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1、LECTURESONETALECOHOMOLOGYJ.S.MILNEAbstract.ThesearethenotesforacoursetaughtattheUniversityofMichi-ganinW89asMath732andinW98asMath776.Theyareavailableatwww.math.lsa.umich.edu/∼jmilne/Pleasesendcommentsandcorrectionstomeatjmilne@umich.edu.v2.01(August9,1998).Firstversionon
2、theweb.ContentsPartI:BasicTheoryNotationsandConventions;References(Homologicalalgebra,Sheaftheory,Algebraicgeometry,Etalecohomology);Comment.1.Introduction.............................................................1AlgebraicTopology1;Briefreviewofsheafcohomology2;Thein
3、adequacyoftheZariskitopology(Thecohomologygroupsarezero,theinversemap-pingtheoremfails,Fibrebundlesaren’tlocallytrivial)4;Etalecohomology5;Comparisonwiththecomplextopology6;Applicationsof´etalecohomol-ogy,(Classicalalgebraicgeometry,Representationtheoryoffinitegroups)7.2.
4、EtaleMorphisms........................................................9Etalemorphismsofnonsingularalgebraicvarieties9;Etalemorphismsofarbitraryvarieties(Etalemorphismsofvarietiesoverarbitraryfields)11;Etalemorphismsofschemes(Flatmorphisms,Unramifiedmorphisms,Etalemorphisms
5、)12;Examples(TheJacobiancriterion,Fields,Dedekinddomains,Standard´etalemorphisms,Normalschemes,Schemeswithnilpo-tents)14;Propertiesof´etalemorphisms15.3.TheEtaleFundamentalGroup........................................18Thetopologicalfundamentalgroup18;The´etalefundamenta
6、lgroup19;VarietiesoverC21;Examples(Thespectrumofafield,Normalvarieties(orschemes),Albanesevariety,Computingthe´etalefundamentalgroup,themostinterestingobjectinmathematics)21.References23.4.TheLocalRingfortheEtaleTopology...............................24Thecaseofvarieties2
7、4;InterludeonHenselianrings27;Schemes29.5.Sites.....................................................................30(TheZariskisiteonX,the´etalesiteonX,variantsofthe´etalesiteonX,theflatsiteonX,thebig´etalesiteonX,thesiteTG,Dictionary.)c1998J.S.Milne.Youmaymakeonecopyo
8、fthesenotesforyourownpersonaluse.iiiJ.S.MILNE6.SheavesfortheEtaleTopology..............................
