Some Researches On Tilting Modules And Frobenius Categories

Tilting theory was introduced and by S.Brenner,MCR Butler,K.Bongartz, D.Happel and CM Ringel and so on.Since it plays a very important role in the representation theory of the Artin algebra,as well as in the research of the dual theory and equivalence theory of modules category,it attracts much scholars' attention.This dissertation includes five chapters altogether,and concentrates on the problems about the tilting modules of the trivial extension algebra and the idempotent completion of Frobenius categories.Chapterâ… gives an introduction to the background and recent development in the tilting theory,the idempotent completion of triangulated categories and Frobenius categories.Meanwhile,sum up the main results of this dissertation.Chapterâ…¡introduces some concepts and properties briefly,such as the tilting adjoint pair,the tilting pair,the trivial extension of a category by an endofunctor, Frobenius categories.Chapterâ…¢discusses when the trivial extension of a tilting adjoint pair is still a tilting adjoint pair,and gets the corresponding corollary when the tilting adjoint pair is determined by a tilting module,i.e.lifting the tilting module to the trivial extension algebra.Chapterâ…£uses the same method to lift exceptional modules and 1-tilting pairs to the ons of the trivial extension algebra.Chapterâ…¤proves that the idempotent completion of an exact category(Frobenius category) is still an exact category(Frobenius category).