^{\ast}$-diagonals”  In _Canad. J. Math._ 38.4, 1986, pp. 969–1008  DOI: 10.4153/CJM-1986-048-0
  * [KV04] Andrew Kresch and Angelo Vistoli  “On coverings of Deligne-Mumford stacks and surjectivity of the Brauer map”  In _Bull. London Math. Soc._ 36.2, 2004, pp. 188–192  DOI: 10.1112/S0024609303002728
  * [Lie08] Max Lieblich  “Twisted sheaves and the period-index problem”  In _Compos. Math._ 144.1, 2008, pp. 1–31  DOI: 10.1112/S0010437X07003144
  * [Lip69] Joseph Lipman  “Rational singularities, with applications to algebraic surfaces and unique factorization”  In _Inst. Hautes Études Sci. Publ. Math._ , 1969, pp. 195–279
  * [LMM21] Christian Liedtke, Gebhard Martin and Yuya Matsumoto  “Linearly Reductive Quotient Singularities”  In _arXiv e-prints_ , 2021, pp. arXiv:2102.01067  arXiv:2102.01067 [math.AG]
  * [LR19] Xin Li and Jean Renault  “Cartan subalgebras in ${\rm C}^{*}$-algebras. Existence and uniqueness”  In _Trans. Amer. Math. Soc._ 372.3, 2019, pp. 1985–2010  DOI: 10.1090/tran/7654
  * [LS14] Christian Liedtke and Matthew Satriano  “On the birational nature of lifting”  In _Adv. Math._ 254, 2014, pp. 118–137  DOI: 10.1016/j.aim.2013.10.030
  * [Lur04] Jacob Lurie  “Tannaka Duality for Geometric Stacks”  arXiv, 2004  DOI: 10.48550/ARXIV.MATH/0412266
  * [Mat21] Siddharth Mathur  “The resolution property via Azumaya algebras”  In _J. Reine Angew. Math._ 774, 2021, pp. 93–126  DOI: 10.1515/crelle-2021-0002
  * [Mil80] James S. Milne  “Étale cohomology” 33, Princeton Mathematical Series  Princeton University Press, Princeton, N.J., 1980, pp. xiii+323
  * [Moe02] Ieke Moerdijk  “Introduction to the language of stacks and gerbes”  arXiv, 2002  DOI: 10.48550/ARXIV.MATH/0212266
  * [Mon93] Susan Montgomery  “Hopf algebras and their actions on rings” 82, CBMS Regional Conference Series in Mathematics  Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1993, pp. xiv+238  DOI: 10.1090/cbms/082
  * [MR01] J.. McConnell and J.. Robson  “Noncommutative Noetherian rings” With the cooperation of L. W. Small 30, Graduate Studies in Mathematics  American Mathematical Society, Providence, RI, 2001, pp. xx+636  DOI: 10.1090/gsm/030
  * [Ols07] Martin Olsson  “A boundedness theorem for Hom-stacks”  In _Math. Res. Lett._ 14.6, 2007, pp. 1009–1021  DOI: 10.4310/MRL.2007.v14.n6.a9
  * [Ols16] Martin Olsson  “Algebraic spaces and stacks” 62, American Mathematical Society Colloquium Publications  American Mathematical Society, Providence, RI, 2016, pp. xi+298  DOI: 10.1090/coll/062
  * [Orl16] Dmitri Orlov  “Smooth and proper noncommutative schemes and gluing of DG categories”  In _Adv. Math._ 302, 2016, pp. 59–105  DOI: 10.1016/j.aim.2016.07.014
  * [Ram69] Mark Ramras  “Maximal orders over regular local rings of dimension two”  In _Trans. Amer. Math. Soc._ 142, 1969, pp. 457–479  DOI: 10.2307/1995367
  * [Rei03] I. Reiner  “Maximal orders” Corrected reprint of the 1975 original, With a foreword by M. J. Taylor 28, London Mathematical Society Monographs. New Series  The Clarendon Press, Oxford University Press, Oxford, 2003, pp. xiv+395  URL: https://mathscinet.ams.org/mathscinet-getitem?mr=1972204
  * [Rev73] Philippe Revoy  “Algèbres de Weyl en caractéristique $p$”  In _C. R. Acad. Sci. Paris Sér. A-B_ 276, 1973, pp. A225–A228
  * [RV89] Idun Reiten and Michel Van den Bergh  “Two-dimensional tame and maximal orders of finite representation type”  In _Mem. Amer. Math. Soc._ 80.408, 1989, pp. viii+72  DOI: 10.1090/memo/0408
  * [Sat12] Matthew Satriano  “The Chevalley-Shephard-Todd theorem for finite linearly reductive group schemes”  In _Algebra Number Theory_ 6.1, 2012, pp. 1–26  DOI: 10.2140/ant.2012.6.1
  * [Sch17] Peter Schauenburg  “The dual and the double of a Hopf algebroid are Hopf algebroids”  In _Appl. Categ. Structures_ 25.1, 2017, pp. 147–154  DOI: 10.1007/s10485-015-9418-7
  * [Spr66] T.. Springer  “Nonabelian $H^{2}$ in Galois cohomology”  In _Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965)_ Amer. Math. Soc., Providence, R.I., 1966, pp. 164–182
  * [SV08] J.. Stafford and M. Van den Bergh  “Noncommutative resolutions and rational singularities” Special volume in honor of Melvin Hochster  In _Michigan Math. J._ 57, 2008, pp. 659–674  DOI: 10.1307/mmj/1220879430
  * [Tot04] Burt Totaro  “The resolution property for schemes and stacks”  In _J. Reine Angew. Math._ 577, 2004, pp. 1–22  DOI: 10.1515/crll.2004.2004.577.1
  * [Van12] M. Van den Bergh  “Non-commutative $\mathbb{P}^{1}$-bundles over commutative schemes”  In _Trans. Amer. Math. Soc._ 364.12, 2012, pp. 6279–6313  DOI: 10.1090/S0002-9947-2012-05469-9
  * [Vis89] Angelo Vistoli  “Intersection theory on algebraic stacks and on their moduli spaces”  In _Invent. Math._ 97.3, 1989, pp. 613–670  DOI: 10.1007/BF01388892
  * [Yek99] Amnon Yekutieli  “Dualizing complexes, Morita equivalence and the derived Picard group of a ring”  In _J. London Math. Soc. (2)_ 60.3, 1999, pp. 723–746  DOI: 10.1112/S0024610799008108