Two Kinds Of Structure Properties Of The Endomorphism Algebras

Tilting theory was viewed as a natural generalization of Morita equivalence and played a significant role in studying the representation theory of Artin alge-bras. Cluster categories and cluster tilting theory were the categorical model of cluster algebras. After that, m-cluster categories and m-cluster tilting algebras were introduced as a generalization of cluster categories. There is a one-to-one correspondence between tilting modules of m-replicated algebras over hereditary algebras with projective dimension at most m and m-cluster tilting objects in m-cluster category. The m-cluster mutation in m-cluster categories can be realized as the mutation sequences of almost tilting modules in m-replicated algebras. These prompt us to further study m-replicated algebraic tilting theory. Tilting theory research mainly contains two aspects:one is the relationship among the tilting modules of the fixed algebra; the other one is the relationship between the representation categories of the endomorphism algebras of tilting modules and the module categories of own algebras. For a faithful almost tilting module, the tilting complements can be found one by one through mutation sequences. There-fore, mutation is important to study the tilting modules and the endomorphism algebras of tilting modules.Let A be a finite dimensional hereditary algebra over an algebraically closed field. There is a close relationship between the bounded derived category Db(A) and m-cluster category of A which is a factor category of Db(A). In order to discuss the relationship between them, Sm is defined as a fundamental domain in m-cluster category. For a silting object in Sm, the silting object can be also regarded as an m-cluster tilting object, the converse is also true. There are plenty of research results about the tilting theory of m-cluster categories. Hence in fundamental domain Sm, the tilting theory of derived categories can be researched by the related content of tilting theory of m-cluster categories.In this dissertation, for a finite dimensional hereditary algebra over an alge-braically closed field, we discuss the related contents of tilting theory about the bounded derived category and the m-replicated algebra on the theoretical basis of the bounded derived category Db(A), m-cluster category Cm(A) and m-replicated algebra A(m).The main contents are the structure properties of the endomorphis-m algebras of silting objects of Db(A) and the endomorphism algebras of tilting modules of A(m).Firstly, we investigate the related contents of tilting theory about the bound-ed derived category Db(A). On the one hand, the tilting quiver of A(m) can be established via the tilting mutations. Similarly, the m-cluster tilting quiver of Cm(A) can be got by the m-cluster mutation triangles. Inspired by the two quiv-ers, we build the silting quiver �'lε of Db(A) by the silting mutation triangles and prove that the �'lε is connected. QT stands for the quiver of the endomorphism algebras of silting objects, we show that QT has no loops and no 2-cycles, and the endomorphism algebra of silting object is a quasi-hereditary algebra with the global dimension is finite.On the other hand, for a basic silting object T of Sm, let T= EndD T be the endomorphism algebra of silting object T, J= add T and W=J* J[1] which is a triangulated category. We can get that the module category W/add (T[1]) is equivalent to mod Γ, which implies that the factor category induced by a triangulated category is equivalent to the module category of the endomorphism algebra of silting object. Therefore, we can obtain many abelian categories from a triangulated category. A basic almost silting object T of Sm has m+1 indecomposable silting complements M0, M1, …, Mm. Let Tj= EndD(Mj(?)T) be the endomorphism algebra of the silting object corresponding to the complement Mj,Jj= add (Mj(?)T) and Wj=Jj*Jj[1] with 0≤j≤ m. Likewise, the module category Wj/add (Mj (?) T)[1] is equivalent to mod Γj. Further, let Sm, be the simple top of the indecomposable projective Tj module HomD(Mj(?)T/Ti,Mj). Then we can get conclusion that the mod Tj/add Smj is equivalent to the Wj,/add (Mj (?)Mj-1(?)T)[1]. If j> m, we haveConsequently, we study the structure properties of the endomorphism alge-bras of tilting modules of A(m).We show the corresponding relationship between tilting modules of A(m) and silting objects of D+. Let T= T1(?)T2(?)…(?)Tn(?)P be a tilting A(m)-module, where P is the direct sum of all indecomposable projective-injective A(m)-modules. Each Ti is indecomposable with deg Ti< deg Ti+1. That is, for each Ti, there are an indecomposable A-module Ti’ and integer ki such that Ti≌Ω-κiTi’ for 0≤κi≤κi+1≤m. Then T’= T1’[κ1](?) T2’,[κ2](?)… (?)Tn’[κn] is a silting object of D+. Similarly, let T= T1(?) T2(?)…(?)Tn be a silting object of D+, where each Ti is indecomposable with deg Ti< deg Ti+1. i.e. for each Ti, there are an indecomposable A-module Ti’ and integer ki such that Ti≌Ti’[ki] for 0≤κi≤κi+1≤m. Therefore, T’=Ω-κ1T1’(?)Ω-κ2T2’(?)…(?)Ω-κnTn’(?)P is a tilting A(m)-module, where P is the direct sum of all indecomposable projective-injective A(m)-modules.A faithful almost tilting A(m)-module M has t+1 nonisomorphic indecom-posable complements. We prove that endomorphism algebras of the tilting mod-ules corresponding to the adjacent complements are derived-equivalence via a BB-tilting module, i.e. the endomorphism algebra EndA(m) (Xi (?) M) is derived-equivalent to the endomorphism algebra EndA(m) (Xi+i (?) M) via a BB-tilting module with 0≤i≤t-1. And we can get the relationship among the t+1 endomorphism algebras, i.e. where each Bi is BB-tilting module. According to invariants of the derived e-quivalence, the endomorphism algebra End^(m) (Xi (?) M) and the endomorphism algebra EndA(m) (Xi+1 (?) M) can keep the finiteness of global dimension and fini-tistic dimension.We also prove that all the endomorphism algebras of tilting modules of can be realized as the iterated endomorphism algebras of BB-tilting modules. That is, for each pair of basic tilting T1, T2, there exists a series of finite di-mensional algebras Ao, ∧0,∧1,..,∧s, which are the endomorphism algebras of some basic tilting A(m)-modules. And for each ∧i, there is a BB-tilting ∧i-module Bi such that ∧0＝EndA(m)T1, ∧i≌End∧i-1Bi-1,1≤i≤s, EndA(m)T2 ≌ End∧sBs.