AR-sequences Of Serre Quotient Categories And Stable Categories Of One-sided Categories

Quotient and localization are two important methods to construct new categoriesfrom the known categories. This paper gives the connection between stable categoriesand localization of categories, researches the AR-sequences of Serre quotientcategories of finitely generated modules over finite dimensional-algebras anddiscusses stable categories of one-sided categories by rigid subcategories.We recall the developments of AR-sequences, AR triangles and constructingabelian categories from triangulated categories, which are the hot issues in therepresentation theory of algebras in recent years. Then we put forward to the researchquestions and give ideas and methods of the research questions.Using the fundamental knowledge in the theory of categories, we prove a fewisomorphism propositions on additive categories which extend the fundamentalisomorphism theorems of groups to the level of categories, and get a connectionbetween the stable categories and the localization of additive categories.Let be a finite dimensional-algebra, be the category of finitely generated-modules and be a thick subcategory of. We discuss the relationship betweenthe AR-sequences of and of by transforming the AR-sequences of tothe AR-triangles of the bounded derived category. We get some necessary andsufficient conditions that the AR-sequences of are induced by theAR-sequences of.Left triangulated categories and right triangulated categories are called one-sidedtriangulated categories, which generalize the notions of triangulated categories andabelian categories. By comprising the difference and connection between one-sidedtriangulated categories and triangulated categories, we prove that some subquotientcategories by rigid subcategories of one-sided triangulated categories are modulecategories over the rigid subcategories, which unifies the previous work oftriangulated categories by Iyama-Yoshino and of exact categories by Demonet-Liu.