拉回正合范畴的若干问题分析

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1、1L福建师范大学李德梅博士学位论文AbstractExactcategoryisnotonlythefoundationofAbeliancategorybutalsoitsnat-uralgeneralization．Sincelast50sor60s，expertsandauthorshavestudiedexactcategories，especiallyintheonesbasedonadditivecategoriescommonlyattributedtoQuillen，whichhave

2、playedallimportantroleinalgebraicK—theory,therepresen-^tationtheory,categorytheory,etc．Theclassofexactcategories，whoseadmissiblemonomorphismsareclosedunderpull-backs，isinvestigatedinthethesis．These-eXauctcategoriescanbecommonlyfoundintorsiontheoryandthe

3、yareexactlydif-ferentfromabeliancategories．Inthisthesis，wecharacterizethelatticestructuresofpull-backexactcategorriesandalsogiveastudyonequivalenceanduniversalpropertitiesofthelocalizationsofmodularpull-backexactcategories．Atthebeginningofthethesis，wema

4、keareviewontheresearches，explorethetrendsrelatedtothisthesis，sumupthegroundworkandgivesomenotationspresentedinthethesis．Thethesisisdividedintofourchapters．Chapterone：Wefirstintroducethenotionofpull-backexactcategoryandgivesomedescriptionsofitsbasicprope

5、rties．Thenwegivesomeexmnplestoillustratetheexistenceoftheseexactcategories．Wealsoshowtheexistenceofnon—abelianpull-backexactcategories．Atlast，westudytherecollementsofpuU-backexactcategories．Chaptertwo：Thelatticestructuresonapull-backexactcategoryareinve

6、sti-gated．Usingadmissiblemonomorphismsandadmissibleepimorphisms，wedefinetwoclassesofmodularlattices(E，V，八)and(E，V，八)，andshowtheexistenceofalatticeisomorphismbetweenthesetwoclasses．Wealsousedistributiveelementstocharacterizelatticecategoryonapullbackexac

7、tcategoryandthenweobtainexactstructuresofalatticecategoryinducedbythoseofapull-backexactcategory．Chapterthree：WeuseSerreclassesanddirectlimitstodefinetwoclassesofIIlocalizationcategoriesofmodularpull-backexactcategories，respectively,andshowthesetwoclass

8、esareequivalent．Chapterfour：Wegiveafurtherstudyon．thelocalizationofmodularpuu-backexactcategoriesunderuniversalproperties．Keywords：exactcategory,(modular)pull-backexactcategory,lattice，Rec-ollement，localization，Serreclass，directl