Simplicity Theory

235 Pages·2013·1.35 MB·English

by  Byunghan Kim

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Simplicity Theory
BYUNGHAN KIM
YonseiUniversity,Seoul,RepublicofKorea
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PREFACE
Thisbookisaboutsimplefirst-ordertheories.Theclassofsimpletheorieswasintroduced
by S. Shelah in the early 1980s [112]. Then several specific algebraic structures having
simpletheorieswerestudiedbyleadingresearchers,notablyE.Hrushovski[23],[26],[55],
[58],[62].Inthemid-90stheauthorestablishedinhisthesisthesymmetryandtransitiv-
ityofnon-forkinginsimpletheories[65],[66]and,withA.Pillay,type-amalgamationover
modelsandLascarstrongtypes[74].Sincethenagreatdealofresearchworkhasbeen
producedonsimplicitytheory,thestudyofsimpletheoriesandstructures.
Theclassofsimpletheoriesproperlycontainsthatofstabletheories.Sincethe1960s
and1970s,Shelah’sstabilitytheory(=thestudyofstabletheories/structures)[113]has
become a main subject of model theory. In particular, geometric stability theory [99],
studying the geometrical nature of forking for stable structures, has turned out to be
a major technical bridge connecting model theory and its applications to number the-
ory and algebraic geometry. Hrushovski’s full resolution of the function-field version of
the Mordell–Lang conjecture in number theory [53] using Zilber’s principle [59] is a
spectacularexample.
Simplicity theory has been developed by generalizing the results in stability theory,
investigating the more complicated nature of simple structures, and revealing the inter-
relationship between stability theory and simple structures. The initial developments in
simplicitytheory,mostlyfocusedongeneralfoundationalissues,aresurveyedin[75],and
anexpositorybookbyF.O.Wagnerwaspublishedin2000[120].
Duringthepastseveralyears,theresearchtrendsinsimplicitytheoryhaveexpandedto
includegeometricsimplicitytheory,whichprimarilyconcernsthecombinatorialgeometric
oralgebraicsidesofsimplicitytheory,paralleltogeometricstabilitytheory.Oneofthemain
goalsofthisnewbook,inadditiontobeingaguidetograduatestudentsstudyingmodel
theory,istointroducethenewresultsongeometricsimplicitytheory.
Throughout this book, some familiarity with basic model theory and stability theory
summarizedinChapter1isassumed.InChapter2basicnotionssuchasdividing,fork-
ing,simplicity,andthetreepropertyareintroduced.Thefundamentaltheoremofforking
isproved,andseveralranknotionsarepresented.Thenon-finiteness(except1)ofthenum-
berofacountablesupersimpletheoryisshown,andsomeexamplesofsimpletheoriesare
given.
Chapter3isdevotedtoLascarstrongtypes,andtypeamalgamationoverLascarstrong
typesisproved.Onegoalistoshowthatbasicindependencerelationstogetherwithtype
amalgamationcharacterizesimplicityandforking.
InChapter4webegintoworkwithhyperimaginaries.Ahyperimaginaryisanequival-
enceclassofatype-definableequivalencerelation.Thesearenecessaryasthecanonical
baseforaLascartypeinasimpletheoryexistsintheformofahyperimaginary.Forking
vi | preface
makessensewithrespecttohyperimaginariesandbasicnon-forkingindependencerela-
tionsandtypeamalgamationcanberestatedintermsofhyperimaginaries.Deeperforking
relationshipsbetweentypesareintroducedandstudiedinthelastsection.
Chapter5addressestheissueofeliminationofhyperimaginaries.TheLascartopolo-
gicalgroupofautomorphismsanditsquotientgroupsareinvestigatedaswell.Atheory
haseliminationofhyperimaginariesifoneachcompletetype,everytype-definableequi-
valence relation is a conjunction of definable equivalence relations. Equivalently, any
hyperimaginaryisinterdefinablewithasequenceofimaginaries.Ifasimpletheoryhaselim-
inationofhyperimaginariesthenLascartypesarestrongtypesandcanonicalbasesexist
asimaginaries.Thatlow(small,resp.)simpletheorieseliminatebounded(finitary,resp.)
hyperimaginaries is shown. Themain resultby Buechler,Pillay, and Wagner is that any
supersimpletheoryeliminateshyperimaginaries.Theresulthasconsequencesinregardto
thedefinabilityandcanonicalbasesofLascartypes.
In Chapter 6, several methods of constructing simple structures, including the
Hrushovski construction, are introduced. These methods are used in finding a counter-
example,orproducingacanonicalunstablesimplestructure(preservingnon-forkinginde-
pendence)fromagivenstablestructure,orusedasatechnicaltoolforobtainingdeeper
results.
Chapter 7 develops the theory of groups in simple structures, most of which were
establishedbyF.O.WagnerfollowingA.Pillay’sinitialwork[98].Thegroupcaneither
be type-definable or hyperdefinable. Commensurativity, a generalization of the group
theoretic notion commensurability, and local connectivity play crucial roles in develop-
ing hyperdefinable group theory. Important results on 1-based groups and supersimple
groups are stated. In particular, the canonical vector space structure found in a min-
imal connected 1-based group will be used in Chapter 9, together with the procedure
forobtainingahyperdefinablegroupanditshyperdefinableactionfromgenericallygiven
data.
Chapter8dealswithgeometricsimplicitytheory.InvestigatingandextendingZilber’s
principle (for stability theory) into simplicity theory is a motivation for this chapter.
Although its full resolution, as in stability theory, is far beyond the reach of current
knowledge,someinitialresultsandpartialanswersareprovidedforω-categoricalcontext.
Chapter9continuesChapter8inanarbitrarycontext.Thegroupconfigurationtheorem
isestablishedundergeneralizedamalgamation.ThenbycombiningresultsfromChapter7,
acanonicalvectorspaceoverthedivisionringofendogeniesishyperdefinablyrecovered
fromanynon-trivialmodularstructure.Generalizedamalgamationnotionsthemselvesare
studiedaswell.Itsevolvementtowardhomologytheoryisaninterestingtopicbutisnot
dealtwithinthisbook.Currenttopicsofresearch,someareevennon-firstorder,going
beyondsimplicity(butanalogoustoit)arenotdealtwitheither.
Itshouldbenotedthatthereisabsolutelynointentionheretomakeanencyclopaedic
presentationofsimplicitytheory,whichnowadaysisanalmostimpossibletaskastoomuch
hasbeendevelopedtocompletelyabsorb(orevenpresent)itall.Theresultsinthisbook
were selected (within the author’s narrow knowledge) mainly to bring the reader up to
speedonthecurrentworkson(geometric)simplicitytheory.
preface | vii
InMarch2012,duringhisvisittoBogotá,Colombia,theauthor’scomputerwasstolen,
inwhichpartoftheLaTeXfileforthisbookwrittenuntilthenhadbeenstored.Noother
savedLaTeXfileswerefound;however,apdffilewasstoredonline.Specialthanksaregiven
toHyeung-JoonKim,SunYoungKim,andJungukLee,whohelpedre-typeacompilable
LaTeXfilefromthepdffile.Partofthisbookwaswrittenduringtheauthor’svisittoMITin
2012andthanksaregivenfortheuseofitsfacilities.Theauthorexpresseshisdeepthanks
tohisfamilyfortheirloveandpatience.ThisbookwassupportedbyNRFofKoreaGrant
2011-812-A00042.
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CONTENTS
1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1
1.1 ModelTheory 3
1.2 Stability 7
1.3 BibliographicalRemarks 11
2 Dividing,Forking,andSimplicity . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.1 DividingandForking 12
2.2 SimplicityandForking 17
2.3 Dϕ,k-RanksandtheTreeProperty 21
2.4 FundamentalTheoremofForking 29
2.5 RanksandSupersimpleTheories 31
2.6 ExamplesofSimpleTheories 42
2.7 BibliographicalRemarks 44
3 LascarStrongTypesandTypeAmalgamation . . . . . . . . . . . . . . . . . 45
3.1 LascarStrongTypes 45
3.2 TypeAmalgamation 47
3.3 CharacterizingSimpleTheories 52
3.4 BibliographicalRemarks 54
4 HyperimaginariesandCanonicalBases . . . . . . . . . . . . . . . . . . . . . 55
4.1 Hyperimaginaries 55
4.2 Non-forkingIndependenceofHyperimaginaries 60
4.3 CanonicalBases 66
4.4 ForkingRelationshipsbetweenTypes 70
4.5 BibliographicalRemarks 80
5 EliminationofHyperimaginaries . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.1 TheLascarGroup 82
5.2 LascarTypesAreStrongTypesinLowTheories 91
5.3 EliminationofFinitaryHyperimaginariesforSmallTheories 92
5.4 EliminationofHyperimaginariesforSupersimpleTheories 96
5.5 Definability 101
5.6 BibliographicalRemarks 107
6 ConstructingSimpleStructures . . . . . . . . . . . . . . . . . . . . . . . . . 108
6.1 ModularityandCM-Triviality 109
6.2 HrushovskiConstruction 113