DECIDABILITY OF THE THEORY OF MODULES OVER COMMUTA

DECIDABILITYOFTHETHEORYOFMODULESOVERCOMMUTATIVEVALUATIONDOMAINSG.PUNINSKI,V.PUNINSKAYA,C.TOFFALORIAbstract.Weprovethat,ifVisane®ectivelygivencommutativevaluationdomainsuchthatitsvaluegroupisdenseandarchimedean,thenthetheoryofallV-modulesisdecidable.1.IntroductionAclassicalstrategytoprovethedecidabilityofthetheoryTRofallmod-ulesoveragivenringRisto`eliminatequanti¯ers',thatis,totranslateuni-formlyanysentence¾inthelanguageofR-modulesintoasimplerequivalentsentence¾0withoutquanti¯ersorwherequanti¯ersareaslessaspossible,suchthatcheckingthetruthof¾0inR-modulesbecomesalmosttrivial.ThisisexactlythewayWandaSzmielewusedtoprovehercapitalresultopeningthislineofresearch[8]:thetheoryofabeliangroups(thatis,Z-modules)isdecidable.AfamousBaur{Monktheorem(see[4,Cor.2.13])givesagoodpushinageneralcase,overanarbitraryringR:everysentenceisequivalentinTRtoabooleancombinationof`invariant'sentences(whichare89sentences,sothatwehaveaneliminationofquanti¯ersdownto89level).Unfortunatelythestructureofinvariantsentencescanbeextremelycomplicated,whichoftenmakesafurthersyntacticalanalysisincrediblyhard.AmoremodernandpowerfulwaytoprovedecidabilityofthetheoryofallmodulesoveraringistousetheZieglerspectrumofR,ZgR,atopologicalspacewhosepointsareindecomposablepure-injectiveR-modules.Agoodaccountofthisapproachandanoverviewofexistingresultscanbefoundin[4,Ch.17]orunpublishedM.Prest'snotes[5].Althoughtheseideashavebeencirculatedforquiteawhile,therearefewexampleswherethisapproachwasputtothefullforce.Theproblemisthattomakeitworkweshould2000MathematicsSubjectClassi¯cation.03B25,13L05,13A18,13C11.Keywordsandphrases.Theoryofmodules,commutativevaluationdomain,decidabil-ity,Zieglerspectrum.The¯rstauthorispartiallysupportedbytheNSFgrantno.DMS-0612720.HealsothanksUniversityofCamerinoforkindhospitality.1collectasubstantialamountofinformationabouttheZieglerspectrumofR,bothaboutpointsandtopology.Evenforrelativelymoderateringsthisisaproblemofscaringcomplexity.IfVisacommutativevaluationdomain,acompleteclassi¯cationofpointsofZgVisknownfrom[9],andasatisfactorydescriptionofthetopologyisalsoavailable(seeforexample[6]).Thusitseemsreasonabletoexpectthatacharacterizationof(countable)commutativevaluationdomainswithadecidabletheoryofmodulesshouldnotbeaveryhardproblem.Forinstance,ifVis¯nite,thenithasa¯niterepresentationtype,henceTVisadecidabletheory.ItfollowsfromSzmielew'sresult[8]that,foreveryprimep,thetheoryofallmodulesoverthelocalizationZ(p)(whichisacommutativevaluationdomain)isdecidable.Aneasygeneralizationofthisresult(thatcanbealsoderivedfrom[1])isthatthetheoryofallmodulesoverane®ectivelygivennoetheriancommuta-tivevaluationdomainisdecidable.Thustheanswerisknownfor(e®ectivelygiven)commutativevaluationdomainswhosevaluegroupisisomorphictotheorderedgroup(Z;+;·)(thatis,fordiscreterankonecommutativeval-uationdomains).Inthispaperweconsideranoppositecase:whenthevaluegroupofarankonecommutativevaluationdomainisdenselyordered(say,orderlyisomorphictotherationals).Additionaldi±cultiesweencounterinthiscasearethattheZieglerspectrumofVisuncountable,andeven(see[7,Thm.12.12])thereexistsasuper-decomposablepure-injectiveV-module.Weshowthatnoneoftheseappearingobstaclesa®ectsdecidability.Namely,weprovethat,ifVisacommutativevaluationdomainwithadenselyor-deredarchimedeanvaluegroup,andVise®ectivelygiveninasensewearegoingtoexplainlater,thenthetheoryofallmodulesoverVisdecidable.Asitshouldbe,theproofofthisresultreliesontheZieglerspectrumapproachasitwasoutlinedin[9]or[4].Wealsohaveinmind(thoughsup-pressinproofs)ageometricalinterpretationofpositive-primitivetypesovercommutativevaluationdomainsasin[7,Ch.12].ThustodecidewhetheragivensentenceholdstrueinthetheoryofallV-modules,weshouldansweraquestionaboutacon¯gurationofrectanglesandlinesontheplane.IfthevaluegroupofVisdenselyorderedandarchimedeanthisapproachprovidesuswithaclearpictureconvertible(thoughwithsometechnicalities)intoaformalproof.Drawingdiagramsbacksmostproofsofthispaper,andwedoubtthattheycouldhavebeenworkedoutorunderstoodotherwise.2Weseparateourproofofdecidabilityintwocases:whentheresidue¯eldofVisin¯niteor¯nite.Theproofinthein¯nitecaseismoreconceptualandreliesmostlyontheusageofZieglertopology.Asitisquitecommon,the¯nitecaseisessentiallymoredi±cult,becauseacombinatoricsof¯niteinvariantscomesinplay.Luckilyweshowthat,ifthevaluegroupofVisdense,then¯niteinvariantsareratherrare,hencetheproofsarestillbearable.Anidealanswerwewouldexpectingeneralcase(thatis,forarbitrarycountablecommutativevaluationdomains)isthefollowing:thetheoryTVofallV-modulesisdecidableifandonlyifsomequestionsinthe¯rstordertheoryofV(asaring)canbeanswerede®ectively.Indeedthisiswhathappensinthedensearchimedeancase,asthecondition`Ve®ectivelygiven'justhasthiscontent.Anyhowweshowthat,ifavaluegroupofVisnon-archimedean,thensomenon¯rst-orderpartsofthetheoryofVcanbeencodedinthe(¯rstorder)theoryofV-modules.Thusthecase,whenthevaluegroupofVisnotarchimedean(ornotdense),appearstobeessentiallymoredi±cultandmayrequiretremendouscombinatoriale®orts.Theauthorsareindebtedtotherefereeforextremelycareful