GALOISMODULESTRUCTUREOFWEAKLYRAMIFIEDCOVERSOFCURVES By SugilLee ADISSERTATION Submittedto MichiganStateUniversity inpartialfulfillmentoftherequirements forthedegreeof MathematicsDoctorofPhilosophy 2020 ABSTRACT GALOISMODULESTRUCTUREOFWEAKLYRAMIFIEDCOVERSOFCURVES By SugilLee Themainthemeofourstudyistheobstructiontotheexistenceofa normalintegralbasis for certainGaloismodulesofgeometricorigin.When ˝ isafinitegroupactingonaprojectivescheme - overSpec Z and F isa ˝ -equivariantcoherentsheafof O - -modules,thesheafcohomology groupsH 8 ( -Œ F ) are ˝ -modules,andoneasksifitsequivariantEulercharacteristic j ( -Œ F ):= Õ 8 ( 1) 8 [ H 8 ( -Œ F )] canbecalculatedusingaboundedcomplexoffinitelygenerated free modulesover Z [ ˝ ] .Thenwe saythatthecohomologyof F hasa normalintegralbasis .Theobstructiontotheexistenceofa normalintegralbasishasbeenofgreatinterestintheclassicalcaseofnumberfields:Asconjectured byFröhlichandprovenbyTaylor,when # / Q isafinitetamelyramifiedGaloisextensionwith Galoisgroup ˝ ,theGaloismodulestructureoftheringofintegers O # isdetermined(uptostable isomorphism)bytherootnumbersappearinginthefunctionalequationsofArtin ! -functions associatedtosymplecticrepresentationsof ˝ .Chinburgstartedageneralizationofthetheoryto someschemeswithtamegroupactionsbyintroducingthereduced projective Eulercharacteristic classes j % ( -Œ F ) .TheseEulercharacteristicsareelementsoftheclassgroupCl ( Z [ ˝ ]) andgive theobstructiontotheexistenceofnormalintegralbasis. Ouraimistogeneralizethetheorytothekindofwildramification,namelyto weaklyramified coversofcurvesoverSpec Z .If # / Q iswildlyramified,then O # isnotafree Z [ ˝ ] -module.Erezshowedthatwhentheorder j ˝ j isodd,thenthedifferentideal D # / Q isa square,andthesquarerootoftheinversedifferentisalocallyfree Z [ ˝ ] -moduleifandonlyif # / Q isweaklyramified.Köckclassifiedallfractionalidealsofweaklyramifiedlocalringsthat havenormalintegralbases.WegeneralizebothoftheresultstocurvesoverSpec Z toconstruct projectiveEulercharacteristicforcertainequivariantsheavesonweaklyramifiedcoversofcurves. Tomyfamily. iii ACKNOWLEDGEMENTS Mypastsevenyearsofgraduateschoolhavebeenacompletere-discoveryofwhoIam.It waslikerunningamarathonjustasmanyseniorcolleaguessaid,andtherehadbeennumerous momentsthatIfeltcompletelytiredandlost.However,thesewonderfulpeoplearoundmehave givenmecourageandhope,remindedmeofhowmuchIwantedtolearnmoreinmathematics. Indeed,nothingwouldhavebeenpossiblewithouttheirwarmth,encouragement,andtheendless support. Firstandforemost,I'mdeeplygratefulformyadvisorGeorgiosPappas.Notonlyhegave meanidentityasamathematicianbyleadingmetothetopicofthethesis,hehasalwaysbeen therebeingunconditionallysupportivetomewhenIdidn'tmeethisinitialexpectationsandfelt completelylostforalongtime.Withhiswisdom,vision,andimmensepatience,Iwasabletogain strengthtofinishmythesisinthisform.George,thankyousomuchforeverything.Ihopetosee youinpersonagainwhenthepandemicsituationisover. IalsothanktherestofthefacultyandstaffatMichiganStateUniversity.RajeshKulkarni, mytemporaryadvisorwhenIjoinedthegraduateprogram,hadgivenmevaluableadvicestoset realisticgoalsinthegraduateschool.Hewasalsoinmydissertationcommitteealongwithother members:MichaelShapiroandAaronLevin.Mydeepestthankstothecommitteeforthework andtime. Teachinghasbecomeanimportantpartofmylifeasamathematician,andIwanttomention TsvetankaSendova,JaneZimmerman,andGabrielNagyforyearsofgreatteachingexperience. Anotherwonderfulgrouptomentionismyfriends/colleaguesthatI'vemetatMichiganState University.ThankyouCharlotteforconversationssincethefirstyearandhelpingmepreparing forthedefense.ThankyouWoongbaeforourcountlessconversationsinmanytopicsofcommon interestincludingmathematics.ThankyouAdam,Ákos,Bosu,Dongsoo,Hitesh,Hyoseon,Jihye, Jiwoong,John,Manousos,Paata,Rami,Reshma,Sami,Seonghyeon,Tyler,Wenzhao,Yewon, andallothersforagoodtime. iv IalsowouldliketothankmyfamilyfortheloveandthesupportthatIhavereceivedsince forever.ThankyouSangjaeLeeandMyeongsookKim,mydearestparentsinKorea,forthe endlessloveandsacrificesyoumadeforme.ThankyouSumoonLeeandByunglokJeon,myonly sisterandbrother-in-law,forencouragingmeandbringinglovelysmilesofHaeunandJaeyoonto ourfamily. v TABLEOFCONTENTS LISTOFFIGURES ....................................... vii KEYTOSYMBOLS ...................................... viii CHAPTER1INTRODUCTION ............................... 1 CHAPTER2BACKGROUND ................................ 9 2.1DiscreteValuationRingsandDedekindDomains..................9 2.2GaloisModules....................................19 2.3AlgebraicGeometry..................................26 CHAPTER3IMPORTANTTHEOREMS .......................... 31 3.1ProjectiveEulerCharacteristic............................31 3.2WeaklyRamifiedExtensions.............................32 CHAPTER4WEAKLYRAMIFIEDCOVERSOFCURVES ............... 34 4.1WeaklyRamifiedCovers...............................34 4.2Example:CyclicAction...............................40 CHAPTER5SQUAREROOTOFTHEINVERSEDIFFERENT ............. 45 BIBLIOGRAPHY ........................................ 48 vi LISTOFFIGURES Figure1.1:Galoiscover.....................................3 Figure2.1:Extensionandthencompletion...........................16 Figure2.2:Subextensionswithrespecttohigherramificationgroups.............17 Figure3.1:Galoiscover.....................................31 Figure4.1:WeaklyramifiedGaloiscoverofcurves......................35 Figure4.2:Galoiscoverofspecialfibres............................36 Figure4.3: % and ˘ ......................................39 Figure4.4:ThecollapseintheexampledescribesthewildramificationfromKummer toArtin-Schreierextension,see[SOS89]formore................41 vii KEYTOSYMBOLS K 0 ( ˝Œ ) theGrothendieckgroupoffinitelygenerated [ ˝ ] -modules K 0 ( [ ˝ ]) theGrothendieckgroupoffinitelygeneratedprojective [ ˝ ] -modules K 0 ( [ ˝ ]) red theclassgroupoffinitelygeneratedprojective [ ˝ ] -modules Cl ( [ ˝ ]) theclassgroupoffinitelygeneratedlocallyfree [ ˝ ] -modules H 8 ( -Œ F ) the 8 'thsheafcohomologygroup ^ H 8 ( ˝Œ" ) the 8 'thTatecohomologygroup j ( ˝Œ-Œ F ) theequivariantEulercharacteristic j % ( ˝Œ-Œ F ) theprojectiveequivariantEulercharacteristic O theringofintegersofanumberfield p aprimeideal m themaximalidealofadiscretevaluationring ^ theresiduefieldofadiscretevaluationring ^ " ˚ ˚ -adiccompletionofamodule ˝ 8 the 8 'thramificationgroup [ ˝ ] groupringof ˝ over R rightderivedfunctoroftheglobalsectionfunctor P 1 projectivespaceofdimension1overthespectrumofaring 1 - / . thesheafofrelativedifferentialsof - / . D # / thedifferentidealof # / viii CHAPTER1 INTRODUCTION ThehistoryofthestudyofGaloismodulestructureisdeeplyrootedinthe normalbasistheorem forfiniteGaloisextensions.If # / isafiniteGaloisextensionoffieldswith ˝ = Gal ( # / ) ,the theoremassertsthat # ,asa -vectorspace,hasabasisoftheform f f ( U ) g f 2 ˝ forsome U 2 # whichisthencalleda normalbasis ,i.e., # isafree [ ˝ ] -moduleofrank 1 .Toformulatethe basicproblem,let # / Q beafiniteGaloisextensionwithGaloisgroup ˝ .Thentheringofintegers O # hasanatural Z [ ˝ ] -modulestructure.Theanalogousquestioncanbeasked:Does O # havea normalintegralbasis ,i.e.,isthereanelement U 2O # suchthat f f ( U ) g f 2 ˝ formsa Z -basisof O # ,sothat O # isfreeover Z [ ˝ ] ?Thisexistenceproblemandthenatureoftheglobalobstructions havebeenthecentralthemeofthesubject. Thefirstapproachtotheproblemwasmadevialocalizationwithcompletion.Wecallafinitely generated Z [ ˝ ] -module " locallyfree if,forallprimes ? in Z ,the ? -adiccompletion " ? isfree over Z ? [ ˝ ] .(Thisisequivalentto " being projective ,see[Swa60].)AtheoremofNoether [Noe32]statesthat Theorem1.0.1. With # / Q asabove, O # islocallyfreeover Z [ ˝ ] ifandonlyif # / Q istame,i.e., atmosttamelyramifiedatevery ? . Thustameramificationisclearlynecessaryfortheexistenceofanormalintegralbasis.Thenatural questionisthentoaskifitissufficient,andifnot,whataretheglobalobstructions.Explicit examplesfor # / Q tamewithnonormalintegralbasescanbefoundin[Mar71]. Thequestioncanberephrasedintermsofacalculationoftheclass ( O # ) Z [ ˝ ] of O # inthe classgroupCl ( Z [ ˝ ]) oftheGrothendieckgroupK 0 ( Z [ ˝ ]) offinitelygenerated projective modules over Z [ ˝ ] .TheclassgroupCl ( Z [ ˝ ]) isafiniteabeliangroupwhichisdefinedasthequotient ofK 0 ( Z [ ˝ ]) bythesubgroupgeneratedbytheclassofthefreemodule Z [ ˝ ] .If " isafinitely generatedlocallyfree Z [ ˝ ] -module,thentherankof " anditsclass ( " ) Z [ ˝ ] 2 Cl ( Z [ ˝ ]) determine 1 " uptostableisomorphism:If # isanotherfinitelygeneratedprojective Z [ ˝ ] -module,thenthe classesof " , # inCl ( Z [ ˝ ]) andtheirrankscoincideifandonlyif " Z [ ˝ ] ˙ # Z [ ˝ ] . Moreover,formost ˝ ,e.g.ofoddorderorabelian,astableisomorphismoflocallyfreemodules impliesanisomorphismbytheSwan-Jacobinskitheorem([Swa86];seealso[Frö83]),i.e.,wehave ``cancellation''. Themostimportantdiscoverythatprovidesthepower,depth,andinterestofthetheoryis theconjecturemadebyFröhlichandprovedbyTaylorin[Tay81].Itstatesthat,forafinite Galoisextensionofnumberfields # / ,if # / istame,thentheclass ( O # ) Z [ ˝ ] isequalto anotherinvariant , # / Q inCl ( Z [ ˝ ]) whichCassou-Noguèshaddefinedin[CN78]usingtheArtin root-numbersofsymplecticrepresentationsoftheGaloisgroup ˝ .Theroot-number , ( j ) ofa character j of ˝ isthecomplexconstantofabsolutevalue1appearinginthefunctionalequationof theextendedArtin ! -function ~ ! ( BŒj )= ~ ! ( BŒ# / Œj ) (withEulerfactorsatinfinity,see[Frö83]): ~ ! ( BŒj )= ~ ! (1 BŒ j ) , ( j ) ( j ) 1 2 B where j isthecomplexconjugaterepresentation, ( j ) isapositiveconstant.When j isreal valued,thentherootnumbersareknowntobe 1 .Theserootnumbersinfluencetheexistenceof non-trivialzerosoftheDedekindzetafunctions: ~ ! ( BŒj ) ( j ) B 2 iseithersymmetricorasymmetric about B = 1 2 dependingonwhether , ( j )=1 or = 1 .If j isacharacterofGal ( # / ) ,thelater caseimpliesinturnthattheDedekindzetafunctionof # vanishesat B = 1 2 ,see[Arm71]. TheconnectionbetweenGalois-structureinvariantsandArtin ! -functionshasbeenabasic themeinresearchonGaloisstructuresincethen.Forexample,astheArtinrootnumbersof symplecticcharactersare 1 , ( O # ) Z [ ˝ ] = , # / is2-torsionbyFröhlichandTaylor[Tay81],and indeed (2 Œ j ˝ j ) ( O # ) Z [ ˝ ] =0 Ł (1.1) Thuswhentheorder j ˝ j ofthegroupisodd,then O # isstablyfree;butcancellationholdswhen j ˝ j isodd,thereforeweknowthat O # isactuallyafree Z [ ˝ ] -module. ThestudyoftheGaloismodulestructureoftheringofintegers O # naturallyextendstoa geometricsituation.Suppose - isasmoothprojectivevarietydefinedover Q withanactionofa 2 finitegroup ˝ .Forany ˝ -equivariantcoherentsheaf F of O - -modules,i.e.,acoherentsheafof O - -modulesequippedwitha ˝ -actioncompatiblewiththe ˝ -actionon - ,theequivariantEuler characteristic j ( -Œ F )= j ( ˝Œ-Œ F ) isthevirtualrepresentation j ( -Œ F ):= Õ 8 ( 1) 8 [ H 8 ( -Œ F )] Ł Itiswell-known(see[BK82])that,iftheactionisfree, j ( -Œ F ) isamultipleoftheregularrepre- sentation Q [ ˝ ] ,i.e.,thecomplexR -Œ F ) inthederivedcategoryofcomplexesof Q [ ˝ ] -modules isisomorphictoaboundedcomplexoffinitelygeneratedfree Q [ ˝ ] -modules.Supposenowthat thevariety - isextendedtoaregularprojectivescheme - 0 whichisflatoverSpec Z andtheaction of ˝ extendedto - 0 aswell.Fora ˝ -equivariantcoherentsheafof O - -modules F on - 0 ,wesay that Definition. Thecohomologyof F hasa normalintegralbasis ifthereexistsaboundedcomplex offinitelygenerated free Z [ ˝ ] -moduleswhichisisomorphictoR -Œ F ) inthederivedcategory ofcomplexesoffinitelygenerated Z [ ˝ ] -modules. Unlikethepreviouscaseover Q ,itisnottrueingeneralthatthecohomologyof F hasanormal integralbasis.Theclassicalexampleof O # isreconstructedunderthesesettingswhen - 0 is thespectrumof O # inaGaloisextensionofnumberfields # / Q withtheGaloisgroup ˝ ,and F = O - 0 = g O # . Fromnowon,weconsiderthegeneralsettingforgeometricGaloismodulestructureasfol- lowing:Fora˝nitegroup ˝ ,weconsidera ˝ -cover c : - ! . ofprojectiveflatschemesover Spec Z .If F isa ˝ -equivariantcoherentsheafof O - -modules,thenthesheafcohomologygroups - - / ˝ c ˝ Figure1.1:Galoiscover. H 8 ( -Œ F ) arefinitelygeneratedmodulesoverthegroupring Z [ ˝ ] .ThustheequivariantEuler 3 characteristic j ( -Œ F ) canbede˝nedasearlierintheGrothendieckgroupK 0 ( ˝Œ Z ) offinitely generated Z [ ˝ ] -modules.Themainobjectiveistocalculatetheobstructiontotheexistenceofa normalintegralbasisofthecohomologyof F . Thesolutiontotheproblemdependsonmanygeometricinvariantsof c : - ! . and F ,and thefoundationofthestudyhasbeendevelopedwhen c isassumedtobeatmosttamelyramified, extendingtheclassicalresultson ( O # ) Z [ ˝ ] . Chinburgintroducedthegeneralframework([Chi94];seealso[CE92],[CEPT96])inthe followingsense:When c : - ! . = - / ˝ istamelyramified,thenforany ˝ -equivariant coherentsheaf F of O - -moduleson - ,thecomplexR -Œ F ) isquasi-isomorphictoabounded complexoffinitelygenerated projective Z [ ˝ ] -modules.Infact,onecantakesuchboundedcomplex % tobeoffree Z [ ˝ ] -modulesexceptforthelastprojectiveterm,thustheclassofthelastterm inCl ( Z [ ˝ ]) istheobstructiontotheexistenceofanormalintegralbasisforthecohomologyof F .TheEulercharacteristic Í 8 ( 1) 8 [ % 8 ] ofsuchaboundedcomplexinK 0 ( Z [ ˝ ]) doesnotdepend onthechoiceoftheboundedcomplex,andiscalledthe projective Eulercharacteristic j % ( -Œ F ) . ThismapstotheequivariantEulercharacteristic j ( -Œ F ) inK 0 ( ˝Œ Z ) underthenaturalforgetful functor K 0 ( Z [ ˝ ]) ! K 0 ( ˝Œ Z ) j % ( -Œ F ) 7! j ( -Œ F ) calledtheCartanhomomorphism,givingusmuchmoreinformationontheGaloismodulestructure oftheequivariantEulercharacteristic. UnderstandingtheseprojectiveEulercharacteristicshasbeenthemainproblemofthetheory ofgeometricGaloisstructure.In[Chi94]and[CEPT96],undersomeadditionalhypotheses,the projectiveEulercharacteristicofaversionofthedeRhamcomplexofsheaveson - (which generalizestheclassicalobstruction ( O # ) Z [ ˝ ] if - = Spec O # )wascalculatedusing n -factorsof Hasse-Weil-Artin ! -functionsforthecover - ! . .In[Pap98],Pappasconsideredthegeneral F onthe``relativecaseoverSpec Z withafreeactionof ˝ byintroducingthetechniqueof 4 cubicstructures.Hegeneralizestheequation(1.1)byshowing (2 Œ j ˝ j ) j % ( -Œ F )=0 (1.2) whenalltheSylowsubgroupsof ˝ areabelian.Here, j % ( -Œ F ) istheclassof j % ( -Œ F ) in Cl ( Z [ ˝ ]) calledthe reduced projectiveEulercharacteristic.Latertheabelianrestrictionon(1.2) wasovercomewithastrongerresultwiththeintroductionofAdamsoperationsin[Pap15].This provesacaseofaconjectureonunramifiedcoversthatthereareintegers = (thatdependsonlyon therelativedimensionof - over Z )and X (thatdependsonlyontheorderof ˝ )suchthat ( =Œ j ˝ j ) X j ? ( -Œ F )=0 Œ see[Pap08]and[CPT09]formore. Unliketherichprogressinnon-ramifiedandtamelyramifiedcovers,littleisknownaboutthe simplestkindofwildramificationcalled weaklyramified .TheGaloisgroup ˝ ofafiniteGalois extensionoflocalfields ! / withrespecttoadiscretevaluation E ! hasafinitechainofnormal subgroups ˝ = ˝ 1 D ˝ 0 D ˝ 1 D ˝ 2 D givenby ˝ = = f f 2 ˝ j E ! ( f ( G ) G ) = +1 8 G 2O ! g calledthe = 'thramificationgroup.Wesay ! / is weaklyramified (resp. tamelyramified , unramified )if ˝ = = f 1 g for = =2 (resp. = =1 , = =0 )andthecorrespondingresiduefield extensionsareseparable.AGaloisextensionofnumberfields # / is weaklyramified ifits localizationsatallprimesin # areweaklyramified.Themainresultofthisthesisgeneralizessome aspectsofthetheoryofgeometricGaloismodulestructuretoweaklyramifiedcoversofcurves. Turningbacktotheclassicalexample,Noether'stheoremassertsthatwhentheramification of # / Q iswild,then O # doesnotpossessanormalintegralbasis.However,therecouldbe otherambiguousideals,i.e.,fractionalidealsthatarealsostableunder ˝ ,thatarefreeover Z [ ˝ ] . Regardlessoftheramificationtype,Ullomin[Ull70]showedthatifanambiguousidealin # 5 islocallyfreeover Z [ ˝ ] ,thenallthesecondramificationgroupsaretrivial.Thustheclassical examplesofnormalintegralbasesareallboundtoweaklyramifiedcases.Thefirstgeneralresult onlocallyfreeambiguousidealsover Z [ ˝ ] otherthantheringofintegersitselfwasgivenbyErez in[Ere91].WhentheorderoftheGaloisgroup ˝ isodd,thenbyHilbert'svaluationformulaon thedifferentideal D ( # / Q ) ,thereexistsacanonicalambiguousideal ( # / Q ) calledthe square rootoftheinversedifferent whosesquareistheinverseof D ( # / Q ) .Erez'stheoremstatesthat Theorem1.0.2. ( # / Q ) islocallyfreeover Z [ ˝ ] ifandonlyif # / Q isweaklyramified. Notethat,sincetheorder j ˝ j isassumedtobeodd, Q [ ˝ ] satisfiestheEichlerconditioninthe Swan-Jacobinskitheorem,thustheclass ( ( # / Q )) Z [ ˝ ] determinestheGaloisstructureof ( # / Q ) upto Z [ ˝ ] -isomorphism. In[Köc04],Köckclassifiedallambiguousidealsofweaklyramifiedextensionsthatarelocally freeover Z [ ˝ ] .Suppose ! / isafiniteGaloisextensionoflocalfieldswithGaloisgroup ˝ , and m ! denotethemaximalidealoftheringofintegersof ! .Thenthemaximaltamelyramified subfield ! C of ! over determinesthewildinertiagroup ˝ 1 = Gal ( ! / ! C ) ,and Theorem1.0.3. Thefractionalideal m 1 ! forsome 1 2 Z isfreeover O [ ˝ ] ifandonlyif ! / is weaklyramifiedand 1 1 j ˝ 1 j . Inthesamepaper,thisresultisthengeneralizedtothegeometricGaloismodulecaseofdimension 1,namelyweaklyramifiedcoverofcurves c : - ! - / ˝ overanalgebraicallyclosedfield : with ˝ -equivariantinvertiblesheaves O - ( ˇ ) .ThesesheavesadmitprojectiveEulercharacteristics definedinK 0 ( : [ ˝ ]) . OurmainobjectiveistostudythisGaloismodulestructureproblemwhen c isaweakly ramifiedcoverofcurvesoverSpec Z ,thefirsthigherdimensionalcase.Oncethemeaningof ``weakisclarifiedonfinitecoversofschemesofdimension2(whichisdiscussedin Chapter4),thenaturalquestioniswhetherwecanstudythecohomologiesof ˝ -equivariantsheaves usinganobstructionwell-definedinCl ( Z [ ˝ ]) .TheequivariantEulercharacteristic j ( -Œ F ) is inK 0 ( ˝Œ Z ) ,andtheCartanhomomorphismK 0 ( Z [ ˝ ]) ! K 0 ( ˝Œ Z ) isnotsurjectiveingeneral. 6 OurfirstresultistodefinetheprojectiveEulercharacteristic j ? foraclassof ˝ -equivariant invertiblesheaveson - usingChinburg'scriteriongivenin[Chi94].ThisgeneralizesKöck'swork toschemesofdimension2(TheexactstatementofthetheoremisgiveninChapter4). Theorem1.0.4. Let c : - ! - / ˝ beaweaklyramifiedcoverofcurvesover Z withafinite actionof ˝ .Suppose F = O - ( ˇ ) isaninvertiblesheafof O - - ˝ -moduleson - corresponding toahorizontaldivisor ˇ .Supposetherestriction ˇ \ - ? tothespecialfibre - ? overeach primedivisor ? oftheorderof ˝ isgivenbya ˝ -equivariantWeildivisor Í G 2 - ? = G [ G ] where = G 1 j ˝ GŒ 1 j .Thenthederivedcomplex R -Œ F ) isisomorphicinthederivedcategory toaboundedcomplex % offinitelygeneratedprojective Z [ ˝ ] -modules.ItsEulercharacteristic Í 8 ( 1) 8 [ % ] 8 inK 0 ( Z [ ˝ ]) isindependentofchoicesanddefinestheprojectiveEulercharacteristic j % ( -Œ F ) . Thesecondmainresultisthattheconditionimposedonthe ˝ -equivariantinvertiblesheaves inthetheoremisnottoostrong,thattherecanonicallyexistsasheafsatisfyingtheconditionwhen - ! . isweaklyramifiedrelativecoverofcurvesoverSpec Z .Thecanonicalexistenceisa generalizationofthesquarerootoftheinversedifferent ( # / Q ) introducedbyErez(Thisis discussedinChapter5). Theorem1.0.5. Thereexistsaninvertiblesheaf F on - suchthat F 2 isthetorsion-freepartof thequotientsheaf O - / Ann 1 - / . ) bytheannihilatorofthesheafofrelativedifferentials 1 - / . and whichsatisfiestheassumptionsoftheprevioustheorem,sothattheprojectiveEulercharacteristic j % ( -Œ F ) exists. Thethesisisorganizedasfollows.InChapter2,wereviewthebackgroundmathematicsthat willbeusedthroughoutthethesis.Forexample,wediscusssomefactsaboutramifiedextensionsof discretevaluationrings,Grothendieckgroups,projectivityandcohomologicaltriviality,quotient schemebyafinitegroup,and ˝ -equivariantsheaves.Chapter3isanextensiontoChapter 2,discussingmoreontheoremsofChinburg,Köck,andErezthatwillbeuseddirectlyinour mainresults.InChapter4,weintroduceourfirstmainresultwhichgivesasufficientcondition 7 for ˝ -equivariantinvertiblesheavesonweaklyramifiedrelativecurvestohaveprojectiveEuler characteristic.Wealsodemonstrateanexamplewhichnaturallyleadstothetopicofthenext chapter.Chapter5discussesthecanonicalexistenceofsheavesthatmeetthehypothesisofthe theoreminChapter4,generalizingthesquarerootoftheinversedifferentthatErezintroduced. 8 CHAPTER2 BACKGROUND Thischapterprovidesasuccinctreviewofmathematicsthatwillbeusedthroughoutthis thesis.Wefirstrecallabasicclassificationoframificationofdiscretevaluationringsusinghigher ramificationgroups.ThesecondpartreviewsgroupcohomologyandtheGrothendieckgroup K 0 ( Z [ ˝ ]) offinitelygeneratedprojective Z [ ˝ ] -modules.Inthelastsection,wereviewgroup actionsonschemesandtheEulercharacteristicofequivariantsheaves. Throughoutthetext, ˝ alwaysdenotesa˝nitegroup,andallringsareassumedtobecommu- tativeandhaveidentity. 2.1DiscreteValuationRingsandDedekindDomains Dedekinddomains arefundamentalobjectsinalgebraicnumbertheoryandsmoothcurves. Thissectionheavilyrelieson[Ser79],andproofsofbasicresultsaremostlyomitted. DiscreteValuationRings. Let beafield, themultiplicativegroupofnon-zeroelements of .A discretevaluation of isasurjectivehomomorphism E : ! Z suchthat E ( G + H ) Inf ( E ( G ) ŒE ( H )) for GŒH 2 Ł Here E isextendedto bysetting E (0)=+ 1 . Theset O ofelements G 2 suchthat E ( G ) 0 isasubringof calledthe valuationring of E .Ithasauniquemaximalideal,namelytheset m ofall G 2 suchthat E ( G ) ¡ 1 .Anelement c 2 m with E ( c )=1 generatesthemaximalideal m ,andsuchanelementiscalleda uniformizer of O (orof E ).Thefield ^ = O / m iscalledtheresiduefieldof O (orof E ). A discretevaluationring ( O Œ m ) (orsimply O ) isaprincipalidealdomain O withexactlyone non-zeromaximalideal m .Thenthefieldofquotients of O isequippedwithadiscretevaluation E derivedby m :For G 2 , E ( G )= = where G 2 m = and G 8m = +1 . Adiscretevaluationringischaracterizedbythefollowingproposition: 9 Proposition2.1.1. Let beanoetherianintegraldomain.Then isadiscretevaluationringif andonlyifthetwofollowingconditionsaremet: i) isintegrallyclosed. ii) hasauniquenon-zeroprimeideal. DedekindDomains. Let O beanoetherianintegraldomain. Proposition2.1.2. Foranoetherianintegraldomain O ,thefollowingareequivalent: i) Foreverynon-zeroprimeideal p 2 Spec O ,thelocalization O p isadiscretevaluationring. ii) O isintegrallyclosedandofdimension 1 . If O satisfiestheseequivalentconditionsandisofdimension 1 ,itiscalleda Dedekingdomain . If isanintegraldomainwiththefieldoffractions ,a fractionalideal ˚ of isasub- -moduleof finitelygeneratedover .If ˜ isanotherfractionalidealof ,theproductideal ˚˜ isgeneratedbyproductsofelementsof ˚ and ˜ .Onesays ˚ is invertible ifthereexistsafractional ideal ˜ suchthat ˚˜ = . Proposition2.1.3. InaDedekinddomain,everyfractionalidealisinvertible. Thenon-zerofractionalidealsofaDedekinddomainformagroupundermultiplicationcalledthe idealgroup ofthering. Let O beaDedekinddomain.Foreachnon-zeroprimeideal p of O ,thelocalization O p defines adiscretevaluationofthefieldoffractions denotedby E p . Proposition2.1.4. If G 2O , G < 0 ,thenonlyfinitelymanyprimeidealscontain G . Corollary2.1.4.1. Forevery G 2 ,thenumbers E p ( G ) arealmostallzero,i.e.,zeroexceptfor afinitenumber. 10 If ˚ isafractionalidealand p isanon-zeroprimeidealof O ,thentheimage ˚ p of ˚ in O p has theform ˚ p =( p O p ) E p ( ˚ ) definingthevaluation E p ( ˚ ) oftheideal ˚ at p . Proposition2.1.5. Everyfractionalideal ˚ of O canbewrittenuniquelyintheform: ˚ = Ö p p E p ( ˚ ) Œ where E p ( ˚ ) areintegersalmostallzero. ExtensionsofDedekindDomains. Let O beaDedekinddomainwithitsfieldoffractions . Let # beaseparableextensionof anddenoteby O # the integralclosure of O in # ,i.e.,theset ofelementsof # thatareintegralover O . Proposition2.1.6. Thering O # isafinitelygenerated O -moduleandDedekind. If p isanon-zeroprimeidealof O ,and q isaprimeidealof O # suchthat q \O = p ,we say q isover p andwrite q j p .Thefractionalideal p O # decomposesintoprimeidealsof O # : p O # = Ö q j p q 4 q where 4 q = E q ( p O # ) 0 .Theinteger 4 q iscalledthe ramificationindex of q intheextension # / . When q isover p ,theresiduefield _ = O # / q isafinitefieldextensionoftheresiduefield ^ = O / p .Thedegree 5 q ofextension _ / ^ iscalledthe residuedegree of q . Proposition2.1.7. Let p beanon-zeroprimeidealof O .Thenthering O # / p O # isan O / p -algebraofdegree = =[ # : ] isomorphictotheproduct Ö q j p O # / q 4 q Ł Wehavetheformula = = Õ q j p 4 q 5 q Ł 11 If q j p and E q isthevaluationinducedby q ,thenfor G 2 , E q ( G )= 4 q E p ( G ) .Wesaythe valuation E q extends E p withindex 4 q .Conversely: Proposition2.1.8. Let F beadiscretevaluationof # extending E p forsome p 2 Spec O with index 4 .Thenthereis q 2 Spec O # , F = E q with 4 q = 4 . GaloisExtensions. Let # / beasbeforeandassumefurtherthattheextensionisGalois. Proposition2.1.9. If # / isGaloisand q j p ,theintegers 4 q Œ5 q dependonlyon p .Ifwedenote themby 4 p Œ5 p ,andif A p denotesthenumberofprimeideals q over p ,then = = 4 p 5 p A p Ł Let ˝ = Gal ( # / ) .Foranon-zeroprimeideal q 2 Spec O # ,the decompositiongroup ˝ q is thesubgroupof ˝ fixing q : ˝ q := f f 2 ˝ j f ( q )= q g Ł For q j p ,denote 4 p , 5 p , A p by 4 , 5 , A respectively.Thenwehaveextensionsoffields: # # ˝ q 4 5 A where # ˝ q isthefixedgroupof # bythedecompositiongroup ˝ q . Nowconsidertheresiduefieldextension _ / ^ at q .Thereisanaturalhomomorphism ˝ q ! Gal ( _ / ^ ) Œ andthekernel ˚ q ofthismorphismiscalledthe inertiagroup of q .Thisgivesextensions: 12 # # ˚ q # ˝ q A where # ˚ q / # ˝ q isaGaloisextensionwithGaloisgroup ˝ q / ˚ q . Proposition2.1.10. Theresidueextension _ / ^ isnormal,andthenaturalhomomorphism ˝ q ! Gal ( _ / ^ ) issurjective. Thedefiningshortexactsequenceoftheinertiagroup ˚ q isthusgivenas 0 ! ˚ q ! ˝ q ! Gal ( _ / ^ ) ! 0 Ł Completion. Suppose isacommutativeringwithidentity,and ˚ isanidealof .Wedenote by ˚ = the = 'thpoweroftheideal ˚ .Thentherearenaturalhomomorphisms / ˚ / ˚ 2 / ˚ 3 Ł Thismakes ( / ˚ = ) aninversesystemofrings,anditsinverselimitring ^ ˚ = lim / ˚ = iscalled the completionof withrespectto ˚ orthe ˚ -adiccompletionofA .Foreach = ,wehaveanatural map ! / ˚ = ,andbytheuniversalpropertyoftheinverselimit,weobtainahomomorphism ! ^ ˚ . Similarly,if " isan -module,wedefine ^ " ˚ = lim " / ˚ = " ,andcallitthe ˚ -adiccompletion of " .Ithasanaturalstructureof ^ ˚ -module.Whennoconfusionarises,wewouldskip ˚ and denote ^ ˚ , ^ " ˚ by ^ , ^ " ,respectively. Westatesomeimportantpropertiesof ˚ -adiccompletionwithoutproofs(seeII. x 9in[Har77]). 13 Proposition2.1.11. Let beanoetherianring,and ˚ anidealof .Then: i) ˚ -adiccompletioncommuteswith˝nitedirectsums(thisholdstruewithoutnoetherianas- sumptionon ); ii) ^ ˚ = lim ˚ / ˚ = isanidealof ^ .Forany = ,thepower ^ ˚ = = ˚ = ^ ,and ^ / ^ ˚ = ˙ / ˚ = ; iii) if " isafinitelygenerated -module,then ^ " ˙ " ^ ; iv) thefunctor " 7! ^ " isanexactfunctoronthecategoryoffinitelygenerated -modules; v) ^ isaflat -module; vi) ^ isanoetherianring; vii) if ( " = ) isaninversesystemwhereeach " = isafinitelygenerated / ˚ = -module,andforall == 0 , 0 ! ˚ = " = 0 ! " = 0 ! " = ! 0 isexact,then " = lim " = isafinitelygenerated ^ -module,andforeach = , " = ˙ " / ˚ = " . Inmostofcasesofourstudy,weconsiderthe m -adiccompletionforadiscretevaluationring ( Œ˚ )=( O Œ m ) . CompleteFields. Considerafield equippedwithadiscretevaluationring ( O Œ m ) .The m -adiccompletionof O isadiscretevaluationring ^ O withthemaximalideal ^ m .Itsfieldof fractions ^ isthecompletionof withrespecttoanultrametricabsolutevalueon definedby, forsomerealnumber 0 0 1 , jj G jj = 0 E ( G ) for G < 0 Œ jj 0 jj =0 Ł Thetopologydeterminedbythismetricdoesnotdependonthechoiceof 0 .Thefield is complete if ^ = . 14 Acompletefield isa localfield ifitsresiduefield ^ isfinite.Alocalfieldofcharacteristiczero iseither Q ? ,thecompletionof Q forthetopologydefinedbythe ? -adicvaluation,ortheLaurent powerseriesinaformalvariable.Alocalfieldwithfinitecharacteristic ?¡ 0 isisomorphicto thefield ˙ (( ) )) offormalLaurentpowerseriesforsomefinitefield ˙ .When isalocalfield,a canonicalwaytochoosethenumber 0 istotake 0 = @ 1 ,where @ isthecardinalityofthefinite residuefield ^ . ExtensionofaCompleteField. Let beafieldequippedwithadiscretevaluationring O . Suppose iscompleteinthetopologydeterminedby O .Let ! / beafiniteextensionoffields, and O ! theintegralclosureof O . Proposition2.1.12. Undertheaboveassumptions, i) O ! isadiscretevaluationring; ii) O ! isafree O -moduleofrank = =[ ! : ] ; iii) ! iscompleteinthetopologydeterminedbythemaximalidealof O ! . Corollary2.1.12.1. If ! / isseparable,then O ! isafinitelygeneratedfree O -module. Thereisauniquevaluation E ! of ! extendingthevaluation E of . Corollary2.1.12.2. Forevery G 2 ! , E ! ( G )= 1 5 E ( # ! / ( G )) where 5 istheresiduedegree. ExtensionandCompletion. Supposeafield isthefractionfieldofadiscretevaluationring ( O Œ m ) whichisnotassumedtobecomplete,andthereisafiniteseparableextension # / with O # theintegralclosureof O in # .Then m decomposesintoaproductofprimes q 8 of O # as m O # = A Ö 8 =1 q 4 8 8 Ł Each q 8 j m hasaresiduedegree 5 8 =[ O # / q 8 : O / m ] .Let ^ # 8 , ^ bethecompletionsof # , inthetopologiesinducedby q 8 , m ,respectively. 15 # ^ # 8 ^ Figure2.1:Extensionandthencompletion. Proposition2.1.13. For ^ # 8 , ^ asgivenabove, i) [ ^ # 8 : ]= = 8 = 4 8 5 8 ; ii) Theramificationindexof ^ # 8 / ^ is 4 8 ,andtheresiduedegreeof ^ # 8 / ^ is 5 8 ; iii) thecanonicalhomomorphism # ^ ! Î 8 ^ # 8 isanisomorphism. Corollary2.1.13.1. If # / isGaloiswithGaloisgroup ˝ ,andif ˝ q 8 denotesthedecomposition groupof q 8 j m ,theextension ^ # 8 / ^ isGaloiswithGaloisgroup ˝ q 8 . Thus # ^ isa = =[ # : ] -dimensionalvectorspaceover ^ .Thefollowingproposition showsthesimilarfor O # O ^ O . Proposition2.1.14. Withthesamehypothesesandnotationasabove,let O # 8 betheringof valuationwithrespectto q 8 j m .Thenthecanonicalhomomorphism O # O ^ O ! Ö 8 ^ O # 8 isanisomorphism. Corollary2.1.14.1. Let " beafinitelygenerated O # -module.Consider " asan O -modulevia thenaturalinclusion O !O # .Let ^ " denotethe m -adiccompletionof " .Then ^ " ˙ Ö 8 ^ " q 8 Ł 16 HigherRamificationGroups. Let ! / beafiniteGaloisextensionoffieldswithcompatible non-trivialdiscretevaluations.Let ˝ = Gal ( ! / ) ,andassumethattheresiduefieldextension is separable withcharacteristic ?¡ 0 .Denotethecorrespondingringsofintegersby O ! / O , maximalidealsby m ! / m ,andresiduefields _ / ^ ,respectively. Forintegers = 1 ,the = 'thramificationgroup ˝ = isdefinedby ˝ = := f f 2 ˝ j forall G 2O ! Œf ( G ) G 2 m = +1 ! g Ł Thus ˝ 1 = ˝ ; ˝ 0 = ˚ isthe inertiasubgroup ,thekernelofthenaturalquotient 0 ! ˚ ! ˝ ! Gal ( _ / ^ ) ! 0 Ł Recallthatthe wildinertiasubgroup in ˚ isGal ( ! / ! C ) where ! C isthemaximaltamelyramified subextensionin ! / .Indeed, ˝ 1 isthewildinertiagroupwhichisalsoaSylow ? -subgroupof ˝ 0 ,where ˝ 0 / ˝ 1 isfinitecyclicoforder 4 C primeto ? bythecanonicalinjectioninto : given thechoiceofauniformizerof O ! . ! _ ! C ! ˚ ^ B = _ ^ ? < 4 C 1 5 5 Figure2.2:Subextensionswithrespecttohigherramificationgroups. Theramificationgroupsformachain ˝ D ˝ 0 D ˝ 1 D ˝ 2 D ofnormalsubgroupsof ˝ .Forsufficientlybig = , ˝ = istrivial.Theextension ! / iscalled weakly ramified ( tamelyramified , unramified ),if ˝ = istrivialfor = =2 ( = =1 , = =0 ,respectively)and _ / ^ isseparable. 17 Let O # / O beanextensionofDedekinddomainswithafiniteGaloisextension # / .Fora non-zeroprimeideal q of O # and p = q \O ,let ^ # q , ^ p denotethecompletionswithrespectto correspondingdiscretevaluationsof # , ,respectively.Wesay # / is weaklyramified ( tamely ramified , unramified )at q if ^ # q / ^ p is weaklyramified ( tamelyramified , unramified ).If # / is weaklyramified ( tamelyramified , unramified )atallnon-zeroprimeideal q 2 Spec O ! ,then # / iscalled weaklyramified ( tamelyramified , unramified ). Hilbert'sFormula. Let beacompletediscretevaluedfieldunder E ! and ! / finiteGalois with ˝ = Gal ( ! / ) .Thenthedifferentideal D ( ! / ) canbedeterminedbytheramification groups(cf.[Ser79],IV.1Proposition4). Proposition2.1.15. If D ( ! / ) denotesthedifferentof ! / ,then E ! D ( ! / )= Õ 8 0 ( j ˝ 8 j 1) Ł NormalIntegralBasis. Let O # / O beanextensionofDedekinddomainswiththeextensionof fieldsoffractions # / asbefore.Suppose # / isGaloiswith ˝ = Gal ( # / ) .Wesawthat O # isafinitelygenerated O -module.Ifthereisanelement U 2O # suchthatthesetofitsconjugates f f ( U ) g f 2 ˝ formsa O -basisof O # ,wesay O # hasa normalintegralbasis .Thushavinga normalintegralbasismeans O # isafree O [ ˝ ] -module.Foraprime p 2 Spec O ,wesay O # is free at p ifthe p -adiccompletion ^ O #Œ p isfreeover ^ O Œ p [ ˝ ] .Wesay O # is locallyfree ifitisfree atallprimes p of O simultaneously. Assumefurtherthat # / isanextensionofnumberfields,i.e.,finiteextensionsof Q .Tobe afree O [ ˝ ] -module,itiscertainlynecessarythatthemoduleshouldbelocallyfree.Noether's theorem(cf.[Noe32])partofwhichgoesbacktoSpeicer(cf.[Spe16])relatestheexistenceof normalintegralbasistoramification. Theorem2.1.1 (Noether) . O # islocallyfreeover O [ ˝ ] ifandonlyif # / istame. 18 2.2GaloisModules Theterm Galoismodule isusedasasynonymfor ˝ -module,i.e., Z [ ˝ ] -module.Let bearing and " afinitelygeneratedmoduleoverthegroupring [ ˝ ] forsomefinitegroup ˝ .Wesay " has a normalintegralbasis over if " haselements U 1 ŒŁŁŁŒU = suchthattheset f f ( U 1 ) ŒŁŁŁŒf ( U = ) g f 2 ˝ isan -basisof " ,i.e., " isfreeoverthegroupring [ ˝ ] . Wesay " is free at p 2 Spec if " p isafree p [ ˝ ] -module.Wesay " is locallyfree over ifitisfreeatall p 2 Spec .When = Z ,wesimplysay " hasanormalintegralbasisoris locallyfreewithoutreferringto Z . Local-globalmethodsoriginatedinnumbertheory(Hasseprinciple)areextendedtoGalois moduletheory,andoneofthemostimportantresultsisSwan'stheorem:Suppose isaDedekind domainwhosefieldoffractionshascharacteristiczero.Let ˝ beafinitegroupsuchthatnorational primedividingtheorderof ˝ isaunitin .Thenfinitelygeneratedprojective [ ˝ ] -modulesare locallyfreeover [ ˝ ] andalsolocallyfreeaftercompletion,i.e.,forallnon-zeroprime p 2 Spec , ^ " p isisafree ^ p [ ˝ ] -module([Swa60]).TheconversefollowsreadilyfromTheorem7.3.29in [BK00].Thustheterms``projective"and``locallyfree"willbeusedinterchangeablythroughout therestofthetext. AnimportantcorollarytoSwan'stheorem(Corollary6.4in[Swa60])isasfollowing: Proposition2.2.1. Let ˝ beafinitegroup, alocalintegraldomain, itsfieldoffractions.Let % , & befinitelygeneratedprojectivemodulesover [ ˝ ] .If % ˙ & as [ ˝ ] -modules,then % ˙ & . TateCohomologyGroups. Wefirstrecallthebasicdefinitionsandresultsonthehomologyand cohomologyofgroups.A ˝ -module " is projective ifthefunctor ˛>< ˝ ( "Œ ) isexact, injective ifthefunctor ˛>< ˝ ( Œ" ) isexact. The ˝ -module " is induced ifithastheform Z [ ˝ ] Z - forsomeabeliangroup - .Every 19 ˝ -moduleisaquotientofaninducedmodule,canonicallygivenbythesurjection Z [ ˝ ] " 0 ! " where " 0 isjust " withitsabeliangroupstructure.Dually,a ˝ -module " iscalled co-induced if ithastheformHom Z ( Z [ ˝ ] Œ- ) forsomeabeliangroup - .Each ˝ -module " embedscanonically intheco-induced ˝ -moduleHom Z ( Z [ ˝ ] Œ" 0 ) .Whenthegroup ˝ isassumedtobefinite,the notionsofinducedandco-inducedmodulescoincide,andthisisourcase. Let " ˝ bethesubmoduleof " consistingoftheelementsfixedby ˝ ;itisthelargestsubmodule of " onwhich ˝ actstrivially.If 5 : " ! " 0 isamorphismof ˝ -modules,then 5 maps " ˝ to " 0 ˝ ,thuswecanspeakofthefunctor " ˝ .Itisaleftexactadditivefunctor.Bydefinition,the rightderivedfunctors ofthefunctor " ˝ arethecohomologygroupsof ˝ withcoefficientsin " , denotedbyH 8 ( ˝Œ" ) , 8 0 .Notethat " ˝ = Hom ˝ ( Z Œ" ) where Z isconsideredasa ˝ -module withtrivialaction.ThenH 8 ( ˝Œ" )= Ext 8 Z [ ˝ ] ( Z Œ" ) sinceExt 8 Z [ ˝ ] ( Z Œ ) arethederivedfunctors ofthefunctorHom ˝ ( Z Œ )= Hom Z [ ˝ ] ( Z Œ ) . Let ˇ" bethesubgroupof " generatedby f ( G ) G , G 2 " , f 2 ˝ .Thequotient " / ˇ" will bedenotedby " ˝ ;itisthelargestquotientmoduleof " onwhich ˝ actstrivially.Thefunctor " ˝ isarightexactadditivefunctor.Bydefinition,its leftderivedfunctors arethehomology groupsof ˝ withcoefficientsin " ,denotedbyH 8 ( ˝Œ" ) for 8 0 .Wehave " ˝ = Z Z [ ˝ ] " , henceH 8 ( ˝Œ" )= Tor Z [ ˝ ] 8 ( Z Œ" ) asthederivedfunctors. Inthegroupalgebra Z [ ˝ ] ,theelement Í f 2 ˝ f willbecalledthe norm andbedenoted # .For every ˝ -module " , # definesanendomorphismof " bytheformula # ( G )= Õ f 2 ˝ f ( G ) Ł Let ˚ ˝ denotethe augmentationideal of Z [ ˝ ] ,thesetoflinearcombinationsofthe f 1 , f 2 ˝ . Thenobviously ˚ ˝ " ˆ Ker ( # ) andIm ( # ) ˆ " ˝ Ł 20 SinceH 0 ( ˝Œ" )= " / ˚ ˝ " andH 0 ( ˝Œ" )= " ˝ ,itfollowsthat # de˝nesaninducedhomo- morphism # : H 0 ( ˝Œ" ) ! H 0 ( ˝Œ" ) Ł Supposethat 0 ! ! ! ˘ ! 0 isanexactsequenceof ˝ -modules.Thenthediagram H 1 ( ˝Œ˘ ) H 0 ( ˝Œ ) H 0 ( ˝Œ ) H 0 ( ˝Œ˘ ) 0 0 H 0 ( ˝Œ ) H 0 ( ˝Œ ) H 0 ( ˝Œ˘ ) H 1 ( ˝Œ ) # # # ˘ iscommutative,andthereisacanonicalhomomorphism Ker ( # ˘ ) ! CoKer ( # ) bythesnakelemma.Moreover(cf.[CE56],V.10),theabovediagramgivesalongexactsequence ! H 1 ( ˝Œ˘ ) ! Ker ( # ) ! Ker ( # ) ! Ker ( # ˘ ) ! CoKer ( # ) ! CoKer ( # ) ! CoKer ( # ˘ ) ! H 1 ( ˝Œ ) ! Ł ThisleadstotheTatecohomologygroups ^ H 8 ( ˝Œ" ) whicharedefinedas ^ H 8 ( ˝Œ" )= H 8 ( ˝Œ" ) if 8 1 ^ H 0 ( ˝Œ" )= Coker ( # ) ^ H 1 ( ˝Œ" )= Ker ( # ) ^ H 8 ( ˝Œ" )= H 8 1 ( ˝Œ" ) if 8 2 Ł RestrictionandCorestriction. If ˛ 6 ˝ isasubgroup,a ˝ -module " inheritsthenatural ˛ -modulestructurebyrestrictingthegroupactionto ˛ .Wewritethis ˛ -moduleasRes ˝ ˛ ( " ) . Clearly " ˝ ˆ " ˛ ,andthisinduces restriction homomorphisms Res : H 8 ( ˝Œ" ) ! H 8 ( ˛Œ" ) Ł Alsothisinducedactionof ˛ yields " ˛ ! " ˝ whichleadsto corestriction homomorphisms Cor : H 8 ( ˛Œ" ) ! H 8 ( ˝Œ" ) Ł 21 Nowgivenatransversaloftheleftcosets ˝ / ˛ in ˝ ,anelement G of " ˛ canbemappedto " ˝ bytakingthe norm of G , Õ f 2 ˝ / ˛ f ( G ) Ł Thisinduces corestriction homomorphisms Cor : H 8 ( ˛Œ" ) ! H 8 ( ˝Œ" ) and restriction homomorphisms Res : H 8 ( ˝Œ" ) ! H 8 ( ˛Œ" ) Ł ThesenaturallyextendtotheTatecohomologygroupsas restriction homomorphisms Res : ^ H 8 ( ˝Œ" ) ! ^ H 8 ( ˛Œ" ) and corestriction homomorphisms Cor : ^ H 8 ( ˛Œ" ) ! ^ H 8 ( ˝Œ" ) Ł Wehavethefollowingrestriction-corestrictionformula(cf.[Ser79],VIII.2.Proposition4). Proposition2.2.2. If = =[ ˝ : ˛ ] ,thenCor Res = = . Corollary2.2.2.1. If = istheorderof ˝ ,thenallthegroups ^ H 8 ( ˝Œ" ) areannihilatedby = . Shapiro'sLemma. Let " bean ˛ -module, ˛ asubgroupof ˝ .Thenthe induced and coinduced ˝ -modulesaredefinedby Ind ˝ ˛ ( " )= Z [ ˝ ] Z [ ˛ ] "Œ Coind ˝ ˛ ( " )= Hom ˛ ( Z [ ˝ ] Œ" ) Ł AfundamentaltoolincalculationisShapiro'slemma(cf.[Wei94],Lemma6.3.2). Proposition2.2.3 (Shapiro) . H 8 ( ˝Œ Ind ˝ ˛ ( " )) ˙ H 8 ( ˛Œ" ) ;and H 8 ( ˝Œ Coind ˝ ˛ ( " )) ˙ H 8 ( ˛Œ" ) . 22 Proposition2.2.4. Iftheindex [ ˝ : ˛ ] isfinite, Ind ˝ ˛ ( " ) ˙ Coind ˝ ˛ ( " ) . Corollary2.2.4.1. If " isaprojective ˝ -module,thenallTatecohomologygroups ^ H 8 ( ˝Œ" ) vanish. DoubleCosetFormula. AnotherusefultoolisMackey'sdoublecosetformula(cf.[Ser77], 7.3Proposition22).Let ˛ 1 , ˛ 2 betwosubgroupsof ˝ .If , isarepresentationof ˛ 1 ,weset + = Ind ˝ ˛ 1 ( , ) .ThedoublecosetformuladeterminestherestrictionRes ˝ ˛ 2 ( + ) .Chooseasetof representatives ( forthe ( ˛ 1 Œ˛ 2 ) doublecosetsof ˝ ,i.e., ˝ isthedisjointunionof ˛ 1 B˛ 2 ,for B 2 ( .For B 2 ( ,let ˛ 1 ŒB = B˛ 1 B 1 \ ˛ 2 6 ˛ 2 .Let , B bethe ˛ 1 ŒB -modulewiththeunderlying set , andtheactionof G 2 ˛ 1 ŒB givenby B 1 GB 2 ˛ 1 . Proposition2.2.5 (Mackey) . Res ˝ ˛ 2 Ind ˝ ˛ 1 ( , ) ˙ B 2 ( Ind ˛ 2 ˛ 1 ŒB ( , B ) ProjectivityandCohomologicalTriviality A ˝ -module " iscalled cohomologicallytrivial if, foreverysubgroup ˛ of ˝ andevery = 2 Z , ^ H = ( ˛Œ" )=0 .ByCorollary2.2.4.1,projective Z [ ˝ ] -modulesarecohomologicallytrivial.ArefinedstatementoftheconverseisgivenasTheorem 7in[Ser79],IX.5(seealsoProposition1.3in[Köc04]): Proposition2.2.6. Let beaDedekinddomainand " bea [ ˝ ] -module.Then " isprojective over [ ˝ ] ifandonlyif " isprojectiveover andcohomologicallytrivial. Asaresult,cohomologicaltrivialityandprojectivitywouldoftenbeequivalentinwhatfollows. Checkingcohomologicaltrivialitywillbeourmaintoolinfindinglocallyfree Z [ ˝ ] -modules. HerearesomegeneralstatementsdescribinglocallyfreeGaloismodulesintermsofgroup cohomology.Firstwerecallthenotionoftheprojectivedimensionofamodule.Givenamodule " ,a projectiveresolution of " isaninfiniteexactsequenceofmodules ! % = !! % 2 ! % 1 ! % 0 ! " ! 0 23 withallthe % 8 projective.Everymodulepossessesaprojectiveresolutionmadeoutoffreemodules, buttheresolutioncouldbeinfinite.Thelengthofafiniteresolutionisthesubscript = suchthat % = isnon-zeroand % 8 =0 for 8¡= .If " admitsafiniteprojectiveresolution,the projective dimension of " istheminimallengthamongallfiniteprojectiveresolutionsof " . Proposition2.2.7 ([Chi94]) . Suppose isaringand " isan [ ˝ ] -module. i) If isafield,then " iscohomologicallytrivialfor ˝ ifandonlyif " isprojectivefor [ ˝ ] . ii) Suppose isaDedekinddomain.Then " iscohomologicallytrivialfor ˝ ifandonlyif " hasprojectivedimensionatmostoneasan [ ˝ ] -module. iii) Suppose isaDedekinddomainandthattheimageofthenaturalmorphism Spec ! Spec Z containstheprimedivisorsoftheorderof ˝ .Afinitelygenerated [ ˝ ] -module " isprojective ifandonlyifitislocallyfree. GrothendieckGroupsof Z [ ˝ ] -modules. Ourproblemsareoftenstatedintermsof Grothendieck groups .Let ' bearingand C acategoryofleft ' -modules.The Grothendieckgroup of C isthe abeliangroupdefinedbygeneratorsandrelationsasfollows:Agenerator [ ] associatedwitheach 2C ,andtherelation [ ]=[ ]+[ ˘ ] isassociatedwitheachexactsequence 0 ! ! ! ˘ ! 0 where ŒŒ˘ 2C Ł ThetwomostcommonexamplesaretheGrothendieckgroupK 0 ( ˝Œ Z ) offinitelygenerated Z [ ˝ ] -modulesandtheGrothendieckgroupK 0 ( Z [ ˝ ]) offinitelygenerated projective Z [ ˝ ] -modules. Whenworkingwithacategoryof ˝ -modules,elementsofGrothendieckgroupsarealsocalled virtualrepresentations .Forexample,givenanexactsequenceofmodules 0 ! " 1 ! " 2 ! " 2 ! " 3 ! " 4 !! " = ! 0 inanyoftheaboveexamplesofGrothendieckgroups,wehaveacorrespondingvirtualrepresen- tationgivenbyanalternatingsum [ " 1 ]=[ " 2 ] [ " 3 ]+[ " 4 ] [ " 5 ]+ +( 1) = [ " = ] Ł 24 AnequalityinK 0 ( Z [ ˝ ]) givesastableisomorphism:If [ " ]=[ " 0 ] inK 0 ( Z [ ˝ ]) ,thereis = 0 , " Z [ ˝ ] = ˙ " 0 Z [ ˝ ] = .TheSwan-Jacobinskitheoremstatesthatwheneverthe Eichlerconditionissatisfied(cf.p.178in[Swa86];seealsop.50in[Frö83]),stableisomorphism impliesisomorphism,i.e.,holds.Thisisthecasewhenevernoneofthesimple componentsof Q [ ˝ ] aretotallydefinitequaterniondivisionalgebrasoveratotallyrealfield,e.g., for ˝ ofoddorderor ˝ abelian.Therearemoregroupswiththecancellationlaw,butitdoes notholdforallgroups(e.g. ˝ = ˛ 32 ,see[Swa79]).Nevertheless,cancellationoftenholds,and knowingthat [ " ] isequaltoafree Z [ ˝ ] -moduleinK 0 ( Z [ ˝ ]) isastrongapproximationtothe existenceofanormalintegralbasis. Motivatedbythis,weconsiderthe reduced GrothendieckgroupK 0 ( Z [ ˝ ]) red whichisthe quotientofK 0 ( Z [ ˝ ]) bythesubgroupgeneratedby [ Z [ ˝ ]] .ByProposition2.2.7,K 0 ( Z [ ˝ ]) red is identifiedwiththe locallyfreeclassgroup Cl ( Z [ ˝ ]) of Z [ ˝ ] whichisobtainedasfollowing:Let # beanumberfieldand A anorderin # [ ˝ ] ,i.e.,asubringof # [ ˝ ] with 1 2 A .A locallyfree A -module " isafinitelygenerated A -modulesothat,forallprimedivisors p of # ,the A p -module " p isfree.Itsrank A ( " ) isdefinedastherankofthefree # [ ˝ ] -module " O # # spannedby " . Thisrankisfinite,andisalsotherankof " p over O p [ ˝ ] forall p . TheGrothendieckgroup K 0 ( A ) oflocallyfree A -modulesistheabeliangroupwithgenerators [ " ] correspondingtothe A -isomorphismclassesoflocallyfree A -modules " andwithrelations [ " 1 " 2 ]=[ " 1 ]+[ " 2 ] Ł Themap N ! K 0 ( A ) whichtakes = intotheclass [ A = ] extendstoahomomorphism Z ! K 0 ( A ) , andwedefinethelocallyfreeclassgroupCl ( A ) tobeitscokernel.Foranalternativedescription ofCl ( A ) called"Homdescription",seeI.2,[Frö83]. Theclassgroupisknowntobeafiniteabeliangroup,andwewilldenotetheimageof [ " ] 2 K 0 ( Z [ ˝ ]) inCl ( Z [ ˝ ]) by ( " ) Z [ ˝ ] . Thenaturalforgetfulfunctor K 0 ( Z [ ˝ ]) ! K 0 ( ˝Œ Z ) 25 calledtheCartanhomomorphism.Thismorphismisneitherinjectivenorsurjectiveingeneral. Whenstudyinganarbitraryfinitelygenerated ˝ -module " ,knowingthat [ " ] 2 K 0 ( ˝Œ Z ) isinthe imageoftheCartanhomomorphismisahugeadvantageinanalyzingitsGaloismodulestructure. Givenaring ,theGrothendieckgroupsK 0 ( [ ˝ ]) andK 0 ( ˝Œ ) aredefinedsimilarly.We alsomentionthatCT ( [ ˝ ]) denotestheGrothendieckgroupoffinitelygenerated [ ˝ ] -modules whicharecohomologicallytrivial.When isaDedekinddomain,theforgetfulhomomorphism K 0 ( [ ˝ ]) ! CT ( [ ˝ ]) isanisomorphismbyProposition2.2.7. MoreonCohomologicalTriviality. Herewelistacouplemoretheoremsoncohomologyfora finitegroupwhichprovidebackgroundtoProposition2.2.6and2.2.7(see[Ser79]forproofsand more). Proposition2.2.8. Let ˝ bea ? -groupandlet " bea ˝ -modulewithout ? -torsion.Thefollowing conditionsareequivalent: i) " iscohomologicallytrivialfor ˝ , ii) ^ H 8 ( ˝Œ" )=0 fortwoconsecutivevaluesof 8 , iii) the F ? [ ˝ ] -module " / ?" isfree. Proposition2.2.9. Let ˝ beafinitegroup, " a Z -free ˝ -module,and ˝ ? aSylow ? -subgroupof ˝ foreachprimenumber ? .Thefollowingconditionsareequivalent: i) " is Z [ ˝ ] -projective, ii) Foreveryprimenumber ? ,the ˝ ? -module " satisfiestheequivalentconditionsofProposition 2.2.8. 2.3AlgebraicGeometry WeintroducebasicnotionsinalgebraicgeometryrequiredfordescribinggeometricGalois modulesandEulercharacteristicsofequivariantsheaves. 26 QuotientSchemebyaFiniteGroup. Let - beaschemeand ˝ Aut ( - ) afinitegroupacting on - .Weareinterestedinwhenwecanformthequotientscheme - / ˝ .Thefollowingstatement canbefoundin[Mum70]. Proposition2.3.1. Supposetheorbitofanypointiscontainedinanaffineopensubsetof - .Then thereisapair ( .Œc ) where . isaschemeand c : - ! . isamorphismsuchthat (i) Asatopologicalspace, ( .Œc ) isthequotientof - fortheactionof ˝ ; (ii) themorphism c : - ! . is ˝ -invariant,andif ( c O - ) ˝ denotesthesubsheafof c O - of ˝ -invariantfunctions,thenaturalhomomorphism O . ! ( c O - ) ˝ isanisomorphism. Thepair ( .Œc ) isuniquelydetermineduptoisomorphismbytheseconditions.Themorphism c is finiteandsurjective.Wedenote . by - / ˝ ,andithasthefunctorialproperty:forany ˝ -invariant morphism 5 : - ! / ,thereisauniquemorphism 6 : . ! / suchthat 5 = 6 c . Inparticular,if - ! Spec Z isprojective,thenthequotientscheme - / ˝ exists.Throughout thetext,weassumethatascheme - withafinitegroupaction ˝ admitsaquotientscheme c : - ! . . Welistafewcorollariesonpropertiesof c and - / ˝ . Corollary2.3.1.1. Thequotientmap c : - ! . isopen. Proof. Suppose * - isopen.Then c ( * ) isopenifandonlyif c 1 ( c ( * )) isopen.Thelater isjusttheunionoftheorbitsof * under ˝ . Corollary2.3.1.2. If - isoffinitetypeover Z and . = - / ˝ exists,thenboth - and . are noetherian. Proof. Sincethestructuremorphism - ! Spec Z isoffinitetypeandSpec Z isnoetherian, - isalsonoetherian.Wecheckthat . isquasi-compactandlocallynoetherian.Suppose f + 8 g 8 2 ˚ isanopencoveringof . .Thisgivesanopencovering f * 8 = c 1 ( + 8 ) g 8 2 ˚ of - ,andsince - is quasi-compact,wecanchooseafinitesubcovering f * 8 g 8 2 ˜ .Since c isopenandsurjectivewith 27 + 8 = c ( * 8 ) , f + 8 g 8 2 ˜ isthenafinitesubcoveringof f + 8 g 8 2 ˚ .Alsoforeach H 2 . ,thereisanopen affineneighborhoodSpec suchthat c 1 ( Spec )= Spec isnoetherianand = ˝ ˆ . Since c isfinite,bythetheoremofEakin-Nagata(Theorem3.7(i)in[Mat87]), isalsonoetherian, hence . islocallynoetherian. Corollary2.3.1.3. Suppose - isoffinitetypeover Z and . = - / ˝ exists.If F isacoherent sheafof O - -modules,then c F isacoherentsheafof O . -modules. Proof. Thisfollowsfromthat - , . arebothnoetherianand c isafinitemorphism(seeII.Ex. 5.5in[Har77]). PurityofBranchLocus. Suppose G 2 - mapsto H 2 . = - / ˝ underthecanonicalprojection c : - ! . .Denotethecorrespondingextensionoflocalringsby O -ŒG / O .ŒH ,maximalideals by m G / m H .Throughoutthetext,weassumethatallresiduefieldextensionsareseparable.The extensionoflocalrings O -ŒG / O .ŒH is unramified if m H O -ŒG = m G in O -ŒG .Thiscoincideswith thepreviouslydefinednotionofunramifiedextensionif O -ŒG / O .ŒH arediscretevaluationrings. Theprojection c is unramified at G if O -ŒG / O .ŒH isunramified. Let * bethelargestopensubschemeof - suchthattherestriction c j * is étale ,i.e., c isflat andunramifiedateverypointin * .Thecomplement - * isnaturallyequippedwiththeclosed subschemestructuredefinedbytheannihilatorofthesheaf 1 - / . ofrelativedifferentials,andwe callthissubscheme ' c the ramificationlocusof c .Since c isfinite,theimage c ( ' c ) isclosed, anditsuniquereducedinducedclosedsubschemestructureiscalled c the branchlocusof c .If both - and . areregular,then c isflat(Remark3.11in[Liu02]). Often,theirreduciblecomponentsof ' c (resp. c )areallofcodimension1.Thispropertyis called purityoframification (resp. branch )locus. Proposition2.3.2 (Purityofbranchlocus) . Let 5 : - ! . beamorphismoflocallynoetherian schemes.Let G 2 - andset H = 5 ( G ) .Suppose i) O -ŒG isnormal; 28 ii) O .ŒH isregular; iii) 5 isquasi-finiteat G ; iv) 38< ( O -ŒG )= 38< ( O .ŒH ) 1 ; v) forall G 0 thatspecializeto G with 38< ( O -ŒG 0 )=1 , 5 isunramifiedat G 0 . Then 5 isétaleat G . EquivariantEulerCharacteristic. Let - beaschemeoveranoetherianring with ˝ ˆ Aut ( - ) afinitegroup.An O - - ˝ - module F on - (or ˝ - equivariant sheaf)isasheafof O - -moduleshavinganactionof ˝ whichiscompatiblewiththeactionof ˝ on O - inthe followingsense:Suppose G 2 - and f 2 ˝ .Let f ( G ) betheimageof G under f .Theactionof f on O - and F giveshomomorphismsofstalks O -Œf ( G ) !O -ŒG and F f ( G ) ! F G ;bothofthese homomorphismswillalsobedenotedby f ,and f ( 0 < )= f ( 0 ) f ( < ) forall 0 2O -Œf ( G ) and < 2 F f ( G ) .Ifsuch F isaquasi-coherent(resp.coherent,locallyfree) O - -modulesheaf,then F iscalledaquasi-coherent(resp.coherent,locallyfree) O - - ˝ -module.Anexampleofalocally free O - - ˝ -moduleofrank1on - canbegivenas O - ( ˇ ) where ˇ = Í = / / isa ˝ -equivariant divisoron - ,i.e., = f ( / ) = = / forall f 2 ˝ and / 2 ˇ . If F isan O - - ˝ -moduleon - ,thenthesheafcohomologygroupsH 8 ( -Œ F ) , 8 0 , are Z [ ˝ ] -modulesinanaturalway.Supposeeachcohomologygroupisafinitelygenerated Z [ ˝ ] -module.Thisholdsif,forinstance, F isacoherent O - - ˝ -moduleon - and - isprojec- tiveoverSpec (III.Theorem5.2in[Har77]).Wede˝nethe equivariantEulercharacteristic j ( -Œ F )= j ( ˝Œ-Œ F ) tobethevirtualrepresentation j ( -Œ F ):= Õ ( 1) 8 [ H 8 ( -Œ F )] intheGrothendieckgroupK 0 ( ˝Œ ) ofallfinitelygeneratedmodulesoverthegroupring [ ˝ ] . Nowassumethat - isprojectiveoverSpec .Asseenbefore,thequotient c : - ! . then exists, . = - / ˝ .Wewillcallsuchquotienta Galoiscover .An O . [ ˝ ] - module G on . isjust 29 a O . - ˝ -moduleon . where ˝ actstriviallyon . .Inmostofourapplications, G = c F fora coherent O - - ˝ -module F .Inthiscase,since c isfiniteandthusaffine,H 8 ( -Œ F )= H 8 ( .Œc F ) forall 8 2 Z ,and c F iscoherentbyCorollary5,therefore j ( -Œ F )= j ( .Œc F ) inK 0 ( ˝Œ Z ) . Byan [ ˝ ] - moduleon . wemeanasheafof [ ˝ ] -moduleson . .Thecategoryof [ ˝ ] -modules on . hasenoughinjectives(cf.[Chi94]).Wesaythecohomologyof F hasa normalintegralbasis over ifthereexistsaboundedcomplexoffinitelygeneratedfree [ ˝ ] -modulesthatisisomorphic toR -Œ F ) inthederivedcategoryofcomplexesof [ ˝ ] -modules.Anequalitybetween j ( -Œ F ) andamultipleof [ [ ˝ ]] inK 0 ( ˝Œ ) isoftentooweaktostudytheexistenceofnormalintegral basis,thusonehopestohaveabetterapproximationintheGrothendieckgroupK 0 ( [ ˝ ]) offinitely generatedprojective [ ˝ ] -modules.Asnotedin2.2,theCartanhomomorphism K 0 ( [ ˝ ]) ! K 0 ( ˝Œ ) isnotsurjectiveingeneral,andthefoundationofourstudyisbasedoncharacterizing O - - ˝ -modules F thatadmita projectiveEulercharacteristic j % ( -Œ F ) inK 0 ( [ ˝ ]) whichwillbe introducedinthenextchapter. 30 CHAPTER3 IMPORTANTTHEOREMS Inthischapter,werecalltwoimportantresults,Chinburg'scriterionfortheexistenceof projectiveEulercharacteristicsandKöck'sclassificationofprojectivefractionalidealsinweakly ramifiedextensionofdiscretevaluationrings. 3.1ProjectiveEulerCharacteristic Let c : - ! . beaGaloiscoveroveranoetherianring withafinitegroup ˝ asin2.3. - . Spec c ˝ ˝ trivial Figure3.1:Galoiscover. LetK + ( .ŒŒ˝ ) (resp.K + ( Œ˝ ) )bethecategoryofcomplexesof [ ˝ ] -moduleson . (resp. [ ˝ ] -modules)whichareboundedbelow.Morphismsinthesecategoriesarehomotopyclassesof morphismsofcomplexes.Amorphismisaquasi-isomorphismifitinducesisomorphismsincoho- mology.The derivedcategories D + ( .ŒŒ˝ ) andD + ( Œ˝ ) arethelocalizationsofK + ( .ŒŒ˝ ) andK + ( Œ˝ ) ,respectively,withrespecttothemultiplicativesystemsofquasi-isomorphismsin thesecategories.Thereareenoughinjectivesinthecategoryofsheavesof [ ˝ ] -moduleson . (III. Proposition2.2in[Har77];also[Chi94]).Hencetheglobalsectionfunctor hasaright-derived functorR + : D + ( .Œ Z Œ˝ ) ! D + ( Z Œ˝ ) . Theorem3.1.1 (Chinburg,Theorem1.1in[Chi94]) . Suppose G 2 K + ( .ŒŒ˝ ) hasthefollowing properties: i) G isaboundedcomplexof [ ˝ ] -moduleson . , 31 ii) eachtermof G isaquasi-coherent O . [ ˝ ] -module, iii) eachstalkofeachtermof G isa ˝ -modulewhichiscohomologicallytrivialfor ˝ ,and iv) thecohomologygroupsof R + ( G ) arefinitelygenerated [ ˝ ] -modules. Then R + ( G ) isisomorphicin D + ( Œ˝ ) toaboundedcomplex % offinitelygenerated [ ˝ ] -moduleswhicharecohomologicallytrivialfor ˝ .TheEulercharacteristic j ( % )= Í ( 1) 8 [ % 8 ] 2 CT ( [ ˝ ]) dependsonlyon G ,andwillbedenoted j R + ( G ) . ByProposition2.2.7,thenaturalforgetfulhomomorphismK 0 ( [ ˝ ]) ! CT ( [ ˝ ]) isan isomorphismwhen isaDedekinddomain.Wecallthepreimageof j R + ( G ) underthe forgetfulhomomorphismthe projectiveEulercharacteristic of G denotedby j % ( .Œ G ) .Indeed, modulesofaboundedcomplex % ofprojectivefinitelygenerated [ ˝ ] -modulesquasi-isomorphic toR + ( G )= R .Œ G ) canbechosentobefreeexceptthelastterm.Theclassofthelastmodulein Cl ( [ ˝ ]) is theobstruction j % ( .Œ G ) totheexistenceofanormalintegralbasis,i.e.,toR .Œ G ) beingrepresentedbyaperfectcomplexof [ ˝ ] -modules.In[CE92],ChinburgandErezshowed thatwhen c : - ! . istameinthesensethattheorderoftheinertiasubgroupateveryclosedpoint G 2 - isrelativelyprimetothecharacteristicoftheresiduefield,thenforall ˝ -equivariantcoherent sheaves F on - ,theprojectiveEulercharacteristic j % ( -Œ F )= j % ( .Œc F ) iswell-defined. 3.2WeaklyRamifiedExtensions RecallthatwhenaGaloisextensionofnumberfields # / withGaloisgroup ˝ istame,then O # isprojectiveover O [ ˝ ] byNoether'scriterion.Anadditiontothat,in[Ull70],Ullomshowed thatwhen # / istame,allambiguousidealsin # (fractionalidealsof # thatare ˝ -modules) areindeedprojectiveover O [ ˝ ] .ThesamequestioncanbeaskedforweaklyramifiedGalois extensions. Let ! / beafiniteGaloisextensionoflocalfieldswith ˝ = Gal ( ! / ) andpositiveresidue characteristic ? ,andwrite O ! / O , m ! / m , _ / ^ forthecorrespondingextensionsofdiscrete 32 valuationrings,maximalideals,andresiduefieldsasin2.1.In[Köc04],Köckclassifiedall ambiguousidealsinthelocalfield ! thatarefreeover O [ ˝ ] . Theorem3.2.1 (Köck,Theorem1.1in[Köc04]) . Let 1 2 Z .Thenthefractionalideal m 1 ! of ! is freeover O [ ˝ ] ifandonlyif ! / isweaklyramifiedand 1 1 j ˝ 1 j . AnimportantcanonicalexampleofalocallyfreeambiguousidealinaweaklyramifiedGalois extension # / ofnumberfieldsisthe squarerootoftheinversedifferent ( # / ) consideredby Erezin[Ere91].Hereweassume [ # : ] isodd.ByProposition2.1.15,theorderofthedifferent ideal D ( # / ) ataprime p in # canbecalculatedintermsoftheordersofthehigherramification groupsas E p ( D ( # / ))= Õ 8 0 ( j ˝ p Œ8 j 1) where ˝ p isthedecompositiongroup ˝ p = f f 2 ˝ j f ( p )= p g Ł When # / isfurtherassumedtobeweaklyramifiedsothatall j ˝ p Œ 2 j =1 ,thisshowsthatthere isanideal ( # / ) in # with ( # / ) 2 = D ( # / ) 1 whichjustifiesthename.Since D ( # / ) isstableundertheactionof ˝ ,sois ( # / ) .Bythe criterionofKöck, ( # / ) isprojectiveover $ [ ˝ ] . 33 CHAPTER4 WEAKLYRAMIFIEDCOVERSOFCURVES 4.1WeaklyRamifiedCovers Let beaDedekinddomainofcharacteristic0withafiniteflatstructuremorphismSpec ! Spec Z .Let - beaproperflatregularcurveover ,i.e., - hasdimension2andallfibreshave dimension1.ByTheorem3.16, x 8.3in[Liu02], - isprojective.Thus,forafinitegroupaction ˝ ˆ Aut ( - ) ,wehaveaGaloiscover c : - ! . withthequotientscheme . = - / ˝ inthesense of2.3.Foreach G 2 - and H = c ( G ) ,wefurtherassumethattheresiduefieldextension : ( G )/ : ( H ) isseparable.Thedecompositiongroup ˝ G = f f 2 ˝ j f ( G )= G g of G actson O -ŒG andonthe stack F G forany O - - ˝ -module F on - .If O -ŒG / O .ŒH isanextensionofdiscretevaluationrings, foreachinteger 8 1 ,the 8 'thramificationgroup ˝ GŒ8 at G iswell-definedforall 8 1 .Recall thatanextension O -ŒG / O .ŒH iscalled weaklyramified if ˝ GŒ 2 istrivial. WewillsaythattheGaloiscover c : - ! . ofcurvesoverSpec is weaklyramified ifthe followingextraconditionsaremet: i) . isalsoregular, ii) thebranchlocus c of c ishorizontali.e.,eachirreduciblecomponentof c surjectsonto Spec viathestructuremorphism, iii) foreachrationalprime ? dividingtheorderof ˝ andeach p 2 Spec over ? viathestructure morphismSpec ! Spec Z ,thespecialfibre . p intersectseachirreduciblecomponentof c transversallyandatsmoothpoints,and iv) atallsuchsmoothpoints H 2 . p and G 2 - p which c ( G )= H ,theextensionofdiscrete valuationrings O - p ŒG / O . p ŒH isweaklyramified. 34 . Spec genericpoint * p . ? c Figure4.1:WeaklyramifiedGaloiscoverofcurves. Remark. OurdefinitionofweaklyramifiedGaloiscoverisstill tamelyramified relativeto c inthesenseofDefinition2.1in[Chi94]becauseofthediscretevaluationcondition:Sincethe branchlocus c isassumedtobehorizontaland - , . areregular,foreach H 2 c ofcodimension1 and G 2 - over H , O -ŒG / O .ŒH isatamelyramifiedextensionofdiscretevaluationrings.However, wedonotrequire c tohavenormalcrossingswhichisanadditionalcrucialconditionfortame coversgiveninDefinition2.5inChinburg'spaper. Ourfirstmainresultisasufficientconditionforaninvertible O - - ˝ -module F tohavea well-definedprojectiveEulercharacteristic j % ( -Œ F ) . Theorem4.1.1 (L,2020) . Let c : - ! . beaweaklyramifiedcoverofcurvesover witha finiteactionof ˝ ˆ Aut ( - ) .Suppose F = O - ( ˇ ) isaninvertiblesheaf O - - ˝ -moduleon - correspondingtoa ˝ -equivarianthorizontaldivisor ˇ .Assumethat,foreach ? dividingtheorder of ˝ and p 2 Spec over ? ,thepull-back ˇ \ - p tothespecialfibre - p isgivenbyaWeildivisor Í G 2 - p = G G of - p where = G 1 j ˝ GŒ 1 j .(If - p isnotsmoothat G ,take j ˝ GŒ 1 j tobe 1 .) Theneachstalkofthedirectimagesheaf G = c F iscohomologicallytrivialfor ˝ . Proof. Take G = c F andlet H = c ( G ) 2 . forsome G 2 - .If c istamelyramifiedat G ,thenbyTheorem2.7in[Chi94], G H iscohomologicallytrivialfor ˝ .Thusitremainstoshow cohomologicaltrivialityat H 2 c ofcodimension2,over ( ? ) 2 Spec Z where ? dividestheorder 35 of ˝ .Noticethatsince G H isaflatmoduleover Z andthustorsion-free,byProposition2.2.9,it isenoughtoshowthat,forallrationalprimes @¡ 0 , G H iscohomologicallytrivialfortheSylow @ -subgroup ˝ @ of ˝ .For @ notdividingtheorderof ˝ , ˝ @ istrivialandcohomologicaltriviality isobvious.For @ < ? dividingtheorderof ˝ , @ isaunitin O .ŒH inducinganautomorphism ^ H 8 ( ˛Œ G H ) @ ˙ ^ H 8 ( ˛Œ G H ) forallsubgroups ˛ ˝ @ .SincetheorderofafinitegroupannihilatesitsTatecohomologygroups byCorollary2.2.2.1, G H iscohomologicallytrivialfor ˝ @ .Theremainingcaseisfor ˝ ? ,the Sylow ? -group,sosuppose ˝ = ˝ ? .Wehaveashortexactsequence 0 ! G H ? ! G H ! G H / ? G H ! 0 whichinducesalongexactsequenceoftheTatecohomologygroups ! ^ H 8 ( ˛Œ G H ) ? ! ^ H 8 ( ˛Œ G H ) ! ^ H 8 ( ˛Œ G H / ? G H ) ! ^ H 8 +1 ( ˛Œ G H ) ! forallsubgroups ˛ 6 ˝ .Sinceeach ^ H 8 ( ˛Œ G H ) isannihilatedbyapowerof ? ,asintheprevious case,itisenoughtoshowthat G H / ? G H iscohomologicallytrivialfor ˝ .Thiscanbedoneby restricting G H tothespecialfibre . p .Thespecialfibres - p , . p formaweaklyramifiedcover ofcurves c p : - p ! . p withitsramificationlocus(resp.branchlocus)exactlyinducedbythe intersectionof ' c (resp. c )and - p (resp. . p )in - (resp. . )because 1 - p / . p = 1 - / . O - F ? . If G 0 2 - p mapsto G and H 0 2 . p mapsto H viabasechange,byassumption, - p and . p are - p - . p . 8 c p c 8 Figure4.2:Galoiscoverofspecialfibres. smoothat G 0 , H 0 .Takingthe p -adiccompletion,wehaveaweaklyramifiedextensionofdiscrete valuationrings ^ O - p ŒG 0 / ^ O . p ŒH 0 .Say ^ < G 0 isthemaximalidealof ^ O - p ŒG 0 .Againbytheassumption, 36 8 ^ F G 0 = ^ < 1 G 0 where 1 = = G 0 1 j ˝ G 0 Œ 1 j .ThereforebyTheorem3.2.1, 8 ^ F G 0 isafree ^ O . ? ŒH 0 [ ˝ G 0 ] -module. Denote 8 F and 8 G by F 0 , G 0 ,respectively.Since \ c p O - 0 H 0 = c p ( G 0 )= H 0 ^ O - 0 ŒG ,wehave G 0 H 0 = Ind ˝ ˝ G 0 F 0 G 0 = Z [ ˝ ] Z [ ˝ G 0 ] F 0 G 0 .Henceforallintegers 8 ,Shapiro'slemmaimpliesthat ^ H 8 ( ˝Œ Z [ ˝ ] Z [ ˝ G 0 ] F 0 G 0 ) ˙ ^ H 8 ( ˝ G 0 Œ F 0 G 0 ) Ł Thus ^ H 8 ( ˝Œ G 0 H 0 )=0 forall 8 .ByapplyingMackey'sdoublecosetformulaforallsubgroupsof ˝ , weconcludethat G 0 H 0 = G H / ? G H iscohomologicallytrivialfor ˝ . Corollary4.1.1.1. Let c : - ! . , F beasintheabovetheorem.Thenthereisawell-defined projectiveEulercharacteristic j % ( ˝Œ-Œ F ) 2 K 0 ( [ ˝ ]) whichmapstotheequivariantEuler characteristic j ( ˝Œ-Œ F ) 2 K 0 ( ˝Œ ) viatheCartanhomomorphism K 0 ( [ ˝ ]) ! K 0 ( Œ˝ ) . Proof. Since c isfiniteandthusaffine,wehaveH 8 ( -Œ F )= H 8 ( .Œc F ) forall 8 2 Z .Now G = c F and . satisfythehypothesesofTheorem3.1.1: G isacoherentsheafon . ,andsince . isprojective,Theorem5.2,III.in[Har77]impliesthateachH 8 ( .Œ G ) isafinitelygenerated [ ˝ ] -module.Moreover,Theorem4.1.1givescohomologicaltrivialityateachstalk.Therefore,by Theorem3.1.1,thereexistsaboundedcomplex % offinitelygenerated [ ˝ ] -moduleswhichare cohomologicallytrivialfor ˝ ,andthatitsEulercharacteristic j ( % )= Í ( 1) 8 [ % ] 8 inCT ( [ ˝ ]) is determinedby F andismappedtotheequivariantEulercharacteristic j ( ˝Œ.Œ G )= j ( ˝Œ-Œ F ) viatheforgetfulfunctorCT ( [ ˝ ]) ! K 0 ( Œ˝ ) . Wereproducethisconstructionherejusttohaveaself-containedproof:Let U beafiniteopen affinecoverof . .The…echcomplex ˘ ( .Œ U Œ G ) of G isisomorphictoR + ( .Œ G ) .Eachterm of ˘ ( .Œ U Œ G ) isadirectsumof [ ˝ ] -modulesoftheform G ( * ) where * istheintersectionof finitelymanyelementsof U .Since . isseparatedoverSpec , * isaffine,say * = Spec . Wefirstshowthateach G ( * ) iscohomologicallytrivial.Weshowedthateachstalk G H at H 2 Spec iscohomologicallytrivialintheprevioustheorem.Sincethelocalring O .ŒH = H isa flat -module,forallsubgroups ˛ ˆ ˝ , ^ H 8 ( ˛Œ G H )= ^ H 8 ( ˛Œ G ( * )) H =0 Ł 37 Sincethisholdsforall H 2 Spec , G ( * ) iscohomologicallytrivialfor ˝ . Since G iscoherent,thecohomologygroupsof ˘ ( .Œ U Œ G ) arefinitelygenerated [ ˝ ] -modules. Wefirstconstructacomplex % offinitelygenerated free [ ˝ ] -modules(possiblynotbounded below)togetherwithaquasi-isomorphismofcomplexes % 7! ˘ ( .Œ U Œ G ) bythefollowingusualinductiveprocedure(cf.,LemmaIII.12.3,[Har77]).Let 0 ! ˘ 0 3 0 ! ˘ 1 3 1 ! ˘ 2 ! 3 = ! ˘ = +1 ! bethe…echcomplex ˘ ( .Œ U Œ G ) .Foralarge # ,wehave ˘ = # =0 .Take % # =0 .Now supposetheinductivehypothesis:For 8¡= ,wehaveamorphismofcomplexes % : % = +1 % = +2 ˘ : ˘ = ˘ = +1 ˘ = +2 6 q = +1 6 = +1 q = +2 6 = +2 3 = suchthat,for 8¡= +1 ,theinducedhomologygroupsareisomorphic 8 ( % ) ˙ 8 ( ˘ ) andker q = +1 ! = +1 ( ˘ ) issurjective.Wewillconstruct % = bythefollowingsteps:Choose afinitesetofgeneratorsof 8 ( ˘ ) ,say f G 1 ŒŁŁŁŒ G A g from f G 1 ŒŁŁŁŒG A gˆ ker 3 = .Alsoconsider 6 1 ( im 3 = ) ˆ % = +1 .Since % = +1 isassumedtobefinitelygeneratedoverthenoetherianring , thesubmodule 6 1 ( im 3 = ) ˆ % = +1 isalsofinitelygenerated.Chooseafinitesetofgenerators H A +1 ŒŁŁŁŒH B of 6 1 ( im 3 = ) ˆ % = +1 .Thentheirimages 6 ( H 8 ) 2 im 3 = canbeliftedto G 8 2 ˘ = , 8 = A +1 ŒŁŁŁŒB . Take % = tobethefree [ ˝ ] -moduleofrank B withgenerators 4 1 ŒŁŁŁŒ4 B anddefine q = : % = ! % = +1 by f 4 1 ŒŁŁŁŒ4 A g!f 0 g 4 8 A +1 7! H 8 Ł 38 Alsodefine 6 = : % = ! ˘ = by 4 8 7! G 8 forall 8 .Thenwehaveacommutativediagram % : % = % = +1 % = +2 ˘ : ˘ = ˘ = +1 ˘ = +2 6 q = 6 = q = +1 6 = +1 q = +2 6 = +2 3 = suchthat = +1 ( % ) ! = +1 ( ˘ ) isanisomorphism,andthatker q = ! = ( ˘ ) issurjectiveas desired. Theconstructedcomplex % offinitelygeneratedfree [ ˝ ] -modulesisboundedabovebutnot necessarilybelow.However,the…echcomplex ˘ isbounded,say ˘ 8 = f 0 g for 8 0 . % 2 % 1 % 0 0 0 ˘ 0 q 2 q 1 Figure4.3: % and ˘ . Replace % 1 by % 1 / im q 2 andthelowerdimensions % 8 for 8 1 by f 0 g .Thenwestill haveaquasi-isomorphism 6 : % ! ˘ ,butthelowestterm % 1 mightnotbefreeanymore.Since 6 isaquasi-isomorphism,themappingcylinder ! of 6 isanexactboundedcomplex([Chi94]). Furthermore,sinceallofthetermsof ˘ arecohomologicallytrivial,atmostonetermof ! is notcohomologicallytrivialfor ˝ .However,theexactnessnowforcesallofthetermsof ! tobe cohomologicallytrivialfor ˝ ,fromwhichitfollowsthatthesameistrueforallofthetermsof % . ByProposition2.2.7,CT ( [ ˝ ]) isisomorphictoK 0 ( [ ˝ ]) .ThuswehavetheprojectiveEuler characteristic j % ( ˝Œ-Œ F ) 2 K 0 ( [ ˝ ]) Œ j % ( ˝Œ-Œ F ) 7! j ( % ) underthisisomorphism. 39 4.2Example:CyclicAction Let ? beanoddprimeandconsider = Z [ Z ] foraprimitive ? -throotofunity Z .Then ? = D _ ? 1 forsomeunit D where _ = Z 1 isauniformizingparameteroftheuniqueprime p of above ? .Let - = P 1 = Proj ( [ - 0 Œ- 1 ]) and ˝ beacyclicgroupoforder ? actingon - generatedby f ( - 0 )= Z- 0 + - 1 Œf ( - 1 )= - 1 Ł Since ˝ isafinitegroup,thequotientscheme - / ˝ oftheprojectivespaceexists.Since - is normal, - / ˝ isalsonormal.Considerthefinitemorphism c : - ! . = P 1 ofdegree ? givenby c ([ - 0 : - 1 ])= ( _- 0 + - 1 ) ? - ? 1 _ ? : - ? 1 Ł Thisisinvariantundertheactionof ˝ ,thereforeitfactorsthroughthecanonicalprojection - = P 1 - / ˝ . = P 1 c q bytheuniversalpropertyofthequotientscheme - / ˝ .Sincethecanonicalprojectionisalsoa finitemorphismofdegree ? , q isafinitebirationalmorphismofintegralschemeswhere - / ˝ is normal.Thisimpliesthat q isthenormalizationof . whichisanisomorphismas . isalready normal. Theramificationlocus ' c consistsoftwoirreduciblecomponentswhicharethetwodivisors givenby ( _- 0 + - 1 ) and ( - 1 ) .For ( _- 0 + - 1 )=( f ( - 0 ) - 0 ) , f ( _- 0 + - 1 )= _ ( Z- 0 + - 1 )+ - 1 = Z ( _- 0 + - 1 ) Ł Thebranchlocus c isasintheFigure4.2 40 1 =[1:0] [1: _ ? ] Spec ( _ ) . Q . p c Figure4.4:ThecollapseintheexampledescribesthewildramificationfromKummerto Artin-Schreierextension,see[SOS89]formore. Wefirstseethat c isweaklyramified:At p =( _ ) ,thecoverofthespecialfibres - p ! . p isramifiedonlyat ( - 1 ) as _ 0 p .Considertheaffinepatchwhere - 0 =1 ,writing - 1 / - 0 = G 1 .Atthelocalring O - p ŒG 1 ,wehave f ( G 1 ) G 1 = G 1 G 1 +1 G 1 = G 2 1 G 1 +1 . 0 ( G 1 ) 3 Ł Thereforethesecondramificationgroupistrivialatallpointsofallfibres. Considerthefollowing ˝ -equivariantdivisor ˇ =(1 ? ) ( - 1 )+(1 ? ) ( _- 0 + - 1 ) supportedontheramificationlocus.Since ? isassumedtobeodd, ˇ /2 isalsoawell-defined divisoron - .Atapoint G 2 - over ( _ ) 2 Spec ,thetwocomponentsof ' c mergeto ( - 1 ) ,and ˇ /2 restrictsto ˇ /2 j - p = 1 ( - 1 ) j - p Ł Thisshowsthattheinvertible O - - ˝ -module F := O - ( ˇ /2) satisfiestherestrictionhypothesis giveninTheorem4.1.1.Asarguedintheproofofthetheorem, G = c F hascohomologically trivialstalks. Let * 0 = Spec [ G 0 ] , * 1 = Spec [ G 1 ] formanopenaffinecover U of P 1 where G 0 = - 0 / - 1 , 41 G 1 = - 1 / - 0 ,andconsiderthe…echcomplex ˘ of F givenby U : F ( * 0 )=( _G 0 +1) (1 ? )/2 [ G 0 ] Œ F ( * 1 )=( _ + G 1 ) (1 ? )/2 G (1 ? )/2 1 [ G 1 ] Œ F ( * 0 \ * 1 )=( _ + G 1 ) (1 ? )/2 G (1 ? )/2 1 [ G 1 Œ 1/ G 1 ] Ł AdirectcomputationshowsthatH 0 ( -Œ F ) isafree -moduleofrank ? generatedby 1 ( _G 0 +1) ( ? 1)/2 G 8 0 for 8 =0 ŒŁŁŁŒ? 1 .ThefirstcohomologygroupH 1 ( -Œ F ) vanishesas F ( * 0 \ * 1 ) ,asan -module,isgeneratedby ( _ + G 1 ) (1 ? )/2 G (1 ? )/2 1 G = 1 forallintegers = ,and ( _ + G 1 ) (1 ? )/2 G (1 ? )/2 1 G = 1 = 8 > > > >< > > > > : ( _ + G 1 ) (1 ? )/2 G (1 ? )/2 1 G = 1 2 F ( * 1 ) Œ if = 0 ( _G 0 +1) (1 ? )/2 G ? 1 = 0 2 F ( * 0 ) Œ otherwise. ThustheequivariantEulercharacteristic j ( ˝Œ-Œ F ) isjust [ H 0 ( -Œ F )] inK 0 ( ˝Œ ) .Since H 8 ( -Œ F ) ˙ H 8 ( .Œ G ) forall 8 , j ( ˝Œ-Œ F )=[ H 0 ( .Œ G )] 2 K 0 ( ˝Œ ) Ł TofinditsprojectiveEulercharacteristicinK 0 ( [ ˝ ]) ,firsttakeanopencover + = f + 0 Œ+ 1 g of . = P 1 whereeach + 8 isstableundertheactionof ˝ .Forexample,take + 0 = Spec [ G 0 ] and + 1 = Spec [ G 1 Œf ( G 1 ) ŒŁŁŁŒf ? 1 ( G 1 )] .Theintersection + 0 \ + 1 isalsoaffine,andletSpec denoteanyof + 0 , + 1 ,or + 0 \ + 1 .Thecorresponding…echcomplexusedincomputingthesheaf cohomologyof G isthen ˘ 0 = G ( + 0 ) G ( + 1 ) ˘ 1 = G ( + 0 \ + 1 ) whicharefinitelygenerated [ ˝ ] -modules.Wehaveashortexactsequenceof [ ˝ ] -modules 0 ! H 0 ( .Œ G ) ! ˘ 0 ! ˘ 1 ! 0 Ł 42 Ateach H 2 Spec ,wesawthat G H iscohomologicallytrivial.Since H isaflat -module, theTatecohomologygroupsovertensorproductsgive ^ H 8 ( ˝Œ G H )= ^ H 8 ( ˝Œ G ( Spec )) H =0 forall 8 .Sincethisistrueforall H 2 Spec ,weconcludethat G ( Spec ) iscohomologically trivialfor ˝ .Thusthetermsof ˘ arealsocohomologicallytrivialfor ˝ ,andsoisH 0 ( .Œ G ) . SinceH 0 ( .Œ G ) isalsoafinitelygeneratedfree -module,weconcludethattheprojectiveEuler characteristic j % ( ˝Œ-Œ F ) 2 K 0 ( [ ˝ ]) isgivenbytheclass [ H 0 ( -Œ F )] intheGrothendieck groupK 0 ( [ ˝ ]) . SincetherankofH 0 ( -Œ F ) over is ? ,itwillbeinterestingtoseeiftheclassof j % ( ˝Œ-Œ F ) , theobstructiontotheexistenceofanormalintegralbasisofthecohomologyof F ,istrivialin Cl ( Z [ ˝ ]) . Proposition4.2.1 (L,2020) . H 0 ( -Œ F ) isafree [ ˝ ] -module,so j % ( -Œ F )=0 . Proof. Forsimplicity,denote G 0 by G andreplaceH 0 ( -Œ F ) bythe -modulegeneratedby f 1 ŒGŒŁŁŁŒG ? 1 g whichis [ ˝ ] -isomorphictoH 0 ( -Œ F ) .WeclaimthatH 0 ( -Œ F ) hasan [ ˝ ] -basis givenby U = 1 ? ? 1 Õ 8 =0 ( _G +1) 8 Ł Toseethat U belongstoH 0 ( -Œ F ) ,write ? 1 Õ 8 =0 ( _G +1) 8 = ? 1 Õ 8 =0 8 Õ 9 =0 8 9 _ 9 G 9 = ? 1 Õ 9 =0 ? 1 Õ 8 = 9 8 9 _ 9 G 9 = ? 1 Õ 9 =0 ? 9 +1 _ 9 G 9 wherethelastequalityisfrom ? 1 Õ 8 = 9 8 9 = ? 9 +1 Ł Sincethecoefficient _ ? 1 ofthelasttermisalsodivisibleby ? ,thewholesumisdivisibleby ? , hence U 2 H 0 ( -Œ F ) . Weobservethechangeof -basesfrom f 1 ŒGŒŁŁŁŒG ? 1 g to f UŒf ( U ) ŒŁŁŁŒf ? 1 ( U ) g canbe brokenintothreestepsof -lineartransformationsbetween -bases, beingthefractionfieldof 43 :firstfrom f 1 ŒGŒŁŁŁŒG ? 1 g to f 1 Œ_G +1 ŒŁŁŁŒ ( _G +1) ? 1 g ,thento f ?UŒf ( ?U ) ŒŁŁŁŒf ? 1 ( ?U ) g , thenfinallyto f UŒf ( U ) ŒŁŁŁŒf ? 1 ( U ) g .Since ( _G +1) 9 = 9 Õ 8 =0 9 8 _ 8 G 8 Œ thecorrespondingmatrixtothefirstlineartransformationistriangularandthedeterminantis Î ? 1 9 =0 _ 9 = _ ? ( ? 1)/2 .ThematrixforthesecondlineartransformationistheVandermonde matrix " = 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 11 1 1 1 ZZ 2 Z ? 1 1 Z 2 ( Z 2 ) 2 ( Z 2 ) ? 1 Ł Ł Ł Ł Ł Ł Ł Ł Ł Ł Ł Ł Ł Ł Ł 1 Z ? 1 ( Z 2 ) ? 1 ( Z ? 1 ) ? 1 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 wheredeterminantisgivenby Î 1 89 ? ( Z 9 1 Z 8 1 )= D 0 _ ? ( ? 1)/2 forsomeunit D 0 2 . Thelastlineartransformationhasdeterminant ? ? =( D _ ? 1 ) ? .Thecompositionofthesethree transformationshasaunitdeterminantasall _ 'scancelout. Wewillseeinthenextchapterthatthechoiceof ˇ and ˇ /2 intheexampleissomewhatmore canonicalthantheotherpossiblechoicesofinvertiblesheavessatisfyingtherestrictionhypothesis ofTheorem4.1.1. 44 CHAPTER5 SQUAREROOTOFTHEINVERSEDIFFERENT Let c : - ! . = - / ˝ beweaklyramifiedasinthepreviouschapterwith ˝ ofoddorder .In thischapterwediscussthecanonicalexistenceofaninvertiblesheafon - verifyingtheconditions ofTheorem4.1.1. Lemma5.0.1. Let bearingflatover Z , ˚ , ˜ idealsof .Supposethat / ˚ , / ˜ are Z -torsion free.If ˚ Q = ˜ Q ˆ Q ,then ˚ = ˜ . Proof. Weshowthatoneiscontainedintheother,say ˚ ˜ .Let G 2 ˚ .Since ˚ Q = ˜ Q , G 1= Õ 8 0 9 = 9 2 ˜ Q Ł Byclearingthedenominatorsof = 9 ,thereisaninteger 0 with 0G 2 ˜ .Since 0G 0 in / ˜ and / ˜ istorsion-free, G 2 ˜ . Theorem5.0.2 (L,2020) . Thereexistsaninvertiblesheaf F on - suchthat F 2 isthetor- sion-freepartofthequotientsheaf O - / Ann 1 - / . ) .Here, Ann 1 - / . ) istheannihilatorofthe sheafofrelativedi˙erentials 1 - / . .The O . [ ˝ ] -module c F hascohomologicallytrivialstalks, andsotheprojectiveEulercharacteristic j % ( -Œ F ) iswell-de˝ned. Remark. Thisgeneralizes thesquarerootoftheinversedifferent discussedattheendof3.2, see[Ere91].When # / isanodddegreeGaloisextensionofnumberfieldswithGaloisgroup ˝ ,thenbyaformulaofHilbert(Proposition4onp.64in[Ser79]),theorderofthedifferent ideal D ( # / ) atevery p 2 Spec O # isalwayseven,hencethereexistsanidealwhosesquareis D ( # / ) 1 . Proof. ConsidertheannihilatoridealsheafAnn 1 - / . ) O - ofthesheafofrelativedi˙er- entials 1 - / . .Thenthe Z -torsion-freepartof O - / Ann 1 - / . ) isagainaquotientsheafof O - whichdeterminesaclosedsubscheme / 1 of - flatoverSpec Z .Wetake I tobetheidealsheaf 45 of / 1 .Ontheotherhand,weassumethattherami˝cationlocus ' c ishorizontal,solet G be thegenericpointofanirreduciblecomponentof ' c whichisofcodimension1.Let H = c ( G ) . Sinceweassumethatallresiduefieldextensionsareseparable,byProposition12onp.57in [Ser79],thediscretevaluationring O -ŒG canbegivenby O .ŒH [ ) ]/( 5 ( ) )) where 5 ismonic.Then Ann 1 - / .ŒG )=( 5 0 ( C )) where C istheimageof ) in O -ŒG .Thisisthesameasthedi˙erentideal of O -ŒG over O .ŒH byProposition14onp.59in[Ser79],andsincetherami˝cationistameatthe horizontal G ,byProposition13onp.58in[Ser79],itsorderis 4 G 1 where 4 G istherami˝cation indexat G .Using 4 G foreachirreduciblecomponent f G g2 ' c ,considerthedivisor ˇ = Õ G 2 ' c ofcodim1 (1 4 G ) f G g Ł Let / 2 betheclosedsubschemeof - withthestructuresheaf O / 2 determinedby O - / O - ( ˇ ) . Byconstruction,Supp ( O - / I )= Supp ( O - / O - ( ˇ ))= ' c asaset.Overthegenericfibre, O - Q ( ˇ )= I Q as I Q = Ann 1 - Q / . Q ) iswithout Z -torsionandwedefined ˇ bytheclosureof thedivisorcorrespondingtoAnn 1 - Q / . Q ) O - Q .Thustheflatclosedsubschemes / 1 , / 2 are identicalbyLemma5.0.1,andweconcludethat I = O - ( ˇ ) . Notethatsincetheorderof ˝ isodd, 1 4 G in ˇ isalwayseven,thus ˇ /2 isawell-defined divisoron - .Since ˇ is ˝ -equivariant,wecantake F = O - ( ˇ /2) tobeour O - - ˝ -module. Wecheckthat F satisfiestheintersectionhypothesisofTheorem4.1.1.Let G beapointin theintersectionof ˇ /2 and - p thespecialfibrewhere p 2 Spec isover ? and ? dividesthe orderof ˝ .Byassumption, - p issmoothat G ,andwehaveanextensionofdiscretevaluationrings O - p ŒG / O . p ŒH where H istheimageof G underthecanonicalprojectionofspecialfibres - p / . p . Wecancomputethevaluationof ˇ /2 at G byfirstcomputingthevaluationofAnn 1 - p / . p ) G . Thiscanbecomputedusingthehigherramificationgroups ˝ GŒ8 andaformulaofHilberton thevaluationofthedifferentidealoflocalextensions:Sincethevaluationsremainunchanged aftertakingcompletion,assume O - p ŒG / O . p ŒH isaweaklyramifiedextensionofcompletediscrete valuationringsover F ? 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