As this is best illustrated by example we begin by studying descent of morphisms Consider a scheme and an open covering of Now for any scheme to give a morphism it is equivalent to give morphisms which agree on intersections which is to say that ID: 87162
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Proof.Since isinjective,weconsiderAasasubsetofB.Weproceedinthreestages:1.Supposethatthereexistsasectiong:B!AsuchthatgjA=IdA.ThenwecanwriteB=AIwhereI=kergisaB-module.ThenB AB=(A AA)(A AI)(I AA)(I AI)andfori2Iwehaved0(i)=i 1�1 i.Now,sinced0allofAinitskernel,itfollowsthatA=kerd0,asdesired.Alternativeproof:Considerthemaph:=g id:B AB!B.Thend0(b)=0impliesthat0=hd0(b)=b�g(b),whichshowsb=g(b)2A.2.SupposethatA!Cisafaithfully atextension.Then(B AB) AC=(B AC) C(B AC):Thus,ifwetensor(*)withCoverA,weobtain0// C // B ACd0// (B AC) C(B AC):SinceCisfaithfully at,itsucestoprovethatthislattersequenceisexact.Thus,wecanreplacethepair(A;B)with(C;B AC).3.Finally,consideranarbitraryA!B.NowweapplythepreviousreductionwithC=B.ThenwegetthepairorringsB,!B AB;b!b 1;andwecanconstructasectionbysettingg(b b0)=bb0.Butthisputsusincase(1),whichcompletestheproof. Infact,usingthesameproofmethod,onecanprovethefollowingfairlyvastgeneralizationofthepreviouslemma:Lemma1.3.Let :A!Bbeafaithfuly atmapofrings,andletMbeanyA�module.Then0// M// M ABd0// M AB 2// dr�2// M AB r2 Z0beananeopenneighborhoodofz,letY0=h�1(Z0)andconsider (h�1(Z0))X.Thisisopen,as isanopenmap,soletX0beananeopenneighborhoodcontainedwithinit.NowaswecanworklocallyonX,replaceX;Y;ZbyX0;Y0;Z0.ThuswecanreducetothecasewhereX;Zareane.3.WriteY=[i2IViwhereViisaneopen.SinceXisaneandquasi-compact,thereisanitesubsetKIsuchthatX=[k2K (Vk):LetJbeanynitesubsetofIcontainingK,andwriteYJ:=tj2JVj.Thisisane,sowecanwriteYJ=SpecB.Likewise,letX=SpecAandZ=SpecC.Now,thesequenceHom(X;Z)!Hom(YJ;Z)Hom(YJXYJ;Z)canberewrittenhom(C;A)!hom(C;B)hom(C;B AB):Sincethehomfunctorisalwaysleft-exact,thatthissequenceisexactfollowsfromlemma1.2.Thus,thereexistsauniquemapg:X!Zinducingh:YJ!Z.SinceJwasanarbitrarynitesubsetandgisunique,itfollowsthatg =h,asdesired. Excercise1.4.LetX;Zbeschemes.ProvethatFZ(U):=hom(U;Z)isasheafonXfl.2DescentofModulesandAneSchemesLetA!Bbeafaithfully atmorphismofrings,andletMbeanA-module.ThenM0:=M ABisaB-module.Moreover,wecandenetwoB ABmodules,givenbyM0 ABandB AM0.Notethatwhiletheunderlyingsetsofthesetwomodulesareclearlythesame,theactionsofB ABareverydi erent.Nontheless,inthiscasewehaveanisomorphism M:B AM0=M0 ABgivenby M(b (m b0))=(m b) b0:4 where (m)=m 1and(m)= (1 m).Thecocycleconditionimpliesthatthediagramcommuteswitheitherthetoporbottomarrows,hencetheleftdownwardmap identiestheirkernels.SinceBisfaithfully atoverA,theupperkernelisB Mandbylemma1.3thebottomkernelisM0.Thisprovestheclaim. Wearenowreadytoproveadescenttheoremforschemes:Theorem2.2.LetY!Xbefaithfully at.SupposeZ0!Yisananescheme,and :YXZ0!Z0XYisanisomorphismofYXYschemes,suchthat 1;3and 2;3 1;2inducethesameisomorphismYXYXZ0!Z0XYXY:Thenuptoisomorphism,thereexistsauniquepair(Z; )consistingofananeX-schemeZ!Xandanisomorphism :ZXY!YofY-schemes,suchthatundertheidentication ,themap onYXZXYbecomesthenaturalmap(y1;z;y2)!(z;y1;y2).Proof.Bytheuniquenessclaim,wemayworklocallyonXandY,wlogX=SpecA;Y=SpecB;Z0=SpecC.Butthenthetheoremfollowsimmediatelyfromlemma2.1ifweletM0betheB-moduleC. Excercise2.3.Let :Z!Xbeamorphism,and :Y!Xbeafaithfully atmorphism,and Y:ZXY!Xthebasechangeof along .ForeachofthefollowingpropertiesP,provethat isamorphismoftypePi Yis:OpenImmersionUnramiedFiniteTypeFiniteFlatEtaleFaithfullyFlatClosedImmersion6 ofFto(YXY)EcanbethusidentiedwitheitherFZ0XYorFYXZ0,sowegetanisomorphismofYXY-scheme :YXZ0!Z0XY.Moreover,restrictingtoYXYXYshowsthat 1;3= 2;3 1;2.Thusbytheorem2.2wegetaschemeZ!Xandanidentication :ZXY!Z0whichmakes intothenatural` iptheco-ordinates'map.Nowtheorem1.1togetherwiththesheafconditionforFgivesusanidenticationF=FZ.Finally,thatZisitselfetalefollowsfromexcercise2.3. Asanimmediatecorollarytotheorem3.2togetherwiththeequivalenceofcategoriesbetweenniteetalecoversand1(X)setsproveninlecture6,wehavethefollowingCorollary3.3.Let xbeageometricpointofX.Thecategoryoflocallyconstantsheavesofsets(resp.abeliangroups)isnaturallyisomorphictothecategoryof1(X;x)-sets(resp.modules).8