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1、1AneVarietiesWewillbeginfollowingKempf'sAlgebraicVarieties,andeventuallywilldothingsmorelikeinHartshorne.Wewillalsousevarioussourcesforcommutativealgebra.Whatisalgebraicgeometry?Classically,itisthestudyofthezerosetsofpolynomials.Wewillnow xsomenotation.kwillbesome xedalgebraicallyclose
2、d eld,anyringiscommutativewithidentity,ringhomomorphismspreserveidentity,andak-algebraisaringRwhichcontainsk(i.e.,wehavearinghomomorphism:k!R).PRanidealisprimei R=Pisanintegraldomain.AlgebraicSetsWede neanen-space,An=kn=f(a;:::;a):a2kg.1niAnyf=f(x;:::;x)2k[x;:::;x]de nesafunctionf:An
3、!k:1n1n(a1;:::;an)7!f(a1;:::;an).ExerciseIff;g2k[x1;:::;xn]de nethesamefunctionthenf=gaspolynomials.De nition1.1(AlgebraicSets).LetSk[x1;:::;xn]beanysubset.ThenV(S)=fa2An:f(a)=0forallf2Sg.AsubsetofAniscalledalgebraicifitisofthisform.e.g.,apointf(a1;:::;an)g=V(x1 a1;:::;xn an).Exercises
4、1.I=(S)istheidealgeneratedbyS.ThenV(S)=V(I).2.IJ)V(J)V(I).P3.V([I)=V(I)=V(I).4.V(IJ)=V(IJ)=V(I)[V(J).De nition1.2(ZariskiTopology).Wecande neatopologyonAnbyde ningtheclosedsubsetstobethealgebraicsubsets.UAnisopeni AnnU=V(S)forsomeSk[x1;:::;xn].Exercises3and4implythatthisisatopolo
5、gy.TheclosedsubsetsofA1arethe nitesubsetsandA1itself.De nition1.3(IdealofaSubset).IfWAnisanysubset,thenI(W)=ff2k[x1;:::;xn]:f(a)=0foralla2WgFacts/Exercises1.VW)I(W)I(V)2.I(;)=(1)=k[x1;:::;xn]3.I(An)=(0).1De nition1.4(AneCoordinateRing).WAnisalgebraic.ThenA(W)=k[W]=k[x1;:::;xn]=I(W)
6、Wecanthinkofthisastheringofallpolynomialfunctionsf:W!k.De nition1.5(RadicalIdeal)p.LetRbearingandIRbeanideal,thentheradicalofIistheidealI=ff2R:fi2Iforsomei2NgpWecallIaradicalidealifI=I.ExercisepIfIisanideal,thenIisaradicalideal.Proposition1.1.WAnanysubset,thenI(W)isaradicalideal.pProo
7、f.WehavethatI(W)I(W).pSupposef2I(W).Thenfi2Iforsomei.Thatis,foralla2W,fi(a)=0.Thus,f(a)m=0=f(a).Andso,f(a)2I.Exercises1.Sk[x1;:::;xn],thenSI(V(S)).2.WAnthenWV(I(W)).3.WAnisanalgebraicsubset,thenW=V(I(W)).pp4.Ik[x1;:::;xn]isanyideal,thenV(I)=V(I)andII(V(I))Theorem1.2(Nul
