; TeX output 1999.09.17:13267 YRXQ cmr12CHAPTER1nNff cmbx12CommutativeffandNoncommutativeffAlgebraicGeometry"N cmbx12Intro`ductionThroughoutMLwrewill xabase eld' msbm10K.lThereadermayconsideritasrealnumbSersorcomplexnrumbSersoranryotherofhismostfavorite elds.A.fundamenrtal/BandpSowerfultoSolforgeometryistoassociatewitheacrhspaceg cmmi12Xethet-algebraoffunctions!", cmsy10OUV(X)fromXtothebase eld(ofcoSecienrts). oThedreamofgeometryisthatthisconstructionisbijectivre,i.e.'Rthattwodi erentspacesareEmappSedtotrwoEdi erentfunctionalgebrasandthateachalgebraisthefunctionalgebraofsomespace.ActuallyRthespacesandthealgebraswillformacategoryV.pThereareadmissiblemaps. FVoruhalgebrasitisquiteclearwhatthesemapswillbSe.FVorspacesthisislessjobrvious,partlyduetothefactthatwedidnotsayclearlywhatspacesexactlyare.)Thenthe*@ cmti12drffeamzofgeffometrywrouldbSethatthesetwocategories,ęthecategoryof(certain)spacesandthecategoryof(certain)algebras,aredualtoeacrhother.AlgebraicgeometryV,noncommrutativegeometryV,andtheoreticalphysicshaveasabasis%thisfundamenrtalidea, thedualityoftwocategories, thecategoryofspaces(statespaces/inphrysics)andthecategoryoffunctionalgebras(algebrasofobservXables)inphrysics.WVewillpresentthisdualityinthe1.chapter.CertainlythetypSeofspacesaswrellasthetypSeofalgebraswillhavetobSespeci ed.TheoreticalIphrysicsusesthecategoriesofloScallycompactHausdor spacesandof;commrutativeCܞ2K cmsy8-algebras.jA7famoustheoremofGelfand-Naimarksaysthatthesecategoriesaredualsofeacrhother.(Ane)algebraicgeometryusesadualitrybSetweenthecategoriesofanealgebraicscrhemesandof(reduced) nitelygeneratedcommutativealgebras.TVo9sgetthewholeframewrorkofalgebraicgeometryoneneedstogotomoregen-eral`zspacesbrypatchinganespacestogether.WOnthealgebrasidethisamountstoconsideringlshearvesofcommutativealgebras.,WVeshallnotpursuethismoregeneralapproacrhtoalgebraicgeometryV,sincegeneralizationstononcommutativegeometryarestillinthestateofdevrelopmentandincomplete.Noncommrutativegeometryuseseither(imaginary)noncommrutativespacesandnotQnecessarilycommrutativeQalgebrasor(imaginary)noncommrutativeQspacesandnotnecessarilycommrutativeCܞ2-algebras.WVecwilltakreanapproachtothedualitybSetweengeometryandalgebrathatheavilyuses-$functorialtoSols,=especially-$represenrtablefunctors.TTheane(algebraic)spaces]+o cmr91*7 &e261.pCOMMUT:ATIVE!ANDNONCOMMUT:ATIVE!ALGEBRAICGEOMETR:YYwre)usewillbSegivenintheformofsetsofcommonzerosofcertainpSolynomials,PuwherethezeroscanbSetakreninarbitrary(commutative)K-algebrasB.Soananespacewillconsistofmanrydi erentsetsofzeros,depSendingonthechoiceofthecoSecientalgebraB.WVe rstgivreashortintroSductiontocommutativealgebraicgeometryinthissetupanddevrelopadualitybSetweenthecategoryofane(algebraic)spacesandthecate-goryof( nitelygenerated)commrutativealgebras.Thenwrewilltransferittothenoncommutativesituation.=ThefunctorialapproachtoalgebraicgeometryisnottoSooftenusedbutitlendsitselfparticularlywrelltothestudyyofthenoncommrutativeysituation.LEveninthatsituationoneobtainsspace-likeobjects.Thecrhapterwillclosewitha rststeptoconstructautomorphism\groups"ofnoncommrutativespaces.*SincetheconstructionofinrversespresentsspSecialproblemswrewillonlyconstructendomorphism\monoids"inthischapterandpSostponethestudyofinrvertibleendomorphismsorautomorphismstothenextcrhapter.Arttheendofthechapteryoushoulds2# #knorwhowtoconstructananeschemefromacommutativealgebra,# #knorwhowtoconstructthefunctionalgebraofananescheme,# #knorwwhatanoncommutativespaceisandknowexamplesofsuch,# #understand88andbSeabletoconstructendomorphismquanrtummonoidsofcer- #tainnoncommrutativespaces,# #understand,Awhry/endomorphismquantummonoidsarenotmadeoutofendo- #morphismsofanoncommrutativespace. p7 &eA:1.pTHE!PRINCIPLESOFCOMMUT:ATIVE!ALGEBRAICGEOMETR:Y93Y5Pn1.FnThePrinciplesofCommutativeAlgebraicGeometryWVeIwillbSeginwithsimplestformof(commrutative)Igeometricspacesandseeadualitry$bSetweentheseverysimple\spaces"andcertaincommutativealgebras.UThisexampleľwillshorwhowtheconceptofafunctionalgebracanbSeusedtoful llthedream ofgeometryinthissituation. ItwillalsoshorwthefunctorialmethoSdsthatwill\bSeappliedthroughoutthistext.vItisaparticularlysimpleexampleofadualitryas$menrtionedintheintroSduction.RTThisexamplewillnotbeusedlateron,CsowrewillonlyskretchtheproSofsforsomeofthestatements.gSExample1.1.1.oQConsiderasetofpSoinrtswithoutanyadditionalgeometricstruc-ture.{So/thegeometricspaceisjustaset.WVeinrtroSducethenotionofitsalgebraoffunctions.LetR&XCbSeaset. o[ThenK22cmmi8X ¹:=9Map(XJg;K)isaK-algebrawithcomponenrtwiseaddition[andmrultiplication:U+V Ogn9)UR= K2he(fG)+ OK2h(gn9).Considerthehomomorphismo:URK2X =Q K2Y 4!K2XY,Cde nedbryW(fg gn9)(x;y)UR:=fG(x)gn9(y).XIn,ordertoobtainauniquehomomorphism,de nedonthetensorproSductwrehavetoshowthatW20z:URK2X }*_K2Y 4!K2XYPisabilinearmap:&W20%V(f+fG208;gn9)(x;y)UR=(fő+}fG208)(x)gn9(y)UR=(fG(x)+f208(x))gn9(y)UR=fG(x)gn9(y)+fG208(x)g(y)UR=(W20%V(f;gn9)+W20(fG208;gn9))(x;y)givres?theadditivityinthelefthandargument. TheadditivityintherighthandargumenrtandthebilinearityischeckedsimilarlyV.kOnecancheckthatRǹisalwaysinjectivre.8IfX+orYare nitethenAŹisbijective.BWAsvaspSecialexamplewreobtainamultiplicationrUR:K2X NK2X2up r4!"K2XX2(ppmsbm8K-:Aacmr6p!5!0WK2Xwhere::X4![ X=˺X inSetWisthediagonalmap(x):=(x;x).ʗFVurthermorewre{get#aunitË:URK2fg2QK-:;cmmi6p^!'@ K2X Awhere:XF4!fgistheuniquemapinrtotheoneelementset.OneXvreri eseasilythat(K2X;n9;r)XisaK-algebra.TwropropSertiesareessentialhere,theassoSciativitryandtheunitofKandthefactthat(XJg;;)isa\comonoid"inthecategorySetӋ:FwX+XnX+XXl32fd&H`O line10-'|{Ycmr81Y&XYiX+XM{fd?ά-ˍȕHKʟǠ*Ffe~Ǡ?` H=ʟǠ*FfepǠ?o#|17 &e461.pCOMMUT:ATIVE!ANDNONCOMMUT:ATIVE!ALGEBRAICGEOMETR:YY Ԡt XԠX+X:2fdnά-ζ vzҟǠ@fez8Ǡ??;np(kSm1X.X$ܔP$ P$?_P$攴P$ P$?^P$P$P$?]Py Py qҟǠ@fe#Ǡ?ꬽ Մ1h0X+X˲fgXPF԰_=~XPF԰_=X+fg:t32fd1pά-r V13RSince+K2-isafunctorthesetrwo+diagramscarryorver+tothecategoryVecEandproSducetherequireddiagramsforaK-algebra.FVorgqamapfq:)X:4!^~YwreobtainahomomorphismofalgebrasK2f ֹ:K2Y zt4!K2XbSecausethediagramsO ɱ㍑mUK2Y e K2Yɱ㍒Z(K2YYˌ{fd* `ά-wi#lK2X 1 K2Xi#͍\K2XX32fd(ά-Íw1㎎1㍒-#K2Y{fd9ά-xK-:##,AҬ>UH'<w ׁ w w w S̟>S̟> #K2Y#5lK2Xܞ32fd"€ά-VK-:fuycommrute.ThrusG(K -:URSet4!+fK-cAlgisaconrtravXariantfunctor.ByLthede nitionoftheset-theoretic(cartesian)proSductwreknowthatK2X ͠=!u cmex10Q UWXrK.0This`idenrtitydoSesnotonlyholdonthesetlevel,աitholdsalsoforthealgebraRJstructuresonK2X 1resp.8Q7XK.WVenorwconstructaninversefunctor謍}RSpSec":URK-cAlg!4!3`SetFI:FVorJ~eacrhpSointxq2X<thereJ~isamaximalidealmx eofQ՟X\Kde nedbymx AX:=qff@p2Map+Q(XJg;K)jfG(x)=0g.IfYIXJ̹isa nitesetthentheseareexactlyallmaximalidealsofQHVXeK.QTVoshorwthisweobservethefollowing.QThesurjectivehomomorphismpxk:Q UWXrKi4!Khaskrernelmx ?shencemxisamaximalideal.\IfmiQXmKisamaximalidealC{anda샹=( 1;:::ʚ; nP)2mC{thenforanry iQ]6=0weget(0;:::ʚ;0;1id;0;:::;0)=L(0;:::ʚ;0; 1 i p ;0;:::;0)( 1;:::; nP) 2mb1hencethei-thfactor0 :::´K:::0ofQ,UXIKisinm.2RSotheelemenrtsaUR2mmusthaveatleastonecommoncompSonent jX=qN0sincem6=K.-gButmorethanonesucrhacompSonentisimpSossiblesincewewrouldvgetzerodivisorsintheresidueclassalgebra..ThusmB=mx wherevx2Xgisthejӹ-thelemenrtsoftheset.(g7 &eA:1.pTHE!PRINCIPLESOFCOMMUT:ATIVE!ALGEBRAICGEOMETR:Y95YOnewcaneasilyshorwmorenamelythattheidealsmxarepreciselyallprimeidealsofMap(XJg;K).With)eacrhcommutativealgebraAwecanassoSciatethesetSpec(A)ofallprimeideals*ofA.IThatde nesafunctorSpSec:}K-Alg^4!.:SetA#ٹ.Appliedtoalgebrasoftheform~K2X Mx=/QFXKwitha nitesetXthisfunctorrecorvers~XasXP!r԰:Z=SpSec,z6(K2X).Thrusthedreamofgeometryissatis edinthisparticularexample.pThe*abSorveexampleshowsthatwemayhopSetogainsomeinformationonthespace(set)XkbryknowingitsalgebraoffunctionsK2X qandapplyingthefunctorSpSectoit.FVor nitesetsandcertainalgebrasthefunctorsK2-}EandSpSecactuallyde neacategorydualitryV.8Wearegoingtoexpandthisdualitrytolargercategories.WVe)shallcarrysomegeometricstructureinrtothesetsXandwillstudythecon-nectionbSetrweenthesegeometricspacesandtheiralgebrasoffunctions. UNFVorthispurpSosewrewilldescribesetsofpoinrtsbytheircoSordinates.ExamplesarethecircleortheparabSola.MoregenerallythegeometricspaceswrearegoingtoconsideraresoEcalledanescrhemesdescribSedbypSolynomialequations.=WVewillseethatsuchgeometricspacesarecompletelydescribSedbrytheiralgebrasoffunctions.HeretheYVonedaLemmawillplaryacentralr^ ole.WVetwill,Zhorwever,taketadi erenrtapproachtofunctionsalgebrasandgeometricspaces,0than"onedoSesinalgebraicgeometryV.We"usethefunctorialapproacrh,whichlends Nitselftoaneasieraccesstotheprinciplesofnoncommrutative NgeometryV.Wewill/ede negeometricspacesascertainfunctorsfromthecategoryofcommrutativealgebras.Xtothecategoryofsets.Thesesetswillharve.Xastronggeometricalmeaning.TheN&functorswillassoSciatewitheacrhalgebraAthesetofpoinrtsofa\geometricvXarietry",wherethepSointshavecoSordinatesinthealgebraA.pDe nition1.1.2.vaùTheYfunctorAUR=A21V:K-cAlg!4!3`SetI ܹ(theYunderlyingfunctor)thatassoSciateswitheacrhcommutativeK-algebraAitsspace(set)ofpSoints(elements)A꨹iscalledtheane35line.Lemma1.1.3.g5QThe35functor\aneline"isarffepresentable35functor.,- cmcsc10Proof.@_ByάLemma2.3.5therepresenrtingobjectisK[x].7Observethatitisuniqueuptoisomorphism.Ecffxff ̟ff ̎ ̄cffDe nition1.1.4.vaùThefunctorA22 j:FfK-cAlg#4!7BSetO:\thatassoSciateswitheacrhcommrutativealgebraAthespace(set)ofpSoinrts(elements)oftheplaneA22iscalledtheane35plane.Lemma1.1.5.g5QThe35functor\aneplane"isarffepresentable35functor.Proof.@_SimilarDtoLemma2.3.9therepresenrtingobjectisK[x1;x2].ThisDalgebraisuniqueuptoisomorphism.Ņcffxff ̟ff ̎ ̄cffLetp1(x1;:::ʚ;xnP);:::;pmĹ(x1;:::;xnP)UR2K[x1;:::;xnP]bSeafamilyofpolynomials.WVeMwranttoconsiderthe(geometric)vXarietyofzerosofthesepSolynomials.atObserve87 &e661.pCOMMUT:ATIVE!ANDNONCOMMUT:ATIVE!ALGEBRAICGEOMETR:YYthatrKmarynotcontainsucientlymanyzerosforthesepSolynomials.#Thuswearego-ing9toadmitzerosinextension eldsofKormoregenerallyinarbitrarycommrutativeK-algebras.In2zthefollorwingrathersimplebuildupofcommutativealgebraicgeometryV,nthereadershouldcarefullyvrerifyinwhichstatementsandproSofsthecommutativityisreally]needed.MostofthefollorwingwillbSeverballygeneralizedtonotnecessarilycommrutativealgebras.XDe nition1.1.6.vaùGivrenQ~asetofpSolynomialsfp1;:::ʚ;pmgURK[x1;:::ʚ;xnP].ThefunctorXthatassoSciateswitheacrhcommutativealgebraAthesetXӹ(A)ofzerosoftheNpSolynomials(pidڹ)inA2n uiscalledananealgebrffaicvarietyNorananescheme(inqA2nP)withde ningpSolynomialsp1;:::ʚ;pmĹ.RTheelemenrtsinXӹ(A)arecalledtheA-pffoints꨹ofXӹ.Theorem1.1.7.pThe aneschemeX˸inA2n5withde ningpffolynomialsp1;:::ʚ;pmis35arffepresentable35functorwithrffepresenting35algebrae{ OUV(Xӹ)UR:=K[x1;:::ʚ;xnP]=(p1;:::ʚ;pmĹ);cffalled35theanealgebraofthefunctorX.Proof.@_First9wreshowthattheaneschemeXU :8K-cAlg%/4!9lSetRwiththede ningKpSolynomialsp1;:::ʚ;pm isafunctor.XLetfm:gnA4!iBQbeKahomomorphismofcommrutative#`algebras.TheinducedmapfG2n ,:A2n ^-4!-B2n fde nedbryapplicationoffontthecompSonenrtsrestrictstoXӹ(A)@!A2n astX(fG):X(A)4!X(B).ֻThistmapiswrell-de nedforlet(a1;:::ʚ;anP)UR2Xӹ(A)bSeazeroforallpolynomialsp1;:::ʚ;pm Jthenpidڹ(fG(a1);:::ʚ;f(anP))=fG(pi(a1;:::ʚ;anP))=fG(0)=0Lforallihencef2nO(a1;:::ʚ;anP)=(fG(a1);:::ʚ;f(anP))a 2B2n 4isazeroforallpSolynomials.MThrusXӹ(fG)a :X(A)4!ۥX(B)iswrell-de ned.8FVunctorialityofX{isclearnorw.Norw3weshowthatXisrepresentablebyOUV(Xӹ)UR=K[x1;:::ʚ;xnP]=(p1;:::ʚ;pmĹ).Ob-servrethat(p1;:::ʚ;pmĹ)denotesthe(two-sided)idealinK[x1;:::ʚ;xnP]generatedbythepSolynomialstp1;:::ʚ;pmĹ.WVeknorwthateachn-tupSel(a1;:::ʚ;anP)UR2A2nĹuniquelytdeter-minesanalgebrahomomorphismfN:wOK[x1;:::ʚ;xnP]4!+AbryfG(x1)wO=a1;:::ʚ;fG(xnP)=anP.F(TheDgpSolynomialringK[x1;:::ʚ;xn]inK-cAlg"ֹisfreeorverDgthesetfx1;:::ʚ;xng,orUaK[x1;:::ʚ;xnP]togetherwiththeemrbSedding :fx1;:::ʚ;xnPg4!/K[x1;:::ʚ;xnP]Uaisacounivrersal[[solutionoftheproblemgivenbytheunderlyingfunctorA':K-cAlg"4!Set-andJthesetfx1;:::ʚ;xnPgbP2SetK3.)OThisJhomomorphismofalgebrasmapspSolyno-mialsfp(x1;:::ʚ;xnP)inrtofG(p)(=p(a1;:::ʚ;anP).CHencef(a1;:::ʚ;anP)fisacommonzeroof[thepSolynomialsp1;:::ʚ;pm `ifandonlyiffG(pidڹ)UR=pi(a1;:::ʚ;anP)=0,xi.e. Op1;:::ʚ;pmareHinthekrerneloffG.ThishappSensifandonlyiffvXanishesontheideal(p1;:::ʚ;pmĹ)orinotherwrordcanbSefactorizedthroughtheresidueclassmape`Jl:URK[x1;:::ʚ;xnP]4!1K[x1;:::ʚ;xnP]=(p1;:::ʚ;pmĹ)ThisinducesabijectionMor*K--2@cmbx8cAlgHY(K[x1;:::ʚ;xnP]=(p1;:::ʚ;pmĹ);A)UR3fQ7!(fG(x1);:::ʚ;f(xnP))UR2Xӹ(A):IF7 &eA:1.pTHE!PRINCIPLESOFCOMMUT:ATIVE!ALGEBRAICGEOMETR:Y97YNorwitiseasytoseethatthisbijectionisanaturalisomorphism(inA).4샄cffxff ̟ff ̎ ̄cffIfnopSolynomialsaregivrenfortheaborveconstruction,GthenthefunctorunderthisconstructionZ;istheanespaceA2n ofdimensionn.BygivingpSolynomialsthefunctorXbSecomes]asubfunctorofA2nP,zbecauseitde nessubsetsXӹ(A)URA2nP(A)=A2n. Bothfunctorsarerepresenrtablefunctors.TheembSeddingisinducedbythehomomorphismofalgebras:URK[x1;:::ʚ;xnP]4!1K[x1;:::ʚ;xnP]=(p1;:::ʚ;pmĹ).Problem1.1.1.R1.(Determinebtheanealgebraofthefunctor\unitcircle" #Sן21 ]inA22.2. #Determinetheanealgebraofthefunctor\unitsphere"Sן2n1"KinA2nP.3. #LetX^denotetheplanecurvreyË=URx22.8,ThenXisisomorphictotheaneline.4. #Let96Y5̹denotetheplanecurvrexyIA=1.$ThenYisnotisomorphictotheane #line.3(Hinrt:xmAnnisomorphismK[x;x21 \|]eG4!K[yn9]sendsxtoapSolynomialp(y) #whicrhmustbSeinvertible.qConsiderthehighestcoSecientofp(yn9)andshow #thatp(yn9)UR2K.8ButthatmeansthatthemapcannotbSebijectivre.)5. #LetKdF=CbSethe eldofcomplexnrumbSers. Showthattheunitfunctor #UL\: xK-cAlg"4!4SetL zinULemma2.3.7isnaturallyisomorphictotheunitcircle #functor4RSן21r۹.(Hinrt:4ThereisanalgebraisomorphismbSetweentherepresenting #algebrasK[e;e21 \|]andK[c;s]=(c22j+s221).)6. #2(DLetKbSeanalgebraicallyclosed eld."Letpbeanirreduciblesquarepolyno- #mialsinK[x;yn9].&LetZbSetheconicsectionde nedbrypwiththeanealgebra #K[x;yn9]=(p).Shorw,thatZ isnaturallyisomorphiceithertoXeortoY)(from #parts3.8resp.4.Remark1.1.8.j6Anem#algebrasofanescrhemesare nitelygeneratedcommu-tativreԑalgebrasandanysuchalgebraisananealgebraofsomeanescheme,sinceAPUR԰n:=K[x1;:::ʚ;xnP]=(p1;:::ʚ;pmĹ)(HilbSertbasistheorem).The fpSolynomialsp1;:::ʚ;pm *arenotuniquelydeterminedbrytheanealgebraofananescrheme."NoteventheidealgeneratedbythepSolynomialsinthepolynomialringK[x1;:::ʚ;xnP]isuniquelydetermined.AlsothenrumbSerofvXariablesx1;:::ʚ;xn B_isnotuniquelydetermined.The(K-pSoinrts( 1;:::ʚ; nP)UR2Xӹ(K)(ofananeschemeXX(withcoSecientsinthebase eldK)arecalledrffationalpoints.prTheydonotsucetocompletelydescribSetheanescrheme.LetforexampleKP=RѹthesetofrationalnrumbSers. \IfXandYgareanescrhemes6withanealgebrasOUV(Xӹ)UR:=K[x;yn9]=(x22+;y22i׹+1)6andOUV(Y)UR:=K[x]=(x22+1)then[3bSothscrhemeshavenorationalpSoints. TheschemeY,Uhowever,has[3exactlytrwo5complexpSoinrts(withcoecienrtsinthe eldCofcomplexnumbSers)andthescrheme:XF hasin nitelymanycomplexpSoints,henceXӹ(C)Z6P԰= Y(C).ThisdoSesnotresultfromtheemrbSeddingsintodi erentspacesA22 Wresp.>9A21.InfactwealsohaveOUV(Y)UR=K[x]=(x22+1)P԰n:=K[x;yn9]=(x22+1;yn9),-socYcanbSeconsideredasananescrhemeinA22.`$7 &e861.pCOMMUT:ATIVE!ANDNONCOMMUT:ATIVE!ALGEBRAICGEOMETR:YYSinceЧeacrhaneschemeXzisisomorphictothefunctorMorK-cAlg8=(OUV(Xӹ);-)wewillhenceforthidenrtifythesetwofunctors,thusremovingannoyingisomorphisms.eDe nition1.1.9.vaùLetK-A denotethecategoryofallcommrutative nitelygenerated(oranecf.Ԣ1.1.8)K-algebras.AnanealgebrffaicvarietyisarepresenrtablefunctorK-A 7(A;-)UR:K-A j4!-ISet@,. \TheanealgebraicvXarietiestogetherwiththenatural[transformationsformthecffategory*0ofanealgebrffaicvarieties[VVar(K)orverK.]ThefunctorthatassoSciateswitheacrhanealgebraAitsanealgebraicvXarietyrepresenrtedbyAisdenotedbySpSecx:URK-A j4!-IVVar@AH(K),SpSec&(A)=K-- NA !(A;-).BytheYVonedaLemmathefunctorBSpSec:URK-A j4!-IVVar@AH(K)isananrtiequivXalence(orduality)ofcategorieswithinversefunctor=AO:URVVarQ(K)UR4!1K-A 7:AnanealgebraicvXarietryiscompletelydescribSedbyitsanealgebraOUV(Xӹ).וThusthedreamofgeometryisrealized.Arbitrary^(notnecessarily nitelygenerated)commrutative^algebrasalsode nerepresenrtablefunctors(de nedonthecategoryofallcommutativealgebras).Thuswealso|harve\in nitedimensional"vXarietieswhichwewillcallgeffometricUspacesoranevarieties.8WVedenotetheircategorybryGeom"P(K)andgetacommutativediagramPB9K-cAlgٰGeom X(K)t32fd*ά- T덍PH԰.0=jnoHK-A HĄVVar (K){fd3QPά-ɪSpSecHŸǠ*Ffe1Ǡ?HŸǠ*Ffe#Ǡ?VWVe}calltherepresenrtablefunctorsXH9:fK-cAlg""4!3SetJ(geffometricMspaces}oranevarieties,Band0therepresenrtablefunctorsX:LK-A d4!.=SetFanesuschemesoranealgebrffaic35varieties.8ThisisanotherrealizationofthedreamofgeometryV.The:HgeometricspacescanbSeviewredassetsofzerosinarbitrarycommutativeK-algebras[B$ofarbitrarilymanrypSolynomialswitharbitrarilymanyvXariables.AThefunctionalgebraofX{willbSecalledtheane35algebrffaofXinbSothcases.eExample1.1.10.uQA'somewhat'lesstrivialexampleisthestatespaceofacircularpSendulumb(oflength1). 9 ThelocationisinL+=f(a;b)2A22ja225+1b22 묹=1g,themomenrtumuisinM6=URfp2Aguwhichisastraightline.SothewholegeometricspaceforthepSendulumis(LM@)(A)X=f(a;b;p)ja;b;pX2A;a22k+b22=1g.>ThisgeometricspaceisrepresenrtedbyK[x;yn9;z]=(x22j+y221)since|(LraM@)(A)UR=f(a;b;p)ja;b;pUR2A;a 22e+rab 2V=1gP԰n:=K-cAlgo(K[x;yn9;z]=(x 22e+ray 21);A):eThe4#trwoantiequivXalencesofcategoriesabSovegiverisetothequestionforthefunc-tionyalgebra.9IfarepresenrtablefunctorX%=URK-cAlgo(A;-)isviewedasgeometricsets r7 &eA:1.pTHE!PRINCIPLESOFCOMMUT:ATIVE!ALGEBRAICGEOMETR:Y99Yofu8zerosofcertainpSolynomials,i.e.asspaceswithcoordinatesinarbitrarycommruta-tivre%algebrasB,4(plusfunctorialbSehavior),4thenitisnotclearwhytherepresentingalgebraȬAshouldbSeanrythinglikeanalgebraoffunctionsonthesegeometricsets.-ItisnotevrenclearwherethesefunctionsshouldassumetheirvXalues.OOnlyifwecanshorw`thatAcanbSeviewedasareasonablealgebraoffunctions,|KweshouldtalkabSoutaLBrealizationofthedreamofgeometryV.]ButthiswillbSedoneinthefollorwingtheo-rem.WVehgwillconsiderfunctionsasmaps(coSordinatefunctions)fromthegeometricsetDiXӹ(B)tothesetofcoSordinatesB,ZmapsthatarenaturalinB.F#SucrhcoSordinatefunctionsarejustnaturaltransformationsfromX{totheunderlyingfunctorA.Theorem1.1.11.wXLffetјXkbeageometricspacewiththeanealgebraAUR=OUV(Xӹ).Then[APUR԰n:=Nat#@(X;A)asK-algebrffas,where[AUR:K-35cAlg":N!3 SetJ:xistheunderlyingfunc-torDoraneline.DTheisomorphismAPUR԰n:=Nat#@(X;A)inducffesanaturaltransformationAXӹ(B)UR!B;(naturffal35inB).Proof.@_FirstYwrede neanisomorphismbSetweenthesetsAandNatT(X;A).BecauseofX=MorJ K-cAlg9H(A;-)=:K-cAlgo(A;-)andA=MorJ K-cAlg9H(K[x];-)=:K-cAlgo(K[x];-)theYVonedaLemmagivresus1]Nat(X;A)UR=Nat(K-cAlgo(A;-);K-cAlg(K[x];-))PUR԰n:=K-cAlg(K[x];A)UR=A(A)P԰n:=Aonthesetlevrel. Let?:A4!h Nat.(X;A)denotethegivenisomorphism. isde ned\bry(a)(B)(p)(x)0:=p(a). By\theYVonedaLemmaitsinverseisgivenby21 \|( h:=UR ((A)(1)(x).Nat%@(X;A)carriesanalgebrastructuregivrenbythealgebrastructureofthecoSef- cienrts.۔FVor acoSecientalgebraB,.saB-pSoinrtp:A4!|Bin Xӹ(B)=K-Algo(A;B),and ; j2%Nate(X;A)wrehave (B)(p)%2A(B)=B. -Hence( !+ʒ O)(B)(p):=( (B)+ O(B))(p)i= (B)(p)+ O(B)(p)4Mand(  O)(B)(p)i:=( (B) O(B))(p)i= (B)(p) O(B)(p)makreNat+Q(X;A)analgebra.Let-abSeanarbitraryelemenrtinA.7oBytheisomorphismgivenabSovethisele-menrtMinducesanalgebrahomomorphismga Z:VK[x]4!AMmappingxontoa.ThisalgebraӾhomomorphisminducesthenaturaltransformation(a):XW4!A. !Onthe}B-levrelitisjustthecompSositionwithgaϹ,i.e._(a)(B)(p)#=(K[x]h HgaJ ?|!)Ah q"pJ ?|!B). L~Since2sucrhahomomorphismiscompletelydescribSedbytheimageofxweget](a)(B)(p)(x)=p(a).TVocomparethealgebrastructuresofAandNat(X;A)leta;a202 A. WVeharve(a)(B)(p)(x)=p(a)and(a20:)(B)(p)(x)=p(a209),hence(a0k+a209)(B)(p)(x)=p(a0k+a20)=p(a)0k+p(a20)=(a)(B)(p)(x)0k+(a20)(B)(p)(x)=((a)(B)(p)*+(a209)(B)(p))(x)UR=((a)(B)*+(a209)(B))(p)(x)UR=((a)*+(a209))(B)(p)(x).Analogouslywreget(aa209)(B)(p)(x)V=p(aa20)=p(a)p(a20)=((a)(a20))(B)(p)(x),andWthrus(auU+a209)P=(a)uU+(a20)Wand(aa20)P=(a)uU(a20).HenceWadditionandmrultiplicationinNat(X;A)arede nedbytheadditionandthemultiplicationofthevXaluesp(a)+p(a209)resp.8p(a)p(a20).WVe{CdescribSetheaction n9(B)Kt:A Xӹ(B)Kt4!uBIof{CAonXӹ(B).LetpKt:A4!uBbSe3aB-poinrtinK-cAlgo(A;B)t3=Xӹ(B).2FVor3eacrhat32A3theimage(a)t3:X64!A ٠7 &e1061.pCOMMUT:ATIVE!ANDNONCOMMUT:ATIVE!ALGEBRAICGEOMETR:YYisnanaturaltransformationhencewrehavemaps n9(B)m:AXӹ(B)m4!B*tsucrhnthat n9(B)(a;p)<=p(a).wMFinally"eacrhhomomorphismofalgebrasf;:(natural8inB).&InparticulartheanealgebraAcanbSeviewredasthesetoffunctionsfromthesetofB-poinrtsXӹ(B)intothe\base"ringRB(functionswhicrharenaturalinB).RInthissensethealgebraAmaybSeconsid-eredasfunctionalgebraofthegeometricspaceXӹ.@>RHD6Xӹ(B)GǠ*Ffez̟Ǡ?3f1X(BI)_commrutes. I WVewillshowthisresultlateronfornoncommutativealgebras. I TheunivrersalpropSertyimpliesthatthefunctionalgebraAofangeometricspaceXmisuniqueuptoisomorphism.Let-XbSeangeometricspacewithfunctionalgebraAǹ=OUV(Xӹ).oIf-p:A4!Kis arationalpSoinrtofXӹ,Ri.e.ahomomorphismofalgebras,thenImp(p)?=K henceKer(p)isamaximalidealofAofcoSdimension1._Conrverselylet/%n eufm10mbeamaximalideal&ofAofcoSdimension1thenthisde nesarationalpoinrtpH:A4!A=mP԰0=K.IfKisalgebraiclyclosedandmanarbitrarymaximalidealofA,mthenA=misa nitelygeneratedK-algebraanda eldextensionofK,henceitcoincideswithK.iThrusthecoSdimension|ofmis1.ThesetofmaximalidealsofAiscalledthemaximal͌spffectrumSpSecmĹ(A).Thisistheapproacrhofalgebraicgeometrytorecoverthegeometricspaceof(rational)pSoinrtsfromthefunctionalgebraA.ǐWVewillnotfollowthisapproachsinceitdoSesnoteasilyextendtononcommrutativegeometryV.:Problem1.1.2.nRLetX{bSeananescrhemewithanealgebraAUR=K[x1;:::ʚ;xnP]=(p1;:::ʚ;pmĹ): Π7 &eA:1.pTHE!PRINCIPLESOFCOMMUT:ATIVE!ALGEBRAICGEOMETR:Y411YDe ne[\coSordinatefunctions"qi &:LXӹ(B)4!R%B'awhicrh[describethecoordinatesofB-pSoinrtsandidentifythesecoSordinatefunctionswithelementsofA.NorwwewillstudymorphismsbSetweengeometricspaces.Theorem1.1.13.wXLffetXN"{A2r 4aandYA2s ;bffeanealgebraicvarietiesandletUR:X%!Y/bffe35anaturaltransformation.fiThentherearepolynomialseYp1(x1;:::ʚ;xrb);:::;psn<(x1;:::;xrb)UR2K[x1;:::;xrb];such35thatLP(A)(a1;:::ʚ;arb)UR=(p1(a1;:::ʚ;arb);:::;psn<(a1;:::;arb));s2forallA.2K-33A Wandall(a1;:::ʚ;arb).2Xӹ(A),Ji.e.themorphismsbffetweenanealgebrffaic35varietiesareofpolynomialtype.Proof.@_LetmOUV(Xӹ)$=K[x1;:::ʚ;xrb]=IandOUV(Y)=K[y1;:::ʚ;ysn<]=Jr. -/FVorA2K-Alg̨and09(a1;:::ʚ;arb)z2Xӹ(A)letfy:K[x1;:::ʚ;xrb]=Ip4! AwithfG(xidڹ)=aibSetheIhomomorphismobtainedfromXӹ(A)P԰=DK-Algo(K[x1;:::ʚ;xrb]=I;A).VTheInaturaltransformationpisgivrenbycompSositionwithahomomorphismgË:URK[y1;:::ʚ;ysn<]=Jq4!K[x1;:::ʚ;xrb]=I+hencewreget(A)UR:K-cAlgo(K[x1;:::ʚ;xrb]=I;A)UR3fQ7!fGgË2K-cAlg(K[y1;:::ʚ;ysn<]=J:;A):SincegXisdescribSedbrygn9(yidڹ)UR=pi(x1;:::ʚ;xrb)2K[x1;:::ʚ;xrb]wreget eʍi(A)(a1;:::ʚ;asn<)UR=(fGgn9(y1);:::ʚ;fgn9(ysn<))i=UR(fG(p1(x1;:::ʚ;xrb));:::;fG(psn<(x1;:::;xrb)))i=UR(p1(a1;:::ʚ;arb);:::;psn<(a1;:::;arb)):ȍ %cffxff ̟ff ̎ ̄cffAnanalogousstatemenrtholdsforgeometricspaces.Example1.1.14.uQTheisomorphismbSetrweentheaneline(1.1.2)andthepara-bSolaeisgivrenbytheisomorphismfn:&K[x;yn9]=(yl~x22)4!gaK[z],fG(x)=z,fG(yn9)=z22thathastheinrversefunctionfG21 {(z)z=x.=OntheanescrhemesA,theaneline,andP,theparabSola,theinducedmapisf:>A(A)3a7!(a;a22)2P(A)resp.fG21͹:URP(A)3(a;b)7!a2A(A).;7  /%n eufm10-2@cmbx8,- cmcsc10+o cmr9*@ cmti12(ppmsbm8' msbm10!u cmex10 q% cmsy6K cmsy8!", cmsy10;cmmi62cmmi8g cmmi12Aacmr6|{Ycmr8N cmbx12Nff cmbx12XQ cmr12O line10B