Recollements And Silting Theory

The recollement of triangulated categories was introduced by Beilinson,Bernstein,and Deligne in order to decompose derived categories of sheaves into two parts.It should be noted that it is difficult to establish the recollements of triangulated categories in general.Many scholars studied how to bulid recollements of triangulated categories,one of ways is using the theory of exact model structures;another useful tool is the infinitely generated tilting theory.As an extension of tilting theory,silting theory can be viewed as an adaptation of tilting theory to triangulated categories.Therefore,it is meaningful to study how to use the silting theory to produce recollements of triangulated categories.This thesis mainly investigates how to use the theory of exact model structures and silting to establish the recollements of triangulated categories.This thesis is divided into five chapters.In Chapter 1,main results and preliminaries are stated.In Chapter 2,we establish the recollements among homotopy categories of model categories associated to triangular matrix rings.Let A.B be two rings and T=(AO MB)with M an A-Bbimodule.Suppose that we are given two complete hereditary cotorsion pairs in A-Mod and B-Mod respectively.We define two cotorsion pairs in T-Mod and show that both of these cotorsion pairs are complete and hereditary.If we are given two cofibrantly generated model structures MA and MB on A-Mod and B-Mod respectively,then using the result above,we investigate when there exists a cofibrantly generated model structure MT on T-Mod and a recollement of Ho(MT)relative to Ho(MA)and Ho(MB).In Chapter 3,we establish the recollements of triangulated categories associated to functor categories over self-injective quivers.Let R be a ring and Q a self-injective quiver,and let Q,RMod be the category of k-linear functors from Q to the left R-module category R-Mod.Given a cotorsion pair in left R-module category,we construct four cotorsion pairs in Q,RMod and investigate when these cotorsion pairs are hereditary and complete.Moreover,under mild assumptions,we show that there exists a recollement among homotopy categories induced by these cotorsion pairs and the homotopy category of model categories constructed by Holm and J?rgensen.In Chapter 4,we show that there exist recollements of derived categories of dg-algebras induced by a good(co)silting dg-module.Let U be a dg-A-module and B the endomorphism dg-algebra of U.We show that if U is a good silting object,then there exists a dg-algebra C and a recollement among the derived categories D(C,d),D(B,d)and D(A,d).Conversely,we show that the existence of such a recollement implies goodness of silting objects.In order to deal with both silting and cosilting dg-modules consistently,the notion of weak silting dg-modules is introduced.Thus similar results for the good cosilting dg-modules are obtained.In Chapter 5,we investigate relations between silting and cosilting modules in Mod-R and Mod-Rm,where m is a maximal ideal of R.We study the localization properties of silting modules and the colocalization properties of cosilting modules,and then we construct a one-to-one correspondence between equivalence classes of cosilting R-modules C and equivalence classes of compatible families F of cosilting Rm-modules.