Non-symmetric orthogonal geometry of Grothendieck

arXiv:alg-geom/9409005v127Sep1994Non-symmetricorthogonalgeometryofGrothendieckringsofcoherentsheavesonprojectivespacesA.L.Gorodentsev12AlgebraSectionoftheSteklovMathematicalInstituteGSP-1Vavilova42,Moscow,Russiae-mail:[e-mail]-August19941ThispaperwasstartedattheUniversityofStockholmonJune1994andfinishedattheMax-Plank-Institutf¨urMathematikonAugust19942InMoscowautorissupportedbythefoundationPROMATHEMATICA(France)andtheJ.Sorrosfoundation(USA)AbstractInthispaperweconsiderorthogonalgeometryofthefreeZ-moduleK0(Pn)withrespecttothenon-symmetricunimodularbilinearformχ(E,F)=X(−1)νdimExtν(E,F).WecalculatetheisometrygroupofthisformanddescribeinvariantsofitsnaturalactiononK0(Pn).Alsoweconsidersomegeneralconstructionswithnon-symmetricunimodularforms.Inparticular,wediscussorthogonaldecompositionofsuchformsandtheactionofthebraidgrouponasetofsemiorthonormalbases.Weformulatealistofnaturalarithmeticalconjecturesaboutsemiorthogonalbasesoftheformχ.Non-symmetricOrthogonalGeometryofK0(Pn)1§1.Introduction.1.1.ThehelixtheoryandtheproblemofdescriptionofexceptionalsheavesonPn.Thehelixtheoryisacohomologytechniquetostudyderivedcat-egoriesofcoherentsheavesonsomealgebraicvarieties.Itappearsfirstin[GoRu]and[Go1]asthewaytoconstructtheexceptionalbundlesonPn,i.e.locallyfreesheavesEsuchthatdimExt0(E,E)=1,Exti(E,E)=0∀i≥1Sincethenthehelixtheorywasdevelopedinthecontextofgeneraltriangulatedcategoriesin[Go2],[Go3],[Bo1],[Bo2].[BoKa].Themainideaofthistheoryistoconsiderexceptionalbasesofatriangulatedcategory,i.e.collectionsofobjects{E0,E1,...,En}thatgeneratethecategoryandhavethefollowingpropertiesdimHom0(Ei,Ei)=1,Homν(Ei,Ei)=0∀ν6=1Homμ(Ei,Ej)=0∀μand∀ij.Thesimplestexampleofasuchcollectionisthecollection{O,O(1),...,O(n)}ofinvertiblesheavesonPn.Themainproblemistodescribeallsuchcollections.Themostimportantfactinthestudyofthisproblemisthatthereexistsanactionofthebraidgrouponthesetofexceptionalcollectionsofagivenlength.Trans-formationsofexceptionalcollectionsbygeneratorsofthebraidgrouparecalledmutations.Themutationsmakepossibletoconstructaninfinitesetofexceptionalcollectionstartingfromagivenone(see[Go1],[Go2]).ThusadescriptionofallexceptionalsheavesonPnsplitsintothreesteps.Wehavetoprovethefollowingthreeconjectures.1.1.1.CONJECTURE.AnyexceptionalobjectintheboundedderivedcategoryofcoherentsheavesonPnisquasiisomorphictoashiftedimageofanexceptionallocallyfreesheaf.1.1.2.CONJECTURE.Anyexceptionalcollection(inparticular,eachexceptionalobjectitself)intheboundedderivedcategoryofcoherentsheavesonPncanbeincludedinanexceptionalbasisofthederivedcategory.1.1.3.CONJECTURE.InboundedderivedcategoryofcoherentsheavesonPnthebraidgroupactstransitivelyonexceptionalcollectionsofanygivenlength.AllthesethreeconjecturesholdonP2(see[GoRu],[Go2]),andthethirdconjectureholdsonP3forexceptionalcollectionsofmaximallength(generatingthederivedcategory,–see[No]).Discussionoftheseproblemsinthegeneralcontextandthesurveyofcorrespondingresultsseein[Go4].2A.L.Gorodentsev1.2.Subjectofthispaper.Inthispaperweconsideranarithmeticalanalogofproblemsformulatedabove.LetusconsidertheGrothendieckgroupK0(Pn)asafreeZ-moduleoffiniterankwithnon-symmetricunimodularbilinearformχ(E,F)=X(−1)νdimExtν(E,F).1.2.1.DEFINITION.Acollectionofvectors{e1,e2,...,ek}⊂K0(Pn)iscalledexceptionalorsemiorthonormaliftheGrammatrixoftheformχatthiscollectionisuppertriangularwithunitsonthemaindiagonal.Obviously,anyexceptionalcollectionofsheavesproducesansemiorthonormalcollectionofvectorsinK0.IfwearegoingtoworkonlyintermsofK0andχ,thenwecannotdistinguishsuchcollectionsandtheirimageswithrespecttotheactionofisometriesoftheformχ.1.2.2.DEFINITION.Z-linearoperatorϕ:K0(Pn)−→K0(Pn)iscalledisometricifχ(v,w)=χ(ϕv,ϕw)∀v,w.ThegroupofallisometricoperatorsisdenotedbyIsomandiscalledtheisometrygroup.In§3,§4wewillprovethatthisgroupisanunipotentAbelianalgebraicgroupofdimension[(n+1)/2].Ithastwoconnectedcomponents,andthecomponentoftheidentityisadirectsumofstandard1-dimensionaladditivegroups.WewriteexplicitformulasforthenaturalactionofisometriesonK0(Pn)anddescribeinvariantsofthisaction.Allthismaybeconsideredasthefirststepinthedirectionofthefollowingconjecture.1.2.3.CONJECTURE.Avectoresuchthatχ(e,e)=1represents(uptotheactionofisometries)aclassofexceptionalsheafifandonlyifitcanbeincludedinsomesemiorthonormalbasisofK0.In§2weconsidersomegeneralconstructionsofnon-symmetricorthogonalge-ometry.Inparticular,wedefineanactionofthebraidgrouponthesetofallsemiorthonormalcollectionsofagivenlength,andintroducethenotionsofthecanonicaloperatorandthecanonicalalgebraofgivenbilinearform,whichplayanimportantroleinthegeneralclassificationofbilinearnon-degenerateforms.Wediscussthisclassification(overanalgebraicallyclosedfieldofcharacteristiczero)in§3andgiveageneralapproachtotheproblemslikeonesconsideredin[Ru].Thispartofpaperisthefirstlittlestepindirectionofthefollowingconjecture,whichreformulatethemainconjectureofhelixtheoryintermsoflinearalgebra.1.2.4.CONJECTURE.AnysemiorthonormalbasisofK0(Pn)maybeobtainedfromanyotheronebychangingsignsofbasicvectorsandtheactionofbraidgro