Galois descent - PDF document

GALOISDESCENTKEITHCONRAD1.IntroductionLetL=Kbeaeldextension.AK-vectorspaceWcanbeextendedtoanL-vectorspaceL KW,andWembedsintoL KWbyw7!1 w.Underthisembedding,whenW6=0aK-basisfeigofWturnsintoanL-basisf1 eigofL KW.PassingfromWtoL KWiscalledascent.Intheotherdirection,ifwearegivenanL-vectorspaceV6=0,wemayaskhowtodescribetheK-subspacesWVsuchthataK-basisofWisanL-basisofV.Denition1.1.ForanL-vectorspaceV,aK-subspaceWsuchthataK-basisofWisanL-basisofViscalledaK-formofV.Forcompleteness,whenV=0(sothereisnobasis),weregardW=0asaK-formofV.ThepassagefromanL-vectorspaceVtoaK-formofViscalleddescent.Whetherwecandescendisthequestionofllinginthequestionmarkinthegurebelow.LL KWVK WOO ?OO Example1.2.AK-formofLnisKnsincethestandardK-basisofKnisanL-basisofLn.Example1.3.AK-formofMn(L)isMn(K)sincethestandardK-basisofMn(K)isanL-basisofMn(L).Example1.4.AK-formofL[X]isK[X]sincetheK-basisf1;X;X2;:::gofK[X]isanL-basisofL[X].Example1.5.EveryL-vectorspaceVhasaK-form:whenV6=0,pickanyL-basisfeigofVanditsK-spanisaK-formofVsincetheei'sarelinearlyindependentoverKandthusareabasisoftheirK-span.WhenK=RandL=C,ascent(passingfromWtoC RW)istheprocessofcomplexicationanddescentisrelatedtoconjugationsoncomplexvectorspaces:anR-formofacomplexvectorspaceisthexedsetofaconjugation.OurdenitionofaK-forminvolvesachoiceofbasis.Let'scheckthischoicedoesn'treallymatter:Theorem1.6.LetVbeanonzeroL-vectorspaceandWbeanonzeroK-subspaceofV.Thefollowingconditionsareequivalent:(1)anyK-basisofWisanL-basisofV,(2)someK-basisofWisanL-basisofV.1 2KEITHCONRAD(3)theL-linearmapL KW!Vgivenbya w7!awisanisomorphismofL-vectorspaces.Proof.(1))(2):Obvious.(2))(3):SupposetheK-basisfeigofWisanL-basisofV.ThentheL-linearmapL KW!Vgivenbya w7!awsends1 eitoeisoitidentiesL-basesoftwoL-vectorspaces.Thereforethismapisanisomorphism.(3))(1):SupposeL KW=VasL-vectorspacesbya w7!aw.ForanyK-basisfeigofW,f1 eigisanL-basisofL KWandthereforeundertheindicatedisomorphismthevectors1
ei=eiareanL-basisofV.ThesecondpropertyofTheorem1.6ishowwedenedaK-form.TherstpropertyshowstheconceptofaK-formisindependentofthechoiceofbasis.Thethirdpropertyisthe\right"denitionofaK-form,1althoughtheotherpropertiesarearguablyabetterwaytounderstandwhattheconceptisallabout(oreventorecognizeitinconcretecaseslikeExamples1.2,1.3,and1.4.)IntheC=R-case,R-formsofacomplexvectorspaceareparametrizedbytheconjugationsonV.Generalizingthis,wewillseethatwhenL=KisaniteGaloisextension,wecanparametrizetheK-formsofanL-vectorspaceVbykeepingtrackofhowGal(L=K)canactina\semilinear"wayonV.2WewillndthatforanysemilinearactionofGal(L=K)onanonzeroL-vectorspaceV,thereisanL-basisofGal(L=K)-invariantvectors:thatmeans(v)=vforall2Gal(L=K).Referencesonthismaterialare[1,pp.295{296],[3,Chap.17],and[5,pp.66{68].2.GaloisdescentonvectorspacesFromnowonwesupposeL=KisaniteGaloisextensionandwriteG=Gal(L=K).WewillintroduceanorganizedwayforGtoactonanL-vectorspace(whichwewillcallaG-structure),whereGinteractsinareasonablewaywithscalarmultiplicationbyL.Denition2.1.ForanL-vectorspaceVand2G,a-linearmapr:V!VisanadditivefunctiononVsuchthat(2.1)r(av)=(a)r(v)forallainLandvinV.WhenistheidentityautomorphismofL,risL-linear.ForgeneralinG,risK-linear(takea2Kin(2.1)),butitisnotquiteL-linear;theeectofscalingbyLonVis\twisted"bywhenrisapplied.Whenthereferencetoin(2.1)isnotneeded,thelabelsemilinearisused,butitshouldbekeptinmindthatasemilinearmapisalwaysrelativetoachoiceofeldautomorphismofL.Example2.2.IfVisacomplexvectorspaceand:C!Ciscomplexconjugation,a-linearmaponVisaconjugate-linearmap.Example2.3.OnLn,r(a1;:::;an)=((a1);:::;(an))is-linear.Example2.4.OnMn(L),r(aij)=((aij))is-linear. 1Amongotherthings,thethirdpropertymakessenseforthezerovectorspacewithoutneedingaseparatedenitioninthatcase.2TheconceptofaK-formdoesnotrequireL=KtobeGalois,butintheGaloiscasewecansayalotaboutalltheK-forms. GALOISDESCENT3Example2.5.OnL[X],r(Pdi=0aiXi)=Pdi=0(ai)Xiis-linear.Example2.6.WhenWisaK-vectorspace,wecanapplytothe\rstcomponent"inL KW:thefunctionr= idWmappingL KWtoitselfbyr(a w)=(a) wonsimpletensorsis-linear,sincer(a0(a w))=r(a0a w)=(a0a) w=(a0)(a) w=(a0)((a) w),sor(a0t)=a0r(t)foralltinL KWbyadditivity.Sincer(1 w)=1 wandtheconditionr(a w)=(a) wisequivalenttor(a(1 w))=(a)r(1 w),thissemilinearactionofGonL KWistheuniqueonethatxes1 Wpointwise.Denition2.7.AG-structureonanL-vectorspaceVisasetoffunctionsr:V!V,oneforeachinG,suchthatris-linear,r1=idV,andrr0=r0.WhenVisgivenaG-structure,wesayGactssemilinearlyonV.Example2.8.WhenK=RandL=C,soG=Gal(C=R)istheidentityandcomplexconjugation,todescribeaG-structureonacomplexvectorspaceVweonlyneedtodescribethemapr:V!Vassociatedtocomplexconjugation,sincebyDenition2.1themapassociatedtotheidentityofGhastobetheidentity.Theconditionsrmustsatisfyare:risadditive,r(zv)= zr(v),andr2=idV.ThisisnothingotherthanaconjugationonV,soachoiceofGal(C=R)-structureonacomplexvectorspaceisthesameasachoiceofconjugationonit.Example2.9.ThemapsrinExamples2.3,2.4,2.5,and2.6,asrunsoverG,areaG-structureonLn,Mn(L),L[X],andL KW.TheG-structureinExample2.6iscalledthestandardG-structureonL KW.Example2.10.If':V!V0isanL-vectorspaceisomorphismandVhasaG-structurefrg,thereisauniqueG-structurefr0gonV0compatiblewith':r0(v0)='(r'�1(v0)):V'// r V0r0 V'// V0Justasitissimplertowritegroupactionsonsetsasgxinsteadofg(x)(wheregisthepermutationonXassociatedtog),itissimplertowriter(v)asjust(v).Inthisnotation,theequationr(av)=(a)r(v)becomes(av)=(a)(v).SoaG-structureonanL-vectorspaceVisawayofmakingthegroupGactsemilinearlyonV:each2Gis-linearonV,idL2GactsasidV,and(0(v))=(0)(v)foralland0inGandv2V.OnacomplexvectorspaceVthereisaone-to-onecorrespondencebetweenR-formsofVandconjugationsonV.WewillgeneralizethiscorrespondencetoonebetweenK-formsofanL-vectorspaceandG-structuresonthevectorspace.(ThisisageneralizationsinceExample2.8saysconjugationsonacomplexvectorspacearebasicallythesamethingasGal(C=R)-structuresonthevectorspace.)Firstweneedseverallemmas.Lemma2.11.LetVbeanL-vectorspacewithaG-structureandletV0beanL-subspacethatispreservedbyG:forall2G,(V0)V0.ThenthequotientvectorspaceV=V0hasaG-structuredenedby(v+V0)=(v)+V0. 4KEITHCONRADProof.WecheckthattheactionofGonV=V0iswell-dened:ifv1v2modV0thenv1�v22V0,soforany2Gwehave(v1�v2)2(V0)V0.Thus(v1)�(v2)2V0,so(v1)(v2)modV0.Thateachacts-linearlyonV=V0isclear,becausetherelevantconditionsarealreadysatisedonVandthusworkoutoncosetrepresentatives.Furtherdetailsarelefttothereader.Lemma2.12.LetAbeanabeliangroupand1;:::;n:A!Lbedistincthomomor-phisms.ForanL-vectorspaceVandv1;:::;vn2V,if1(a)v1+


+n(a)vn=0foralla2Athenallviare0.Proof.ThespecialcaseV=Listhelinearindependenceofcharacters.Itisleftasanexercisetorereadtheproofofthatspecialcaseandgeneralizetheargument.WhenVisanL-vectorspacewithaG-structure,thexedsetofGinVisVG=fv2V:(v)=vforall2Gg:ThisisaK-subspace.WhenK=RandL=C,soaG-structureonVisachoiceofconjugationconV,VG=fv2V:c(v)=vg.Lemma2.13.LetVbeanL-vectorspacewithaG-structure.DeneacorrespondingtracemapTrG:V!VbyTrG(v)=X2G(v):ThenTrG(V)VG,andwhenv6=0inVthereisa2LsuchthatTrG(av)6=0.Inparticular,ifV6=0thenVG6=0.Proof.ToshowthevaluesofTrGareinVG,forany02G0(TrG(v))=X2G0((v))=X2G(0)(v)=X2G(v)=TrG(v):Toshowifv6=0thatTrG(av)6=0forsomea2L,weprovethecontrapositive.Assumeforaxedv2VthatTrG(av)=0foralla2L.Then0=X2G(av)=X2G(a)(v)foralla2L.ByLemma2.12withA=L,every(v)is0.Inparticular,at=idLwegetv=0.NextisourmainresultlinkingK-formsandG-structures.WewillusethetensorproductdescriptionofaK-form(fromTheorem1.6):aK-subspaceWofanL-vectorspaceVisaK-formexactlywhenthenaturalL-linearmapL KW!VisanisomorphismofL-vectorspaces.Theorem2.14.LetVbeanL-vectorspace.ThereisabijectionbetweenthefollowingdataonV:(1)K-formsofV,(2)G-structuresonV.Inbrief,thecorrespondencefrom(1)to(2)isW L KWwithitsstandardG-structureandthecorrespondencefrom(2)to(1)isV VG. GALOISDESCENT5Proof.ThisisclearifV=0sincef0gistheonlyK-form(eventheonlyK-subspace)andthereisonlyoneG-structure.SofromnowontakeV6=0.IfwestartwithaK-formWofV,so':L KW!Vbya w7!awisanL-linearisomorphism,wegetaG-structureonVbyusing'totransportthestandardG-structurefrgonL KW(Example2.6)toaG-structurefWg2GonV(Example2.10).WesimplyinsistthediagramsL KW'// r VW L KW'// Vcommuteforall2G.Explicitly,setW Xiaiwi!=Xi(ai)wi;whereai2Landwi2W.(Thewell-denednessofthisformula,wherethewi'sareanyelementsofW,dependson'beinganisomorphismofL-vectorspaces.)Conversely,ifVhasaG-structurethentheK-subspaceVGturnsouttobeaK-formonV:theL-linearmapf:L KVG!Vbya w7!awisanisomorphism.Toshowfisone-to-one,supposef(t)=0forsomet2L KVG.Writetasasumofsimpletensors,sayt=Pai wi.Thisnitesumcanbearrangedtohavewi'sthatarelinearlyindependentoverK:hereweneedtoknowVG6=0(Lemma2.13).Then0=f(t)=Paiwi.WewillshowK-linearlyindependentvectorsinVGareL-linearlyindependentinV,henceallaiare0andthiswouldmeant=0sofisinjective.ToproveeveryK-linearlyindependentsetinVGisL-linearlyindependent,assumeotherwise:thereisaK-linearlyindependentsetinVGthatisL-linearlydependent:suchadependencerelationlookslike(2.2)a1w1+


+anwn=0;wherew1;:::;wn2VGandtheai'sinLarenotall0.Take(2.2)tobeanontrivialL-linearrelationamongK-linearlyindependentvectorsinVGofleastlength(leastnumberofterms).Theneveryaiisnonzeroandn2.Byscaling,wemaysupposean=1.Applying2Gto(2.2),weget(2.3)(a1)w1+


+(an)wn=0:Subtract(2.3)from(2.2):(a1�(a1))w1+


+(an�(an))wn=0:Thelasttermis0sincean=1,sothisL-linearrelationhasn�1terms.Bytheminimalityofn,suchanL-linearrelationamongK-linearlyindependentvectorsw1;:::;wn�1hastobethetrivialrelation:ai�(ai)=0fori=1;2;:::;n�1.SoeachaiisxedbyallinG,henceai2Kfori=1;2;:::;n�1.Alsoan=12K.Butthatmeans(2.2)isanontriviallineardependencerelationamongthewi'soverK,thatisimpossible(thewi'sarelinearlyindependentoverK).Sowehaveacontradiction,whichprovesfisone-to-one.Toshowfisonto,welookattheimagef(L KVG),whichisanL-subspaceofV.ThisimageisstableundertheactionofG:onsimpletensorsa winL KVG,(f(a w))=(aw)=(a)(w)=(a)w=f((a) w); 6KEITHCONRADsobyadditivityofonVwehave(f(L KVG))f(L KVG).Thereforethequotientspace V:=V=f(L KVG)inheritsaG-structurefromVby( v)= (v)(Lemma2.11).Forv2V,TrG(v)2VGf(L KVG),soon VwehaveTrG( v)= TrG(v)= 0.Sinceallelementsof Vhavetrace 0, Vhastobe 0byLemma2.13.(Anonzeroelementof VwouldhaveanonzeroL-multiplewithnonzerotrace.)Since V= 0,V=f(L KVG),sofisonto.IntheC=R-case,thesurjectivityoff:C RVc!VforanycomplexvectorspaceVwithaconjugationcisbasedontheequation(2.4)v=v+c(v) 2+iv�c(v) 2i;wherethevectors1 2(v+c(v))and1 2i(v�c(v))arexedbyc.Theformula(2.4)canbegeneralizedtotheL=K-caseusingTrG:iff1;:::;dgisaK-basisofLthentherearef1;:::;dginLsuchthatv=PjjTrG(jv)forallv2V.Thatwouldgiveasecondproofofsurjectivityoff.ItremainstocheckthatourcorrespondencesbetweenK-formsofVandG-structuresonVareinversesofoneanother.K-formtoG-structureandback :PickaK-formWofV.ThecorrespondingG-structurefW:2GgonVisgivenbyW(Piaiwi):=Pi(ai)wiforai2Landwi2W.WeneedtocheckthesubspaceVG=fv2V:W(v)=vforall2GgassociatedtothisG-structureisW.CertainlyWVG.ToshowVGW,pickaK-basisfeigofW.SinceWisaK-formofV,anyv2Vhastheformv=Paieiwithai2L(allbutnitelymanycoecientsaiare0).ThenW(v)=Pi(ai)ei.IfW(v)=vforall2GthenXi((ai)�ai)ei=0forall2G.Theei'sarelinearlyindependentoverL(becausetheyarethebasisofaK-formofV),so(ai)�ai=0forallaiandall2G.ThusallaiareinK,sov2W.ThusVGW.3G-structuretoK-formandback :GivenaG-structureonV,whichwewritesimplyasv (v),thecorrespondingK-formisVG.WehavetocheckthattheisomorphismL KVG!Vgivenbya w7!awtransportsthestandardG-structureonL KVGtotheoriginalG-structurewestartedwithonV(ratherthantosomeotherG-structureonV).UndertheisomorphismL KVG!V,atensorPiai wiinL KVGisidentiedwithPiaiwiinV,andthestandardG-structureonthetensorproductisgivenby(Piai wi)=Pi(ai) wi(forall2G),whichgoesovertoPi(ai)wiinVbytheisomorphism.Sincethewi'sareinVG,Pi(ai)wi=(Piaiwi),sotheisomorphismfromL KVGtoVdoesidentifythestandardG-structureonL KVGwiththeoriginalG-structureonV.Thedown-to-earthcontentofTheorem2.14isthatwhenGactssemilinearlyonV,thereisaspanningsetofVoverLconsistingofG-invariantvectors.UsingthisiscalledGaloisdescent.Remark2.15.Theorem2.14istruewhenL=KisaninniteGaloisextensionandthesemilinearactionGV!Viscontinuous,whereGhasitspronitetopologyandVhasthediscretetopology.See[2,Lemma5.8.1]. 3ThatVGWcanalsobeprovedusingthenormalbasistheorem. GALOISDESCENT7Corollary2.16.IfVhasaG-structure,K-independentvectorsinVGareL-independent.Proof.AK-independentsubsetofVGcanbeextendedtoaK-basisofVG,whichisanL-basisofV(sinceVGisaK-formofV),soitislinearlyindependentoverL.Corollary2.17.ForanyK-vectorspaceW,withL KWgivenitsstandardG-structure,(L KW)G=1 W.Proof.Exercise.Example2.18.LetXbeanitesetonwhichtheGaloisgroupG=Gal(L=K)acts(bypermutations).ThenthespaceV=Map(X;L)ofallfunctionsX!LisanL-vectorspaceunderpointwiseoperations.AbasisofVoverLisfx:x2Xg.ThegroupGactssemilinearlyonV:foranyfunctionf:X!LinVand2G,dene(f):X!LsothediagramXf//  L X(f)// Lcommutes.Thismeans(f)((x))=(f(x))forallx2X,so(f)(x)=(f(�1(x))):Thisequationdenes(f)asafunctionX!L.Let'scheck:V!Vis-linear.Fora2L,wewanttocheckthat(af)=(a)(f).Well,ateachx2X,((af))(x)=((af)(�1(x)))=(af(�1(x)))=(a)(f(�1(x)))=(a)((f)(x));so(af)=(a)(f)asfunctionsinMap(X;L).Check(x)=(x).WhatisVG?ItisnaturaltoguessthatVGequalsMap(X;K),whichistheK-spanoffx:x2Xg.Asasmallpieceofevidence,Map(X;K)isaK-subspaceofVwithK-dimension#XanddimK(VG)=dimL(V)=#Xtoo.However,thedelta-functionsxlieinMap(X;K)and(x)=xonlywhen(x)=x.Thereforeallthex'sareinVGonlywhenGactstriviallyonX.SoVGisnotMap(X;K)ifGactsnontriviallyonX.Infact,VG=ff:(f)=fg=ff:(f(�1(x)))=f(x)forall;xg=ff:(f(x))=f((x))forall;xg;soVGconsistsofthefunctionsX!LthatcommutewiththeG-actionsonXandL.Functionssatisfying(f(x))=f((x))forall2Gandx2XarecalledG-maps(theyrespecttheG-actions).BecauseVGspansVoverL,everyfunctionX!LisanL-linearcombinationofG-mapsX!L.IfGactstriviallyonXthenVG=ff:(f(x))=f(x)forall;xg=Map(X;K),soaG-mapX!LinthiscaseisjustafunctionX!K. 8KEITHCONRADWeconcludethissectionbyshowinghowtointerpretTheorem2.14intermsofrepre-sentations.(Ifyoudon'tknowrepresentationsofnitegroups,youmaywanttoskiptherestofthissection.)Firstlet'srecallhowyoucoulddiscoverthattheringC[G]shouldberelevanttorepresentationsofGoncomplexvectorspaces.Let'sstareattheformulas(2.5)()(cv)=c()(v)and()(()v)=()(v)forarepresentationofGonacomplexvectorspaceV(herec2Candv2V).Nowabstracttheseformulasbywritingsomeneutralsymbolefor()andtakingawayv:introducetheC-vectorspaceC[G]=L2GCewithmultiplicationrules(2.6)ec=ceandee=einspiredby(2.5).NowC[G]isanassociativeringwithidentity,andevenaC-algebrasinceCcommuteswithallthebasisvectorse.WecanconsideranyrepresentationofGonVasawayoflettingthebasisvectorseactasC-linearmapsV!V,byev=()(v),andtheconditions(2.5)and(2.6)saypreciselythatthismakesVintoa(left)C[G]-module.Conversely,anyleftC[G]-modulestructureonVprovidesarepresentationofGonVbyfocusingattentiononhowthee'sinsideC[G]actonV(whichdoesn'tloseanyinformationsincethee'sspanV).Nowlet'slookatG-structuresonanL-vectorspaceV,whereG=Gal(L=K),withthegoalofndinganabstractringwhosemodulestructuresonanyL-vectorspaceVarethesamethingasG-structuresonV.AnyG-structureonVprovidesuswith-linearmapsr:V!Vforall2G,whichmeansther'sareadditiveand(2.7)r1(v)=v;r(cv)=(c)r(v);andr(r(v))=r(v)forallc2Landv2V.Nowlet'swipevoutoftheseequationsandturnthemapsrintobasisvectorse.DenetheL-vectorspaceC(G)=L2GLeanddeclaremultiplicationinC(G)tobegivenbytherules(2.8)e1=1;ec=(c)e;andee=eforandinGandc2L.4ThismakesC(G)anassociativeringwithidentitye1(check!)andC(G)isaK-algebra(sinceec=cewhenc2K).IfGisnottrivial(thatis,L6=K),ecisnotceforc2L�K,soC(G)isnotanL-algebra(inthesamewaythequaternionsHareanR-algebrabutnotaC-algebraeventhoughCH).Themultiplicationrules(2.8)inC(G)areanabstractformofthewayaG-structurebehaves,asdescribedin(2.7):aG-structureonVisthesamethingasaleftC(G)-modulestructureonV,whereeactsonVasthe-linearmapr.Explicitly,ifwehaveaG-structurefrgonVthenmakeVintoaC(G)-modulebytheformula X2Gae!(v)=X2Gar(v)and,conversely,ifVhasaC(G)-modulestructurethenfocusingonhowthebasisvectorseactonVgivesusaG-structure(checkspecicallywhyeacheisa-linearmaponVfromthedenitionofaC(G)-modulestructureonV).Theorem2.14thereforesays,intermsofC(G),thattheK-formsofVareessentiallythesamethingastheC(G)-modulestructuresonV. 4UnlikethegroupringC[G],theconstructionofC(G)dependsessentiallyonGbeingaGaloisgroupsinceweuseitsactiononLinthedenitionofthemultiplicationin(2.8). GALOISDESCENT9ThemostbasicexampleofaC(G)-moduleisL,onwhichG=Gal(L=K)actsbyitsverynatureasaGaloisgroupandthisextendstoaC(G)-modulestructureviatheformula(P2Gae)(x)=P2Ga(x)forallx2L.ThateveryG-structurehasanassociatedK-formtellsussomethingaboutC(G)-submodulesofaC(G)-module.LetVbeanonzeroC(G)-module(whichisaconcisewayofsayingVisanonzeroL-vectorspacewithaG-structure).TheexistenceofaK-formmeansVhasanL-basisfvigofG-invariantvectors:V=Li2ILviand(vi)=viforall2G.ThelineLviispreservedundertheactionofGandL,henceundertheactionofC(G)=L2GLe.ThereforeeachLviisaC(G)-submodule,andvibeingG-invariantmakesLviisomorphictoLasC(G)-modulesbythenaturalmapxvi7!x.(Warning:aC(G)-linearmapisnotL-linearsincethee'sdon'tcommutewithL,unlessGistrivial.)ThusallnonzeroC(G)-modulesaredirectsumsofcopiesofLasaC(G)-module.ThisinfactisanotherwayofthinkingabouttheexistenceofK-forms.Indeed,supposeweknew(bysomeothermethod)thateverynonzeroC(G)-moduleisadirectsumofC(G)-submodulesthatareeachisomorphictoLasaC(G)-module.ThenforanynonzeroL-vectorspaceVwithaG-structure,viewVasaC(G)-moduleandbreakitupasLi2IViwhereVi=LasC(G)-modules.Letfi:L!VibeaC(G)-moduleisomorphismandsetvi=fi(1).Thenforany2G,(vi)=(fi(1))=fi((1))=fi(1)=vi,soviisaG-invariantvector.SinceLviViandbothareL-vectorspacesofthesameniteK-dimension(becausefiisaK-linearisomorphism,forgettingalittlestructureintheprocess),Lvi=Vi.NowthedirectsumdecompositionV=Li2ILvirevealsaK-formforV,namelyW=Li2IKvi.3.ApplicationstoVectorSpacesOurrstapplicationofGaloisdescentistosystemsoflinearequations.IftheequationshavecoecientsinKandthereisanonzerosolutionoverLthenthereisalsooneoverK.Theorem3.1.Foranyhomogeneoussystemoflinearequationsinnunknownswithco-ecientsinK,thesolutionsinLnareL-linearcombinationsofthesolutionsinKn.Inparticular,ifthereisanonzeroL-solutionthenthereisanonzeroK-solution.Proof.WritethesystemoflinearequationsintheformAx=0,whereAisanmnmatrixwithentriesinK(mbeingthenumberofequations).LetVLnbetheL-solutionsofthesystem:V=fv2Ln:Av=0g.Thereisastandardsemilinear(coordinatewise)actionofGonLn,andbecauseAhasentriesinKtheG-actionpreservesV:ifv2Vthen(v)2Vbecause(Av)=A((v))and(0)=0.SowegetacoordinatewiseG-structureonV,andbyTheorem2.14VisspannedoverLbyitsG-xedsetVG=V\(Ln)G=V\Kn,whicharethesolutionstoAx=0inKn.Theorem3.1istruewithoutL=KbeingGalois:foraK-linearmapA:Kn!Km,themap1 A:Ln!Lmsatisesim(1 A)=L
im(A).Nextwedescribehowdescentbehavesonsubspaces:ifV0VandwehaveaG-structureonV,whendoesV0haveaK-forminVG?TheanswerisconnectedtothepreservationofV0bytheG-structureonV.Theorem3.2.LetVbeanL-vectorspacewithaG-structure.ForanL-subspaceV0V,thefollowingconditionsareequivalent:(1)V0hasanL-spanningsetinVG,(2)(V0)V0forall2G, 10KEITHCONRAD(3)(V0)=V0forall2G,(4)V0hasaK-forminVG.Whenthesehold,theonlyK-formofV0inVGisV0\VG.IfdimL(V0)1,theseconditionsarethesameasdimK(V0\VG)=dimL(V0).Proof.EverythingisobviousifV0=0,sowemaytakeV06=0.(1))(2):SupposeV0=PLviwherevi2VG.Thenfor2G,(V0)P(L)vi=PLvi=V0.(2))(1):Suppose(V0)V0forall2G.ThenthegivenG-structureonVisaG-structureonV0,sobyTheorem2.14V0hasanL-spanningsetin(V0)GVG.(2))(3):Using�1inplaceof,wehave�1(V0)V0soV0(V0).Thus(V0)=V0forall2G.(3))(2):Obvious.(3))(4):TheG-structureonVisaG-structureonV0,so(V0)GisaK-formofV0and(V0)GVG.(4))(2):ForaK-formW0ofV0inVG,V0=LW0,so(V0)L(W0)=LW0=V0.Whentheseconditionshold,(V0)GisaK-formofV0,and(V0)G=V0\VG.SupposeW0isanyK-formofV0inVG.WewanttoshowW0=(V0)G.ThenaturalmapL KW0!V0givenbya w07!aw0isanL-vectorspaceisomorphism,andthetransportedG-structureonV0fromthestandardG-structureonL KW0throughthisisomorphismisW0(aw0)=(a)w0fora2Landw02W0.SinceW0VG,intermsoftheoriginalG-structureonVwehave(a)w0=(a)(w0)=(aw0)=VG(aw0)=(V0)G(aw0),soW0=(V0)G.Bytheone-to-onecorrespondencebetweenK-formsandG-structuresonV0,W0=(V0)G.NowassumedimL(V0)isnite.WewanttoshowthefourconditionsareequivalenttodimK(V0\VG)=dimL(V0).Wewillshowthisdimensionconstraintisequivalentto(1).Letd=dimL(V0).Suppose(1)holds.IfV0hasanL-spanningsetinVGithasanL-basisinVG,sayv1;:::;vd.Thenforv02V0,writev0=Pdi=1aiviwithai2L.Ifv02VGtoo,thenforany2G,v0=(v0)=dXi=1(ai)vi;solinearindependenceofthevi'soverLimpliesai=(ai)foralli(and),soai2Kforalli.Thusv02Pdi=1Kvi,soV0\VGPdi=1Kvi.Thereverseinclusionisclear,sodimK(V0\VG)=d=dimL(V0).Conversely,assumedimK(V0\VG)=dimL(V0)andletfv1;:::;vdgbeaK-basisofV0\VG.Thisbasishassizedbecauseofthedimensionconstraint.Thevi'sareaK-linearlyindependentset,sotheyareL-linearlyindependentsinceVGisaK-formofV.ThenPiLvihasL-dimensiondandliesinV0,whoseL-dimensionisd,soPiLvi=V0,whichiscondition(1).Remark3.3.SinceK-independentvectorsinVGareL-independent,forallV0wehavedimK(V0\VG)dimL(V0).ThusthedimensionconditioninTheorem3.2saysV0\VGhasitsbiggestpossibleK-dimension. GALOISDESCENT11Anytwoconjugationsonacomplexvectorspacearerelatedtoeachotherbyanauto-morphismofthevectorspace.Moregenerally,anytwoG-structuresonanL-vectorspacearelinkedtooneanotherbyanautomorphismofthevectorspace:Theorem3.4.Letfrgandfr0gbetwoG-structuresonanL-vectorspaceV6=0.Thereisa'2GL(V)suchthatr0='r'�1forall2G:thediagram(3.1)V'// r Vr0 V'// Vcommutesforall2G.Hereandlater,GL(V)meansautomorphismsofVasanL-vectorspace.Proof.LetW=fv2V:r(v)=vforall2GgbetheK-formofVforfrg.(ItwouldbebadtowritethisasVGsincetherearetwoG-structureswearedealingwithonVandthusthenotationVGwouldbeambiguous.)LetW0=fv2V:r0(v)=vforall2GgbetheK-formofVforfr0g.ThetwodiagramsL KWf// W Vr L KWf// VL KW0f0// W0 Vr0 L KW0f0// Vcommuteforall2G,wherefandf0arethenaturalL-linearmaps(isomorphisms).Sincefandf0areisomorphisms,dimK(W)=dimL(V)=dimK(W0)(thesemightbeinnitecardinalnumbers).ThereforethereisaK-linearisomorphism :W!W0,itsbaseextension1 :L KW!L KW0isanL-linearisomorphism,andthediagramL KW1 // W L KW0W0 L KW1 // L KW0commutesforall2G:onanysimpletensora w,goingalongthetopandrightoralongtheleftandbottomsendsthissimpletensorto(a) (w).NowcombinethethreecommutativediagramstogetthecommutativediagramVf�1// r L KWW 1 // L KW0W0 f0// Vr0 Vf�1// L KW1 // L KW0f0// Vforevery2G.Themapsalongthetopandbottomdon'tinvolve,andareallL-linearisomorphisms.Callthe(common)compositemapalongthetopandbottom',so'2GL(V),andremovethemiddleverticalmapstobeleftwithacommutativediagramoftheform(3.1). 12KEITHCONRADCorollary3.5.LetVbeanL-vectorspacewithtwoK-formsWandW0.LetfWgandfW0gbethecorrespondingG-structuresonV.Thereis'2GL(V)suchthat'(W)=W0andthediagramV'// W VW0 V'// Vcommutesforall2G.Proof.ByTheorem3.4,thereis'2GL(V)suchthatV'// W VW0 V'// Vcommutesforall2G.Itremainstocheck'(W)=W0.WehaveW=fv2V:W(v)=vforall2GgandW0=fv2V:W0(v)=vforall2Gg.Soforw2Wand2G,W0('(w))='(W(w))='(w).Thus'(w)2W0,so'(W)W0.Forw02W0,writew0='(v)withv2V.FromW0(w0)=w0,W0('(v))='(v),so'(W(v))='(v).Since'isinjective,W(v)=vforall,sov2W.ThusW0'(W).Theorem3.6.LetV1andV2beL-vectorspacesequippedwithG-structures.LetW1andW2bethecorrespondingK-formsofV1andV2.AnL-linearmap:V1!V2istheL-linearbaseextensionofaK-linearmap':W1!W2ifandonlyifthediagramV1// W1 V2W2 V1// V2commutesforall2G.Equivalently,theL-vectorspaceHomL(V1;V2)hasaG-structuregivenby():=W2�1W1withcorrespondingK-formHomK(W1;W2).Proof.Exercise.WeconcludethissectionwithareinterpretationofTheorem3.4intermsofmodules,usingtheringC(G)=L2GLeintroducedattheendofSection2.WesawtherethatanynonzeroC(G)-moduleisadirectsumofcopiesofLasaC(G)-module.Bytheinvarianceofdimensionofvectorspaces,thismeansaC(G)-moduleiscompletelydetermineduptoisomorphismbyitsL-dimension.Inotherwords,theendofSection2showsthatanytwoC(G)-modulestructuresonanonzeroL-vectorspaceVareisomorphic:thereisaC(G)-linearmap':V!VthatturnsoneC(G)-modulestructureintotheother.Thisisthesameassaying'isaL-linearautomorphismofVsuchthat'(ev)=e'(v)(becarefulonlytoviewVasaleftL-vectorspace,consideringhowmultiplicationofitwithGinsideofC(G)istwisted),andthatequationforallispreciselytheconclusionofTheorem3.4exceptitisgivenintheterminologyofG-structuresratherthanC(G)-modules.Sowe GALOISDESCENT13alreadyhadaproofofTheorem3.4attheendofSection2andit'salotmoreconceptualthantheproofwewroteoutforTheorem3.4.ThisillustrateshowusefulitcanbetointerpretG-structuresasC(G)-modulestructures.4.ApplicationstoIdealsOurnextsetofapplicationsofGaloisdescentconcernidealsinL[X1;:::;Xn],whichwewillabbreviatetoL[X ].AbasicquestioniswhetheranidealinthisringisgeneratedbypolynomialsinK[X ].LetV=L[X ]andletGactonVbyactingoncoecients.ThisactionisaG-structureonVwithcorrespondingK-formW:=VG=K[X ].ForanyidealIofL[X ],IG=I\K[X ]isanidealinK[X ].SayanidealIL[X ]isdenedoverKifithasageneratingset(asanideal!)inK[X ].Example4.1.InC[X;Y],I=(X+iY2;X�iY2)isdenedoverRsinceI=(X;Y2).Theorem4.2.ForanidealIL[X ],thefollowingconditionsareequivalent:(1)IisdenedoverK,(2)(I)Iforall2G,(3)(I)=Iforall2G.Proof.Itistrivialthat(1)implies(2)sinceageneratingsetofIinK[X ]isnotchangedby.Since(2)isstatedoverall,from�1(I)IforallwegetI(I)forall,so(I)=Iforall,whichis(3).Finally,assuming(3),since(I)=ItheGaloisgroupGactssemilinearlyonIasanL-vectorspace,sobyGaloisdescentonI,IhasaspanningsetasanL-vectorspaceinIG=I\K[X ]K[X ].SinceIisanidealinL[X ],anL-vectorspacespanningsetofIisalsoageneratingsetofIasanideal.HereisanexamplewhereL=Kisnon-GaloisandTheorem4.2breaksdown.LetK=Fp(T),L=Fp(pp T),andI=(X�pp T)inL[X].Then(I)=Iforall2AutK(L)(whichisatrivialgroup),butIhasnogeneratorinK[X].Infact,I\K[X]=(Xp�T),sotheidealinL[X]generatedbyI\K[X]issmallerthanI.Remark4.3.ByHilbert'sbasistheorem,I\K[X ]=(f1;:::;fr)=rXi=1K[X ]fiforsomefi'sinK[X ].IfIisdenedoverKthenbytheproofofTheorem4.2,IisspannedasanL-vectorspacebypolynomialsinI\K[X ]:I=PjLhjwherehj2I\K[X ].EachhjisaK[X ]-linearcombinationofthefi's,soIPri=1L[X ]fiandthereverseinclusionholdssinceIisanideal,soI=Pri=1L[X ]fi.Thatis,anidealinL[X ]denedoverKisnitelygeneratedoverL[X ]byanynitesetofidealgeneratorsofI\K[X ].Thespecialcaseof(3))(1)inTheorem4.2inonevariablecanbeprovedusingHilbert'sTheorem90insteadofGaloisdescent.LetIbeanidealinL[X].SinceL[X]isaPID,I=(f)forsomef2L[X].Then(I)=((f))forall2G.Saying(I)=Iforall2Gisequivalentto(f)=fforsome2L.Thenapplyany2Gtoget((f))=()(f),so()(f)=()f.Also()(f)=f,sof=()f,so=()sincef6=0.Hencethenumbersf:2Ggarea1-cocycleG!L.BythemultiplicativeformofTheorem90,=()=forsome2Landall2G,so(f)=(()=)f,so(f=)=f=forall2G.Thusf=2K[X]and(f=)=(f)asidealsinL[X].So(f)isdenedoverK. 14KEITHCONRADTherelationsbetweenGaloisdescentandcohomologygofurther.LetVbeanL-vectorspacewithaG-structure.A1-cocycleonVisafunctionc:G!Vsuchthatc()=c()+(c()).Example4.4.Fixingv2V,c()=(v)�visa1-cocyclesincec()+(c())=((v)�v)+((v)�v)=()(v)�v=c():TheadditiveformofTheorem90saysall1-cocyclesc:G!Llookliketheexample:c()=()�forsome2L.Let'srecallaproof.Forx2L,sety=P2Gc()(x).Forany2G,(y)=P(c())()(x)=P(c()�c())()(x)=P(c()�c())(x)=y�c()TrL=K(x).ChoosexsoTrL=K(x)6=0.Thenc()=z�(z)forz=y=TrL=K(x).Set=�z,soc()=()�.UsingGaloisdescentwecanextendthisfromcocyclesinLtococyclesinL-vectorspaceswithaG-structure.Theorem4.5.ForanyL-vectorspaceVwithaG-structure,every1-cocycleonVhastheformc()=(v)�vforsomev2V.Proof.WemaysupposeV6=0.LetfvigbeaK-basisofVG,sobyGaloisdescentV=MiLvi:Forany1-cocyclec:G!V,writec()=Pia;ivi,wherea;i2L.Thenthecocycleconditionc()=c()+(c())meansXia;ivi=Xia;ivi+ Xia;ivi!=Xia;ivi+Xi(a;i)vi=Xi(a;i+(a;i))vi;soforeachi,a;i=a;i+(a;i).Thusforeachi,7!a;iisa1-cocycleinL.ByTheorem90,foreachithereisbi2Lsuchthata;i=(bi)�biforall.Sincea;i=0forallbutnitelymanyi,wecanusebi=0forallbutnitelymanyi.Thenc()=Xi((bi)�bi)vi=Xi(bi)vi�Xibivi=(v)�v;wherev=Pibivi.HereisanimportantapplicationofTheorem4.5totheGaloisactiononquotientringsofL[X ].Theorem4.6.LetIL[X ]beanidealdenedoverK,soL[X ]=IhasaG-structureby(f+I)=(f)+I(Lemma2:11).Foreachf2L[X ],thefollowingareequivalent:(1)(f)fmodIforall2G,(2)fgmodIforsomeg2K[X ].Proof.Itistrivialthatthesecondconditionimpliestherst.Forthemoreinterestingreversedirection,assume(f)fmodIforall2G.Denec:G!Ibyc()=(f)�f. GALOISDESCENT15Byacomputation,c()+(c())=c()forallandinG,socisa1-cocycleintheL-vectorspaceI.ByTheorem4.5,c()=(h)�hforsomeh2I,so(f)�f=(h)�hforall2G.Thereforef�hisxedbyall2G,sof�h2K[X ].Setg=f�h,sofgmodIandg2K[X ].EventhoughidealsinL[X ]arenitelygeneratedasideals,theyareinnite-dimensionalasL-vectorspaces(exceptforthezeroideal),soitiscrucialthatTheorem4.5appliestogeneralvectorspaces,notjustnite-dimensionalvectorspaces.TheseGaloisdescentfeaturesonidealsleadtoapplicationsinalgebraicgeometry.Wepresenttwoofthem.Theorem4.7.LetVLdbethezerosetoff1;:::;fr2K[X ].TheidealofallpolynomialsinL[X ]thatvanishinLdwherethefi'svanishhasageneratingsetinK[X ].Equivalently,theidealI(V)=fg2L[X ]:g(P)=0forallP2VgisdenedoverK.NotethatbeforeTheorem4.7we'vewrittenVforavectorspace.NowVisdenotinganalgebraicvariety,soitisdenitelynotavectorspaceingeneral!ThereisnontrivialcontenttoTheorem4.7becausetheidealI(V)neednotbegeneratedbythefi'sthemselves.Forinstance,inC2letVbethezerosetoff1=X31andf2=X21�X1X2.ThenI(V)=(X1)whereastheideal(f1;f2)isstrictlycontainedinX1(anypolynomialintheideal(f1;f2)hasnoX1-term,butX1ofcoursedoes).Proof.ForP2V,fi(P)=0foralli,sofi((P))=0forall2G.Thus(P)2V,so(V)Vforall2G,hence(V)=Vforall2G.ToshowI(V)isdenedoverK,weshow(I(V))I(V)forall2G.Pickf2I(V)L[X ].For2GandP2V,setQ=�1(P)2V.Thenf(Q)=0,andapplyingtothisgives(f)((Q))=0,so(f)(P)=0.Thus(f)2I(V),so(I(V))I(V)forall2I(V).ThereforeI(V)isdenedoverK.Example4.8.IfVCdisthezerosetofsomerealpolynomialsf1;:::;fr,thentheidealI(V)ofcomplexpolynomialsvanishingonVisgeneratedbyrealpolynomials.TheproofofTheorem4.7isnotconstructive,sowedon'tgetamethodtowritedowngeneratorsofI(V)inK[X ]fromtheoriginalpolynomialsfiinK[X ]thatdeneV.Theorem4.9.LetIL[X ]beahomogeneousprimeidealdenedoverK.ThenL(I)G=fg=h:g;h2K[X ];h62I;gandharehomogeneousofequaldegreeg.Proof.Theinclusionisobvious.Weworkouttheinclusion.Supposeg;h2L[X ]arehomogeneousofequaldegree,h62I,andg=h2L(I)G,whichmeans(g=h)=g=hforallinG.Wecanwriteg=h=gk=hkforanynonzerok2L[X ].For2G,sinceh62Iand(I)=I,(h)62I.ThenQ2G(h)62IandtheproductisinK[X ].Sowithoutlossofgenerality,h2K[X ].Then(g=h)=(g)=h,so(g)=ginL(I).Both(g)andgareinL[I]=L[X ]=I,so(g)gmodIforall2G.ThereforeTheorem4.6tellsusgegmodIforsomeeg2K[X ].Sinceg�eg2I,Iisahomogeneousideal,andgisahomogeneouspolynomial,thehomogeneouspartsofegnotofdegreedeggareinI.Thereforewithoutlossofgeneralityegishomogeneousofthesamedegreeasg.Sog=h=eg=hinL(I),andegandhareinK[I]. 16KEITHCONRAD5.ApplicationstoAlgebrasOurnalsetofapplicationsofGaloisdescentistoL-algebras.WeunderstandL-algebratobeusedinthesenseofanyL-vectorspaceequippedwithanL-bilinearmultiplicationlaw.Wewillnotassumethealgebraisassociative.ExamplesofL-algebrasincludeMn(L),L[X ],andaLiealgebraoverL(whichisnotassociative)suchasgln(L).ForanyK-algebraA,L KAisanL-algebra.Denition5.1.AK-formofanL-algebraAisaK-subalgebraAofAsuchthatthenaturalmapL KA!Agivenbyx a7!xaisanisomorphismofL-algebras.AK-formofanL-algebraisaK-formasanL-vectorspace,butit'smorethanthat:thealgebrastructureneedstoberespected.IfAisanL-algebraandAisaK-formofAthenAisassociativeifandonlyifAisassociativeandAisaLiealgebraoverLifandonlyifAisaLiealgebraoverK.Thereasonisthattherelevantproperties(associativityortheJacobiidentity)aretrueonanL-algebraifandonlyiftheyaretrueonanL-basis,andwecanuseaK-basisofAasanL-basisofA.Example5.2.TwoR-formsoftheC-algebraM2(C)areM2(R)andthequaternionsH,viewedinsideM2(C)bya+bi+cj+dk7!(a+bi�c�dic�dia�bi).Bothare4-dimensionalR-algebraswhosestandardR-basisisaC-basisofM2(C).SinceM2(R)andHarenotisomorphicR-algebras(HisadivisionringandM2(R)isnot),dierentK-formsofanL-algebraneednotbeisomorphicK-algebras.Thisisanimportantcontrastwiththelineartheory(Theorem3.4),whereallK-formsofanL-vectorspaceareisomorphicasK-vectorspaces.Whenworkingwithalgebraswehavetokeeptabsonthemultiplicativestructuretoo,andthatcreatesnewpossibilities.AswithK-formsofanL-vectorspace,K-formsofanL-algebracorrespondtoanap-propriatesystemofsemilinearG-actionsonthealgebra.AG-structureonanL-algebraAisacollectionofmapsr:A!Aforall2Gsuchthatrisa-linearK-algebraautomorphism(notL-algebraautomorphism!),ridL=idA,andrr0=r0.Example5.3.TheentrywiseorcoecientwiseG-actionsonMn(L),gln(L),andL[X ]asL-vectorspacesarealsoG-structuresasL-algebras.Example5.4.IfAisaK-algebrathentheL-algebraL KAgetsastandardG-structurebyA Xici ai!:=X(ci) ai:Thisis-linear(AisadditiveandA(ct)=(c)A(t)forallc2Landt2L KA)andalsoAismultiplicative(A(tt0)=A(t)A(t0)),soAisaK-algebraautomorphismofL KA.ThinkingofAjustasaK-vectorspace,bytheproofofTheorem2.14wehave(L KA)G=1 A.IfAisanL-algebrawithaG-structure,thexedsetAGisaK-formofAasvectorspacesbyTheorem2.14.ThespaceAGisalsoaK-algebraandtheL-linearisomorphismL KAG!AisanisomorphismofL-algebras,notjustofL-vectorspaces.SoAGisaK-formofAasanL-algebra.ThusG-structuresleadstoK-forms.Conversely,ifAisanL-algebrawithaK-formA,thenaturalmapL KA!AisanL-algebraisomorphismandthestandardG-structureonL KAinExample5.4canbe GALOISDESCENT17transportedtoaG-structureonAwhosexedsetisA.(ThisisanalgebraanalogueofExample2.10.)SoK-formsonanL-algebraAleadstoG-structuresonA.ItcanbenosurpriseatthispointthatTheorem2.14hasananalogueforalgebras:theK-formsofA(asanalgebra)areinone-to-onecorrespondencewiththeG-structuresonA(asanalgebra).OnejustreadsthroughtheproofofTheorem2.14andchecksthemapsconstructedtherebetweenalgebrasarenotjustsemilinearbutalsomultiplicative.Example5.5.WithrespecttotheirstandardG-structures,theK-formsoftheL-algebrasMn(L),gln(L),andL[X ]areMn(K),gln(K),andK[X ].ForaC-algebraA,aGal(C=R)-structureisdeterminedbyaconjugationonthealgebra.Thisisafunctionc:A!AsuchthatcissemilinearwithrespecttocomplexconjugationonC,isanR-algebraautomorphismofA,andc2=idA.Example5.6.TheC-algebraM2(C)hasR-formsM2(R)andH(embeddedinM2(C)asinExample5.2).TheseareeachthexedpointsofoneofthefollowingtwoconjugationsonM2(C):c =   ;c0 = � �  :Notec0,whosexedsetisH,isnotanextensionoftheusualconjugationonquaternions,asthatdoesn'txallofH(also,theusualquaternionicconjugationreversestheorderofmultiplication).LetAbeanL-algebraandAandA0betwoK-formsofA(asanL-algebra).EachK-formprovidesAwithaG-structure,sayfrgandfr0gforAandA0.CallAandA0equivalentK-formsofAifthereisaK-algebraisomorphism :A!A0commutingwiththeG-actions: (r(a))=r0( (a))foralla2Aand2G.BecauseofexampleslikeM2(R)andHinsideM2(C),notallK-formsofanL-algebraareequivalent,astheK-formsmaynotbeisomorphicK-algebras.TheequivalenceofK-formsofanL-algebracanbedescribedbythealgebraanalogueofTheorem3.4(whereallK-formsareequivalent,usinganobviousnotionofequivalenceforvectorspaceswithG-structure):Theorem5.7.LetAbeanL-algebrawithK-formsAandA0.WritethecorrespondingG-structuresonAasfrgandfr0g.ThenAandA0areequivalentK-formsofAifandonlyifthereisanL-algebraautomorphism'ofAsuchthatr0='r'�1forall2G:thediagramA'// r Ar0 A'// Acommutesforall2G.Proof.(()Ifthereissucha'then'isaK-algebraautomorphismofA,andfora2A,'(a)=r0('(a))forall,so'(a)2A0.If'(a)2A0witha2A,then'(r(a))='(a),sor(a)=aforall,soa2A.Thus'(A)=A0,so'restrictstoaK-algebraisomorphismfromAtoA0.())SupposethereisaK-algebraisomorphism :A!A0suchthat (r(a))=r0( (a))foralla2A.Baseextend to1 ,anL-algebraisomorphismfromL KAtoL KA0.ThesebaseextensionsarebothL-algebraisomorphictoAinanaturalway,so1 can 18KEITHCONRADberegardedasanL-algebraautomorphism'ofA.Itislefttothereadertocheckwiththis'thatthediagramcommutes.ForourlastapplicationofGaloisdescent,weworkoutthestructureoftheL-algebraL KL,wherescalingbyLcomesintherstcomponent:c(x y)=cx yonsimpletensors.For2G,letLbeLasaringwithL-scalinggivenbythetwistedrulec
x=(c)x.ThenLisanL-algebra.SetA=Y2GL;whoseelementsarewrittenintuplenotationas(x).ThisisanL-algebrausingcompo-nentwiseoperations.Inparticular,scalingbyLonAisgivenbyc
(x)=(c
x)=((c)x):Let2GactonAby((x))=(x)=(y);wherey=x.ToshowthisisaG-structureonA,itiseasytoseethatactsadditivelyandmultiplicativelyonA,andtheidentityofGactsonAastheidentitymap.For1and2inG,1(2((x)))=1((y))wherey=x2=(z);wherez=y1=x12.So1(2((x)))=(x12)=(12)((x)):Thuscomposingtheactionsof1and2onAgivestheactionof12.Also(c
(x))=(((c)x))=(y);wherey=()(c)x=((c))xand(c)(((x)))=(c)((x))=((c)
x)=(((c))x)=(y);soacts-linearlyonA.ThexedsetAGforthisG-structureonAisf(x):((x))=(x)forall2Gg,whichamountstox=xforalland,soallthecoordinatesarethesameelement,sayx,ofL:AG=f(x):x2Lg.ThusL=AGasK-algebrasbyx7!(x).(ThismapisK-linearbutnotL-linearif#G�1.)SinceL=AGasK-algebras,L KL=L KAGasL-algebrasby 7! ().ByGaloisdescent,L KAG=AasL-algebrasby ()7!
()=(
)=(()),so(5.1)L KL=A=Y2GL GALOISDESCENT19by 7!(()).Foranapplicationof(5.1)toaproofofthenormalbasistheorem,see[4].References[1]N.Jacobson,\LieAlgebras,"Dover,NewYork,1979.[2]J.H.Silverman,\TheArithmeticofEllipticCurves,"Springer-Verlag,NewYork,1986.[3]W.Waterhouse,\IntroductiontoAneGroupSchemes,"Springer-Verlag,NewYork,1979.[4]W.Waterhouse,TheNormalBasisTheorem,Amer.Math.Monthly86(1979),212.[5]D.J.Winter,\TheStructureofFields,"Springer-Verlag,NewYork,1974.