Grothendieck's tohoku paper

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1、SomeaspectsofhomologicalalgebraTranslationof:Surquelquespointsd'algµebrehomologiqueAlexandreGrothendieck1TranslatedbyMarciaL.BarrandMichaelBarrJanuary3,20111TheessentialcontentofChapters1,2,and4,andpartofChapter3wasdevelopedinthespringof1955duringaseminarinhomologicalalgebraat

2、theUniversityofKansas.ReceivedMarch1,1957.ContentsIntroductioniii0.1Contentofthearticle...............................iii0.2Applications....................................iv0.3Omissions.....................................ivTranslator'sprefacevi1Generalitiesonabeliancategorie

3、s11.1Categories.....................................11.2Functors......................................41.3Additivecategories................................51.4Abeliancategories................................61.5In¯nitesumsandproducts............................81.6Categories

4、ofdiagramsandpermanenceproperties..............101.7Examplesofcategoriesde¯nedbydiagramschemes..............111.8Inductiveandprojectivelimits.........................131.9Generatorsandcogenerators...........................141.10Injectiveandprojectiveobjects...................

5、......151.11Quotientcategories................................172Homologicalalgebrainabeliancategories202.1@-functorsand@¤-functors............................202.2Universal@-functors...............................212.3Derivedfunctors.................................242.4Spect

6、ralsequencesandspectralfunctors....................262.5Resolventfunctors................................303Cohomologywithcoe±cientsinasheaf363.1Generalremarksonsheaves...........................36p3.2De¯nitionoftheH(X;F)...........................39©iiiCONTENTS3.3CriteriaforA

7、cyclicity..............................403.4Applicationstoquestionsofliftingofstructuregroups............433.5Theexactsequenceofaclosedsubspace....................493.6Onthecohomologicaldimensionofcertainspaces...............503.7TheLerayspectralsequenceofacontinuousfunction..

8、...........543.8ComparisonwithCechcohomology.................