Brauer groups of 1-motives

Contents lists available atScienceDirect

Journal

of

Pure

and

Applied

Algebra

www.elsevier.com/locate/jpaa

Brauer

groups

of

1-motives

Cristiana Bertolin∗, Federica Galluzzi

DipartimentodiMatematica,UniversitàdiTorino,ViaCarloAlberto10,Italy

a r t i cl e i n f o a b s t r a c t

Articlehistory:

Received 14 October 2020 Received in revised form 16 March 2021

Available online 8 April 2021 Communicated by C.A. Weibel

MSC:

14F22; 16H05

Keywords:

Gerbes on a stack

Azumaya algebras over a stack Brauer group of a stack 1-motives

Overanormalbasescheme,weprovethegeneralizedTheoremoftheCubefor 1-motivesandthatatorsionclassofthegroupH2

´

et(M,Gm,M) ofa1-motiveM ,whose

pull-backviatheunitsection: S_{→ M is}zero,comesfromanAzumayaalgebra. Inparticular,wededucethatoveranalgebraicallyclosedfieldofcharacteristiczero, allclassesofH2

´

et(M,Gm,M) comefromAzumayaalgebras.

©2021ElsevierB.V.Allrightsreserved.

Contents

0. Introduction . . . 2

1. Recallonsheaves,gerbesandPicardstacksonastack . . . 5

2. GerbeswithAbelianbandonastack . . . 7

2.1. Homologicalinterpretationofgerbesoverasite . . . 7

2.2. Gerbesonastack . . . 8

2.3. 2-descentofGm-gerbes . . . 8

3. TheBrauergroupofalocallyringedstack . . . 10

4. GerbesandAzumayaalgebrasover1-motives . . . 12

5. ThegeneralizedtheoremoftheCubefor1-motivesanditsconsequences . . . 14

5.1. Itsconsequences . . . 14

5.2. Itsproof . . . 15

6. Cohomologicalclassesof1-motiveswhichareAzumayaalgebras . . . 17

Appendix A. AcommunicationfromP.Deligneon2-descenttheoryforstacks . . . 20

References . . . 21

* Corresponding author.

E-mailaddresses:cristiana.bertolin@unito.it(C. Bertolin), federica.galluzzi@unito.it(F. Galluzzi).

https://doi.org/10.1016/j.jpaa.2021.106754

0. Introduction

Grothendieck has definedthe Brauer groupBr(X) of ascheme X asthe group of similarityclasses of Azumayaalgebras overX.In[25,I,§1] heconstructedaninjectivegrouphomomorphism

δ : Br(X)−→ H2_et´(X, Gm) (0.1)

from the Brauer group of X to the étale cohomology group H2 ´

et(X,Gm). This homomorphism is not in

generalbijective,aspointed outbyGrothendieckin[25,II, §2],wherehefoundaschemeX whoseBrauer group is a torsion group but whose étale cohomology groupH2

´

et(X,Gm) is not torsion. However, if X is

quasi-compacttheelementsofδ(Br(X)) aretorsionelementsofH2 ´

et(X,Gm),andsoGrothendieckaskedin

[25] thefollowingquestion:

Question.Foraquasi-compactschemeX,istheimageofBr(X) viathehomomorphismδ (0.1) thetorsion subgroup H2

´

et(X,Gm)Tors ofH2´et(X,Gm)?

Grothendieck showed thatif X is regular, theétale cohomology group H2 ´

et(X,Gm) is atorsion group,

and sounderthishypothesisthequestioniswhethertheBrauergroupofX isallofH2 ´

et(X,Gm).

Thefollowing well-knownresultsarerelated tothis question:IfX hasdimension ≤ 1 orifX isregular and of dimension ≤ 2, then the Brauer group of X is all of H_et2_´(X,Gm,X) ([25, II, Cor 2.2]). Gabber

(unpublishedtheorem)showedthattheBrauergroupofaquasi-compactandseparatedschemeX endowed

with an ampleinvertible sheaf isisomorphic toH2 ´

et(X,Gm)Tors. Adifferent proof ofthis resultwas found

bydeJong(see[18]).

In[20] GiraudintroducedgerbesinthegeneralsettingofnonabeliancohomologyfollowingGrothendieck’s ideas:inparticularheprovedthatgerbesgiveageometricaldescriptionofclassesofthegroupH2_(X,G

m).

TheaimofthispaperistoextendGrothendieck’stheoryofBrauergroupsto1-motives,usinggerbesas fundamentaltools.

Inparticular,

• westudygerbes onstackswhicharenotseparated;

• westudyAzumayaalgebras andBrauergroupsforstacks whicharenotseparated;

• weapply theaboveresultsto 1-motivesusingthedictionary betweenlengthtwo complexes ofabelian sheaves and Picardstacks developed byDeligne in[17, Exposé XVIII,§1.4]. Remarkthat thePicard stacksassociated to1-motivesarenotalgebraic inthesense of[30] sincetheyarenotquasi-separated.

Weproceed inthefollowingway:

LetS beasite.InSection1weassociatetoastackingroupoidsX overS thesiteS(X),whichallowsus to studythenotionofsheafandgerbeonastack.

In Section2we provethefollowing homologicalinterpretationof F -gerbes,with F anabelian sheafon asiteS:the Picard2-stackGerbeS(F ) ofF -gerbesis equivalent(as Picard2-stack)to thePicard2-stack

associated tothecomplexF [2],whereF [2]= [F → 0→ 0] withF indegree-2:

GerbeS(F ) ∼= 2st(F [2]) (0.2)

(Theorem 2.2). In particular, for i = 2,1,0,we have an isomorphism of abelian groups between the i-th

classifyinggroupGerbei_S(F ) andthecohomologicalgroupHi_{(S,F ).TheequivalenceofPicard2-stacks(0.2)}

containsthefollowingclassicalresult:elementsofGerbe2_S(F ),whichareF -equivalenceclassesofF -gerbes,

are parametrized bycohomologicalclasses ofH2_{(S,F ).Always inSection2, applying [20, ChpIV] tothe}

abeliansheaf onthesiteS(X).WefinishSection2proving theeffectivenessofthe2-descentofGm-gerbes

withrespecttoafaithfullyflatmorphismofschemeswhichisquasi-compactorlocallyoffinitepresentation (Theorem2.7).

Let Set´ be the étale site on an arbitrary scheme S and let X = (X,OX) be a locally ringed S-stack

with associatedétalesiteS´et(X).InSection3we recallthenotionof theBrauergroupBr(X) ofX and in

Theorem3.4 weestablishaninjectivegrouphomomorphism

δ : Br(X) −→ H2_´et(X, Gm,X), (0.3)

whichextendsGrothendieck’sgrouphomomorphism(0.1) tolocallyringedS-stacks.

LetM = [u: X→ G] bea1-motivedefinedoveraschemeS,withX anS-groupschemewhichis,locally fortheétaletopology,aconstantgroupschemedefinedbyafinitelygeneratedfreeZ-module,G anextension of an abelian S-scheme by anS-torus, and finally u: X → G a morphism of S-groupschemes. Since in [17,ExposéXVIII,§1.4] DeligneassociatestoanylengthtwocomplexofabeliansheavesaPicardstack,in Section4we candefine theBrauergroupofthe1-motiveM astheBrauergroupBr(M) oftheassociated PicardstackM andbyTheorem3.4wehaveaninjectivegrouphomomorphismδ : Br(M)→ H2

´

et(M,Gm,M).

At theend of Section 4we provethe effectivenessof the descent of Azumaya algebras and of Gm-gerbes

withrespect tothequotientmapι: G→ [G/X]∼=M (Lemma 4.2andLemma4.3).

Denote by sij := M×SM→ M×SM×SM the map which inserts the unitsection  : S → M of M

intothek-thfactorfork∈ {1,2,3}− {i,j}.If isaprimenumberdistinctfromtheresiduecharacteristics of S, we say that the 1-motive M satisfies the generalized Theorem of the Cube for the prime  if the homomorphism  (i,j)∈{1,2,3}s∗ij : H2et´(M3, Gm,M3)() −→  H2 ´ et(M2, Gm,M2)() 3 x −→ (s∗12(x), s∗13(x), s∗23(x))

is injective,where () denotes the-primary component(Definition5.1). We startSection5 studying the consequencesofthegeneralizedTheoremoftheCubefor1-motives.InCorollary5.6weshowthatifthebase schemeisconnected,reduced,normalandnoetherian,extensionsofabelianschemesbysplittorisatisfythe generalizedTheoremoftheCubeforanyprime distinctfromtheresiduecharacteristicsofS (Corollary5.6). Then,asaconsequenceoftheeffectivenessofthe2-descentofGm-gerbes withrespect tothequotientmap

ι : G → [G/X] ∼=M (Lemma 4.3), we get Theorem 5.7: 1-motives, which are defined over aconnected, reduced, normal and noetherian scheme S, and whose underlying tori are split, satisfy the generalized Theorem ofthe Cubeforany prime distinctfrom the residuecharacteristics ofS. Note thatin[8, Thm 5.1] S.Brochardand thefirstauthor provetheTheorem oftheCube(involvingtheH1(M,Gm,M) instead

ofH2(M,Gm,M))for1-motives,andin[9] theauthorsshowthatthesheafofdivisorialcorrespondencesof

extensionsofabelianschemes bytoriisrepresentable.

InSection6weinvestigateGrothendieck’sQuestion for 1-motivesandouransweriscontainedin Theo-rem6.2 whichstatesthatifM = [u: X → G] is1-motivedefinedoveranormalandnoetherianscheme S

and iftheextension G underlyingM satisfies thegeneralizedTheorem ofthe Cubeforaprimenumber

distinctfromtheresiduecharacteristicsofS,thenthe-primarycomponentofthekernelofthe homomor-phismH2_´et(): H2_´et(M,Gm,M)→ Het2´(S,Gm,S) inducedbytheunitsection: S→ M of M ,iscontainedin

theBrauergroupofM :

kerH2_´et() : H2_et´(M, Gm,M)−→ H2et´(S, Gm,S)



()⊆ Br(M).

Weprovethisresultasfollows:firstweshowthistheoremforanextensionofanabelianschemebyatorus using Hoobler’sTheorem[27, Thm3.3] (Proposition6.1). Then,thanksto theeffectivenessofthe descent of Azumaya algebras and of Gm-gerbes with respect to the quotient map ι : G → [G/X] ∼= M, we get

therequiredstatementforM . WefinishSection6givingapositiveanswerto Grothendieck’sQuestion for 1-motives(andsoinparticularforsemi-abelianvarieties)overanalgebraicallyclosedfieldofcharacteristic zero(Corollary6.3).

Inthelastyears,severalauthorshaveworkedwiththeBrauergroupofstacks(seeforexample[1],[19], [31])butmostofthem focus onalgebraicor separatedstacks.Moreoverthetechniques used inthis paper are rather differentfrom the onesused in[1], [19], [31]. Since thePicardstack associated to a1-motiveis notquasi-separated,werecallthetheoryofBrauergroupof stacks.

Animportantroleinthispaperisplayedbythe2-descenttheoryofgerbesforwhichweaddanAppendix. Acknowledgment

We are very grateful to Pierre Deligne for his comments on the first version of this paper and for his communicationon 2-descenttheory forstacks (seeAppendix).We wouldliketo thankalso therefereefor theveryusefulcomments.

Notation Stacklanguage

Here werefermainlyto [20].LetS beasite.A stack overS isafiberedcategoryX overS such that • (Gluing conditionon objects)descentiseffectiveforobjectsinX,and

• (Gluing condition on arrows)forany objectU of S andfor everypairof objectsX,Y of thecategory X(U),thepresheafofarrowsArr_X(U)(X,Y ) ofX(U) isasheafoverU .

Forthenotionsofmorphisms ofstacks (i.e.catesianfunctors)andmorphismofcartesian functorswerefer to [20, Chp. II 1.2]. An equivalence (resp. isomorphism) of stacks F : X → Y is a morphism of stacks whichisanequivalence(resp.isomorphism)offiberedcategoriesoverS,thatisF (U ):X(U)→ Y(U) isan equivalence (resp.isomorphism)ofcategories foranyobjectU ofS.A stackingroupoids overS isastack X overS suchthatforanyobjectU ofS thecategoryX(U) isagroupoid,i.e.acategoryinwhichallarrows are invertible.Recall that2-morphismsof stacks ingroupoidsare automaticallyinvertible.From now on, allstackswill be stacksingroupoids.

A gerbe overthesiteS isastackG over S suchthat

• G islocally notempty:foranyobjectU ofS,thereexistsacovering{φi: Ui→ U}i∈I forwhichtheset

ofobjectsofthecategoryG(Ui) isnotemptyforalli∈ I;

• G islocally connected:foranyobjectU ofS andforeachpairofobjectsg1 andg2ofG(U),thereexists

acovering{φi : Ui→ U}i∈I ofU suchthattheset ofarrowsfrom g1|Ui tog2|Ui inG(Ui) isnotempty

foralli∈ I.

A morphism (resp. isomorphism) of gerbes isjustamorphism(resp.isomorphism)ofstacks whosesource and target aregerbes,anda2-morphismof gerbesis amorphism ofcartesian functors. An equivalenceof gerbes isanequivalenceofstacks.

A strictlycommutativePicardstack overthesiteS (justcalledaPicardstack)isastackP overS endowed withamorphismofstacks⊗:P×SP→ P,calledthegrouplawofP,andtwonaturalisomorphismsa and

c, expressingtheassociativityand thecommutativityconstraintsofthegrouplawofP, suchthatP(U) is astrictly commutativePicardcategory forany objectU ofS (see[17] 1.4.2 formoredetails).An additive functor (F,):P1 → P2 between two Picardstacks is amorphism ofstacks F :P1 → P2 endowed with

a naturalisomorphism : F (a⊗_P1b) ∼= F (a)⊗P2 F (b) (forall a,b ∈ P1) whichis compatible with the naturalisomorphismsa andc underlyingP1 andP2.

A strict 2-category (just called 2-category) A = (A,C(a,b),Ka,b,c,Ua)a,b,c∈A is given by the following

data: aset A ofobjects a,b,c,...;for each ordered pair(a,b) ofobjects of A, acategory C(a,b); for each orderedtriple (a,b,c) of objectsA,acompositionfunctorKa,b,c: C(b,c)× C(a,b)→ C(a,c), thatsatisfies

theassociativity law; foreach objecta, aunitfunctor Ua: 1→ C(a,a) where1 is theterminal category,

thatprovidesaleft andrightidentity forthecompositionfunctor.

A 2-stack overthesiteS isafibered2-categoryX overS (i.e.afamilyof2-categoriesindexedbyobjects ofS,see[15,1.10p.29] formoredetails) suchthat

• 2-descentiseffectiveforobjectsinX (see [15,1.10p.31]),and

• foranyobjectU ofS andforeverypairofobjectsX,Y ofthe2-categoryX(U ),thefiberedcategoryof arrows ArrX(U )(X,Y ) ofX(U ) isastackoverS|U.

For thenotionsof morphisms of 2-stacks(i.e. cartesian 2-functors),morphisms of cartesian 2-functors, modificationsof2-stacksandequivalencesof2-stacks,wereferto[26,ChpI].A 2-stackin2-groupoids over S isa2-stackX overS suchthatforanyobjectU ofS the2-categoryX(U ) isa2-groupoid.From nowon, all2-stackswill be2-stacksin 2-groupoids.

LetS beanarbitraryschemeanddenotebyS thesiteofS foraGrothendiecktopologythatwewillfix later. We will call a stack,a Picardstack, a2-stack over S respectively an S-stack,aPicard S-stack, an

S-2-stack.

1. Recallonsheaves,gerbesandPicardstacksonastack

LetS beasite.LetX beastackoverS.Wealwaysassumethatfibered(2-)categoriescomewithafixed cleavage(see[16,§2,§6]).Delignefurnishedusthefollowing definitionofsiteassociatedto astack.

Definition1.1. The siteS(X) associatedtoX overS isthesitedefinedinthefollowingway:

• the category underlying S(X) consists of theobjects (U,u) with U anobject of S and u an objectof X(U), and ofthearrows(φ,Φ): (U,u)→ (V,v) withφ: U → V amorphism ofS andΦ: φ∗v→ u an

isomorphisminX(U).Wecallthepair(U,u) an openofX withrespectto thechosentopology. • thetopologyonS(X) istheonegeneratedbythepre-topologyforwhichacoveringof(U,u) isafamily

{(φi,Φi): (Ui,ui)→ (U,u)}i suchthatthemorphismofSφi:Ui→ U isacoveringofU .

Definition 1.2.A sheaf(ofsets) F on X isasystem(FU,u,θφ,Φ),where forany object(U,u) ofS(X),FU,u

isasheafonS_|U,andforanyarrow(φ,Φ): (U,u)→ (V,v) ofS(X),θφ,Φ :FV,v→ φ∗FU,uisamorphismof

sheavesonS_|V,suchthat

(i) if (φ,Φ): (U,u) → (V,v) and (γ,Γ) : (V,v)→ (W,w) aretwo arrows of S(X),then γ_∗θφ,Φ◦ θγ,Γ =

θγ◦φ,φ∗Γ◦Φ;

(ii) if(φ,Φ): (U,u)→ (V,v) isanarrowofS(X),themorphismofsheavesφ−1FV,v→ FU,u,obtainedby

adjunctionfromθφ,Φ,isanisomorphism.

Tosimplifynotations,wedenotejust(FU,u) thesheafF = (FU,u,θφ,Φ).Thesetof globalsections Γ(X,F)

ofasheafF onX isthesetoffamilies(sU,u) ofsectionsofF ontheobjects(U,u) ofS(X) suchthatforany

arrow(φ,Φ): (U,u)→ (V,v) ofS(X),resφsV,v= sU,u.

An abeliansheafF onX isasystem(FU,u) verifyingtheconditions(i) and(ii) ofDefinition1.2,wherethe

FU,uareabeliansheavesonS|U.WedenotebyAb(X) thecategoryofabeliansheavesonX.Accordingto[23,

ExpII, Prop.6.7] and[21,Thm1.10.1],thecategoryAb(X) isanabeliancategory withenoughinjectives. Let RΓ(X,−) be theright derivedfunctor of the functor Γ(X,−) : Ab(X)→ Ab of global sections (here

Ab isthecategoryofabeliangroups).Thei-thcohomologygroupHi_RΓ(X,−)_{ofRΓ(X,−) isdenotedby}

Hi_(X,−).

A stack onX is astack Y overS endowed with amorphismof stacks P :Y→ X (calledthe structural morphism)suchthatforanyobject(U,x) ofS(X) thefiberedproductU×x,X,P Y isastackoverS|U.

A gerbe on X is stack G over S endowed with a morphism of stacks P :G → X (called the structural morphism) such that for any object (U,x) of S(X) the fibered product U ×x,X,P G is a gerbe over S|U.

A morphism of gerbes on X isa morphism of gerbes which is compatible with the underlying structural morphisms.

Let F : X→ Y bea morphismof S-stacks and let G be agerbe on Y. The pull-back of G via F isthe fiberedproduct

F∗G := X ×F,Y,PG (1.1)

of X andG viathemorphismF :X→ Y andthestructural morphismP :G→ Y underlyingG (see[7,Def 2.14] forthedefinitionoffiberedproductofS-stacks).

A Picard stack on X is a stack P over S endowed with a morphism of stacks P : P → X (called the structural morphism),withamorphismofstacks⊗:P×P,X,P P→ P,andwithtwo naturalisomorphisms

a and c,suchthatU×x,X,P P isaPicardstackoverS|U forany object(U,x) of S(X).

A Picard 2-stack on X isa2-stack P over S endowed with amorphism of 2-stacksP : P → X (called

the structuralmorphism - herewe seeX asa2-stack),with amorphismof 2-stacks⊗: P×P,X,P P → P,

and withtwonatural2-transformationsa andc,suchthatU×x,X,P P isaPicard2-stackover S|U forany

object (U,x) ofS(X) (formoredetailssee [7, §1] or[6]). An additive2-functor (F,λF): P1→ P2 between

two Picard2-stacks onX is givenbyamorphism of 2-stacksF : P1→ P2 andanatural2-transformation

λF: ⊗P2◦F

2 _{→ F ◦ ⊗}

P1, which are compatible with the structural morphisms of 2-stacks P1 : P1 → X and P2: P2→ X andwiththenatural2-transformationsa andc underlying P1 andP2. An equivalenceof

Picard 2-stacks onX is anadditive 2-functorwhose underlyingmorphism of 2-stacksis anequivalence of 2-stacks.

Denote by 2P icard(X,S) the category whose objects are Picard 2-stacks on X and whose arrows are isomorphism classesof additive2-functors.Applying [36, Cor 6.5] to thesiteS(X),we havethe following equivalence ofcategories

2st :D[−2,0](S(X)) −→ 2Picard(X, S), (1.2) where D[−2,0]_{(S(X)) is the derivedcategory of length three complexes of abelian sheaves on X. Via this}

equivalence, Picard2-stacks(resp.Picardstacks)onX correspondtolengththree(resp.two)complexesof abelian sheaves on X.Therefore, the theory of Picardstacks is included inthe theory of Picard2-stacks. Wedenote by[ ] theinverseequivalence of2st.

IfP isaPicardstackoverasiteS wedefineits classifyinggroupsPifori= 1,0 inthefollowingway:P1is thegroupofisomorphismclassesofobjectsofP andP0_{isthegroupofautomorphismsoftheneutralobject}

e of P. We define the classifying groupsPi _{for i = 2,1,0 of aPicard 2-stack P over asite S recursively:}

P2 _{is thegroup of equivalence classes of objects of P , P}1 _=Aut1

(e) andP0 _{= Aut}0

(e) where Aut(e) is thePicardstackofautomorphismsoftheneutralobjecte ofP .Explicitly,P1 _{isthegroupofisomorphism}

classesofobjectsofAut(e) andP0_{isthegroupofautomorphismsoftheneutralobjectofAut(e).Wehave}

the followinglinkbetweenthe classifyinggroupsPi _{andthecohomology groupsH}i_{(S,[P ]) ofthecomplex}

[P ] associated toP via(1.2):Pi∼_{= H}i−2_{(S,[P ]) for i= 0,1,2.}

If two Picard 2-stacks P and P are equivalent as Picard 2-stacks, then their classifying groups are isomorphic: Pi∼_{= P} i_{fori= 2,1,0.Theinverseaffirmationisnottrueas explainedin[3,Rem1.3].}

Let S be anarbitrary scheme anddenote by S thesite ofS for aGrothendiecktopology.Let X bean

2. GerbeswithAbelianband ona stack

LetF beanabeliansheafonasiteS.DenotebyGerbeS(F ) thefibered2-categoryofF -gerbesoverS.

Lemma2.1. The fibered2-category GerbeS(F ) of F -gerbesis aPicard2-stackoverS.

Proof. By[15,§2.6] the2-descentiseffectiveforobjectsofGerbeS(F ).Moreover,morphismsofgerbes are

just morphisms of stacks and so by [15, Examples 1.11 i)], the gluing condition on arrows of GerbeS(F )

is satisfied.Thus, thefibered2-category GerbeS(F ) is infacta2-stackover S.In[20, ChpIVProp 2.4.1

(i)] Giraud hasdefinedthecontracted productof gerbes (seeinparticular [20] Example2.4.3forthe case of gerbes bound by abelian sheaves). He also showed that this contracted product satisfies associativity and commutativity constraints (see [20, ChpIV Cor 2.4.2 (i) and (ii)]). Hence we canconclude that the contractedproductofF -gerbesendows the2-stackofF -gerbes withaPicardstructure. 

2.1. Homologicalinterpretationof gerbes overasite

Let F be an abelian sheaf ona site S. The classifyinggroups Gerbei_S(F ) for i = 2,1,0 of the Picard 2-stackGerbeS(F ) are

• Gerbe2_S(F ),theabeliangroupofF -equivalenceclassesofF -gerbes;

• Gerbe1_S(F ),theabeliangroupofisomorphismclassesofmorphismsofF -gerbesfromaF -gerbetoitself. • Gerbe0S(F ),theabeliangroupofautomorphisms ofamorphismof F -gerbesfromaF -gerbetoitself.

Theorem 2.2.Let F be an abelian sheaf on a site S. Then the Picard 2-stack GerbeS(F ) of F -gerbes is

equivalent(asPicard2-stack)tothePicard2-stackassociatedtothecomplexF [2],whereF [2]= [F → 0→ 0]

with F indegree -2:

GerbeS(F ) ∼= 2st(F [2]).

In particular, for i= 2,1,0, we have an isomorphism of abelian groups between thei-th classifying group

GerbeiS(F ) andthecohomologicalgroupHi(S,F ).

Proof. It is a classical result that via the equivalence of categories stated in [17, Exposé XVIII, Prop 1.4.15], the complex F [1] corresponds to the Picard stack Tors(F ) of F -torsors: Tors(F ) = 2st(F [1]). A higherdimensionalanalogueof thenotionoftorsorunderan abeliansheafis thenotionof torsorundera Picard stack,which was introduced byBreen in[13, Def 3.1.8] and studied by the first author in [5, §2] (remark thatinfactin[7] thefirst authorintroducesthenotionof torsorunderaPicard 2-stack,seealso [4], [2] and [10]). Hence we have the notion of Tors(F )-torsors. The contracted product of torsors under aPicard2-stack, introduced in[7, Def2.11], endows the2-stack T ors(Tors(F )) ofTors(F )-torsors with a Picardstructure,and by [7, Thm0.1] this Picard2-stack T ors(Tors(F )) corresponds, viatheequivalence ofcategories(1.2),to thecomplex[Tors(F )][1]:

T ors(Tors(F )) = 2st(F [2]). (2.1)

In[15,Prop2.14] BreenconstructsacanonicalequivalenceofPicard2-stacksbetweenthePicard2-stack GerbeS(F ) ofF -gerbes andthePicard2-stackT ors(Tors(F )) ofTors(F )-torsors:

This equivalence and the equality (2.1) furnish the expected equivalence GerbeS(F ) ∼= 2st(F [2]). The

classifyinggroupsofthePicard2-stackGerbeS(F ) are therefore

Gerbei_S(F ) ∼= Hi−2(S, F [2]) = Hi(S, F ). 

Remark 2.3. Via the cohomological interpretation (2.1) of torsors under the Picard stack of F -torsors,

the equivalence of Picard 2-stacks (2.2) is the geometrical counterpart of the canonical isomorphism in cohomology H2_{(S,F )}∼_{= H}1_{(S,F [1]).}

2.2. Gerbeson astack

LetX beastackoverasiteS anddenotebyS(X) thesiteassociatedtoX.Applying[20,ChpIV] tothe siteS(X),wegetthenotionofF-gerbesonthestackX,withF anabeliansheafonX.Werecallbrieflythis notion.

An F-gerbe is agerbe G onX such thatfor any object (U,x) of S(X) the fiberedproduct U ×x,X,P G

is aFU,x-gerbe over S|U (here P :G→ X isthe structural morphism): inparticular foreach i indexinga

covering{Ui→ U}i ofU ,itexistsanobjectgi of(U×x,X,P G)(Ui) andanisomorphismFU,x|Ui → Aut(gi)

of sheaves of groups on Ui (see [15, Def 2.3]). Considernow an F-gerbeG and an F-gerbe G on X. Let

u : F → F a morphism of abelian sheaves on X. A morphism of gerbes m : G → G is an u-morphism if u is compatiblewith themorphism ofbands Band(Aut(g)U,x)→ Band(Aut(m(g))U,x) (see [20, ChpIV

2.1.5.1]).Asin[20,ChpIVProp2.2.6] anu-morphismm:G→ G isfullyfaithfulifandonlyifu:F → F is anisomorphism,inwhichcasem is anequivalence ofgerbes.IfG andG aretwoF-gerbesonX,instead of idF-morphism G→ G weusetheterminologyF-equivalenceG→ G ofF-gerbes onX.

F-gerbesonX buildaPicard2-stackonX,thatwedenote by GerbeS(X)(F)

whose group law is given by the contracted product of F-gerbes on X ([20, Chp IV 2.4.3]). Its neutral element isthestackTors(F) ofF-torsorsonX,whichiscalledthe neutralF-gerbe.ApplyingTheorem 2.2

to theabeliansheafF onthesiteS(X) (seeDefinition1.2)weget Corollary 2.4.We havethefollowingequivalenceof Picard2-stacks

GerbeS(X)(F) ∼= 2st(F[2]).

In particular,Gerbei_S(X)(F)= H∼ i(X,F) fori= 2,1,0.

Hence, F-equivalence classes of F-gerbes on X, which are the elements of the 0th-homotopy group Gerbe2_S(X)(F),areparametrizedbycohomologicalclassesofH2_(X,F).

Remark 2.5.GerbeS(X)(F) is a Picard S(X)-2-stack. Via the structural morphism F : X → S, we

can view GerbeS(X)(F) also as a Picard S-2-stack GerbeS(F). In this case we have that GerbeS(F) ∼=

2st(τ_≤0RF_∗(F[2])) whereτ_≤0 isthegoodtruncation indegree0.Wewillnotusethisfactinthepaperand therefore weomittheproof.

2.3. 2-descentof Gm-gerbes

Wefinishthissectionprovingtheeffectivenessofthe2-descentofGm-gerbeswithrespecttoafaithfully

the semi-localdescriptionofgerbes donebyBreenin[16,§4] and[14,§2.3],thatwerecallonlyinthecase ofGm-gerbes.DenotebyTors(Gm) thePicardstackofGm-torsors.AccordingtoBreen,tohaveaGm-gerbe

G overasiteS isequivalentto havethedata 

(Tors(Gm,U), Ψx), (ψx, ξx)



x∈G(U),U∈S (2.3)

indexedbytheobjectsx oftheGm-gerbeG (recallthatG islocallynotempty),where

• Ψx : G|U → Tors(Gm,U) isanequivalenceofU -stacksbetweentherestrictionG|U toU oftheGm-gerbe

G and theneutralgerbe Tors(Gm,U).This equivalenceisdeterminedbytheobjectx inG(U),

• ψx = pr∗1Ψx◦ (pr2∗Ψx)−1 : Tors(pr2∗Gm,U)→ Tors(pr∗1Gm,U) is an equivalence ofstacks over U×S U

(here pri : U×SU → U are theprojections),which restrictsto theidentity when pulledback viathe

diagonalmorphism Δ: U → U ×SU ,and

• ξx : pr∗23ψx◦ pr∗12ψx ⇒ pr∗13ψx is aisomorphismofcartesian S-functors betweenmorphisms ofstacks

on U×SU ×SU (here prij : U ×S U×SU → U ×SU are thepartialprojections), whichsatisfies the

compatibilitycondition

pr₁₃₄∗ ξx◦ [pr∗34ψx∗ pr∗123ξx] = pr∗124ξx◦ [pr234∗ ξx∗ pr∗12ψx] (2.4)

whenpulledbacktoU×SU×SU×SU := U4(hereprijk: U4→ U ×SU×SU andprij : U4→ U ×SU

are thepartial projections.See[12,(6.2.7)-(6.2.8)] formoredetails).

Therefore,theGm-gerbe G maybereconstructedfromthelocaldata(Tors(Gm),Ψx)xusingthetransition

data(ψx,ξx).Wecalltheequivalencesofstacks Ψx the localneutralizations oftheGm-gerbe G definedby

the local objectsx∈ G(U).The transition data (ψx,ξx) are infact2-descentdata. SeeAppendix forthis

reconstructionofaGm-gerbevialocalneutralizationsand2-descentdata.

In§5wewill needthesemi-localdescriptionofaGm-equivalence classof aGm-gerbewhich consistsin

the following data: afamily(Tors1_(G

m,U))U∈S ofgroupsof isomorphism classesof Gm-torsors,bijections

Tors1_(pr∗

2Gm,U)→ Tors1(pr∗1Gm,U) of their pull-backson U×SU via theprojectionspri : U×SU → U,

and compatibilityconditionsonthe pull-backonU×SU ×SU of these bijections(here weuse theabove

notations).

Remark2.6. Inthispaper, Breen’ssemi-localdescriptionofgerbesallowsusto reduceofonethedegreeof thecohomologygroupsinvolved:insteadofworkingwithgerbes,whicharecohomologyclassesofH2_(S,G

m),

wecanworkwith torsors,whicharecohomologyclassesofH1(S,Gm).

Theorem 2.7.Let p: S → S beafaithfully flat morphismof schemeswhich isquasi-compact or locallyof finite presentation.Tohave aGm,S-gerbeoverS isequivalent tohaveatriple

(G, ϕ, γ)

where G is a Gm,S-gerbe over S and (ϕ,γ) are 2-descent data on G with respect to p : S → S. More

precisely,

• G isaGm,S-gerbeover S,

• ϕ : p∗₁G → p∗₂G is an equivalenceof gerbes over S×SS, where pi : S×S S → S are the natural

projections,

• γ : p∗₂₃ϕ◦p∗₁₂ϕ⇒ p∗₁₃ϕ isanaturalisomorphismoverS×SS×SS,wherepij : S×SS×SS→ S×SS

such thatover S×SS×SS×SS thecompatibility condition

p∗₁₃₄γ◦ [p∗₃₄ϕ∗ p∗₁₂₃γ] = p∗₁₂₄γ◦ [p∗₂₃₄γ∗ p∗₁₂ϕ] (2.5)

issatisfied,wherepijk: S×SS×SS×SS×SS → S×SS×SS andpij : S×SS×SS×SS → S×SS

are thepartialprojections.

Under this equivalence, the pull-back p∗ : GerbeS(Gm,S) → GerbeS(Gm,S) is the additive 2-functor

which forgetsthe2-descentdata:p∗(G,ϕ,γ)=G.

Proof. Let (G,ϕ,γ) be a triplet as in the statement. According to Appendix, the data (ϕ,γ) satisfying

theequality(2.5) are2-descentdataforthegerbe G.AsobservedinLemma2.1,thefibered2-categoryof Gm-gerbes buildsa2-stack(that is,inparticular,the2-descentis effectiveforobjects),and soG withits

2-descentdatacorresponds toaGm,S-gerbe G overS. 

3. TheBrauer groupofalocallyringedstack

LetX beastackover asiteS andletS(X) beitsassociated site.

A sheafofrings A onX isasystem(AU,u) verifyingtheconditions(i) and (ii) ofDefinition1.2,where

the AU,u are sheaves of ringson S|U. Considerthe sheaf of rings OX on X given by the system (OX U,u)

with O_{X U,u} thestructural sheafofU .ThesheafofringsO_Xis the structuralsheafofthe stackX andthe pair (X,O_X) isa ringed stack. An O_X-module M is asystem(MU,u) verifying the conditions(i) and (ii)

of Definition1.2,where theMU,u aresheavesof OU-modulesonS|U.AnOX-algebra A isasystem(AU,u)

verifyingtheconditions(i) and(ii) ofDefinition1.2,wheretheAU,uaresheavesofOU-algebrasonS|U.An

OX-moduleM is offinitepresentation iftheMU,uaresheavesofOU-modulesoffinitepresentation.

NowletS beanarbitraryschemeandletS´etbetheétalesiteonS.LetX= (X,OX) bea locallyringed

S-stack, i.e. aringedstack suchthat,forany object(U,u) oftheassociated étalesiteS´et(X),andfor any

sectionf ∈ OX U,u(U ),thereexistsa covering{(Ui,ui)→ (U,u)}i∈I of(U,u) suchthatforanyi∈ I either

f|(Ui,ui)or(1− f)|(Ui,ui)is invertibleinΓ(Ui,OX Ui,ui)

An Azumaya algebra over X is an OX-algebra A= (AU,u) of finite presentation as OX-module which

is, locally for the topology of S´et(X), isomorphic to a matrixalgebra, i.e. for any open (U,u) of X there

exists acovering{(φi,Φi): (Ui,ui)→ (U,u)}i inS´et(X) suchthatAU,u⊗OU,uOUi ∼= Mri(OUi,ui) forany i.

Azumayaalgebras overX,thatwe denoteby

Az(X),

build anS-stack onX by[22,ExposéVIII 1.1, 1.2] (seealso[30, (3.4.4)]).TwoAzumayaalgebras A and A _{overX are Brauer-equivalent ifthereexisttwolocallyfreeOX-modulesE andE} _{offiniteranksuchthat}

A ⊗OXEndOX(E) ∼=A

_⊗

OXEndOX(E

_).

The above isomorphism defines an equivalence relation because of the isomorphism of O_X-algebras End_OX(E)⊗_OXEnd_OX(E) ∼= End_OX(E⊗_OX E). We denote by [A] the equivalence class of an Azumaya algebraA overX.Theset ofequivalenceclassesofAzumayaalgebraisagroupunderthegrouplawgiven by[A][A]= [A⊗OXA].A trivialization ofanAzumayaalgebraA overX isacouple(L,a) withL alocally freeOX-moduleanda: End_OX(L)→ A anisomorphismofsheavesofOX-algebras.AnAzumayaalgebraA is trivial ifitexistsatrivializationofA.TheclassofanytrivialAzumayaalgebraistheneutralelementof theabovegrouplaw. Theinverseofaclass[A] istheclassA0withA0theoppositeO_X-algebraofA. Definition 3.1.Let X= (X,O_X) bealocally ringedS-stack.The Brauer group ofX,denotedbyBr(X),is thegroupofequivalenceclassesofAzumayaalgebrasover X.

Br(−) is afunctorfrom thecategory of locally ringedS-stacks (objectsare locally ringedS-stacks and arrows are isomorphism classes of morphisms of locally ringed S-stacks) to the category Ab of abelian groups. Remark that the above definition generalizes to stack the classical notion of Brauer group of a scheme:infactifX isalocallyringedS-stack associatedtoanS-scheme X,then Br(X)= Br(X).

Consider the following sheaves of groups on X: the multiplicative group Gm,X, the linear general

group GL(n,X), and the sheaf of groups PGL(n,X) on X defined by the system (PGL(n,X)U,u) where

PGL(n,X)U,u= Aut



Mn(OX U,u)(automorphisms ofMn(OX U,u) as asheafof OX U,u-algebras). Wehave

thefollowing

Lemma3.2. Assumen> 0. Thesequenceof sheavesof groupsonX

1−→ Gm,X−→ GL(n, X) −→ PGL(n, X) −→ 1 (3.1)

isexact.

Proof. Itisenoughto showthatforanyétaleopen(U,u) ofX,therestrictiontotheétalesiteofU ofthe sequence 1→ GmU,u → GL(n)U,u → PGL(n)U,u → 1 is exactand this follows by [32, IV,Prop. 2.3. and

Cor2.4.]. 

LetLf(X) betheS-stackonX oflocallyfreeO_X-modules.LetA beanAzumayaalgebraoverX.Consider themorphismofS-stacks onX

End : Lf(X) −→ Az(X), L −→ End_OX(L) (3.2)

Following [20, ChpIV2.5],letδ(A) be the fiberedcategory over Set´ oftrivializations ofA defined inthe

followingway:

• for any U ∈ Ob(S´et), the objects of δ(A)(U) are trivalizations of A|U, i.e. pairs (L,a) with L ∈

Ob(Lf(X)(U)) anda∈ Isom_UEnd_OX(L),A_|U,

• for any arrow f : V → U of S´et, the arrows of δ(A) over f with source (L,a) andtarget (L,a) are

arrows ϕ:L→ L ofLf(X) overf suchthatAz(X)(f)◦ a= a◦ End(ϕ), withAz(X)(f):A_|V → A_|U. Since Lf(X) and Az(X) are S-stacks onX, δ(A) isalso anS-stack on X (see [20,ChpIV Prop2.5.4(i)]). ObservethatthemorphismofS-stacksEnd : Lf(X)→ Az(X) islocallysurjectiveonobjectsbydefinitionof Azumayaalgebra.Moreover,itislocallysurjective onarrowsbyexactnessofthesequence(3.1).Therefore as in[20, ChpIVProp2.5.4(ii)], δ(A) isagerbe over X,called the gerbeof trivializationsof A.Forany object(U,u) ofS´et(X) themorphismofsheavesofgroupsonU

(Gm,X)U,u= (O∗_X)U,u−→



Aut(L, a)_U,u,

that sends a section g of (O∗_X)U,u to the multiplication g· − : (L,a)U,u → (L,a)U,u by this section, is

an isomorphism. This means that the gerbe δ(A) is infact a Gm,X-gerbe.By Corollary 2.4 we canthen

associatetoanyAzumayaalgebraA overX acohomologicalclassinH2 ´

et(X,Gm,X),denotedbyδ(A),which

isgivenbytheGm,X-equivalenceclassofδ(A) inGerbe2S(Gm,X).

Proposition 3.3. An Azumaya algebra A over X is trivial if and only if its cohomological class δ(A) in

H2 ´

et(X,Gm,X) iszero.

Proof. The AzumayaalgebraA istrivialifand onlyifthe gerbeδ(A) admitsaglobalsection ifand only ifitscorrespondingclassδ(A) is zeroinH2

´

Theorem 3.4.The morphism

δ : Br(X) −→ H2_et´(X, Gm,X)

[A] −→ δ(A)

is aninjective grouphomomorphism.

Proof. Let A,B betwoAzumayaalgebrasover X.Forany U ∈ Ob(Set´),themorphismofgerbes

δ(A)(U) × δ(B)(U) −→ δ(A ⊗_OXB)(U)

((L, a), (M, b)) −→ (L ⊗_OXM, a ⊗_OXb)

is a+-morphism,where+: Gm,X× Gm,X→ Gm,X isthegrouplawunderlyingthesheafGm,X.Therefore

δ(A) + δ(B) = δ(A ⊗_OXB) (3.3)

inH2 ´

et(X,Gm,X).Thisequalityshowsfirstthatδ(A) = −δ(A0) andalsothat

[A] = [B] ⇔ [A ⊗_OXB0] = 0Prop⇔ δ(A ⊗3.3 _OXB0_{) = 0}(3.3⇔ δ(A) + δ(B) 0_{) = 0}⇔ δ(A) = δ(B)

Theseequivalencesprovethatthemorphismδ : Br(X)→ H2 ´

et(X,Gm,X) iswell-definedandinjective.Finally

always fromtheequality(3.3) wegetthatδ is agrouphomomorphism.  4. GerbesandAzumayaalgebrasover1-motives

LetM = [u: X→ G] bea1-motivedefinedoveraschemeS,denotebyM itsassociatedPicardS-stack

undertheequivalenceconstructedin[17,ExposéXVIII,Prop1.4.15] anddenotebyS(M) thesiteassociated to thestackM asinDefinition1.1.

Definition 4.1.

(1) The S-stackofAzumayaalgebrasoverthe1-motiveM istheS-stack ofAzumayaalgebrasAz(M) over M.

(2) The Brauergroupof the1-motiveM istheBrauergroupBr(M) ofM.

(3) AGm-gerbeonthe1-motiveM isaGm,M-gerbeonM (i.e.aGm,M-gerbe onthesiteS(M)).

By [30, (3.4.3)] theassociated Picard S-stack M is isomorphic to the quotient stack [G/X] (whereX

acts on G viathegivenmorphismu: X→ G). Notethatingeneralitisnotalgebraic inthesenseof [30] becauseitisnotquasi-separated.However thequotientmap

ι : G−→ [G/X] ∼=M

is representable, étale and surjective. The fiber product G×[G/X]G is isomorphic to X ×S G. Via this

identification, the projections qi : G×[G/X]G → G (for i = 1,2) correspond respectively to the second

projection p2 : X×S G→ G and to the map μ : X ×SG → G given by the action (x,g)→ u(x)g. We

canfurtheridentifythefiberproduct G×[G/X]G×[G/X]G withX×SX×SG andthepartialprojections

q13,q23,q12: G×[G/X]G×[G/X]G→ G×[G/X]G respectivelywiththemapmX×idG: X×SX×SG→ X×SG

p23: X×S X×SG→ X ×SG.The effectivenessof thedescentof Azumayaalgebras with respect to the

quotientmapι: G→ [G/X] isprovedinthefollowingLemma(see[35,(9.3.4)] forthedefinitionofpull-back ofO_M-algebras):

Lemma4.2. Thepull-backι∗: Az(M)→ Az(G) isanequivalenceofS-stacksbetweentheS-stackofAzumaya algebrasonM andtheS-stackofX-equivariantAzumayaalgebrasonG.Moreprecisely,tohaveanAzumaya algebra A onM isequivalent tohave apair

(A, ϕ)

whereA isanAzumayaalgebraonG andϕ: p∗₂A→ μ∗A isanisomorphismofAzumayaalgebrason X×SG

that satisfies(uptocanonicalisomorphisms) thecocyclecondition

(mX× idG)∗ϕ =  (idX× μ)∗ϕ  ◦(p23)∗ϕ  . (4.1)

Under this equivalence, the pull-back ι∗ : Az(M) → Az(G) is the morphism of stacks which forgets the descent datum:ι∗(A,ϕ)= A.

Proof. Since theassertionislocal forthetopologyonSet´(M),itsufficesto proveitforany open(U,u) of

M,whereU isanobjectofSet´ andx isanobject ofM(U).Thedescentofquasi-coherentmodulesisknown

for themorphism ιU : G×ι,M,xU → U obtained bybase change (see[30, Thm(13.5.5)]). The additional

algebrastructuredescendsby[29,IIThm3.4].FinallytheAzumayaalgebrastructuredescendsby[28,III, Prop2.8].SinceanAzumayaalgebraonM isbydefinitionacollectionofAzumayaalgebrasonthevarious schemesU ,thegeneralcasefollows. 

Now weprovealso theeffectivenessof the2-descentof Gm-gerbes underthe quotientmap ι : G→ M,

usingtheresultofSection2.3.

Lemma4.3. To haveaGm,M-gerbeG on M isequivalent tohave atriplet

(G, ϕ, γ)

whereG isaGm,G-gerbeon G and(ϕ,γ) are 2-descentdata onG withrespecttoι: G→ [G/X],that is

• ϕ: p∗2G → μ∗G isanequivalenceof gerbeson X×SG, • γ :(idX× μ)∗ϕ  ◦(p23)∗ϕ  ⇒ (mX× idG)∗ϕ isanaturalisomorphismonX×SX×SG∼= G×[G/X] G×[G/X]G,

whichsatisfies thecompatibilitycondition

p∗₁₃₄γ◦ [p∗₃₄ϕ∗ p∗₁₂₃γ] = p∗₁₂₄γ◦ [p∗₂₃₄γ∗ p∗₁₂ϕ] (4.2)

when pulled back to X ×S X ×S X ×S G ∼= G×[G/X]G×[G/X]G×[G/X]G := G4 (here prijk : G4 →

G×[G/X]G×[G/X]G andprij : G4→ G×[G/X]G are thepartialprojections).

Proof. AGm,M-gerbe on M isby definition acollection of Gm,U-gerbes over the various objects U of S.

Hence itis enough to prove thatfor any object U of S andany object x of M(U), the 2-descentof Gm

-gerbeswith respecttothemorphismιU : G×ι,M,xU → U obtainedbybase changeiseffective.Butthisis

5. ThegeneralizedtheoremoftheCubefor1-motivesanditsconsequences

WeusethesamenotationofthepreviousSection.WedenotebyM3=M×SM×SM (resp.M2=M×SM)

the fiberedproduct of 3 (resp. 2) copies of M. Since any Picard stack admits a global neutral object, it exists aunitsectiondenotedby: S→ M.Considerthemap

sij :=M ×SM → M ×SM ×SM

whichinsertstheunitsection: S→ M intothek-thfactorfork∈ {1,2,3}− {i,j}.If isaprimenumber and H isanabeliangroup,H() denotesthe-primarycomponentofH.

Definition 5.1. LetM be a1-motive definedover ascheme S. Let  bea primenumberdistinct from the residue characteristicsof S.The1-motiveM satisfiesthe generalizedTheoremof theCubefortheprime

 ifthenaturalhomomorphism  s∗_ij : H2_et´(M3, Gm,M3)() −→  H2_´et(M2, Gm,M2)() 3 x −→ (s∗₁₂(x), s∗₁₃(x), s∗₂₃(x)) (5.1) is injective. 5.1. Itsconsequences

Proposition 5.2. LetM be a1-motive satisfying thegeneralizedTheorem oftheCubeforaprime distinct from theresiduecharacteristicsofS.LetN :M→ M bethemultiplication byN onthePicardS-stack M. Then foranyy∈ H2_´et(M,Gm,M)() we havethat

N∗(y) = N2y + N

2_{− N}

2 

(−id_M)∗(y)− y. (5.2)

Proof. First we prove that given three contravariant functors F,G,H : P → M, we have the following equality foranyy inH2

´

et(M,Gm,M)()

(F + G + H)∗(y)− (F + G)∗(y)− (F + H)∗(y)− (G + H)∗(y) + F∗(y) + G∗(y) + H∗(y) = 0. (5.3) Letpri:M× M× M→ M theprojectionontotheithfactor.Putmi,j= pri⊗ prj :M× M× M→ M and

m= pr1⊗ pr2⊗ pr3:M× M× M→ M,where⊗ isthelawgroupofthePicardS-stack M.Theelement

z = m∗(y)− m∗_1,2(y)− m∗_1,3(y)− m∗_2,3(y) + pr₁∗(y) + pr∗₂(y) + pr₃∗(y)

of H2_´et(M3,Gm,M3)() iszerowhenrestrictedto S× M× M,M× S × M andM× M× S (thisrestriction is obtainedinsertingtheunitsection: S→ M).Thus itiszeroinH2

´

et(M3,Gm,M3)() bythegeneralized Theorem oftheCubefor. Finally,pullingbackz by(F,G,H):P→ M× M× M weget(5.3).

Now, settingF = N,G= id_M,h= (−id_M) weget

N∗(y) = (N + id_M)∗(y) + (N− id_M)∗(y) + 0∗(y)− N∗(y)− (id_M)∗(y)− (−id_M)∗(y). Werewritethisas

Ifwedenotez1= y andzN= N∗(y)−(N −idM)∗(y),weobtainzN +1= zN+ y + (−idM)∗(y).Byinduction,

wegetzN = y + (N− idM)(y + (−idM)∗(y)).From theequalityN∗(y)= zN+ (N− idM)∗(y) wehave

N∗(y) = zN+ zN−1+· · · + z1,

andthereforewearedone. 

Corollary5.3. LetM be a1-motive satisfying thegeneralizedTheorem of theCubefor aprime. Then,if  = 2, then_{-torsion elementsof H}2

´

et(M,Gm,M) arecontained in

ker(n_M)∗: H2_´et(M, Gm,M)−→ H2´et(M, Gm,M)

andif = 2,they arecontainedin

ker(2n+1_M )∗: H2_et´(M, Gm,M)−→ H2et´(M, Gm,M)



.

Proof. Theresultfollowsby(5.2). 

5.2. Its proof

Wefinishthissection bysearchingthehypothesisweshouldputonthebasescheme S inorderto have that the 1-motive M = [X → G] satisfiesu the generalized Theorem of the Cube. From now on we will switch freely between the two equivalent notion of invertible sheaf L on the extension G and Gm-torsor

Isom(OG,L) onG.TheextensionG fitsinto thefollowing shortexactsequence

0−→ T −→ G−→ A −→ 0π

The pull-back of gerbes defined in (1.1) allows us to define an homomorphism of abelian groups π∗ : Gerbe2_S(Gm,A)→ Gerbe2S(Gm,G).

Proposition 5.4. LetS bea normal scheme. LetG bean extension of an abelian S-scheme A byGmr.The

pull-backπ∗: Gerbe2_S(Gm,A)→ Gerbe2S(Gm,G) issurjective.

Proof. DenotebyTorsRig(Gm,G) thePicardS-stackof Gm-torsorsonG withrigidificationalongtheunit

section G : S → G. Because of this unit section G, the group of isomorphism classes of Gm-torsors

over G withrigidificationis canonicallyisomorphicto thequotient of thegroupofisomorphismclasses of Gm-torsors overG bythegroupofisomorphismclassesofGm-torsorsover S:

TorsRig1

(Gm,G) ∼=Tors1(Gm,G)/Tors1(Gm,S). (5.4)

DenotebyCub(G,Gm) thePicardS-stackofGm-torsorsonG withcubicalstructureandbyCubi(G,Gm)

fori= 1,0 itsclassifyinggroups.RoughlyspeakingaGm-torsoronG iscubicalifitsatisfiestheTheorem

of the Cube, for details see [11, Def 2.2]. In [11, Prop 2.4], Breen proves the Theorem of the Cube for extensionsofabelianschemesbytoriwhicharedefinedoveranormalscheme,thatistheforgetfuladditive functorCub(G,Gm)→ TorsRig(Gm,G) isanequivalenceof PicardS-stacks.Inparticular

Cub1_{(G, G}

m) ∼=TorsRig1(Gm,G). (5.5)

Withourhypothesis,by[33,ChpI,Rem7.2.4],anyGm-torsorsonG withcubicalstructurecomes froma

π∗:Tors1(Gm,A)−→ Tors1(Gm,G). (5.6)

Now let G be a Gm,G-gerbe on G. Breen’s semi-local description of gerbes (2.3) asserts that to have G

is equivalent to have the local data (Tors(Gm,G,U),Ψx)x∈G(U),U∈Sf ppf endowed with the transition data

(ψx,ξx). Let y = π(P (x))∈ A(U), where P :G → G isthestructural morphism of G. Theequivalence of

U×SU -stacksψx:Tors(pr2∗Gm,G,U)→ Tors(pr∗1Gm,G,U) inducesabijectionbetweentheclassifyinggroups

ψx1:Tors1(pr∗2Gm,G,U)→ Tors1(pr1∗Gm,G,U).Because ofthesurjection (5.6),alltorsors overG comefrom

torsors overA uptoisomorphisms,andso wehaveabijection

ψ1_y:Tors1(pr₂∗Gm,A,U)→ Tors1(pr1∗Gm,A,U)

such that ψ1

y = (π∗)∗ψ1x. By pull-back via (5.6), the isomorphism of cartesian S-functors ξx : pr∗23ψx◦

pr₁₂∗ ψx ⇒ pr13∗ ψx givesξy1 = (π∗)∗ξx1 : pr23∗ ψ1y◦ pr∗12ψy1 ⇒ pr13∗ ψy1 over U ×S U ×SU , which satisfies an

analogueoftheequality(2.4).Thelocaldata(Tors1_(G

m,A,U))y=π(P (x)),x∈G(U)endowedwiththetransition

data (ψ1

y,ξ1y),whicharedefinedonisomorphismclassesofGm-torsors,furnishaGm,A-equivalenceclass G

of aGm,A-gerbe onA suchthatπ∗(G)=G. 

Wegiveanimmediateapplicationofthisresultinthecaseofextensionsover afieldk.

Corollary 5.5.Let G be an extension of an abelian variety by a torus over a field k. Then Br(G) ∼= H2

´

et(G,Gm,G).

Proof. ByGabber’s unpublishedresult[18],ifA is anabelianvarietydefinedover afieldk,then Br(A)∼= H2

´

et(A,Gm,A).Wehavethefollowingcommutativediagram

Br(A) π∗ ∼ = H2 ´ et(A, Gm) π∗ Br(G) δ H 2 ´ et(G, Gm) (5.7)

where π : G→ A isthe surjectivemorphism ofvarieties underlyingG andπ∗ denotes thepull-back maps ofAzumayaalgebrasandcohomologicalclasses.By[25,II,Prop1.4] thecohomologicalgroupsH2

´

et(G,Gm)

and H2 ´

et(A,Gm) aretorsiongroupsandsoTheorem2.2and Proposition5.4imply thatπ∗: H2´et(A,Gm)→

H2 ´

et(G,Gm) issurjective. Hencetheinjectivehomomorphismδ onthebottomrowissurjective too. 

LetGibeanextensionofanabelianS-schemeAibyanS-torusfori= 1,2,3,anddenotebyGi : S→ Gi

its unit section. Let sG

ij := Gi ×S Gj → G1×S G2×S G3 be the map obtained from the unit section

G

k : S→ Gk after the base changeGi×SGj → S (i.e.the mapwhich insertsGk : S→ Gk into thek-th

factorfork∈ {1,2,3}− {i,j}).

Corollary 5.6. LetS beaconnected, reduced,normal andnoetherian schemeand letGi bean extension of

an abelian S-scheme Ai by Grim fori= 1,2,3.Let  beaprime distinct fromthe residuecharacteristics of

S. Then,with theabovenotation, thenaturalhomomorphism



sG_ij∗: H2_´et(G1×SG2×SG3, Gm)() −→ (i,j)∈{1,2,3}H2´et(Gi×SGj, Gm)()

x −→ (sG∗₁₂(x), sG∗₁₃(x), sG∗₂₃(x)) (5.8)

Inparticular,ifthebaseschemeisconnected,reduced,normalandnoetherian,anextensionofanabelian S-scheme by Gmr satisfies the generalized Theorem of the Cube for any prime  distinct from the residue

characteristicsof S.

Proof. LetpG_ij : G1×SG2×SG3→ Gi×SGj betheprojectionmapsfori,j = 1,2,3 andletαGbethemap

αG _: 

(i,j)∈{1,2,3}H2´et(Gi×SGj, Gm)() −→ Het2´(G1×SG2×S G3, Gm)()

((y2, y3), (y1, y3), (y1, y2)) −→ i,j=1,2,3pGij∗(yi, yj))

(5.9)

AnalogouslywedefinethemapsαA_andpA

ij.Weobservethattheinjectivityofthemap



sG∗_ij isequivalent tothesurjectivityofthemapαG_{([34,Rempage55]).Ifwedenoteπ}

i: Gi→ Ai thesurjectionsunderlying

the extensions Gi (for i = 1,2,3), (πi × πj)◦ pijG = pAij ◦ (π1 × π2× π2) and so we have the following

commutativediagram  (i,j)∈{1,2,3}H2´et(Ai×SAj, Gm)() αA H2 ´ et(A1×SA2×SA3, Gm)()  (i,j)∈{1,2,3}H2et´(Gi×SGj, Gm)() αG H2_et´(G1×SG2×SG3, Gm)() (5.10)

where the vertical arrows, which are the pull-backs induced by the πi, are surjective by Proposition 5.4

and Corollary 2.4. The top horizontal arrow is surjective since under our hypotheses abelian S-schemes

satisfythegeneralizedTheoremoftheCube(see[27,Cor2.6]).Hencewecanconcludethatalsothedown horizontalarrowissurjective,i.e.sG_ij∗ isinjective. 

Using the effectivenessof the 2-descent forGm-gerbes viathe quotient map ι: G → M (Lemma4.3),

from theabovecorollaryweget

Theorem5.7. 1-motives,whicharedefinedoveraconnected,reduced,normalandnoetherianschemeS,and whose underlying toriare split, satisfy the generalizedTheorem of the Cube forany prime  distinct from theresiduecharacteristics ofS.

6. Cohomologicalclassesof1-motiveswhichareAzumayaalgebras

LetS beascheme.Wewill needthefinitesiteonS:firstrecallthatamorphismof schemesf : X→ S

is saidto be finite locallyfree if it is finite andf_∗(OX) is alocally freeOS-module.In particular, by[24,

Prop(18.2.3)] finiteétalemorphismsarefinitelocallyfree.ThefinitesiteonS,denotedSf,isthecategory

offinite locally freeschemesoverS, endowedwiththetopologygenerated from thepretopologyfor which theset ofcoveringsofafinite locallyfree schemeT over S istheset ofsinglemorphismsu: T → T such thatu isfinite locally freeand T = u(T) (set theoretically). Thereis amorphismof siteτ : Sf ppf → Sf.

IfF isasheaffortheétaletopology,thenwedefineas in[27,§3]

F (T )f :=

y∈ F (T ) | there is a covering u : T→ T in Sf with F (u)(y) = 0

 i.e.F (T )f aretheelementsof F (T ) whichcanbe splitbyafinite locallyfreecovering.

Proposition 6.1.Let G be an extension of an abelian scheme by a torus, which is defined over a normal and noetherian scheme S, and which satisfies thegeneralized Theorem of the Cube fora prime number  distinct from thecharacteristics of S.Then the -primary component of thekernel of the homomorphism

H2 ´

et() : H2et´(G,Gm,G)) → H2et´(S,Gm,S) induced by the unit section  : S → G of G, is contained in the

Brauer groupof G:

kerH2_´et() : H2_et´(G, Gm,G))−→ H2et´(S, Gm,S)



()⊆ Br(G). Proof. In order to simplify notations we denote by ker(H2

´

et()) thekernel of the homomorphism H2et´() :

H2 ´

et(G,Gm,G)→ H2et´(S,Gm,S).Weidentifyétaleandfppfcohomologiesfor thesmoothgroupschemesμn

and Gm.

(1) First we showthat H2_f(G,τ∗μn) isisomorphic to H2

´

et(G,μn)_f. Bydefinition, R1τ_∗μ_n is the sheaf

onG associatedtothepresheafU → H1(Uf ppf,μn).ThislattergroupclassifiestorsorsinU_{f ppf} underthe

finitelocallyfreegroupschemeμn.Sinceanyμ_n-torsoristrivializedbyusingitselfasanf -cover,wehave

that R1_τ

∗μn = (0). TheLeray spectralsequence for themorphism ofsites τ : S_{f ppf} → S_f (see [32, page

309])givesthentheisomorphism

H2_f(G, τ_∗μn) ∼= kerH2_et´(G, μ_n)−→ Hπ 0_f(G, R2τ_∗μ_n)= H2_et´(G, μ_n)_f

where the map π is the edge morphism which can be interpreted as the canonical morphism from the presheafU → H2

´

et(Uf ppf,μn) totheassociatedsheafR2τ_∗μ_n.

(2) Nowwe provethatH2 ´

et(G,μ∞)f mapsontoker(Het2´())().Let x bean elementof ker(H2´et()) with

n_{x= 0 for some n. The filtration on the Leray spectral sequence for τ : S}

f ppf → Sf and the Kummer

sequence givesthe following exactcommutativediagram

Pic(G) π  d H0 f(G, (R1τ∗Gm)n) d 0 H2 ´ et(G, μn)_f H2 ´ et(G, μn) π i H0 f(G, R2τ∗μn) 0 H2_´et(G, Gm)f H2´et(G, Gm) n H0_f(G, R2τ_∗Gm) H2 ´ et(G, Gm,G) (6.1) where (R1_τ

∗Gm)n is the cokernel of the multiplication by n. Since nx = 0, we can choose an y ∈

H2 ´

et(G,μn) such thati(y)= x. ByCorollary 5.3, theisogeny2n : G→ G is afinite locally freecovering

whichsplitsx,thatisx∈ H2 ´

et(G,Gm)f.Moreoveri((l2n)∗y)= (l2n)∗i(y)= 0 andsothereexistsanelement

z ∈ Pic(G) suchthatd(z)= (l2n)∗y.Inparticular d(π(z))= π((l2n)∗y) is anelementof H0_f(G,R2τ∗μn).

BytheTheoremoftheCubefortheextensionG (see[11,Prop2.4]),wehavethat(l2n₎∗_{z = l}2n_z _forsome

z∈ Pic(G),whichimpliesthatπ(z)= 0 in(R1_τ

∗Gm)n(G_f).Fromtheequalityπ((l2n)∗y)= d(π(z))= 0

follows π(y)= 0,whichmeansthaty isanelement ofH2 ´

et(G,μn)_f.

(3) Here we show that ker(H2 ´

et())() ⊆ τ∗H2f(G,Gm). By the first two steps, H2f(G,τ∗μn) maps

onto ker(H2_´et())(). Since H_f2(G,τ∗μn) ⊆ H2

f(G,τ∗Gm), we can then conclude that ker(H2et´())() ⊆

τ∗H2_f(G,Gm).

(4)Letx beanelementofker(H2 ´

et()) withnx= 0 forsomen.Usingthemorphismofsitesτ : Sf ppf →

Sf,in[27,Lem3.2] Hooblerbuiltacommutativediagramwhereupperandlowerlinesgiverisetospectral

sequencescomparing ČechandsheafcohomologyforSf ppf andSf.Usingthisdiagram heshowedthatthe

group τ∗H2