| In 1988,Keller and Vossieck introduced the notion of the silting object when they studied t-structures on bounded derived categories of representation-finite hered-itary algebras.As the generalization of tilting objects,silting objects were studied sporadically in the following years.Until 2012,the silting mutation introduced by Aihara and Iyama overcomes the shortcoming of tilting theory and attracts more attention.And the latest researches also show that silting objects are closely related to support τ-tilting modules and cluster tilting objects.The doctoral dissertation fo-cuses on topics related to the silting theory of triangulated categories,which includes the following six chapters.At the beginning of this dissertation,we introduce the development history of the silting theory,and introduce the research content and framework.In the first chapter,we review the main concepts and some known conclusions involved in this dissertation.In the second chapter,we study the relations between silting objects and hered-itary triangulated categories.We give the the concept of the strong global dimension of the triangulated category with respect to the silting object.We show that the triangulated category is hereditary if and only if the strong global dimension of the triangulated category is finite,which generalizes Happel-Zacharia’s result.Further-more,we give the bounds of strong global dimensions under triangle equivalences.Meanwhile,we construct the higher cluster tilting subcategories by using the silting objects.In the third chapter,we consider the problem of the preservation of two-term silting objects under the silting mutation.By investigating the properties of the t-structures induced by the bilateral Bongartz’s complements of the two term presilting objects,we give a necessary and sufficient condition for the silting mutation to keep two term silting objects.And we show that the left and right Bongartz’s complements are connected by a triangle.In the fourth chapter,we investigate the problem of HRS-tilting in the wide subcategory.We construct a wide subcategory of the heart of t-structure induced by the given silting object.We prove that the wide subcategory is Morita equivalent to module category for some explicitly constructed finite dimensional algebra by determining the projective generator.Moreover,we restrict the HRS-tilting to the wide subcategory,and show that there is a one-to-one correspondence between the torsion pairs in wide subcategory and the intermediate t-structures between the two t-structure induced by the bilateral Bongartz’s complements of the two term presilting objects.In the fifth chapter,we study derived equivalences between generalized matrix algebras.For a given generalized matrix algebra,the tilting complexes for the alge-bras on the diagonal can be lifted to being the tilting complex for the generalized matrix algebra by using some adjoint functor pairs.Meanwhile we determine its endomorphism algebra which is derived equivalent to the given one.As application-s,we establish various derived equivalences between the typical generalized matrix algebras. |
