monomorphisms in categories of log schemes

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1、MonomorphismsinCategoriesofLogSchemesShinichiMochizukiNovember2014Abstract.Inthepresentpaper,westudycategory-theoreticpropertiesofmonomor-phismsincategoriesoflogschemes.Thisstudyallowsonetogiveapurelycategory-theoreticreconstructionofthelogschemethatgaverisetothecategoryundercon-sideration.Wealsoobt

2、ainanalogousresultsforcategoriesofschemesoflocallyfinitetypeovertheringofrationalintegersthatareequippedwith“archimedeanstruc-tures”.Suchreconstructionswerediscussedintwopreviouspapersbytheauthor,butthesereconstructionscontainedsomeerrors,whichwerepointedouttotheau-thorbyC.NakayamaandY.Hoshi.Theseerr

3、orsrevolvearoundcertainelementarycombinatorialaspectsoffandecompositionsoftwo-dimensionalrationalpolyhedralcones—i.e.,ofthesortthatoccurintheclassicaltheoryoftoricvarieties—andmayberepairedbyapplyingthetheorydevelopedinthepresentpaper.Contents:§0.NotationsandConventions§1.GeneralitiesonMonomorphisms

4、andMinimalPoints§2.TheSchemeStructureofSubmonicLogSchemes§3.SeamlessPartitionsofOrientableLogSchemes§4.Category-theoreticRepresentationofArchimedeanStructuresAppendixIntroductionThepurposeofthepresentpaperistostudy,insomedetail,variousaspectsofthestructureofcategoriesoflogschemesthatrevolvearoundthe

5、behaviorofmonomorphismsinsuchcategories.Thisstudyleadsnaturallytoapurelycategory-theoreticreconstructionofthelogschemethatgaverisetothecategoryunderconsideration.Ourmainresultisthefollowing[cf.Theorem3.8,(iii)].TheoremA.(Category-theoreticreconstructionoflogschemes)Forlogi=1,2,letXibealocallynoether

6、ianfslogscheme[cf.thediscussionentitled“Logschemes”in§0].Fori=1,2,weshallwriteSchlog(Xlog)forthei2010MathematicsSubjectClassification.Primary14A15;Secondary14G40.Keywordsandphrases.Logscheme,scheme,category,anabeliangeometry,archimedeanstructure.TypesetbyAMS-TEX12SHINICHIMOCHIZUKIlogcategoryofnoether

7、ianfslogschemesoffinitetypeoverXiandmorphismsoffinitetype[cf.thediscussionatthebeginningof§1formoredetails].Letloglog∼loglogΦ:Sch(X1)→Sch(X2)bean[arbitrary!]equivalenceofcategories.Thenthereexistsauniqu