Some Researches On Recollements Of Triangulated And Abelian Categories

The notion of recollements was introduced by A.Beilinson,J.Bernstein and P.Deligne.Because of their internal geometric meanings and fruitful algebric applications, rec-ollements have attracted many people's attention. This dissertation concentrates onthe preservability of recollements, which ineludes six chapters altogether.The first chapter gives an introduction to the background and recent develop-ment in triangulated categories,derived categories and recollements. Meanwhile,sumup the main results of this dissertation.The second chapter discusses the K-theory of triangulated categories in right(or left) recollements. Proves that if a triangulated category (?) admits a right (or left)recollement relative to (?)' and (?)" then K₀(?) (?) K₀(?)⊕K₀((?)") and K₁^det(?)K₁^det⊕K₁^det((?)").In the third chpater,the relaionship among the dimensions of triangulated cat-egories in right (or left) recollements and the dimension of D^b(A) are given, where Ais a finte-dimensional hereditary k-algebra.Chpater four points out the relationship between idempotent completion andabelianization.Proves that the idempotent completion of a right (or left) recollementof triangulated categories is still a right (or left) recollement, while the abelianizationof a recollement is a right recollement.In the fifth chapter,recollements of module categories of category algebras arestudied.In chapter six,a recollement of the category of sheaves of abelian groups isconstructed.