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cmr10NIKIT*AUUA.KARPENKO `$ -
cmcsc10Abstract.\`W*e|provethe|so-called ':
cmti10Unitary̫Hyp}'erbolicity̬Theorem,a|resultonhypGerbol- $ icityofunitaryinvolutions.Theanalogouspreviouslyknown
resultsfortheorthogonal$ andRsymplecticRinvolutionsareRformalconsequencesoftheunitaryone.pWhiletheorigi-$ nal|proGofsintheorthogonalandsymplectic{caseswerebasedontheincompressibilityof$ generalizedMSeveri-BrauerMvqarieties,OstheproGofintheunitarycaseisbasedontheincom-$ pressibilityUUoftheirW*eiltransfers.l^html: html: XQ cmr121. Q!-
cmcsc10Inroduction WVekrefertok[html:14 html:]forterminologyandbasicfactsconcerningcenrtralsimplealgebraswithinrvolutions.WVe+Fx+Gthefollorwingnotation:$g cmmi12F
isaeld,;nK5=Faseparablequadraticeldextension,AacenrtralsimpleK ܞ-algebra,XanFƹ-linearunitaryinvolutiononA. Preciselyasintheorthogonalandinthesymplecticcase,@arighrtidealI*ofthealgebraA/*is3@ cmti12isotrffopic(withrespSecttothe/)inrvolution/*n9),Kif(I )'!",
cmsy10Io/=}0. fTheinrvolution/*cishypfferbolic,ifVthereexistsUanisotropicidealofreduceddimension(degɹA)=2.Notethatthezreducedzdimensionofarighrtideal(or,moregenerallyV,ofazrighrtA-moSdule)isdenedI`asitsdimensionorverKF(notorverFƹ)dividedbrythedegreedegaAUR:=`p
UT` z & SdimH%2 cmmi8KA7ofA. Inthisnotewreprove(inSectionhtml:4 html:)1 html: html:Theorem1.1(UnitaryHyp` erbolicit yTheorem).-!AssumeothatpcrharbF6=2.ZIfpisnothypfferbolic,thenthereexistsaeldextensionFƟ2(K cmsy80o=FRsuchthatK ܞ20 ):=URKe
FFƟ20!isaeld,the5cffentral5simpleK ܞ20-algebraA20(:=ZHA
F FƟ20is5split,6andtheFƟ20o-linear5unitaryinvolutionn920Ĺ:=URF.:)q% cmsy60
Kon35A20nisstil lnothypfferbolic.1 TheoremWhtml:1.1 html:Wistheunitaryanalogueofthefollorwingresultconcerningorthogonalin-vrolutions: html: html:Theoremk1.2(OrthogonalHyp` erbolicit ykTheorem-[html:12 html:,*ETheorem1.1]).GAssumeV*thatcrhar(FMD6=~2.LffetBbeacentralsimpleF-algebrawithanorthogonalinvolutionW.Ifiscjnotckhypfferbolic,wthenthereckexistsaeldextensionFƟ20o=F1suchthatthecffentralsimpleFƟ20o-algebrffarB 20 /:=B2T
F
FƟ20 isrsplitandtheorthoffgonalinvolutionF.:03onB 20isstil lnothypfferbolic. In6the5casewhentheexpSonenrtofAis2,Theoremhtml:1.1 html:hasbSeendeducedfromTheoremhtml:1.2 html:Din[%html:18 html:].InoursettingtheexpSonenrtofAisarbitrary;|
ourproofisaunitaryadaptationoftheproSofofTheoremhtml:1.2 html:.V ff <