Transcription of Patching and Galois theory - Penn Math
1 Patchingand GaloistheoryDavidHarbater Mathematics,University of PennsylvaniaAbstract:Galoistheoryover (x) is well-understood asa consequenceof Riemann sExistenceTheorem,which classifiesthealgebraicbranchedcoversof of thattheoremusesanalyticandtopologicalmet hods,includingtheabilitytoconstructcover slocallyandtopatch studytheGaloisextensionsofk(x) forotherfieldsk, onewouldlike to have ananalogof Riemann sExistenceTheoremforcurves overk. Such a resultremainsoutof reach, butpartialresultsin thisdirectioncanbe provenusingpatchingmethodsthatareanalogo ustocomplexpatching,andwhich applyin method is formalpatching,in which formalcompletionsof schemesplay theroleof is rigidpatching,in which yieldtherealizationof arbitraryfinitegroupsas Galoisgroupsoverk(x) forvariousclassesof fieldsk, as well as such patchingmethods andtheirrelationshipsto theclassicalapproach over , andshowshow thesemethods provideresultsaboutGaloisgroupsandfundam entalgroups.
2 Supportedin partby NSFG rants DMS9970481andDMS0200045.( , ,2002.)2000 MathematicsSubject Classification. Primary14H30,12F12,14D15;Secondary13B05, 13J05, andphrases: fundamentalgroup,Galoiscovers, Patching ,f ormalscheme,rigidanalyticspace,affinecur ves,deformations, : thismanuscriptSection2: : : : TowardRiemann Shafarevich ,embeddingproblems, sConjectureandembeddingproblemsReference s2 Section1: IntroductionThismanuscriptdiscussespatch ingmethodsandtheirusein thestudyof a particularfocusonRiemann sExistenceThe-oremandtheinverseGaloispro blem,andtheirgeneralizations(bothknownan dconjec-tured).
3 Thisfirstsectionprovidesanintroduction,b eginningwitha briefoverviewofthetopicin providedin thepaper,brieflyindicatingthecontent of each algebraicin its origins,arisingfromthestudyof polynomialequationsandtheirsolvability. Butit hasalwayshadintimateconnectionsto geometry. Thisisevidenced,forexample,whenonespeaks of an icosahedralGaloisextension meaninga fieldextensionwhoseGaloisgroupisA5, thesymmetrygroupof progressin Galoistheoryreliesonconnectionsto geometry, particularlyontheparallelbetweenGaloisgr oupsandthetheoryof coveringspacesandfundamentalgroupsin topology.
4 Thisparallelis morethanan analogy, withthegroup-theoreticandtopologicalappr oachesbeingbrought togetherin thecontextof algebraicgeometry. Therealizationof allfinitegroupsas Galoisgroupsover (x) is anearlyexampleof years,thisapproach hasdrawnheavilyonthenotionofpatching, cut-and-paste constructionsthatbuildcoverslocallyandth encombinethemto formaglobalcover performedonlyforspacesdefinedover , in ordertostudyGaloisgroupsover fieldslike (x).Butby carryingcomplexanalyticmethodsover toothersettings mostnotablyviaformalandrigidgeometry resultsin Galoistheoryhave now beenproven fora broadarray of ringsandfieldsby meansof providesan overviewof thisapproach to Galoistheoryviapatching, key themein bothcontextsisRiemann s ExistenceTheorem.
5 Inthecomplexcase,thatresultprovidesa classificationof thefiniteGaloisextensionsof thefield (x) andmoregenerallyof thefunctionfieldKof any RiemannsurfaceX. (InthecaseK= (x),Xis theRiemannsphere, 1 ). Thisclassificationrelieson thecorrespondencebetweenthesefieldextens ionsandthebranchedcoversofX, andontheclassificationof thebranchedcoversofXwithgivenbranch locusB. Thiscorrespondencebetweenfieldextensions andcoversin turnproceedsby provingtheequivalenceof coversin thealgebraic,analytic,andtopologicalsens es,andthenrelyingontopologyto classifythecoveringspacesof thecomplement of a finitesetB X.
6 In demonstratingthisequivalence,oneregardsb ranchedcoversas beinggivenlocallyover discs,wherethecover breaksupinto a unionof cycliccomponents,andwhere3agreement ontheoverlapsis givenin orderto definethecover globally. Using complexpatching (specifically, Serre sresultGAGA),such ananalyticcover in factarisesfromacover of complexalgebraiccurves,given by proves thateveryfinitegroupis a Galoisgroupover (x)andmoregenerallyoverKas above, andit providesthestructureof theabsoluteGaloisgroupof ,if onefixesa finitesetof pointsB, theapproach showswhichfinitegroupsareGaloisgroupsof coverswiththatbranch locus,andhow thosegroupsfittogetheras quotients of thefundamentalgroupofX thisapproach madeit desirableto carryit over to othersettings,inorderto studytheGaloistheoryof otherfieldsK (x) wherekis a fieldotherthan , or even arithmeticfieldsK.
7 In orderto dothis,oneneedsto carryover thenotionof , ifKis thefunctionfieldof a schemeX( SpecR, whereRis anintegraldomainwhosefractionfieldisK), thenonewouldlike to constructcoversofXlocally, withagreement ontheoverlaps,andthenbe ableto asserttheexistenceof aglobalcover is thatoneneedsanappropriatetopologyonX. Ofcourse,thereis theZariskitopology, butthatis too ,ifUis a Zariskiopensubsetof anirreducibleschemeX, thengivinga branchedcoverV Uis alreadytantamount to givinga cover over allofX, sinceX Uis justa closedsubsetof lower dimension(andonecantake thenormalizationofXinV).
8 Instead,oneneedsa finernotion,which behaves morelike thecomplexmetrictopologyin theclassicalsetting,andwhereonecanspeako f theringof holomorphicfunctions onany opensetin thismanuscript,afterdiscussingtheclassic alformof Riemann sExistenceTheoremviaGAGA forcomplexcurves,we present two refinements of theZariskitopologythatallow patchingconstructionsto take placein many (butnotall) to each other,buttheydevelopedseparately. Each reliesonananalogof GAGA,whoseproof parallelstheproof of Serre realizefinitegroupsas Galoisgroupsover variousfunctionfields,andto show how thesegroupsfit togetherinthetower of allextensionsof thefield(correspondingto informationaboutthestructureof theabsoluteGaloisgroup or of a fundamentalgroup,if thebranch locusis fixed).
9 Underlyingthisentireapproach is theability to passback andforthbetweenalgebraandgeometry. Thisability is basedon therelationshipbetweenfieldextensionsand covers,withGaloisgroupsof fieldextensionscorrespondingto groupsof deck transformationsofcovers,andwithabsoluteG aloisgroupsof fieldsplayinga roleanalogousto fundamentalgroupsof reviewedin ,wherebasictermi-nologyis alsointroduced.(Readerswhoarefamiliarwit hthismaterialmay wishto skip ) thestructureof thispaper as a wishto thankClausLehr,FlorianPop,RachelPries,Ka teStevenson,JakobStix,andOliver Watsonfortheircomments andsuggestionsonearlierversionsof wouldalsolike to , Galoistheorystudiesfieldextensionsby meansof symmetrygroups( ).
10 Coveringspacescanalsobe studiedusingsymmetrygroups( deck transformations).In fact,thetwo thealgebraicsituation,thebasicobjectsof studyarefieldextensionsL K. Tosuch anextension,oneassociatesitssymmetrygrou p, (L/K),consistingof automorphismsofLthatfixall theelements ofK. IfLis a finiteextensionofKof degree[L:K] =n, thentheorderof theGaloisgroupis at mostn; andtheextensionisGaloisif theorderis exactlyn, theextensionis as symmetricas possible.(Forfiniteextensions,thisis equivalent to theusualdefinitionin termsof beingnormalandseparable.)
