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cmcsc10Introduction Themessenrtialldimensionofan\algebraicstructure"isanrumericalinvXari-anrtthatmeasuresitscomplexityV.¯Informally,dtheessentialdimensionofanalgebraicZstructureorver[aZeld"g cmmi12F^ isthesmallestnrumbSerZofalgebraicallyinde-pSendenrtrFparametersrGrequiredtodenethestructureorverarFeldextensionofFn(see["html:1 html:]or[!html:10 html:]). Let(%!",
cmsy10Fx:ag1" cmssi12Fields!E=F-!afSetsXܹbSeafunctor'(an\algebraicstructure")fromthecategoryFields ,=FAYofeldextensionsofFAZandeldhomomorphismsorverFAZtothecategoryofsets.HLetK2fFields!=Fƹ,2gF1(K ܞ)andK |{Y cmr80 ~ϹasubeldofKorver5Fƹ.dWVe4saythatùis4@ cmti12deneffdoverK0F9(and4K0iscalled5aeldofdenitionof)?ifthereexistsanelemenrt0^2[F1(K0)suchthattheimage(0)#2 cmmi8K{of0under themapF1(K0)>G!>HF(K ܞ) coincideswith.The essential;Adimension of@,denotedped0}W&K cmsy8F(),isptheleasttranscendencepdegreetr:deg$RTF+(K0)orverpalleldsofdenitionK0of.8Theessential35dimensionofthefunctorFis$ ed H(F1)UR=supfedWF4()g;wherethesupremrumistakenovereldsK12URFields 9=Fnandallh2URF1(K ܞ). LetDpbSeEaprimeinrtegerandi2V\F1(K ܞ).:Theessential3p-dimension3ŹedWF"pQ()'̍of'RBis>the>minimrumofedWF\(K 'q% cmsy60
퉹)orverall>niteeldextensionsK ܞ20=KRofdegreeprimeFtoFp.ATheessentialp-dimensioned\p#(F1)ofFwܹisFthesupremrumofedWF"pd()orveralleldsK12URFields 9=Feandallh2URF1(K ܞ)(see[#html:14 html:,˕x6]).+ClearlyV,˔ed(F1)UR edpU(F1)forallp. Let GbSe analgebraicgrouporverFƹ.Theessentialddimensiond۹ed$(G) (resp.essential7p-dimensionedypֹ(G))ofGistheessenrtialdimension(resp.ݘessenrtialp-dimension)vtofvuthefunctorG-torrsors*ƹtakingaeldKStothesetofisomorphismclassesofallG-torsors(principalhomogeneousG-spaces)orverK ܞ. If&G=PGL ßn+|9orverFƹ,"thefunctor%G-torsors,xisisomorphictothefunctorAlg
TFY(n)JtakingKaeldKtothesetofisomorphismclassesofcenrtralsimpleK ܞ-algebrasofdegreen.#LetpbSeaprimeinrtegerandletp2r vbSethehighestpSorwer<of<pdividingn.ThenedpkG(u
cmex10 CmAlg*wTF2ƹ(n)GT=URedJpܧG \Alg+TF2(p2rb)G[#html:14 html:,_ Lemma8.5.5].EvreryScentralsimpleE -algebraofdegreepSiscyclicoveraniteeldextensionof:degreeprime:top,^henceedpIG BKAlg*UTF2(p)GT=UR2[#html:14 html:,^Lemma8.5.7].OItwrasprovenfoin["html:11 html:]thatedpupG rAlg+|TF2˹(p22)G3=[1p22m+1andingeneral,edpvMG OAlg+YTF2(p2rb)G[12rAforall38r>6in[#html:14 html:,Th.88.6]. WVeprorvethefollowing: l97o cmr91 *f荠t
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#Theorem.8Lffet35Fbeaeldandpanintegerdierentfromcrhar_(Fƹ).fiThen߆ ed شp G Alg TF l(prb)GTUR(r6 1)pr=
+1:O/Inotherwrords,-wehavethefollowinglowerbSoundfortheessentialdimension #ofPGLoF&뾹(p2rb):hKedtG yPGL |͟F (prb)GTURedJpܧG \PGL5NpF<](pr)GUR(r6 1)pr=
+1:#html: html: ȍ /2. 3 Preliminaries html: html:2.1.Characters.Let4FabSeaeld,FsepaseparableclosureofFand =Gal[(FsepZ[=Fƹ)PtheOabsoluteGaloisgrffoupIfofF.
FVorPa -moSduleM3wrewriteH V2n(FS;M@)forthecohomologygroupH V2n( ;M@). Thecharffacter35groupChB(Fƹ)ofFnisdenedasȉUOHommcont~Mj( ;.
msbm10Q=Z)UR=H V1Z(FS;Q=Z)'H V2Z(FS;Z):FVor))acrharacter)*2Ch:(Fƹ),8set))F()=(FsepZ[)2Ker
y2().dThenFƹ()=Fisacycliceldextensionofdegreeord().8IfURChX(Fƹ)isanitesubgroup,wreset FFƹ()UR=(FsepZ[)\jKer%()!*m;Owhere7the7inrtersectionistakrenover7all>2.The7GaloisgroupG>ܹ=Gal[G ۲Fƹ()=FG isCabSelianandiscanonicallyisomorphictothecrharactergroup38Ch(G)UR=Hom(G;Q=Z)ofG. IfJFƟ20 uuFisasubeldandv2Ch(FƟ20o),6wrewriteFfortheimageofunder!thenaturalmap!Ch%B(FƟ20o)!Ch(Fƹ)and!F()forF(FO).IfCh(F)isanitesubgroup,thenthecrharacterF.:()istrivialifandonlyifUR2.html: html:%Lemma2.1.(Lffet\|;20J{ӹChW(Fƹ)\}betwo\}nitesubgroups.@Supposethat\}foraeldextensionK5=F,
7wehaveK
=I20bK
^inCh!(K ܞ).qThentherffeisanitesubffextension35K ܞ20=FinK5=FsuchthatK 0B۹=UR20K 0 inCh6(K ܞ20
).`Prffoof.#RChoSoset?asetofcrharactersf1;:::ʜ;mggeneratingt@andasetofchar-acters*f20RA1;:::ʜ;20RAmggenerating20sucrhthat(idڹ)K;=z(20RAi)Kffor*alli.BLeti'T=i&