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 scheme X the natural map from the Brauer group equivalence classes of Azumaya algebras to the cohomolog 2 ical Brauer group 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 space 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 algebraic space 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 stack and this stack is algebraic Knowing that an algebraic stack has a presentation as a quotient Z G with G a linear algebraic group 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 equivalence relation 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 orbifolds are quotient stacks Theorem ii all regular Deligne Mumford stacks of dimension 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 quotient stack 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 descenttyp 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 category 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 groupoids 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 sheaf 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 morphism of schemes 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 characteristic 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