The Trivial Extension Of Abelian Categories, Recollements, K-Theory And The Applications

Recollements and the algebraic K-theory are two important branches of the algebra research. According to their internal topologic and geometric meanings, and fruitful algebraic applications, they arouse the cross-development among many subjects and yield a series of profound results and the challenging research. This dissertation includes six chapters altogether, and concentrates on the relation among recollements, K-theory, trivial extensions of abelian categories and so on.In the first chapter, we introduce the background and the recent development of recollements, K-theory and triangulated categories. Then we sum up main results of this dissertation.In the second chapter, we apply category theories to compute K_i-groups of trivial extensions of abelian categories, and then obtain results about K_i-groups of the trivial extension of a ring by a bimodule.In the third chapter, we discuss whether recollements by trivial extensions of abelian categories are recollements or not. Then we get some results about Morita equivalences and idempotent completion categories.In the fourth chapter, we compute K_i- groups of recollements and obtain some results about K_i-groups of comma categories and lower triangular matrixes.In the fifth chapter, we study whether a recollement of abelian categories can induce a new recollement relative to abelian categories and comma categories or not.In the sixth chapter, we discuss the relation among K_i-groups of recollements, t-structures and idempotent completion categories.