The Geometry of Linear Regular Types

arXiv:math/0406543v1[math.LO]26Jun2004THEGEOMETRYOFLINEARREGULARTYPESTRISTRAMDEPIROAbstract.Thispaperisconcernedwithextendingtheresultsof[2]inthecontextofthesolutionsetDofaregularLstpdefinedover∅inasimpletheoryT.In[7],anotionofp-weightisde-velopedforregulartypesinstabletheories.Hereweshowthatthecorrespondingnotionholdsinsimpletheoriesandgiveageo-metricanalysisofassociatedstructuresG(D)andG(D)large,theformerofwhichappearsin[2].WeshowthatDislineariffG(D)andG(D)large(localized,resp)arebothmodularwithrespecttothep-closureoperatorclp.Finally,weshowthatmodularityofG(D)largeprovidesalocalanalogueof1-basednessforthetheoryT.1.PreliminariesForconveniencewewillassumethattheambienttheoryTissuper-simple.In[4],KimshowsthatforkinginsidesimpletheoriessatisfiestheIndependenceTheoremoveramodelM.Inordertoapplytheindependencetheoremoverparameters,thenotionofLascarstrongtypeisintroduced.Asisshownin[1],ifTissupersimple,thenThaseliminationofhyperimaginariesandtheno-tionofLascarStrongTypesimplifiestothefollowing;Lstp(¯a/A)=Lstp(¯b/A)ifftp(¯a/acl(A))=tp(¯b/acl(A))iffstp(¯a/A)=stp(¯b/A)whereacl(A)denotesthealgebraicclosureofAinMeq.WecanthenapplytheIndependenceTheoremforLascarstrongtypes;If¯c⌣|¯dA,Lstp(¯a/A)=Lstp(¯b/A)and¯a⌣|¯cA,¯b⌣|¯dATheauthorwassupportedbytheWilliamGordonSeggieBrownResearchFellowship.12TRISTRAMDEPIROthenthenonforkingextensionsoftp(¯a/A)=tp(¯b/A)toA¯candA¯drespectivelycanbeamalgamated.Givenacompletetypep(¯x,b),wherebdenotesapossiblyinfinitesetofparameters,wedefinetheparallelismclassofp(¯x,b)tobe;B={p(¯x,c):E(c,b)}whereEisthetransitiveclosureoftherelationR(c,b)≡p(¯x,c)∪p(¯x,b)haveacommonnonforkingextensionAsisshownin[6],ifbisanamalgamationbase,thenEisatypedefinableequivalencerelationontp(b).IfTissupersimple,thentheparametersetofstp(a/B)isanamalgamationbaseandwedefinethecanonicalbaseC=Cb(Lstp(a/B))tobetheEclassofacl(B).Again,assumingTsupersimple,Eistheintersectionofdefinableequivalencerelationsontp(acl(B))andwemaytakeCtobeapossiblyinfinitesetofparametersinMeq.NotetheassumptionthatTissupersim-pleisnotcriticalinwhatfollowsprovidedweworkwithMheqinsteadofMeq.Ingeneral,wedonotassumethatourparametersetsarealgebraicallyclosed.Asthenotionofnon-forkingisinvariantunderalgebraicclosureinMeq,weoftenimplicitlyreplaceaparametersetBbyitsalgebraicclosureinMeq,hopingthiswillnotcauseconfusion.Wewillrequirethefollowingfactsaboutcanonicalbasesasgivenin[6],[5]and[8];Fact1.1.1.TheIndependenceTheoremholdsfortherestrictionofaLstpoverAtothebaseC⊂acl(A).LetA⊂Bbesetsand¯aatuple,then;2.¯a⌣|BAiffCb(Lstp(¯a/B))⊂acl(A).Asaconsequence,ifC=Cb(Lstp(¯a/A)),then¯a⌣|ACand¯a⌣|CA3.IfD=Cb(Lstp(¯a/B))and¯a⌣|BA,then,usingthefactthatCandDareamalgamationbases,dcl(C)=dcl(D).THEGEOMETRYOFLINEARREGULARTYPES34.If{¯ai:iω}isaMorleysequenceinLstp(¯a/A),thenC=Cb(Lstp(¯a/A))⊂dcl(¯ai:iω)Definition1.2.ApregeometryisasetSwithaclosureoperationcl:P(S)→P(S)satisfyingthefollowingaxiomsfoundin[7];1.IfA⊆S,thenA⊆cl(A),cl(A)=cl(cl(A)).2.IfA⊆B⊆S,thencl(A)⊆cl(B).3.IfA⊆S,a,b∈S,thena∈cl(Ab)\cl(A)impliesb∈cl(Aa).4.Ifa∈Sanda∈cl(A),thenthereissomefiniteA0⊂Awitha∈cl(A0).Wesaythat(S,cl)ismodularifforA,BfinitedimensionalclosedsubsetsofS,dim(A∪B)=dim(A)+dim(B)−dim(A∩B).Remarks1.3.Anecessaryandsufficientconditionformodularityofapregeometry(S,cl)isthefollowing;Whenevera,b∈S,B⊂Sisclosedandfinitedimensional,dim(ab)=2anddim(ab/B)≤1,thenthereisc∈cl(ab)∩cl(B)withc/∈cl(∅)(*)2.RegularTypesandp-weightLetpbeanon-algebraiccompleteLascarstrongtypeover∅.Recall-ingthedefinitionoforthogonalityinsimpletheories,seeforexample[8],wesaythatpisregularifitisorthogonaltoallitsforkingexten-sions.Lemma2.1.Ifpisregular,therealisationsDofpformapregeometrywiththetheclosureoperationclgivenbycl(A)={x∈p:x6↓A∅}.Proof.Wechecktheaxioms,2istrivialand4followsfromthefinitecharacterofforking.3followsimmediatelyfromforkingsymmetryandalltheworkisinshowingthat1holds,namelywehavetoseethatifA⊂p,a,b1...bnisatupleinpsuchthatbi6↓A∅foreachianda6↓b1,...,bn∅theninfacta6↓A∅.Supposenot,soa4TRISTRAMDEPIROrealisesanonforkingextensionofρtoA.EachbirealisesaforkingextensionofρtoAsobydefinitionofregularity,wemusthavethata⌣|b1A.NowwejustrepeattheargumentwithAb1replacingA,clearlybi6↓Ab1∅fori≥2andagainusingregularitya⌣|b2Ab1,sowegeta⌣|b1b2A.Afternsteps,usingtransitivity,wehavethata⌣|b1...bnAandso,asa⌣|A∅wegeta⌣|b1...bn∅.Thiscontradictstheoriginalhypothesis.GivenasetofparametersA,weletDA={x∈D:x⌣|A∅}anddefineaclosureoperationclAbyclA(B)={x∈DA:x6↓BA}forB⊂DA.Imitatingtheaboveproof,oneeasilychecksthat(DA,clA)formsapregeometrywhichwerefertoasthelocalisationpAofptoA.Givenanypregeometry(S,cl),weusethestandardnotation(S′,cl′),asin[2],fortheassociatedgeometry.ForclosedB⊂S,wehaveanotionofdimensiondim(B).ForclosedsetsB⊂C⊂S,wedefinedim(C/B)=dim(C)−dim(B)andforarbitrarysetsB,C⊂S,wedefinedim(C/B)=dim(cl(C∪B)/cl(B)).RecallthatthisnotionisadditiveandthesameholdsforthelocalisedanaloguedimA.Letp1andp2betypesoverpossiblydifferentsets.Wesaythatp1ishereditarilyorthogonaltop2ifeveryextensionofp1isorthogonaltop2.NowfixaregularcompleteLstppdefinedover∅anddefineaLstpqoveradomainBtobep-simpleif,foralla