Researches On Recollements Of Categories

Beilinson,Bernstein and Deligne first introduced the notion of recollement of triangulated categories in their study of geometry of singular spaces in 1982. A recollement of categories describes one such category as being "glued together" from two others,which plays an important role in many fields of mathematics.This dissertation concentrates on recollements of abelian and triangulated categories. It includes seven parts altogether.In the preface,we give a brief introduction of the background and recent developments related to this dissertation,and make a systemic exposition of our main results.In the first chapter,werecall some foundamentary concepts and basic properties relevant to this dissertation,which give a necessary preparation for the following chapters.In the second chapter,we study the relationship between one-point extensions and recollements of module categories.We prove that for algebras A,B and C, if A-mod admits a recollement relative to B-mod and C-mod,then A[R]-mod admits a recollement relative to S[S]-mod and C-mod,where AIR]and B[S]are the one-point extensions of A by R and of B by S.In particular,we deduce that if two finite dimensional algebras are Morita equivalent,then so are their respective one-point extensions.In the third chapter,we study the relationship between one-point extensions and recollements of derived categories.We prove that for algebras A,B and C,if D-(A-Mod) admits a recollement relative to D-(B-Mod) and D-(C-Mod),then D-(A[M]-Mod) admits a recollement relative to D-(B[N]-Mod) and D-(C-Mod). As a consequence,we obtain the main result of[55],i.e.,if two finite dimensional algebras are derived equivalent,then so are their respective one-point extensions.In the fourth chapter,we study the relationship between quotient categories and recollements.Mainly discuss how a recollement of triangulated categories induces a recollement of Verdier quotient categories.In particular,if U is a localizing (or colocalizing) subcategory of a triangulated category D and V is a triangulated subcategory of U,then U/V is a localizing(or colocalizing) subcategory of D/V and we have a triangulated equivalence(D/V)/(U/V)≌D/U.Similarly a recollement of abelian categories can induce a recollement of Serre quotient categories.In the fifth chapter,we study the relationship between AR-triangles of a triangulated category D and those of quotient category D/U.We get several necessary and sufficient conditions for the image of an AR-triangle of D by the quotient functor Q is an AR-triangle of D/U.Moreover,we get a necessary and sufficient criterion for an AR-triangle of D/U is induced by an AR-triangle of D. Then we apply them to the recollement of triangulated categories and the case of hereditary abelian category.In the sixth chapter,we study how a recollement of triangulated categories induces a recollement of abelian categories.Let D,D' and D" be triangulated categories.Suppose D admits a recollement relative to D' and D".We give some necessary and sufficient criterions for a t-structrue on D induces t-structures on D' and D".Moreover,we take the advantage of the relationship between a t-structure on D and t-structures on D' and D" to show that the heart of the t-structure on D admits a recollement relative to the other two hearts.Thus,we get a few recollements of abelian categories according to a recollement of triangulated categories.