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1、TopicsinAlgebraicGeometryProfessorLucIllusieUniversit´edeParis-SudD´epartementdeMath´ematiquesBˆatiment42591405Orsay,FranceEmail:luc.illusie@math.u-psud.frContents1HomologicalAlgebra11Additiveandabeliancategories,complexes...........12Bicomplexesandcones......
2、................83Homotopycategoryofcomplexes,triangulatedcategories...154Derivedcategories.........................235Derivedfunctors.........................426ThefunctorsRΓ,Rf,Lf∗,⊗L..................50∗ii7RHom,RHom,Ext,Ext....................628CechCohomol
3、ogy.........................68ˇExercises.............................732CohomologyofAffineandProjectiveMorphisms811Serre’sTheoremonAffineSchemes...............812Koszulcomplexandregularsequences.............85r3CohomologyofPwithvaluesinOPr(n)............914Finiten
4、essandvanishingtheorems................945HilbertPolynomial........................1023DifferentialCalculus1111K¨ahlerdifferentialsandderivations...............1112Smoothunramified´etalemorphisms...............1243Smoothness,flatnessandregularity...............141
5、4SmoothnessandDeformations..................1525Serre-GrothendieckGlobalDualityTheorem..........1616SpectralSequences........................1804Formalgeometry1951LocallyNoetherianFormalSchemes...............195iiiCONTENTS2TheComparisonTheorem.................
6、...2023Grothendieck’sexistencetheorem................2174Applicationtoliftingproblems..................2275Serre’sExample..........................235Bibliography245Chapter1HomologicalAlgebra1Additiveandabeliancategories,complexesDefinition1.1.Anadditivecategor
7、yisacategoryAhavingthefollowingproperties:(i)ForanyobjectsL,MofA,thesetofmorphismsHom(L,M)isen-dowedwiththestructureofanabeliangroup,andforanyobjectsL,M,N,thecompositionHom(L,M)×Hom(M,N)→Hom(L,N)isZ-bilinear.(ii)Thereexistsanobjectwhichisbothinitialandfinal,whi
8、chiscalledthezeroobjectanddenotedby0:foranyobjectL,Hom(L,0)=Hom(0,L)={0}.(iii)ForanyobjectsL,MofA,thesumL⊕MandtheproductL×Mexist.Itiseasilycheckedthat,inpresenceof(i)and(ii),(iii)i
