Recollements Of Derived Categories Of Derived Discrete Algebras

In this thesis,we focus on the normal form of recollements of the derived categories of a class of derived discrete algebras.Vossieck proved that a basic connected derived discrete algebra is derived equivalent to either a hereditary algebra of Dynkin type or a one-cycle gentle algebra not satisfying the clock-condition.The latter was shown by Bobi(?)ski,Gei(?) and Skowro(?)ski to be derived equivalent to a gentle algebra ?(r,n,m) with 1?r?n and m?0.The class of algebras we study is ?(r,n,0)with 1?r<n.Precisely,let A=?(r,n,0) and R be a recollement of the derived category D(A) of A by D(B) and D(C),where B and C are finite-dimensional algebras with B or C being the base field K.We prove that R belongs to a stratifying ladder.To prove this,we first show that ifLis a ladder containing R,then L is stratifying if and only if there is an indecomposable projective A-module or a simple A-module in the (?)-orbits of the image of B or C in D(A),where (?) is the derived Nakayama functor.The key to prove the main theorem is then to find out all indecomposable objects which generate recollements and prove that there exists a simple module or projective module in the (?)-orbit of any of these objects.For this,the twist functor arising from cycles of exceptional objects introduced by Broomhead,Pauksztello and Ploog is a useful tool.The general framework of this thesis is as follows:In chapter 1,we introduce the background of recollements and the main result of the thesis.In chapter 2,we collect some preliminaries on triangulated categories,derived categories,recollements and ladders of derived module categories and derived discrete algebras.In chapter 3,we describe the structure of the bound derived category of the algebra A =?(r,n,0)and we will find out two important auto-equivalences of it,which help us to locate the indecomposable projective modules and the simple modules in the (?) components of the Auslander-Reiten quiver.In chapter 4,we will find out all the indecomposable objects,which can generate a recollement of the category D(?(r,n,0)) and construct the corresponding recollements explicitly.Moreover,the main result is proved in this chapter.