HISTORY OF HOMOLOGICAL ALGEBRA

HISTORYOFHOMOLOGICALALGEBRACharlesA.WeibelHomologicalalgebrahaditsoriginsinthe19thcentury,viatheworkofRiemann(1857)andBetti(1871)on\homologynumbers,andtherigorousdevelopmentofthenotionofhomologynumbersbyPoincarein1895.A1925observationofEmmyNoether[N25]shiftedtheattentiontothe\homologygroupsofaspace,andalgebraictechniquesweredevelopedforcomputationalpurposesinthe1930’s.Yethomologyremainedapartoftherealmoftopologyuntilabout1945.Duringtheperiod1940-1955,thesetopologically-motivatedtechniquesforcom-putinghomologywereappliedtodeneandexplorethehomologyandcohomologyofseveralalgebraicsystems:TorandExtforabeliangroups,homologyandcoho-mologyofgroupsandLiealgebras,andthecohomologyofassociativealgebras.Inaddition,Lerayintroducedsheaves,sheafcohomologyandspectralsequences.AtthispointCartanandEilenberg’sbook[CE]crystallizedandredirectedtheeldcompletely.Theirsystematicuseofderivedfunctors,denedviaprojectiveandinjectiveresolutionsofmodules,unitedallthepreviouslydisparatehomologytheories.Itwasatruerevolutioninmathematics,andassuchitwasalsoanewbeginning.Thesearchforageneralsettingforderivedfunctorsledtothenotionofabeliancategories,andthesearchfornontrivialexamplesofprojectivemodulesledtotheriseofalgebraicK-theory.Homologicalalgebrawasheretostay.SeveralneweldsofstudygrewoutoftheCartan-EilenbergRevolution.Theimportanceofregularlocalringsinalgebragrewoutofresultsobtainedbyho-mologicalmethodsinthelate1950’s.ThestudyofinjectiveresolutionsledtoGrothendieck’stheoryofsheafcohomology,thediscoveryofGorensteinringsandLocalDualityinbothringtheoryandalgebraicgeometry.Inturn,cohomologicalmethodsplayedakeyroleinGrothendieck’srewritingofthefoundationsofalge-braicgeometry,includingthedevelopmentofderivedcategories.NumbertheorywasinfusedwithnewresultsfromGaloiscohomology,whichinturnledtothedevelopmentofetalecohomologyandtheeventualsolutionoftheWeilConjecturesbyDeligne.Simplicialmethodswereintroducedinthe1950’sbyKan,DoldandPuppe.Theyledtotheriseofhomotopicalalgebraandnonabelianderivedfunctorsinthe1960’s.Amongitsmanyapplications,perhapsAndre-Quillenhomologyforcom-mutativeringsandhigheralgebraicK-theoryarethemostnoteworthy.SimplicialmethodsalsoplayedamorerecentroleinthedevelopmentofHochschildhomology,topologicalHochschildhomologyandcyclichomology.Thiscompletesaquickoverviewofthehistoryweshalldiscussinthisarticle.Nowletusturntothebeginningsofthesubject.TypesetbyAMS-TEX12CHARLESA.WEIBELBettinumbers,TorsionCoefficientsandtheriseofHomologyHomologicalalgebrainthe19thcenturylargelyconsistedofagradualeorttodenethe\Bettinumbersofa(piecewiselinear)manifold.BeginningwithRie-mann’snotionofgenus,weseethegradualdevelopmentofnumericalinvariantsbyRiemann,BettiandPoincare:theBettinumbersandTorsioncoecientsofatopologicalspace.Indeed,thesubjectdidnotreallymovebeyondthesenumeri-calinvariantsuntilabout1930.Anditwasnotconcernedwithanythingexceptinvariantsoftopologicalspacesuntilabout1945.RiemannandBetti.TherststepwastakenbyRiemann(1826{1866)inhisgreat1857work\TheoriederAbel’schenFunktionen[Riem,VI].LetCbeasystemofoneormoreclosedcurvesCjonasurfaceS,andconsiderthecontourintegralRCXdx+Ydyofanexactdierentialform.RiemannremarkedthatthisintegralvanishedifCformedthecompleteboundaryofaregioninS(Stokes’Theorem),andthisledhimtoadiscussionof\connectednessnumbers.RiemanndenedStobe(n+1)-foldconnectedifthereexistsafamilyCofnclosedcurvesCjonSsuchthatnosubsetofCformsthecompleteboundaryofapartofS,andCismaximalwiththisproperty.Forexample,Sis\simplyconnected(inthemodernsense)ifitis1-foldconnected.Asweshallsee,theconnectnessnumberofSisthehomologyinvariant1+dimH1(S;Z=2).RiemannshowedthattheconnectednessnumberofSwasindependentofthechoiceofmaximalfamilyC.Thekeytohisassertionisthefollowingresult,whichisoftencalled\Riemann’sLemma[Riem,p.85]:SupposethatA,BandCarethreefamiliesofcurvesonSsuchthatAandBformthecompleteboundaryofoneregionofS,andAandCformthecompleteboundaryofasecondregionofS.ThenBandCtogethermustalsoformtheboundaryofathirdregion,obtainedasthesymmetricdierenceoftheothertworegions(obtainedbyaddingtheregionstogether,andthensubtractinganypartwheretheyoverlap).IfwewriteC0toindicatethatCisaboundaryofaregionthenRiemann’sLemmasaysthatifA+B0andA+C0thenB+C0.This,inmodernterms,isthedenitionofadditioninmod2homology!Indeed,theCjinamaximalsystemformabasisofthesingularhomologygroupH1(S;Z=2).Riemannwassomewhatvagueaboutwhathemeantby\closedcurveand\sur-face,butwemustrememberthatthispaperwaswrittenbeforeMobiusdiscoveredthe\Mobiussurface(1858)orPeanostudiedpathologicalcurves(1890).ThereisanotherambiguityinthisLemma,pointedoutbyTonelliin1875:everycurveCjmustbeusedexactlyonce.Riemannalsoconsideredtheeectofmakingcuts(Querschnitte)inS.BymakingeachcutqjtransversetoacurveCj(see[Riem,p.89]),heshowedthatthenumberofcutsneededtomakeSsimplyconnectedequalstheconnectivitynumber.ForacompactRiemannsurface,heshows[Riem,p.97]thatoneneedsanevennumber2pofcuts.Inmodernlanguage,pisthegenusofS,andtheinteractionbetweenthecurvesCjandcutsqjformsthegermofPoincareDualityforH1(S;Z=2).Riemannhadpoorhealth