On the opposite of the category of rings

arXiv:0806.1476v1[math.RA]9Jun2008ONTHEOPPOSITEOFTHECATEGORYOFRINGSRICHARDVALEAbstract.ForeveryringR,weconstructaringedspaceNCSpec(R)andforeveryringhomomorphismR→S,amorphismofringedspacesNCSpec(S)→NCSpec(R).Weshowthatthisgivesafullyfaithfulcontravariantfunctorfromthecategoryofringstoacategoryofringedspaces.IfRisacommutativering,weshowthatSpec(R)embedsnaturallyinNCSpec(R)asadensesubspace.WethenexplainhowthespacesNCSpec(R)maybeglued,andstudyquasicoherentsheavesonthem.Asanexample,wecomputethecategoryofquasicoherentsheavesonaspaceconstructedfromaskew-polynomialringRbyananalogueoftheProjconstruction.1.IntroductionAringedspaceisatopologicalspaceXequippedwithasheafofrings.Itisawell-knownfactfromalgebraicgeometrythatthecategoryofcommutativeringscanbeidentifiedwithacategoryofringedspaces.Moreprecisely,thereisacontravariantfunctorfromthecategoryofcommutativeringstothecategoryofringedspaces,whichassociatestoaringRitsprimespectrumSpec(R)equippedwiththestructuresheafO.Thisfunctorisfaithful.Thespaces(Spec(R),O)arecalledaffineschemes.Thenotionofaffineschemefirstappearedin[Gro60]andotherexpositionsmaybefoundin[Sha94,ChapterV],[Har77]and[EH00].Theaimofthepresentpaperistomakeasimilarconstructionforringswhicharenotnecessarilycommutative.2.ConventionsHerewelistsomeconventionswhichwillbeusedinthispaper..WedenotebyRingsthecategoryofallunitalringsincludingthezeroring.Thatis,alltheringsRwhichweconsiderhaveanidentityelement1Rsatisfyingx1R=1Rx=xforallx∈R.Thezeroringhasonlyoneelement0,anditsidentityelementis10=0.Werequire1ringhomomorphismstopreservetheidentityelements.Thus,thezeroring0isafinalobjectinthecategoryRings,andifRisnotthezeroring,thenthereisnohomomorphism0→R.Definition2.1.LetRbearing.Anelementx∈Risaunitifthereexistsy∈Rwithyx=xy=1R.Notethat0isaunitinthezeroring.Notealsothatifθ:R→Sisaringhomomorphismandu∈Risaunit,thenθ(u)isaunitinS.2.1.Ringedspaces.Asinalgebraicgeometry,weconsidertopologicalspaceswithsheavesofringsonthem.Definition2.2.IfXisatopologicalspacethenwedenotebyΩ(X)thecollectionofallopensubsetsofX.WeregardΩ(X)asaposetorderedbyinclusion,andhencealsoasacategory.Definition2.3.AsheafofringsonatopologicalspaceXisafunctorΩ(X)op→Ringssatisfyingthesheafaxioms.(Forthesheafaxioms,see[Sha94,V.2].)Definition2.4.Aringedspaceisapair(X,OX)consistingofatopologicalspaceXtogetherwithasheafofringsOXonX.Amorphism(X,OX)→(Y,OY)ofringedspacesisapair(φ,φ#)whereφ:X→Yisacontinuousfunctionandφ#:OY→φ∗OXisamorphismofsheavesofringsonY.WedenotethecategoryofringedspacesbyRingedSp.Noticethatourdefinitionofringedspacediffersfromtheusualoneinalgebraicgeometry,inwhichtheringsareassumedtobecommutative.Wewishtoallownoncommutativerings.Wearenowreadytostateouraimmoreprecisely.3.AimWewishtoconstructafaithfulfunctorRingsop→RingedSp.Inotherwords,wewanttoassociatetoeachringRaringedspaceNCSpec(R)inafunctorialway.Thereisoneveryeasysolutiontothisproblem,namelyforeachringR,takeaone-pointspacewiththesheafwhosesectionsovertheuniquenonemptyopensetareR.However,suchaconstructionwouldnotbeveryinteresting.WethereforealsowishourconstructiontocoincidewithSpec(R)whenRhappenstobeacommutativering.Wearenotabletoachievethis,but2wewillshowthatwhenRiscommutative,Spec(R)embedsnaturallyinthecomplementofthegenericpointinNCSpec(R)asadensesubspace.Thestructureofthispaperisasfollows.InSection4,wedescribevariousfunctorialpropertiesofthelocalizationofaringRatasubsetA⊂R,whichisdefinedviaauniversalproperty.WeputthelocalizationsatfinitesubsetstogethertoobtainapartiallyorderedsetL(R)fromaringR.InSection5,weequipL(R)withatopologyandasheafofrings.InSection6,werecallthesoberificationconstructionfromtopology.ThesoberificationofL(R)inheritsasheafofringsfromL(R),andthisisthespacewhichwecallNCSpec(R).InSection7,weproveourmaintheorem,Theorem7.7,whichsaysthatNCSpecisafullyfaithfulfunctorfromRingsoptoasubcategoryofRingedSpwhichwedescribe.InSection8,weexplainhowSpec(R)embedsnaturallyinNCSpec(R)whenRiscommutative.WethengivesomeexamplesinSection9.InSection10,weexplainhowthespacesNCSpec(R)maybegluedalongOrelocalizationsanddefinequasicoherentsheavesonthem.Aftersomecalculations,weshowinProposition10.14,asanexample,thatthecategoryofleftquasico-herentsheavesonaspaceconstructedfromtheskew-polynomialringR=Cλ[x1,...,xn]isequivalenttothecategoryofgradedleftR–modulesmodulotorsion.3.1.Acknowledgements.TheauthorwishestothankSefiLadkaniforprovidingusefulinformationontheAlexandrovtopologyandsheaves.TheauthorwishestothankYuriBerest,GregMuller,MichaelWemyssandGeordieWilliamsonforvaluablecomments.4.LocalizationOurconstructionsusethetechniqueofnoncommutativelocalizationofaring.Inthissection,wecollectthedefinitionandbasicpropertiesofthisconstruction.Definition4.1.LetRbearingandletAbeasubsetofR.AlocalizationofRatAconsistsofaringloc(R,A)togetherwitharinghomomorphismαA:R→loc(R,A)suchthatαA(a)isaunitforalla∈Aandsuchthatthefollowinguniversalpropertyholds.Ifθ:R→Sisaringhomomorphismsuchthatθ(a)isaunitforalla∈A,thenthereexistsauniqueringhomomorphismθ′:loc(R,A)→Ssuchthatthefollowingdiagram3commutes.Rθ((PPPPPPPPPPPPPPPPPPαA//loc(R,A)θ′SForeveryringRan