; TeX output 2001.05.10:1239@፠k1N cmbx12ONٚACTIONSOFINFINITESIMALGROUPSCHEMESTXHK`y cmr10GAET*ANAUURESTUCCIAANDHANS-J@xURGENSCHNEIDER0MqXQ cmr121.- cmcsc10Introduction:{Letg cmmi12kѹbSea eldofcrharacteristicp{>0andH a nite-dimensional.{commrutativeHopfalgebraorverkQŹwithunderlyingalgebra𶍍OHB=URkg[X|{Ycmr81;:::ʜ;X2cmmi8nP]=(XOp-:;cmmi6sAacmr611 N;:::;X p-:snڍn޹);nUR!", cmsy101;s1Vsn1:.{(1.1)x:{Bythestructuretheoremofin nitesimal,thatis niteandconnected.{groupscrhemes(see[W u,14.4]),any nite-dimensional,commutativeand.{loScalHopfalgebraorveraperfect eldhasthisform.:{LetAbSeacommrutativealgebra,طand :|A!A,R HV,arighrtHV-.{comoSdulealgebrastructureonA.:{Inotherwrords,weareconsideringanaction|X+GUR!X.{ofthein nitesimalgroupscrhemerepresentedbyHontheanescheme.{represenrtedbyA.:{FVoranryaUR2A,wecanwriteq9s2(a)UR=a 1+ "u cmex10X 1 K cmsy8in0Didڹ(a) xi+Ptermsofhigherorderinthe2xi;W.{where9thexiʹaretheresidueclassesoftheXiinH,andwhereDi,:URA!.{A;1URin,arekg-linearderivXationsofA.:{LetR:=zA2coH?bSethesubalgebraofAofcoinrvXariantelementsunder.{theBcoactions2.ATheextensionR?AiscalledanHV-Galoisextension.{(see[M =])(andtheprojectionX!"Y,xrepresenrtedbytheinclusion.{R kAiscalledaprincipalhomogeneousspaceforG)ifthecanonical.{mapLA R ;AUR!A HF:;a b7!as2(b);.{isbijectivre.:{InTheorem4.1wreproveaJacobicriterionforHV-Galoisextensions..{FVorloScalalgebrasAthiscriterionsarysthatthefollowingstatements.{areequivXalenrt:.{ ff< B 1991': cmti10MathematicsSubje}'ctClassi cation.Primary:q17B37;UUSecondary:16W30. Keywor}'dsandphrases.HopfUUGaloisextensions,in nitesimalgroupschemes. uUUV*ersionUUofMay10,2001.ThismworkwaspartiallysuppGortedbytheGraduiertenkollegoftheMath.Institut(UniversitatMGunchen). ++o cmr91*@፠kK.{2_GAET:ANA!RESTUCCIAANDHANS-J'ҟUR9GENSCHNEIDER=kHlRURAisanHV-Galoisextension.=kHlThereUareelemenrtsy1;:::ʜ;yn 2 Asuchthatthematrix(Didڹ(yjf ))HlisinrvertibleoverA.7.{ByrenrumbSeringtheyijwecanassumethatforall1mnthema-.{trices'Y(Didڹ(yjf ))1i;jvm&iareinrvertible.Assuming'Ythisstrongercondition.{wrebshowasanapplicationofourcriterionforHV-Galoisextensionsthat.{theelemenrtsnHynn9 q1/č1 uqy n9 nڍn ;0UR i,+ x= 0퍍v i>; x2A" D <(a)D Y(b);ꦹandzD(0;:::\;0)=id :(l_.{FVorXFalli,sletit=(ijJ)1jvn52A,whereXFij Z˹=1,ifXFj=i,andXFij Z˹=0,.{otherwise. 8zWVe1de neDi /=+UD8:ioN;1in:1Thrusthelinearmaps.{Di,:URA!A=arederivXationsofthealgebraA,andforallaUR2A=wrehaveA aks2(a)UR=a 1+ X 1in0Didڹ(a) xi+zX%퍟u 2AÍj j2D (a) x :.{(2.2)-ɍ:{Let*@HV2 ךbSethedualHopfalgebraofHandguN=Pƹ(HV2Z)*@thep-Lie.{algebrafoftheprimitivreelementsinHV2Z.ThusgistheLiealgebraof.{the'groupscrhemerepresentedbyHV.Recallthattheprimitiveelements.{in4HV2 arethe"-derivXationsd:HtZ!kg,'that4isthelinearfunctions.{satisfyingEd(xyn9)UR="(x)d(y)Y+d(x)"(y)Eforallx;yË2URHV.FVoralli,fletdi,:.{H!$WkebSedHthe"-derivXationwithdidڹ(xjf )=ijJ;1i;j*ndH(Kronecrker.{s2).Thenjdd1;:::ʜ;dn isabasisofg.ThecoactionȄ:URA!A HWde nes.{aleftHV2Z-moSdulealgebrastructureonA, ]andtheactionofdijyisgivren.{brythederivXationDidڹ.8ThusR.gUR!Der/(A;A);di,7!URDid;1in;<@፠kK.{6_GAET:ANA!RESTUCCIAANDHANS-J'ҟUR9GENSCHNEIDER.{isamapofp-Liealgebras.8Inparticular,[Did;Djf ];DOSpáip=2L,X'؍UR1lKn\kgDl!;1URi;j%n:.{(2.3) Ս:{In thecasewhens1pA==`x==sn X=1,-mHV2 lis therestrictedenrveloping.{algebra-ofg,yandtheactionisdeterminedbrytheactionofg.! Hencein.{thiscaseA2coH8Ϲ=URA2fDq1*;:::\;Dn7g, ,whereIjA fDq1*;:::\;Dn7g/^:=URfa2AjDidڹ(a)=0forall&/1ing:.{(2.4)̍3.Theadditivegroup:{TheHopfalgebraHawwithunderlyingalgebram퍍L`HaY!=URkg[X1;:::ʜ;XnP]=(XOp-:s11 N;:::;X p-:snڍn޹);nUR1;s1Vsn1;.{(3.1).{andcomrultiplicationgivenby% (xidڹ)UR=xi 1+1 xi;1URin;.{(3.2)[.{willm1bSecalledtheHopfalgebraoftheadditivregroup(wherenand.{s1;:::ʜ;sn are xed).:{NotewthatforanrycommutativealgebraTƹ,kGaϹ(T)eV:=AlgW(Ha;Tƹ)'.{f(t1;:::ʜ;tnP)=2kg2n MXjt2p-:si3=0;1ing,ywheresOthegroupstructureis.{compSonenrtwiseaddition.:{WVeconsidercoactionsofHa onarbitrary,Kpnotnecessarilycommru-.{tativreJaalgebrasA.X HoweverinallthelaterresultsabSoutHV-comodule.{algebras5uAforourgeneralHopfalgebraH"˹satisfying_<(2.1) wreassume.{thatAiscommrutative..{Theorem3.1.Lffet!HabetheHopfalgebrade nedbyף(3.1),%(3.2)1,%and.{A@'arightHV-cffomodule@'algebrawithstructuremap:G[A!Aq Ha..{De ne35thederivationsD1;:::ʜ;Dn ۅby@(2.2)!andletRn=URA2coHal.:{Then35thefollowingarffeequivalent:4(1)HlRnURA35isafaithfully atHa-Galoisextension.4(2)HlTherffearey1;:::ʜ;yn <2Awiths2(yidڹ)=yi 1+1 xi;forallHl1URin:.{Suppffose(2)holds. De neyn92 "=URynn9 q1/č1 uqy2n9 nRAn ; h=( 1;:::ʜ; nP)2A. Then%w7~R HaY!!URA;r6 x J7!rSyn9 c;r2RJ; h2A;.{is]aleftRJ-lineffarandrightHa-colinearisomorphism.8wInparticular,the.{elementsȸyn9 c; h2URA;.{form35anRJ-bffasisofAasaleftR-moffdule.Y.{Prffoof.Re6(1)_)(2):By(1),{A^isaninjectivreHaϹ-comoSdule(seeforexam-.{ple[S,TheoremI]).HencetherighrtHaϹ-colinearmapk;!kA;17!1,.{canbSeextendedtoanHaϹ-colinearmap n:URHaY!!A.8Thenforalli,%l5s2( (xidڹ))UR=( Q id uL)(xi)= (xi) 1+1 xi;O@፠k&eiON!A9CTIONSOFINFINITESIMALGROUPSCHEMES6w7.{sincem йisHaϹ-colinearand (1)4a=1;mandtheclaimfollorwswithyi;:=.{ (xidڹ)forall1URin::{SuppSose(2)holds.Thenwrede neakg-linearmap :`Ha d!Aby.{ (x2 )UR:=yn92 @¹forall h2URA:SinceandP$arealgebramaps,߰andforall.{i,h(xidڹ)UR=xi 1+1 xi;s2(yi)UR=yi 1+1 xi;ύ.{ ĹisrighrtHaϹ-colinear.:{SincePHaasanalgebraisgeneratedbrythegroup-likeelement1and.{theprimitivreelementsxidڹ,kistheonlysimplesubScoalgebraofHa([M =,.{5.5.1]).(Thereforethemap aisconrvolutioninvertible(cf.([M =,55.2.10])..{ThrustheHaϹ-extensionRnURAisHa-cleft,henceHa-Galois,andw7~R HaY!!URA;r6 x J7!rSyn9 c;r2RJ; h2A;.{isbijectivre(see[M =,8.2.4,7.2.3]).cffxff ̟ff ̎ ̄cffǍ.{Example3.2.5WVe.considertheLiealgebracaseoftheadditivregroupYHaY!=URkg[X1;:::ʜ;XnP]=(XOp1;:::;X pڍn):.{InYthiscasecoactionsofHa (aregivrenbyderivXationsD1;:::ʜ;Dn 2.{DerA?(A;A)withzmDidDj\=URDjf Di;ꦹandzDOSpáip==0;1i;j%n;.{whereXK[D J=ōDnS q1/č1[z5E ΍\> 1!ō-$*D2S nRAn-$*[z(՟ ΍a nP!C2; h=UR( 1;:::ʜ; nP);0 i,+ x= 0퍍v i>; x2A" e X e Y:(Qcffxff ̟ff ̎ ̄cff.{Corollary3.3.Assumeqs1ٹ==sn q%=1.!LffetAbeanalgebra,.and.{Ȅ:URA!A Haتacffoaction.FDe nethederivationsD1;:::ʜ;Dn }+by](2.2).{and35letRn=URA2coH}.:{Then35thefollowingarffeequivalent:E4(1)HlRnURA35isafaithfully atHa-Galoisextension.4(2)HlTherffe35arey1;:::ʜ;yn2URAwithDidڹ(yjf )=ij ~forall1i;j%n:.{Suppffose35(2)holds.fiThenw7~R HaY!!URA;r6 x J7!rSyn9 c;r2RJ; h2A;.{is35aleftRJ-lineffarandrightHa-colinearisomorphism.Ǎ.{Prffoof.Re6ThisfollorwsfromTheorem3.1andExample3.2Qcffxff ̟ff ̎ ̄cff`h@፠kK.{8_GAET:ANA!RESTUCCIAANDHANS-J'ҟUR9GENSCHNEIDER:{TheQnextLemmashorwsthatundercertainconditionsthecoinvXari-.{anrta%elementsintheLiealgebracasearethecoinvXariantelementsunder.{someactionoftheadditivregroup.ߩTheideaofthisLemmaappSearssev-.{eralxtimesintheliterature(see[LWs]resp.PQ[RMi]whenthecrharacteristic.{ofthe eldis0resp.8pUR>0.)WVeincludetheproSofforcompleteness.ۍ.{Lemma3.4.y15LffetAbeacommutativealgebra,HnaHopfalgebrasat-.{isfying#ݹ(2.1)!aDwith[s1V=UR=URsn=1, andarightHV-cffomodulealgebra.{structurffe35onAwithderivationsD1;:::ʜ;Dn ۅde nedby@(2.2)g.:{Assumey1;:::ʜ;yn2URAsuchthatthematrix(Didڹ(yjf ))1i;jvn#isinvert-.{ible.:{Then35therffearederivations@1;:::ʜ;@n2URDerk(A;A)35suchthatL6[@id;@jf ]UR=0;@Opái˹=0;@idڹ(yjf )=ijJ;1i;j%n;33and A coH8Ϲ=A f@q1*;:::\;@n7g' : 卑.{In35pffarticular,theelements'ynn9 q1/č1 uqy n9 nڍn ;0UR i,4(3)HlIfC Aisloffcal:_ThereC arey1;:::ʜ;yn2URAsuchthatforall1mn,Hlthe35mmmatrix(Didڹ(yjf ))1i;jvm&EoverAisinvertible.&.{Prffoof.Re6(1)(F)(2):By(1),71Ԝ xlJis(Fintheimageofthecanonicalmap.{A.? R !AL!A HDforall1Lln.|Hencethereareelemenrtsyj V2.{A;1TPj#N@,forwsomeN42TPNsucrhthatthereareajvl \"2A;1j#.{N;1URln,withWh`XōhQy1jvN%[ 2A+ajvlD (yjf ) x J=UR1 xl!;1ln:&ч.{Inparticular,forall1URi;ln,ꨟPU1jvN0 ajvlDidڹ(yjf )=il::{(2))(1):ByTheorem2.2itisenoughtoshorwthattheintegralt.{:=xOp-:s1,n11E:::)x2p-:sn 1RAnBiskintheimageofthecanonicalmapcanV/:At R.{AUR!A HV.:{FVorall1URln,usingo(2.2) &#and(2)wrecompute:.{can@(X ㇍jUVajvl z yjf )UR= X ㇍_jaajvlyj 1+X ㇍j$;iajvlDidڹ(yjf ) xi!b+jGXj$;j j2;ajvlD (yjf ) x "p2= X ㇍_jaajvlyj 1+1 xlp+jGXj$;j j2;ajvlD (yjf ) x : <.{De ning&zlw=URPjfajvl  8yjlBPjajvlyj 1,@and&a2 yl J=URPjfajvlD (yjf )forall.{1URln꨹and h2URA;j j2,wreobtainvcanJ(zl!ȹ)UR=1 xlp+zXj j2a ڍl? x ;1URln: .{WVeclaimthatcan(zOp-:s1,n11Y(*Y$z p-:sn 1ڍnL)UR=1 :s2.{Sincethecanonicalmapisanalgebramap,wrehaveT-.{can@(zOp-:s1,n11Y(*Y$z p-:sn 1ڍnL)%4=URcanF(z1) p-:s1,n1E)Acan;p5(znP) p-:sn 1`%4=L,Y'؍UR1lKn\ %1 xlp+zXj j2a ڍl? x   C]p sl1#:.{WVeseethatthisproSductisasumoftermsoftheforml=,ԟY'؍851lKnQy(1 xl!ȹ) ci?l] ٟXBj j2'Ma ڍl? x   C]di?l;clp+dlw=URp si?l 1;ꦹforall(/1ln:.{ThegeneraltermofthissumcanbSewrittenasW1a x Y;aUR2A; n2N nP;j jL,X'؍1lKn^(clp+2dl!ȹ)=L,X'؍1lKn(p si?l 1)+X'؍1lKndl!: Р@፠kK.{10_GAET:ANA!RESTUCCIAANDHANS-J'ҟUR9GENSCHNEIDER.{OnlyQOthetermsaqp x2 withQO n=UR( 1;:::ʜ; nP)and lwp2si?lȐqp1;1ln,.{canbSenon-zero.'ThereforewreonlyhavetoconsideraBy x2 Pwithj jUR.{P;&1lKnT靹(p2si?l 1),thatiswithdlw=UR0foralllC.8Thisprorvesourclaim.V:{(2))(3)follorwsfromthenextLemma.Qcffxff ̟ff ̎ ̄cff.{Lemma4.2.y15Lffet!Abeacommutativelocalring,=[1ߜn;N 2N!and.{dij 2*A;1in;1j/N@.AssumeKthattherffeareajvl 2*A;1lƹ.{n;1URj%Ntwith35P1jvN0V"dijJajvl ]$=il;35forall1URi;ln.c:{Thentherffeare1j1 <1;sm+16==s1;ꦹforsome5N1mn:.{RecallCfromSection2thatHV2[1] =URkg[HV2p]isaHopfsubalgebraofH1Jwith.{quotienrtHopfalgebra\-z ӍH>(1)F =URHF:=(x2pjx2HV2+ ι):꨹Then>P=sakg[X1;:::ʜ;XmĹ]=(XOp-:s1*!q% cmsy611W;:::;X p-:sm1ڍmjR;althoughthe2nK2.{matrix(Didڹ(Tjf ))1i;jv2#Jisinrvertible.:{2)J8iseasytocrheck.WTVoJ8prove3)notethatforanycommutativeHV-.{comoSdulealgebrastructure:ZA!A HV,HAcommrutativeandlocal.{with(mHD)2p-:s й=0,%andforalla2A,%s2(a)2aVf 1+A mHD,%hence.{(s2(a))2p-:s v=URa2p-:s v 1:nŹThrusinducesanHV-comoSdulealgebrastructureon.{eacrhloScalizationofA.Qcffxff ̟ff ̎ ̄cffO5.Anapplica32tion:{WVeT nallyapplythepreviousresultstothecasewhenAisregular.{loScalorregularlocalandcomplete.%.{Theorem5.1.LffetwHebeaHopfalgebrasatisfyingA(2.1)M,=andAacom-.{mutative@rightHV-cffomodule@algebrawithstructuremapȄ:URA!A HV..{De ne35thederivationsD1;:::ʜ;Dn ۅby@(2.2)!andletRn=URA2coH}.:{AssumethatAisarffegularlocalringofdimensiondwithmaximal.{ideffalĪmA andtherearey1;:::ʜ;yn 2bmA suchthatforall1mn,.{(Didڹ(yjf ))1i;jvn${is35invertible.:{Then35RLisrffegular35local.:{(1)P8AssumemorffeoverP8thats1K= = snP,Wxormorffegenerallythatfor.{all351URin;yn92p-:si!2RJ.:{Then35therffearezn+1;:::ʜ;zd42URRLsuchthat81y1;:::ʜ;ynP;zn+1;:::;zd is35arffegular35systemofpffarameters35ofA;33and;yOn9p-:s11 b;:::ʜ;y n9p-:snڍnU;zn+1;:::;zd is35arffegular35systemofpffarameters35ofRJ:u:{(2)TAssumeinadditionto(1)thattherffegularTlocalring(A;mA)is.{cffomplete,$and[kOA=mA `?isaseparable eldextension.SThenthereis.{a35sub eldkoURFRLof35RsuchthatcF2'p!>RJ=mR2 ӓ'p H !P A=mA;.{and35analgebrffaisomorphismhFƹ[[Y1;:::ʜ:YnP;Zn+1;:::;Zdߨ]]2'pUR$!\xA;.{inducing35analgebrffaisomorphismu"Fƹ[[YOpp-:s11;;:::ʜ:Y pp-:snڍn;Zn+1;:::;Zdߨ]]2'pUR$!\xRJ:ˎ@፠kK.{14_GAET:ANA!RESTUCCIAANDHANS-J'ҟUR9GENSCHNEIDER.{Prffoof.Re6By7[Theorem4.1,[7RnURAisHV-Galois,inparticularAisa nitely.{generated5xfreeRJ-moSdule.{HenceRN¹islocal,Ynoetherianandregular(see.{[Ma9,23.7]).:{Assume1(1).;{FVollorwingtheargumentin[RMi,Theorem5],wre rst.{note9thattheresidueclassesofy1;:::ʜ;yn inmA=m22bA arelinearlyin-.{depSendenrtoverA=mA.P SuppSosePG1jvn.O ajf yj 2m22bA zforsomeaj2A.c.{By2applyingthederivXations@i~oftheproSofofLemma3.4wreseethat.{ai,2URmA Ȍforalli.:{Thrus{Zby4.5RJ=(yOn9p-:s11 b;:::ʜ;y2n9p-:snRAnU)2K'pKo!IA=(y1;:::;ynP){Zisregular,andwret.{canuDextendtheyOn9p-:siáibryzn+1;:::ʜ;zd Ttoaregularsystemofparameters.{ofRJ.*?Theny1;:::ʜ;ynP;zn+1;:::;zdnisaregularsystemofparametersof.{A꨹sinceAandRharvethesameresidue eldbry4.4.:{TVooprorve(2),|wenotethatR;isalsocompleteloScalsinceAis nitely.{generatedandfreeorverA(ThemR-adictopSologyofAisthemA-adic4.{topSologyV,sincef A=mRAisartinian.HenceR@wRb'WRY'WA,andAisfree.{orverRandwfbR$ofthesamerank.8ThrusRn=wbURRJιbyNakXayama).ʍ:{Since:RJ=mR2\&'p x J!a1A=mA ɹisseparableorver:kthereisacoSecienrt eld.{kF9mRbry[Ma9,O_28.3]. eTheclaimfollowsfrom(1)and[Ma9,.{29.7].DՄcffxff ̟ff ̎ ̄cff,S[References.{[DG]HM.UUDemazure,P*.Gabriel,GroupGesAlgGebriquesI,North-Holland,1970. .{[L]C J.OLipman,F*reederivqationsmoGdulesonalgebraicvarieties,Amer.+J.Math.O2"V cmbx1087C (1965),UU874{898..{[Ma]G4H. Matsumura,70CommutativeRingTheory*,70CambridgestudiesinadvqancedC mathematicsUU8,CambridgeUniversityPress,1986..{[M]C S.Montgomery*,Hopfalgebrasandtheiractionsonrings,CBMSLectureNotesC 82,UUAmer.Math.SoGc.,1993..{[MS]GS.YMontgomeryandH.-J.Schneider,ZPrimeidealsinHopfGaloisextensions,C Isr}'aelJ.Math.UU112(1999),187{235..{[RM]I cmmi10n-dimensionalactionsof niteabGelianHopfal-C gebrasΫincharacteristicpQ>0,R}'ev.RoumaineMath.PuresAppl.Ϋ43(1998),C 881{895..{[R*T]FG.RestucciaandA.Tyc,RegularityoftheringofinvqariantsundercertainC actionsAof niteabGelianHopfalgebrasincharacteristicp,xJ.Algebr}'a159(1993),C 347{357..{[S]C H.-J.Schneider,PrincipalhomogeneousspacesforarbitraryHopfalgebras,C Isr}'aelJ.Math.UU72(1990),167{195..{[W]CQ cmmi10K`y cmr10