looijenga Algebraic Geometry lecture notes

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1、AlgebraicGeometryFall2013&Spring2014EduardLooijengaRingsarealwayssupposedtobecommutativeandtopossessaunit1.Aringhomomorphismwillalwaystakeunittounit.Weallowthat1=0,butinthatcasewegetofcoursethezeroringf0g.IfRisaring,thenwedenotethemultiplicativegroupofinvert

2、ibleelements(units)ofRbyR.WesaythatRisadomainifRhasnozerodivisors,equivalently,if(0)isaprimeideal.AnR-algebraisaringAendowedwitharinghomomorphism:R!A,butifisunderstood,thenforeveryr2Randa2A,theproduct(r)aisoftendenotedbyra.WesaythatAis?nitelygeneratedasa

3、nR-algebraifwecan?nda1;:::;aninAsuchthateveryelementofAcanbewrittenasapolynomialintheseelementswithcoef?cientsinR;inotherwords,theR-algebrahomomorphismR[x1;:::;xn]!Awhichsendsthevariablexitoaiisonto.Thisisnottobeconfusedwiththenotionof?nitegenerationofanR-mo

4、duleMresp.whichmerelymeanstheexistenceofasurjectivehomomorphismofR-modulesRn!M.Similarly,a?eldKissaidtobe?nitelygeneratedasa?eldoverasub?eldkifwecan?ndelementsb1;:::;bninKsuchthateveryelementofKcanbewrittenasafractionoftwopolynomialsintheseelementswithcoef?c

5、ientsink.ContentsChapter1.Af?nevarieties51.TheZariskitopology52.Irreducibilityanddecomposition83.FinitenesspropertiesandtheHilberttheorems144.Theaf?necategory185.Thesheafofregularfunctions236.Theproduct257.Function?eldsandrationalmaps278.Finitemorphisms319.D

6、imension3610.Nonsingularpoints3911.Thenotionofavariety47Chapter2.Projectivevarieties511.Projectivespaces512.TheZariskitopologyonaprojectivespace533.TheSegreembeddings564.Projections575.Eliminationtheoryandclosedprojections586.Constructiblesets627.TheVeronese

7、embedding638.Grassmannians659.Multiplicitiesofmodules7010.HilbertfunctionsandHilbertpolynomials74Chapter3.Schemes791.Presheavesandsheaves792.Thespectrumofaringasalocallyringedspace843.Thenotionofascheme884.Formationofproducts945.Elementarypropertiesofschemes

8、956.Separatedandpropermorphisms997.Quasi-coherentsheaves1098.Whenisaschemeaf?ne?1139.Divisorsandinvertiblesheaves11610.Theprojectivesetting12411.Cohomologyofcoherentmodulesonaprojectivescheme127