Let A be a Finitely Generated Graded Algebra. Basi

June30,20051SomeNewMethodsInTheTheoryofCOHEN-MACAULAYALGEBRASbyA.M.GarsiaandN.WallachJune30,20052NotationXn={x1,x2,...,xn}Q[Xn]=⇒Thealgebraofpolynomialsinx1,x2,...,xnVagradedvectorspaceHm(V)=⇒thesubspaceofthehomogeneouselementsofdegreeminV.V=H0(V)⊕H1(V)⊕H2(V)⊕···⊕Hm(V)⊕···TheHilbertseriesofVFV(q)=
m≥0dimHm(V)qmJune30,20053BasicsLetAbeaFinitelyGeneratedGradedAlgebra.A=m≥0Hm(A)Let(1)n1=theorderof“1”asapoleoftheHilbertseriesFA(q).(2)n2=themaximumnumberofalgebraicallyindependentelementsinA.(3)n3=theminimumnumberofelementsq1,q2,...,qnsuchthatdimA/(q1,q2,...,qn)A∞(∗)Fact:n1=n2=n3=nA=the“Krulldimension”ofAIf(*)holdswithn=nAthenq1,q2,...,qnarecalleda“SystemOfParameters”.“S.O.P.”inbrief.June30,20054MoreBasicsLetq1,q2,...,qn∈Abehomogeneousofdegreesd1,d2,...,dnandsupposetheyconstituteaS.O.PforA.LetB=f1,f2,...,fNbeabasisforthequotientA/(q1,q2,...,qn)A∞TheneveryP∈AhasanexpansionoftheformP=N
i=1fiQi(q1,q2,...,qn)withQ1,Q2,...,QN∈Q[y1,...,yn].Inotherwordsthecollectionfiqp11,qp22,...qppn1≤i≤NspansAasavectorspace.InparticularitfollowsthatFA(q)Ni=1qdegree(fi)(1−qd1)(1−qd2)···(1−qdn)June30,20055TheCohenMacaulayPropertyIfintheexpansionP=N
i=1fiQi(q1,q2,...,qn)PuniquelydeterminesthecoefficientsQi(q1,q2,...,qn)thenthecollectionfiqp11,qp22,...qppn1≤i≤NisavectorspacebasisforA.ThisholdstrueifandonlyifwehavethetheequalityFA(q)=Ni=1qdegree(fi)(1−qd1)(1−qd2)···(1−qdn)ThenAisafreemoduleoverQ[q1,q2,...,qn]ofrankNandAissaidtobe“Cohen-Macaulay”.June30,20056AusefulcriterionTheoremALetq1,q2,...,qn∈Abehomogeneousofdegreesd1,d2,...,dnandanS.O.P.forA.LetdimA/(q1,q2,...,qn)A=NwithbasisB=f1,f2,...,fN.Thentheconditionlimq→1(1−qd1)(1−qd2)···(1−qdn)FA(q)=NforcestheequalityFA(q)=Ni=1qdegree(fi)(1−qd1)(1−qd2)···(1−qdn)yieldingthatAisafreemoduleoverQ[q1,q2,...,qn]ofrankNandthereforeAisaCohen-Macaulayalgebra.June30,20057m-Quasi-InvariantsDenoteby“si,j”thetranspositionthatinterchangesxiandxj.Thisgiven,wesetQIm[Xn]=P(x)∈Q[Xn]:(1−sij)P(x)(xi−xj)2m+1∈Q[Xn]∀1≤ij≤n NotethatQ[Xn]=QI0[Xn]⊃QI1[Xn]⊃QI2[Xn]⊃···⊃QIm[Xn]⊃···⊃QI∞[Xn]=SYM[Xn]Notefurtherthatforall1≤ij≤nwehave(1−sij)PQ=((1−sij)P)Q+(sijP)(1−sij)QThuseachQIm[Xn]isanalgebraandinfactalsoanSn-module.June30,20058r-Quasi-SymmetricRecalltheFlorentHivert“local”r-actionofSnsixaixbi+1=xaixbi+1ifa,b≥rxbixai+1otherwise(∗)heresi=si,i+1.Sincewehaves2i=1,sisi+1si=si+1sisi+1,sisj=sjsi∀|i−j|≥2(*)definesanactionofSncalledthe“r-action.”FlorentHivertdefinesr-QSym=P(x):P(x)isinvariantunderther-action WehaveadescendingchainofalgebrasQ[Xn]⊃1-QSym[Xn]⊃2-QSym[Xn]⊃···⊃r-QSym[Xn]⊃···⊃Sym[Xn]Notethat1-QSym[Xn]isthespaceofGessel’s“quasi-symmetricfuntions”.June30,20059Whatiscommontoallthesealgebras?June30,200510Whatiscommontoallthesealgebras?AretheyallCohen-Macaulay?June30,200511Whatiscommontoallthesealgebras?AretheyallCohen-Macaulay?AlmostJune30,200512Whatiscommontoallthesealgebras?QIm[Xn]isafreemoduleoverSym[Xn](ConjecturedbyFelder-VeselovforallCoxeterGroups)(provedbyEtingov-Ginsburg)June30,200513Whatiscommontoallthesealgebras?QIm[Xn]isafreemoduleoverSym[Xn](ConjecturedbyFelder-VeselovforallCoxeterGroups)(provedbyEtingov-Ginsburg)1-QSym[Xn]isafreemoduleoverSym[Xn](ConjecturedbyJ.Y.ThibonandalsobyF.BergeronandC.Reutenauer)(provedbyA.M.G.andN.Wallach)F.BergeronandC.Reutenauerhaveaconjecturedbasisforthequotient1-QSym[Xn]/(e1,e2,...,en)June30,200514Whatiscommontoallthesealgebras?QIm[Xn]isafreemoduleoverSym[Xn](ConjecturedbyFelder-VeselovforallCoxeterGroups)(provedbyEtingov-Ginsburg)1-QSym[Xn]isafreemoduleoverSym[Xn](ConjecturedbyJ.Y.ThibonandalsobyF.BergeronandC.Reutenauer)(provedbyA.M.G.andN.Wallach)F.BergeronandC.Reutenauerhaveaconjecturedbasisforthequotient1-QSym[Xn]/(e1,e2,...,en)r-QSym[Xn]isafreemoduleoverSym[Xn]forallr≥1(ConjecturedbyFlorentHivert)(Stillopeninspiteofsomeprematureannouncements)June30,200515TowardscommonProofTheoremLetAbeafinitelygeneratedgradedalgebrasuchthat(1)A⊆Q[Xn](2)q1,q2,...qn∈AhomogeneousS.O.P.forQ[Xn]ofdegreesd1,d2,...dn(3)WehaveΠ(x)∈Q[q1,q2,...qn]suchthatΠ(x)Q[Xn]⊆A(4)dimA/(q1,q2,...qn)A≤d=d1d2...dnThisgiven,AisfreeoverQ[q1,q2,...qn]ofrankdJune30,200516SketchofProofWebeginbychosingapointa=(a1,a2,...,an)suchthat(1)Π(a)=0(2)Theset[a]=q1(x)=q1(a),q2(x)=q2(a),...,qn(x)=qn(a)hascardinalityd=d1d2···dn.Thisgiven,wedefineJ[a]=P∈A:P(x)=0∀x∈[a]andsetR[a]=A/J[a]WeshownextthatdimR[a]=dJune30,200517ThedimensionofR[a]If[a]=b1,b2,...,bdconstructΦi(x)∈Q[Xn]sothatΦi(x)=1ifx=bi0ifx=bjwithj=iSetΨi(x)=Φi(x)Π(x)Π(a)thenΨi(x)∈AandΨi(x)=1ifx=bi0ifx=bjwithj=iMoreoverforallP(x)∈AwehaveP(x)≡d
i=1P(bi)Ψi(x)(moduloJ[a])ThusΨi(x)di=1isabasisforR[a]consequentlydimR[a]=dJune30,200518MoreQuotientRingsForP(x)∈Q[Xn]leth(P)denotethehomogeneouscomponentofhighestdegreeinP(x)andsetgrJ[a]=h(P):P∈J[a]AandgrR[a]=A/grJ[a]WecaneasilyprovethatdimgrR[a]=dimR[a]=dNownotethatqi(x)−qi(a)∈J[a]=⇒qi(x)∈grJ[a]inparticulargrJ[a]⊃q1,q2,...,qnAandthusd=dimA/grJ[a]≤dimA/q1,q2,...,qnAbutthenhypothesis(4)givestheequalitydimA/q1,q2,...,qnA=dJune30,200519ConclusionsHypothesis(3)givesqdegreeΠ(x)(1−q)nFA(q)1(1−q)nThuslimq→1(1−