FOUNDATIONS OF ALGEBRAIC GEOMETRY .pdf

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1、MATH216:FOUNDATIONSOFALGEBRAICGEOMETRYmath216.wordpress.comOctober23,2012draft!c2010,2011,2012byRaviVakil.ContentsChapter1.Introduction91.1.Goals91.2.Backgroundandconventions11PartI.Preliminaries13Chapter2.Somecategorytheory152.1.Motivation152.2.Categories

2、andfunctors172.3.Universalpropertiesdetermineanobjectuptouniqueisomorphism222.4.Limitsandcolimits312.5.Adjoints342.6.Anintroductiontoabeliancategories382.7.!Spectralsequences47Chapter3.Sheaves613.1.Motivatingexample:Thesheafofdifferentiablefunctions.613.2.

3、Definitionofsheafandpresheaf633.3.Morphismsofpresheavesandsheaves683.4.Propertiesdeterminedatthelevelofstalks,andsheafification723.5.Sheavesofabeliangroups,andOX-modules,formabeliancategories763.6.Theinverseimagesheaf793.7.Recoveringsheavesfroma“sheafonabase”

4、82PartII.Schemes87Chapter4.Towardaffineschemes:theunderlyingset,andtopologicalspace894.1.Towardschemes894.2.Theunderlyingsetofaffineschemes914.3.VisualizingschemesI:genericpoints1024.4.Theunderlyingtopologicalspaceofanaffinescheme1034.5.AbaseoftheZariskitopol

5、ogyonSpecA:Distinguishedopensets1064.6.Topological(andNoetherian)properties1074.7.ThefunctionI(·),takingsubsetsofSpecAtoidealsofA115Chapter5.Thestructuresheaf,andthedefinitionofschemesingeneral1175.1.Thestructuresheafofanaffinescheme1175.2.Visualizingschemes

6、II:nilpotents1205.3.Definitionofschemes1235.4.Threeexamples12635.5.Projectiveschemes,andtheProjconstruction132Chapter6.Somepropertiesofschemes1416.1.Topologicalproperties1416.2.Reducednessandintegrality1436.3.Propertiesofschemesthatcanbechecked“affine-locall

7、y”1446.4.Normalityandfactoriality1496.5.Wherefunctionsaresupported:Associatedpointsofschemes153PartIII.Morphismsofschemes163Chapter7.Morphismsofschemes1657.1.Introduction1657.2.Morphismsofringedspaces1667.3.Fromlocallyringedspacestomorphismsofschemes1687.4

8、.Mapsofgradedringsandmapsofprojectiveschemes1747.5.Rationalmapsfromreducedschemes1767.6.!Representablefunctorsandgroupschemes1817.7.!!TheGrassmannian(initialconstruction)186Chapter8.Usefulclassesofmorphismsof