Extended centres of finitely generated prime algeb

arXiv:0704.2378v1[math.RA]18Apr2007ExtendedcentresoffinitelygeneratedprimealgebrasJasonP.Bell∗DepartmentofMathematicsSimonFraserUniversity8888UniversityDriveBurnaby,BC,[e-mail]†MaxwellInstituteofSciencesSchoolofMathematics,UniversityofEdinburghJamesClerkMaxwellBuilding,King’sBuildingsMayfieldRoad,EdinburghEH93JZ,[e-mail]sification:16P90Keywords:GKdimension,quadraticgrowth,extendedcentre,transcendencedegree.AbstractLetKbeafieldandletAbeafinitelygeneratedprimeK-algebra.WegeneralizearesultofSmithandZhang,showingthatifAisnotPIanddoesnothavealocallynilpotentideal,thentheextendedcentreofAhastranscendencedegreeatmostGKdim(A)−2overK.Asaconsequence,weareabletoshowthatifAisaprimeK-algebraofquadraticgrowth,theneithertheextendedcentreisafiniteextensionofKorAisPI.Finally,wegiveanexampleofafinitelygeneratednon-PIprimeK-algebraofGKdimension2withalocallynilpotentidealsuchthattheextendedcentrehasinfinitetranscendencedegreeoverK.∗ThefirstauthorthanksNSERCforitsgeneroussupport.†ThesecondauthorwassupportedbyGrantNo.EPSRCEP/D071674/1.11IntroductionWeconsiderprimealgebrasoffiniteGKdimension.GivenafieldKandafinitelygeneratedK-algebraA,theGKdimensionofAisdefinedtobeGKdim(A):=limsupn→∞log(dimVn)/logn.IfAisnotfinitelygenerated,thenitsGKdimensionissimplythesupremumoftheGKdimensionsofallfinitelygeneratedsubalgebras.WenotethatinthecasethatAisafinitelygeneratedcommutativealgebra,GKdimensionisequaltoKrulldimension.Forthisreason,GKdimensionhasseengreatuseovertheyearsasausefultoolforobtainingnoncommutativeanaloguesofresultsfromclassicalalgebraicgeometry.FormoreinformationaboutGKdimensionwereferthereadertoKrauseandLenagan[5].SmithandZhang[9]investigatedcentresindomainsoffiniteGKdimen-sion.TheyshowedthatifAisafinitelygeneratednon-PIalgebraoverafieldKwhichisadomainofGKdimensiondandZisthecentreofthequotientdivisionalgebraofA,thentrdeg(Z)≤GKdim(A)−2.PIalgebrasare,insomesense,closetobeingcommutativeandnecessarilyhavelargecentres.Thepurposeofthispaperistoshowthatsimilarresultsholdforprimealgebras.Generalprimealgebrasdonothaveaquotientdivi-sionalgebra.Nevertheless,Martindale[6]showedthataquotientringcanbeformed,whichisnowcalledtheMartindaleringofquotients.Roughlyspeak-ing,thisquotientalgebrais—uptoacertainequivalence—thecollectionofallrightA-modulehomomorphismsf:I→AwhereIissomenonzeroidealofA.ThecentreofthisalgebraiscalledtheextendedcentreofA.InthecasethatAisadomainoffiniteGKdimension,theMartindaleringofquo-tientscoincideswiththeordinary(Goldie)quotientdivisionalgebra.Thereisoneessentialdifficultythatariseswhenstudyingextendedcentresofprimealgebraswhichdoesnotoccurwithdomains:itispossibleforprimealgebrastohaveanonzerolocallynilpotentideal.Asitturnsout,theexistenceofanonzerolocallynilpotentidealcanaffectthetranscendencedegreeoftheextendedcentre.OurmaintheoremisthefollowinggeneralizationofSmithandZhang’s[9,Theorem3]result.Theorem1.1LetKbeafield,letAbeafinitelygeneratedprimeK-algebra,andletZdenotetheextendedcentreofA.IfAisnotPIandhasnononzero2locallynilpotentideals,thentrdeg(Z)+2≤GKdim(A).Asanimmediatecorollary,weobtainthefollowingresult.Corollary1.2LetKbeafieldandletAbeafinitelygeneratedprimeKalgebraofGKdimensionlessthan3.IfAisnotPIandhasnononzerolocallynilpotentideals,thentheextendedcentreofAisalgebraicoverK.InthecasethatAhasquadraticgrowth,weareabletoshowthattheex-istenceoflocallynilpotentidealsdonotpresentanyproblems.RecallthatafinitelygeneratedK-algebraAhasquadraticgrowthifforeveryfinitedi-mensionalK-vectorsubspaceVofAthatcontains1andgeneratesAasaK-algebra,thereexistpositiveconstantC1andC2suchthatC1n2dimVnC2n2foralln≥1.Theorem1.3LetKbeafieldandletAbeaprimeaffineK-algebraofquadraticgrowth.TheneitherAsatisfiesapolynomialidentityortheex-tendedcentreofAisalgebraicoverK.IfanalgebrahasGKdimension2butdoesnothavequadraticgrowth,thentheextendedcentrecaninfactbeverylargeifithasanonzerolocallynilpotentideal.Theorem1.4LetKbeafield.Thenthereisafinitelygeneratednon-PIprimegradedK-algebraofGKdimensiontwowhoseextendedcentrehasinfinitetranscendencedegreeoverK.ThisfactthattheextendedcentrecanhaveinfinitetranscendencedegreeoverthebasefieldinafinitelygeneratedprimealgebraoffiniteGKdimensionissomewhatsurprising,sincethetranscendencedegreeoftheextendedcentreofafinitelygeneratedprimecommutativealgebraoveritsbasefieldisequaltotheKrulldimensionofthealgebra,andinparticularisnecessarilyfinite.WenotethatifKisalgebraicallyclosedfield,Theorem1.3saysthatanon-PIfinitelygeneratedprimeK-algebraofquadraticgrowthhastrivial3extendedcentre.Theextendedcentreisespeciallyusefulinstudyingbothtensorproductsandgradedalgebras,andknowingthattheextendedcentreistrivialoftenallowsonetounderstandtheidealstructureofanalgebra.In§2,Theorems1.1and1.3areproved.Todothisweuseseveralesti-matesaboutalgebrasofGKdimensionatleast2.In§3,weproveTheorem1.4.OurconstructionusesamodifiedexampleofZelmanov[12,11]alongwithanexampleofIrving[4].In§4,wemakesomeremarksaboutextendedcentresandtensorproductsandapplicationstojustinfinitealgebras.2ProofsInthissection,weproveTheorem1.1andT