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BRAUER GROUPS AND QUOTIENT STACKS

DAN EDIDIN BRENDAN HASSETT ANDREW KRESCH AND ANGELO VISTOLI

Abstract A natural question is to determine which algebraic stacks are

quotient stacks In this pap er we give some partial answers and relate it

to the old question of whether for a X the natural map from the

Brauer equivalence classes of Azumaya algebras to the cohomolog

2

ical the torsion subgroup of H X G is surjective

m

Introduction

Quotients of varieties by algebraic groups arise in many situations for in

stance in the theory of mo duli where mo duli spaces are often naturally con

structed as quotients of parameter spaces by linear algebraic groups The

quotient of a scheme by a group need not exist as a scheme or even as an

algebraic and even when a quotient exists the quotient morphism may

not have exp ected prop erties For example if Z and G are smo oth then the

morphism Z Z G need not b e smo oth

To overcome this diculty it is often helpful to consider quotients as stacks

rather than as schemes or algebraic spaces If G is a at group scheme acting

on an Z G must b e separated and nitely presented over some

base scheme with the space Z and the action map dened over this base then

a quotient Z G always exists as a and this stack is algebraic Knowing

that an has a presentation as a quotient Z G with G a linear

say can make the stack easier to study for then the geometry

of the stack is the Gequivariant geometry on the space Z

A natural question is to determine which algebraic stacks are quotient stacks

In this pap er we give some partial answers to this question and relate it to the

old question of whether for a scheme X the natural map from the Brauer

group classes of Azumaya algebras mo dulo an to the

cohomological Brauer group the torsion subgroup of etale H X G is sur

m

jective

Edidin received supp ort from the NSA NSF and the University of Missouri Research

Board while preparing this pap er Hassett and Kresch were partially supp orted by NSF

Postdo ctoral Research Fellowships Hassett received additional supp ort from the Institute

of Mathematical Sciences of the Chinese University of Hong Kong and NSF Vistoli was

partially supp orted by the University of Bologna funds for selected research topics

D EDIDIN B HASSETT A KRESCH AND A VISTOLI

Some quick answers to this natural question are the rst two are folklore

i all are quotient stacks Theorem ii all regular Deligne

Mumford stacks of are quotient stacks Example iii

there exists a DeligneMumford stack normal and of nite typ e over the com

plex numb ers but singular and nonseparated which is not a

Example

In fact the example in iii is a stack with stabilizer group Z at every

p oint it is a gerb e over a normal but nonseparated scheme of dimension

over the complex numb ers Theorem says such a stack is a quotient stack

if and only if a certain class in the cohomological Brauer group asso ciated

with it lies in the image of the map from the Brauer group So iii yields an

example of indep endent interest of nonsurjectivity of the Brauer map for a

nitetyp e normal but nonseparated scheme Corollary This stands in

contrast with the recent result of S Schroer Sch which says that the Brauer

map is surjective for any separated geometrically normal algebraic surface

The pap er is organized as follows In Section we review the denition of

algebraic stacks and state accompanying results relative to quotient stacks

Additional results concern nite covers of stacks by schemes In Section we

review gerb es and Brauer groups and state the result relating the Brauer map

to gerb es b eing quotient stacks Finally in Section we give pro ofs

Acknowledgements The authors thank Andrei Caldararu Bill Graham

and Amnon Yekutieli for helpful discussions They are also grateful to Laurent

MoretBailly and the referee for a numb er of corrections and suggestions

Stacks and quotient stacks

Stacks Here we give a brief review of stacks Some references are DM

Vi and LMB

Stacks are categories b ered in group oids satisfying typ e axioms

the stacks of interest to us will b e algebraic and hence admit descriptions

in the form of group oid schemes First recall that a group oid is a small

C in which all arrows are isomorphisms Write R HomC and

X Ob jC There are two maps s t R X sending a morphism to its

source and target resp ectively a map e X R taking an ob ject to the

identity morphism of itself a map i R R taking a morphism to its inverse

and a map m R R R taking a pair of comp osable morphisms to their

tX s

comp osition Write j t s R X X There are obvious compatibilities

b etween these maps

A group oid scheme consists of schemes R and X dened over a xed base

scheme L together with maps s t e i m satisfying the same compatibility

conditions as ab ove A group oid scheme is called etale resp ectively smo oth

resp ectively at if the maps s and t are etale resp smo oth resp faithfully

at and lo cally of nite presentation The stabilizer of a group oid scheme is

BRAUER GROUPS AND QUOTIENT STACKS

the scheme S j here X X is the diagonal This is a group

X X

scheme over X

Let L b e a xed ground scheme and let F b e a category together with

a functor p F Sch L For a xed Lscheme B let F B denote the

sub category of F consisting of ob jects mapping to B and morphisms mapping

to Roughly a category bered in over L is pair consisting of a

B

category F and a functor p F Sch L such that

i For all Lschemes B F B is a group oid

0

ii For any morphism of Lschemes f B B and any ob ject x F B

 0

there is an ob ject f x in F B unique up to canonical isomorphism together



with a morphism f x x lying over f For the precise denition see eg

DM Sec

A morphism of categories b ered in group oids is simply a functor commuting

with the pro jection functors to Sch L An isomorphism of categories b ered

in group oids is a morphism which is an equivalence of categories

Any contravariant functor Sch L sets determines a category b ered in

group oids We say that a category b ered in group oids over L is represented

by a scheme resp algebraic space if it is equivalent to the functor of p oints

of a scheme resp algebraic space

An imp ortant construction is the b er pro duct Given morphisms f F

F and f F F the b er pro duct F F is the category b ered in

F

group oids dened as follows Ob jects are triples x x where x is an

ob ject of F x is an ob ject of F and f x f x is an isomorphism

lying over an identity morphism of Sch L A morphism is sp ecied by a pair

of morphisms compatible with the induced isomorphism in F

Denition A category b ered in group oids F p is a stack if it satises

two descent prop erties

For ob jects x y in F B the functor Iso x y Sch B sets assigning

B

0  

to a B scheme f B B the set of isomorphisms b etween f x and f y is a

for the etale top ology

F has eective descent for etale morphisms

Denition A morphism of stacks is representable if for any morphism of

0

an algebraic space B F the b er pro duct B F is represented by an

F

algebraic space A morphism is strongly representable if for any morphism of

0

a scheme B F the b er pro duct B F is represented by a scheme

F

Let P b e a prop erty of which is preserved by base

0

change and is lo cal for the smo oth top ology A representable morphism F

F has prop erty P if for all morphisms B F of algebraic spaces the induced

0

morphism B F B has prop erty P

F

Stein factorization holds for algebraic spaces and implies Kn I I that

if f X Y is a separated quasinite morphism of algebraic spaces and if Y

D EDIDIN B HASSETT A KRESCH AND A VISTOLI

is a scheme then X is a scheme Hence a representable separated quasinite

morphism is always strongly representable

Denition A stack F is algebraic or is an Artin stack if

There exists a representable smo oth surjective morphism X F from a

scheme

The diagonal morphism F F F is representable quasicompact and

L

separated

Remark The representability of the diagonal implies that any morphism

from an algebraic space is representable For stacks with quasinite diagonal

any morphism from a scheme is strongly representable

Remark A stack F is called a DeligneMumford stack if there exists an

etale cover of F by a scheme By LMB this happ ens if and only if

the diagonal F F F is unramied A DeligneMumford stack has in

L

particular quasinite diagonal The geometric b ers of the diagonal are group

schemes so if all the residue elds of L have then conversely

any algebraic stack with quasinite diagonal is a DeligneMumford stack

Finally we describ e very briey group oid presentations or atlases of al

gebraic stacks see LMB for a full treatment By denition any algebraic

stack F admits a smo oth surjective map from a scheme X X F is called

a smooth atlas In this case the b er pro duct R X X is an algebraic

F

space However for stacks with quasinite diagonal the diagonal is strongly

representable so R is in fact a scheme The smo oth group oid scheme R  X is

called a presentation for F Conversely any smo oth group oid scheme R  X

with separated nitetyp e relative diagonal R X X determines an al

gebraic stack R  X A theorem of Artin cf LMB says that any

faithfully at group oid scheme R  X with separated nitetyp e relative di

agonal determines an algebraic stack In this case the group oid scheme R  X

is called a faithful ly at presentation for F By Remark an algebraic stack

is a DeligneMumford stack if and only if it has an etale presentation If F is an

algebraic stack with quasinite diagonal then it is relatively straightforward

KM Lemma to show that F has a quasinite faithfully at atlas of

schemes

If the group scheme G acts on the algebraic space Z we assume G at

separated and of nite presentation over the ground scheme L the space Z

should b e an Lspace and the action map Z G Z an Lmorphism then

L

the action determines a group oid Z G  Z This will b e a at atlas for the

L

stack whose b er over any Lscheme T is the category of principal Gbundles

1

As explained in LMB unramied should b e understo o d to mean lo cally of nite

typ e and formally unramied

BRAUER GROUPS AND QUOTIENT STACKS

E T together with Gequivariant morphisms E Z This is an algebraic

stack denoted Z G

As noted ab ove any algebraic space is an algebraic stack the following

result says when the converse holds

Prop osition LMB and Let F be an algebraic stack and

let s t R  X be a faithful ly at presentation for F Then F is an algebraic

space if and only if the map R X X is a monomorphism If we set

S t s then this is equivalent to S X being an isomorphism by

X

either s or t

We call attention to the map S X of Prop osition The b er pro duct of

the diagonal F F F with itself is an algebraic stack I F F The

F F F

pro jection to either factor I F is the stabilizer map and is represented

F

by the stabilizer S X of the group oid space R  X for any atlas X

Results on stacks The rst theorem states that stacks with quasinite

diagonal are nitely parametrized ie admit nite covers by schemes This is

the strongest p ossible result since any nitely parametrized stack must have

quasinite diagonal This result extends results of Vistoli Vi and Laumon

and MoretBailly LMB for DeligneMumford stacks The rst result of this

form of which the authors are aware is due to Seshadri Se Theorem in

the context of group actions on varieties In fact the use of Lemma was

inspired by reading his pap er

Theorem Let F be an algebraic stack of nite type over a Noetherian

ground scheme L Then the diagonal F F F is quasinite if and

L

only if there exists a nite surjective morphism X F from a not necessarily

separated scheme X

Remark Existence of nite scheme covers is an imp ortant ingredient in in

tersection theory on DeligneMumford stacks It is used for instance to dene

prop er pushforward for nonrepresentable morphisms of cycles mo dulo ratio

nal equivalence with Q co ecients General intersectiontheoretic machinery

has recently b een develop ed for Artin stacks whose geometric stabilizers are

ane groups Kr All of intersection theory on DeligneMumford stacks as

in Gi and Vi generalizes to Artin stacks with quasinite diagonal where

Theorem is used to provide nonrepresentable prop er pushforwards

Denition Let F b e a stack of nite typ e over a No etherian base scheme

L We say F is a quotient stack if F is isomorphic to a stack of the form Z G

where Z is an algebraic space of nite typ e over L and G is a subgroup scheme

of the general linear group scheme GL for some n with G at over L nL

D EDIDIN B HASSETT A KRESCH AND A VISTOLI

Remark Every ane group scheme of nite typ e over a eld is a subgroup

scheme of GL so the condition on G in Denition is the natural notion

n

of linear algebraic group over a general No etherian base

0

Remark The quotient Z Z GL G where G acts on Z and acts

L nL

0

by translation on GL exists as an algebraic space and Z GL Z G

nL n

So every quotient stack is a quotient by GL for some n

n

We state two foundational results followed by two results giving sucient

conditions for a stack to b e a quotient stack Recall that f E F is a

pro jective morphism if and only if f factors up to isomorphism as a closed

immersion followed by pro jection E PE F where E is a nitetyp e

quasicoherent sheaf on F and PE denotes its pro jectivization

Lemma Let F be an algebraic stack of nite type over a Noetherian

scheme The fol lowing are equivalent

i F is a quotient stack

ii There exists a vector bund le V F such that at every geometric point

the stabilizer action on the ber is faithful

iii There exists a vector bund le V F and a local ly closed substack V V

such that V is representable and V surjects onto F

Lemma Let E F be a at projective map of stacks of nite type

over a Noetherian base scheme which is surjective If E is a quotient stack

then so is F

Theorem Let F be an algebraic stack of nite type over a Noetherian

scheme and let f X F be a nite cover by a scheme or algebraic space

If the coherent sheaf f O is the quotient of a local ly free coherent sheaf then

 X

F Z GL where Z is an algebraic space In particular if every coherent

n

sheaf on F is the quotient of a local ly free coherent sheaf then F is a quotient

stack

Remark If the ground scheme L is normal and separated and has the

prop erty that every coherent sheaf on L is the quotient of a lo cally free sheaf

eg if L is ane or regular and if F Z GL where Z is a scheme

n

equivariantly emb edded in a regular No etherian separated scheme then the

equivariant resolution theorem of Th implies that every coherent sheaf on F

is the quotient of a lo cally free coherent sheaf on F

Corollary Let F be an algebraic stack of nite type over a Noetherian

scheme If F has a nite at cover by an algebraic space then F is a quotient

stack In particular if F is regular and has a nite cover by a CohenMacaulay

algebraic space then F is a quotient stack

BRAUER GROUPS AND QUOTIENT STACKS

Example Assume the base scheme is a eld or more generally any uni

versally Japanese scheme for instance Sp ec Z Any regular stack of dimension

with quasinite diagonal is a quotient stack

We emphasize the second statement of Corollary b ecause the Deligne

Mumford stacks considered by Mumford in Mu satisfy ii In particular they

are quotient stacks and the intersection pro duct he constructs is a sp ecial case

of the intersection pro duct of EG

Finally if F is DeligneMumford then we have the following result which we

obtained based on conversations with Bill Graham In the characteristic zero

setting this result is familiar from the study of orbifolds

Theorem If F is a smooth DeligneMumford stack of nite type over

the Noetherian base scheme such that the automorphism group of a general

geometric point of F is trivial then F is a quotient stack

Thus any stack which admits a representable morphism to a smo oth Deligne

Mumford stack with trivial generic stabilizers also is a quotient stack

Corollary Let F be a smooth DeligneMumford stack of nite type over

a Noetherian base scheme Assume F has nite stabilizer and suppose the

automorphism group of a general geometric point of F has trivial Then

F is a quotient stack

Recall that an algebraic space Q has quotient singularities if lo cally in the

etale top ology Q is isomorphic to quotients U H where H is a nite group

and U is smo oth By Vi Prop osition any separated scheme of nite typ e

over a eld of characteristic zero with quotient singularities is a mo duli space

for a smo oth stack F which has generically trivial stabilizer so we have the

following consequence

Corollary Any separated scheme of nite type over a eld of character

istic which has at worst quotient singularities is a quotient Q Z G where

Z is a smooth algebraic space and G is a linear algebraic group

Lastly as promised not every DeligneMumford stack is a quotient stack

Example Let Y b e the scheme Sp ec C x y z xy z whose non

reg

singular lo cus is Y Y r fg There is a unique up to isomorphism

reg

nontrivial involution of Y B Z which commutes with the pro jection

reg

Let F b e the stack gotten by glueing two copies of Y B Z map to Y

via this involution Then F is not isomorphic to Z G for any algebraic space

Z and algebraic group G

D EDIDIN B HASSETT A KRESCH AND A VISTOLI

and Brauer groups

In this section we give a brief review of gerb es and Brauer groups and state

our accompanying results References for gerb es are Mi and LMB For

Brauer groups see Gr and Mi

Gerb es In what follows we x a base scheme X assumed No etherian

and we take G to b e a group scheme at separated and of nite typ e over

X The gerb es that arise in the theorem that relates gerb es to Brauer groups

Theorem have G equal to the algebraic torus G or a group of ro ots of

m

unity We only discuss gerb es that are mo deled on some group scheme G

n

over the base

Denition A G over X is a morphism F X with F an algebraic

stack such that there exists a faithfully at map lo cally of nite presentation

0 0 0

X X such that F X BG X

X X

We say the Ggerb e F X is trivial if F BG Note that a gerb e

F X which admits a section x F X satises F B Aut x where

F

Aut x is group scheme or group space Iso x x such a gerb e is called

F F

neutral Nontrivial gerb es are easy to construct much the way one constructs

nontrivial vector bundles or torsors For instance one can glue two copies of

A B Z along a nontrivial involution of A r fg B Z to obtain a

nontrivial Zgerb e over P

Denition Let G and H b e two group schemes over X The sheaf of

band isomorphisms denoted BandG H is the sheacation of the quotient

of the sheaf of group isomorphisms IsoG H by the conjugation action of H

When G H this is the sheaf of outer automorphisms of G which is denoted

OutG

Denition Given a Ggerb e F X the associated torsor of outer auto

morphisms is the sheaf P over X dened as follows Let T b e an X scheme If

there exists an ob ject t F T then we dene P T to b e BandAut t G

F X

T One checks that if t denotes another ob ject in F T then there is a

canonical element BandAut t Aut t obtained by chosing lo cal isomor

F F

phisms of t with t this canonically identies BandAut t G T with

F X

BandAut t G T In general P T is dened as the dierence kernel

F X

0 0 0 0 0 0

P T  P T T with resp ect to any at cover T T such that t F T

T

exists Elements of P T pull back in the obvious fashion

There is an obvious action of OutG on P making it into a torsor This

torsor is classied by some H X OutG This is the rst obstruction

to triviality of F

For the remainder of this section we assume that OutG is a nite at

group scheme over X This is the case when i G is nite group viewed

BRAUER GROUPS AND QUOTIENT STACKS

as a group scheme over Sp ec Z and hence over any base ii G for

n

any p ositive integer n iii G G Now there are two ways to remove the

m

rst obstruction to triviality for a gerb e First one can hop e that is in

the image of H X AutG H X OutG and then use the AutG

0

co cycle to substitute in place of G a new group scheme G lo cally isomorphic

to G For instance if the symmetric group S acts on A r fg by z

sgn z then F A r fgS is nontrivial as a Zgerb e over X A r

fg Its rst obstruction class is the nontrivial element of H X AutZ

0

H X Out Z Twisting we obtain a group scheme G over X and we nd

0

in this example that F BG

The second metho d which do esnt require hoping is to pull back to the

total space of the OutGtorsor So the rst obstruction to triviality vanishes

upon nite at pul lback

Assume our Ggerb e has trivial rst obstruction and let a trivialization of P

b e xed If the center of G is trivial then one can use the stack axioms to glue

lo cal sections of F X to get a section dened over X the co cycle condition

will automatically b e satised In general the obstruction is a co cycle with

values in the center Z of G The class H X Z is the second obstruction

to triviality of F this dep ends on the choice of trivialization of P a dierent

choice will dier by a global section of OutG and the class in H X Z

resulting from the new section is the result of applied to by the obvious

action of OutG on Z

Remark A gerb e F is said to b e banded it is b ecoming standard to trans

late as band the French verb lier by G if the gerb e is endowed with a global

section of the asso ciated torsor of outer automorphisms When G is ab elian

to say that F is banded by G is equivalent to saying that for every X scheme



U and ob ject u F U there is chosen an isomorphism GU Aut u

F

compatible with pullbacks

If G is ab elian then by cohomological machinery the set of isomorphism

classes of gerb es on X banded by G is in bijection with H X G Mi xIV

For G nite and at with at center Z let F X b e a gerb e banded by G

with second obstruction H X Z then the Z gerb e E X asso ciated

with admits a nite at representable morphism to F So such a Ggerb e is

covered by a gerb e banded by G which in turn is covered by a gerb e banded

by the center of G

Prop osition Let G be a nite at group scheme over X Assume that

the center Z and the sheaf of outer automorphisms OutG are nite and at

as wel l Let F X be a Ggerbe Then there exists an OutGtorsor Y X

and a gerbe E Y banded by Z such that E admits a nite at representable

surjective morphism to F

D EDIDIN B HASSETT A KRESCH AND A VISTOLI

Brauer groups Let X b e a No etherian scheme The Brauer group

B r X is the group of Azumaya algebras sheaves of algebras etalelo cally

isomorhic to endomorphism algebras of vector bundles mo dulo the equiva

0 0 0

lence relation E E if E End V E EndV for some pair of vector

0

bundles V and V on X By the SkolemNo ether theorem the rank n Azu

maya algebras on X are classied by H X PGL The exact sequence

n

G GL PGL

m n n

identies the obstruction to a rank n b eing the endomor

phism algebra of a as an element in fact an ntorsion ele

ment of the etale group H X G There is thus determined

m

a homomorphism

B r X H X G

m

It is a fact that this homomorphism is always injective Mi IV Th

0

The cohomological Brauer group denoted Br X is dened to b e the torsion

0 0

subgroup of H X G When X is regular the map Br X Br k X

m

0

is injective where k X denotes the generic p oint of X For a eld Br Br

and the full second cohomology group agree It is only in the presence of

singularities that the cohomological Brauer group may dier from the full

cohomology group H X G

m

The Brauer map is the injective group homomorphism

0

BrX Br X

A ma jor question in the study of Brauer groups is for which schemes X is

the Brauer map an isomorphism The article Ho identies some classes of

schemes for which this is known The Brauer map is known to b e an iso

morphism for ab elian varieties lowdimensional varieties general varieties of

dimension and regular varieties of dimension ane varieties and sepa

rated unions of two ane varieties Recently the Brauer map has b een shown

to b e an isomorphism for separated geometrically normal algebraic surfaces

0

Sch Also known in general is that if Br X is trivialized by a nite at

cover then lies in the image of the Brauer map

Theorem Let X be a Noetherian scheme Let be an element of H X G

m

The fol lowing are equivalent

i lies in the image of the Brauer map

ii There exists a at projective morphism of schemes Y X surjective



such that in H Y G

m

iii The G gerbe with classifying element is a quotient stack

m

Furthermore if n and H X is a preimage of under the map

n

of cohomology coming from the Kummer sequence then conditions i ii

and iii are equivalent to

BRAUER GROUPS AND QUOTIENT STACKS

iv The gerbe with classifying element is a quotient stack

n

Remark Here we are writing Ggerb e with classifying element for

G G or to refer to a gerb e banded by G whose second obstruction to

m n

triviality is H X G Such a gerb e is dened uniquely up to isomorphism

hence the abusive terminology the Ggerb e

Remark In characteristic p or in mixed characteristic the cohomol

ogy groups ab ove are at cohomology groups By Gr I I I sheaf cohomol

ogy with values in G or in when n in invertible is the same in the etale

m n

and at top ologies

Remark Statements i and ii do not involve stacks so the implications

i ii have indep endent interest One direction i ii is wellknown

if is in the image of the b oundary homomorphism H X P GL

m

m m

H X G then pullback to the asso ciated P bundle trivializes the P

m

bundle is the famous BrauerSeveri scheme The other direction ii i

seems to have b een known only as folklore until recently The result now

app ears in the PhD thesis of A Caldararu Ca Prop

Remark The question of whether a general DeligneMumford stack is a

quotient stack is hard even with strong hyp othesis such as smo oth and prop er

over a eld But for gerb es over schemes over a eld of characteristic zero

Lemma can b e used in conjunction with Prop osition and Theorem

to reduce the question to the case of gerb es Indeed by Prop osition

n

any Ggerb e has a nite at representable cover by an ab elian group gerb e

which in turn admits a closed immersion to a pro duct of gerb es

Example then tells us

Corollary Let X be the union of two copies of Sp ec C x y z xy z

0

glued along the nonsingular locus Then the Brauer map BrX Br X is

not surjective So the nonseparated union of two ane schemes need not have

surjective Brauer map

Going the other way Theorem provides an example of a stack with ane

but not quasinite diagonal of nite typ e over a eld which is not a quotient

stack Note that the stack in Example only has quasiane diagonal

Example Let X b e a normal separated surface over a eld if one

wishes C such that H X G contains a nontorsion element Gr I Ib

m

Then the G gerb e F classied by has ane diagonal and is not a quotient

m

stack

2

R Ho obler has p ointed out that it is p ossible to verify directly using cohomological

0

metho ds that the scheme X in this statement satises Br X and Br X Z

D EDIDIN B HASSETT A KRESCH AND A VISTOLI

Proofs of results

Finite parametrization of stacks Here we prove Theorem which

states that that every stack with quasinite diagonal has a nite cover by a

scheme We b egin with an easy but very useful lemma

Lemma Suppose that p F F and p F F are representable

respectively strongly representable morphisms Assume that F is covered by

open substacks U U such that the ber products U F and F U are

F F

representable by algebraic spaces resp schemes Then the ber product is

F F is also represented by an algebraic space resp scheme

F

Proof The inverse images of U F and F U in F F are represented

F F F

by algebraic spaces resp schemes b ecause p and p are representable resp

strongly representable But these inverse images are op en substacks which

cover F F

F

Proof of Theorem Since F is nitely presented over the ground scheme

we may assume that F is obtained by base change from a stack of nite typ e

over Sp ec Z Hence to obtain a cover we may assume that F is of nite typ e

over Sp ec Z Also since the morphism F F is nite and surjective we can

r ed

assume F is reduced By working with each irreducible comp onent separately

we can assume F is integral Finally by normalizing we can assume that F is

normal

k i

Supp ose that F has an op en cover F F such that F has a nite

i i i

cover by a scheme Z The comp osite morphism Z F F is quasinite

i

Thus by Zariskis Main Theorem LMB Theorem the morphism Z F

i

factors as an op en immersion followed by a nite representable map Z Z

i

F Since F is assumed to b e irreducible the nite representable morphism

Z F has dense image so it must b e surjective Set Z Z Z Z

i F F k

The induced map Z F is nite representable and has dense image so it is

surjective Since any nite representable morphism is strongly representable

we can by applying the Lemma conclude that Z is a scheme

Thus to prove the theorem it suces to prove that F has a cover by op en

substacks which admit nite covers by schemes By KM Lemma F

has a quasinite at cover by a scheme V Let V b e an irreducible comp onent

i

of V Once again applying Zariskis Main Theorem the quasinite morphism

0

V F factors as V F F where the rst map is an op en immersion

i i

0

and the second map is nite and by density surjective Replacing F by F

we may therefore assume that F is generically a scheme In particular we can

assume that F has a generic p oint Sp ec K

Let s t R  X b e a smo oth presentation for F Since we are working

lo cally we can assume that X is a normal variety By KM Lemma

the smo oth cover can b e rened to a quasinite at cover by a scheme V

BRAUER GROUPS AND QUOTIENT STACKS

and the morphism V X is the comp osition of a closed immersion and an

etale morphism Again since we are working lo cally we may assume that V is

irreducible In particular we may also assume that V is normal

Since the morphism V F is quasinite it is op en Replacing F by an

op en substack we may assume that V F is surjective Now we construct a

nite cover of F by a scheme The map V F is generically nite so K V

0

is a nite extension of K recall that Sp ec K is the generic p oint of F Let K

0

b e a normal extension of K containing K V Then K is Galois over a eld

00 0

K which is a purely inseparable extension of K Let F b e the normalization

0 0 0 00

of F in K Let U b e the preimage of V in F and for GalK K let

U b e the translate of U under the action of Each U is a scheme Since

normalization commutes with smo oth pullback LMB Lemma we

0 00

may invoke Bour Prop V to deduce that GalK K acts transitively

0 0 0

on the b ers of F F Hence the U cover F so F is a scheme which is a

nite cover of F

As a corollary of indep endent interest we obtain Chows Lemma for stacks

with nite diagonal extending DM Theorem a stack with quasinite

diagonal is separated if and only if the diagonal is nite

Corollary Let F be an algebraic stack of nite type over a Noetherian

ground scheme If the diagonal of F is nite then F admits a proper surjec

tive generical ly nite morphism from a quasiprojective scheme

Stacks which are quotient stacks In this section we give pro ofs of

Lemmas and and from these deduce Theorems and

In Lemma the implication i iii is wellknown if F XG let

m m

G act linearly on some ane space A freely on some op en U A such

that the structure map from U to the base scheme is surjective Now we take

m

V V to b e X U G X A G with the diagonal Gaction Clearly

iii implies ii If V F is a vector bundle of rank n such that at every

geometric p oint the stabilizer action is faithful on the b er then the stabilizer

action on frames is free at every geometric p oint hence the asso ciated frame

bundle P is an algebraic space Prop osition and F P GL This

n

establishes ii implies i and we have proved Lemma

To prove Lemma let E and F b e nitetyp e stacks over a No etherian

ground scheme and let E F b e a at pro jective morphism Let O

denote a relatively ample invertible sheaf on E and for a coherent sheaf E on

E we let E k denote E O k We know that for k suciently large we

i

have R E k for i and hence E k lo cally free these are lo cal

 

assertions and for schemes this is well known

Supp ose E is a quotient stack Then there is a lo cally free coherent sheaf

E on E such that the geometric stabilizer group actions on b ers are faithful

D EDIDIN B HASSETT A KRESCH AND A VISTOLI

Replacing E by E O if necessary the stabilizer actions on b ers of E k for

E

i

each k will b e faithful as well Cho ose k such that R E k for i





and such that the natural map of sheaves E k E k is surjective We



may also supp ose E k is very ample when restricted to the b ers of If we

let F E k then F is a lo cally free coherent sheaf on F such that the



stabilizer group actions on b ers are faithful Indeed if p Sp ec F is a

geometric p oint with stabilizer group then Y E Sp ec is a pro jective

F

scheme with very ample coherent sheaf E k O that is generated by global

Y

sections and since acts faithfully on the b ers of E k it follows that acts

faithfully on H Y E k O Lemma is proved

Y

Now Theorem is proved as follows Let f X F b e a nite cover

of F by a scheme or algebraic space By assumption there is a surjection

of sheaves E f O Let V b e the vector bundle asso ciated with E Then

 X

there is a closed immersion of X into the stack V Since X is representable

and X F is surjective F is a quotient stack by Lemma

Remark If in the situation of Theorem the stack F admits a nite

map to a scheme Q this o ccurs exactly when F has nite stabilizer and hence

has a mo duli space KM and the mo duli space is a scheme then Z the

algebraic space for which we have F Z GL is in fact a scheme The

n

reason that Z is a scheme is as follows Let Y F b e a nite cover of F by

a scheme Then since Z F is ane the b er pro duct Z Y is an ane

F

Qscheme Thus by Chevalleys theorem for algebraic spaces Kn I I I it

follows that Z is an ane Qscheme as well

Finally Theorem is a direct consequence of Lemma provided we

know that the tangent bundles and higher jet bundles of smo oth Deligne

Mumford stacks enjoy faithful actions by the stabilizers of geometric p oints

Prop osition Let s t R  X be an etale presentation of a smooth Deligne

Mumford stack F Let S X be the stabilizer group scheme Assume that

no component of S r eX dominates a component of X Then for some k

S acts faithful ly on the bund le of k jets in X

Proof Let x b e a p oint in X Replacing X by an etale cover if necessary we

may assume the p oints of x all have residue eld equal to the residue

eld of x Then for any r x r ex the induced maps

b b

s t O O

X x Rr

are isomorphisms Thus the comp osite

t

s

b b b

O O O

X x Rr X x

b

gives an automorphism of the completed lo cal O By assumption on

X x

S s t in a neighb orho o d of r R so the automorphism is nontrivial Thus

BRAUER GROUPS AND QUOTIENT STACKS

k

r must act nontrivially on the vector space O m for some k Then

X x

x

there exists k such that the stabilizer group x acts faithfully on the space

of k jets at x

By No etherian induction on X there is a k for which the stabilizer action

on k jets is faithful at all p oints of X

p

Example Let k b e a eld of characteristic p The map z z z

realizes C P as a cyclic cover of P of degree p branched only over innity

So P is the coarse mo duli space of the stack F C Zp where a generator

of Zp acts on C by z w z w w The stabilizer of F acts faithfully

on the tangent bundle everywhere except at the p oint over innity For n

the action on njets is faithful at all p oints

Finally Corollary follows from the following construction Let F b e

a smo oth DeligneMumford stack with nite stabilizer I F There is an

F

op en dense substack F on which the restriction I F of the stabilizer map

is etale Let J b e the closure of I in I then J F is etale since any

F

nite unramied morphism from a scheme to a normal No etherian scheme

is O such that every comp onent dominates the target is etale Then E

J



a lo cally free coherent sheaf F We claim that the total space of the asso ciated

vector bundle has trivial generic stabilizers from which it follows since F

emb eds in any vector bundle as the zero section that F is a quotient stack

Let p Sp ec F b e a general geometric p oint of F with automorphism

group G Then the b er of over p is canonically isomorphic to G and the

action of G over this b er is by conjugation Since G by hyp othesis has trivial

center the generic action on b ers of O is faithful

J



A nonquotient stack We work out Example By Lemma if

we can show the stack F of Example has no nontrivial vector bundles it

follows that F is not a quotient stack

reg

Let Y Sp ec C x y z xy z with nonsingular lo cus Y The nontriv

reg

ial involution i of Y B Z is sp ecied by it suces to say how i acts

reg

on pairs consisting of map T Y and trivial Ztorsor on T

f f

reg reg  reg

iT Y T Z T T Y f A r fg Y

The stack F is the union of two copies of Y B Z glued via i

By MP the scheme Y has no nontrivial vector bundles this fact holds

more generally for any ane toric variety Gu and since Y is normal and

the glueing is over a lo cus whose complement has co dimension the scheme

reg

Y this is the scheme over which F is a gerb e also has no nontrivial vec Y q

Y

tor bundles Every vector bundle on F splits into and eigenbundles

for the stabilizer action so we deduce that the eigenbundle is trivial

We claim the eigenbundle is zero Let F b e a lo cally free coherent

sheaf on F such that the stabilizer action is multiplication by on sections

D EDIDIN B HASSETT A KRESCH AND A VISTOLI

and let F i denote the restriction of F over the ith copy of Y Then

i

there is a given isomorphism



reg reg

i F j F j

F F

reg

Let L denote the pullback to F of the unique torsion invertible sheaf on

reg reg

Y for any lo cally free coherent sheaf F on F such that the stabilizer acts



by we have i F L F Hence

reg reg

L F j F j

Y Y

But this is imp ossible unless F F for otherwise F j i is

i Y

m m reg

free of some rank m and hence we have L O on Y But

reg

Y sits inside A Z with complement of co dimension so this implies

an isomorphism on A Z b etween a free coherent sheaf and a nontrivial

lo cally free coherent sheaf

Gerb es and the Brauer group Here we prove Theorem For i

iii let H X P GL b e the class of an Azumaya algebra representing

n

a given cohomological Brauer group element H X G If P X is the

m

P GL bundle asso ciated with then by the denition of the b oundary map

n

in nonab elian cohomology the gerb e represented by is P GL

n

For iii i we note that if F is the G gerb e asso ciated with then a

m

vector bundle B on F decomp oses into eigenbundles indexed by characters in

b

G Given a faithful stabilizer action on b ers the characters whose eigenbun

m

r _ s

b

dles are nonzero must generate G then the decomp osition of B B

m

for suitable integers r and s has nonzero eigenbundle B F for the unit

character The complement of the zero section of B is a BrauerSeveri scheme

over X and the asso ciated Azumaya algebra represents

As we have remarked i ii is wellknown The implication ii iii

is an immediate consequence of Lemma

Finally supp ose is ntorsion with preimage in H X and let us

n

0

show iii iv Let F b e the G gerb e asso ciated with and let F b e

m

the gerb e asso ciated with There is a natural representable morphism

n

0

F F hence iii iv For the reverse implication let E b e a lo cally

0

free coherent sheaf on F such that the stabilizer action on sections is faith

ful Consider the quasicoherent sheaf F E with its decomp osition into



L

eigensheaves F F We claim each F is lo cally free of nite typ e

b

2G

m

and F if and only if the eigensheaf of E corresp onding to the restriction

of to is nonzero Indeed it suces to verify the claims etale lo cally and

n

the claims hold in the case of trivial gerb es We can cho ose a nite set S

of characters which generates G such that F for every S Then

m

L

F is a lo cally free coherent sheaf on F such that the stabilizer action

2S

on sections is faithful

BRAUER GROUPS AND QUOTIENT STACKS

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D EDIDIN B HASSETT A KRESCH AND A VISTOLI

Department of Mathematics University of Missouri Columbia MO

Email address edidinmathmissouriedu

Department of MathematicsMS Rice University S Main St

Houston TX

Email address hassettriceedu

Department of Mathematics University of Pennsylvan ia Philadelphia PA

Email address kreschmathupennedu

Dipartimento di Matematica Universita di Bologna Bologna Italy

Email address vistolidmuniboit