Degeneracy and decomposability in abelian crossed

arXiv:0809.1395v1[math.RA]8Sep2008DegeneracyandDecomposabilityinAbelianCrossedProductsKellyMcKinnieSeptember8,2008AbstractLetpbeanoddprime.Inthispaperwestudytherelationshipbetweendegener-acyanddecomposabilityinabeliancrossedproducts.Inparticularweconstructanindecomposableabeliancrossedproductdivisionalgebraofexponentpandindexp2.Thealgebraweconstructisgenericinthesenseof[AS78]andhastheprop-ertythatitsunderlyingabeliancrossedproductisadecomposabledivisionalgebradefinedbyanon-degeneratematrix.Thisalgebraalsogivesanexampleofaninde-composablegenericabeliancrossedproductwhichisshowntobeindecomposablewithoutusingtorsionintheChowgroupofthecorrespondingSeveri-Brauervarietyaswasneededin[Kar98]and[McK08].1IntroductionLetFbeafield.AnabeliancrossedproductisacentralsimpleF-algebrawhichcontainsamaximalsubfieldthatisabelianGaloisoverF.LetΔbeanabeliancrossedproductoverF(wewillwritethisasΔ/F)withabelianmaximalsubfieldKandG=Gal(K/F)=hσ1i×...×hσri.Asdetailedin[AS78]or[McK07],foreveryabeliancrossedproductthereisamatrixu=(uij)∈Mr(K∗)andavectorb=(bi)ri=1∈(K∗)rsothatΔisisomorphictothefollowingalgebra.Δ∼=(K/F,G,z,u,b)=M0≤ij≤njKzi11...zirr(1.1)Herenj=|σj|andmultiplicationinthisalgebraisgivenbytheconditionszizj=uijzjzi,znii=biandzik=σi(k)ziforallk∈K.Throughoutthispaperwewillusemulti-indexnotation:form=(m1,...,mr)∈Nrsetzm=zm11...zmrrandσm=σm11...σmrr.Moreover,setum,n=zmzn(zm)−1(zn)−1∈K∗.PropertiesofthematrixudeterminepropertiesoftheabeliancrossedproductΔandhenceonecanstudythealgebraΔbystudyingthematrixu.In[AS78]thenotionofamatrixbeingdegeneratewasdefinedandthisnotionwasfurtherstudiedandextendedin[McK07]and[Mou07].Inthispaperweusetheoriginaldefinitiongivenin[AS78].Thatis,thematrixuisdegenerateifthereexistelementsσm,σn∈Gandelementsa,b∈K∗sothathσm,σniisnoncyclicandum,n=σm(a)a−1σn(b)b−1.1RecallthatifΔisanabeliancrossedproductthenthegenericabeliancrossedproductassociatedtoΔ,whichwewilldenotebyAΔistheabeliancrossedproductAΔ∼=(K(x1,...,xr)/F(x1,...,xr),G,z,u,bx).Herex1,...,xrareindependentindeterminatesandbx={bixi}ri=1.Letpbeaprime.Ap-algebraisacentralsimplealgebraoverafieldofcharacteristicwithp-powerindex.Asthemainresultin[AS78]genericabeliancrossedproductsandnon-degeneracywereusedtoestablishtheexistenceofnon-cyclicp-algebras.Anotherpropertydeterminedbynon-degeneracywasestablishedin[McK08,Theorem2.3.1](inthecasechar(F)=p)and[Mou07,Theorem3.5].Intheseresultsitisshownthatagenericabeliancrossedproductofp-powerdegreedefinedbyanon-degeneratematrixisindecomposable.Recallthatap-powerindexdivisionalgebraDoverafieldFissaidtobedecomposableifthereexistsanisomorphismD∼=D1⊗FD2withind(Di)1.In[McK08,section3.3]examplesofsuchalgebraswithindexpnandexponentpforallp6=2andalln≥2aregiven.Anexampleisalsogiveninthecasep=2andn=3.Inthep6=2exampletheabeliancrossedproductΔisconstructedbygenericallyloweringtheexponentofanabeliancrossedproductwithexponentequaltoindexequaltopn.Thatis,letΔ′beanabeliancrossedproductdefinedbythegroupG∼=(Z/pZ)n,n≥2,withindexandexponentpn.SetY=SB(Δ′⊗p),theSeveri-Brauervariety,andletF(Y)bethefunctionfieldofY.ThensetΔ=Δ′⊗F(Y).Δhasexponentpandindexpnbytheindexreductiontheoremof[SVdB92].Δisshowntobedefinedbyanon-degeneratematrixbystudyingthetorsioninCH2(SB(Δ))([McK08,Prop.3.11]).SinceΔisdefinedbyanon-degeneratematrixAΔ,itsassociatedgenericabeliancrossedproduct,isindecomposable.BecauseofthemethodofconstructionofΔ,by[Kar98,Corollary5.4],theabeliancrossedproductΔisitselfanindecomposabledivisionalgebraofexponentpandindexpn.Inthispaperweconstructanabeliancrossedproduct,ΔFA,whichisdecomposableofindexp2,exponentp(p6=2),andisdefinedbyanon-degeneratematrix.AsmentionedaboveAΔFA,thegenericabeliancrossedproductassociatedtoΔFA,isthereforeindecomposable.Thestrategyistomakeanabeliancrossedproductdecomposableinagenericwayandprovethatthematrixdefiningtheresultingdecomposableabeliancrossedproductisnon-degenerate.Theoutlineofthepaperisasfollows.Insection2weconstructΔFA,adecom-posabledivisionalgebraofindexp2andexponentpwithmaximalabeliansubfieldL(Lemma2.9andCorollary2.17).WestudyΔFAfortherestofthepaper,withourgoalbeingtoshowthatitisdefinedbyanon-degeneratematrix.Thedifficultyinprovingnon-degeneracyofthematrixdefiningΔFAliesinthefactthatthelat-ticeMωusedinthedefinitionofLisnotH1-trivial.Insection3wealleviatethisproblembyconstructinganH1-trivialization,M,ofthelatticeMωandanalyzingitsstructureasamoduleoveragroupring.Insection4westudytheformofelementsinMwhichcouldpossiblymakethematrixdegenerate.Moreover,itisnotedthatitsufficestoprovethematrixisnon-degenerateinthelatticeMsinceMisanH1-trivialmodule.Finallyinsection5weprovethemaintheorem,Theo-rem5.2,whichstatesthatthematrixdefiningtheabeliancrossedproductΔFAisnon-degenerate.AcknowledgmentsTheauthorwouldliketothankDavidSaltmanandAdrianWadsworthforhelpwiththisproject,especiallyfortheirhelpwiththehomologicalformulationofthedegeneracyconditiongiveninsection5.22TheexampleThegoalofthissectionistoconstruct,inaverygenericway,adecomposableabeliancrossedproductofindexp2andexponentp.Thisisdoneusingfieldsgeneratedbygrouplatticesasin[Sal02],[Sal99,section12]