Exact Structures And Tilting Theories

Exact categories and play an important role in homological algebra, representation theory, algebraic geometry, mathematical physics and so on. The concept of exact categories originated from Quillen in 1973. On the other hand, tilting theories aries from the representation theory of finite dimensional algebras, and goes back to the fundamental work by Bernstein, Gelfand and Ponomarev in 1973, later generalized by Brenner and Butler in 1980. For a long time, exact categories and tilting theories have gained some popularity, these theories promote mightly the development of homological algebra and representation theory of algebras. In this paper, we mainly study the following three aspects:exact structures in abelian categories, relative left derived functors of tensor product functors and tilting theories in functor categories.This paper is divided into four chapters.In Chapter 1, main results and preliminaries are stated.In Chapter 2, in an abelian category A with small Ext groups, we show that there exists a one-to-one correspondence between any two of the following:balanced pairs, subfunctors F of ExtA1(-,-) such that A has enough F-projectives and e-nough F-injectives and Quillen exact structures ε with enough ε-projectives and e-nough ε-injectives. In this case, we get a strengthened version of the translation of the Wakamatsu lemma to the exact context, and also prove that subcategories which are ε-resolving and epimorphic precovering with kernels in their right ε-orthogonal class and subcategories which are ε-coresolving and monomorphic preenveloping with cokernels in their left ε-orthogonal class are determined by each other. Then we apply these results to construct some (pre)enveloping and (pre)covering classes and complete hereditary ε-cotorsion pairs in the module category.In Chapter 3, we introduce and study the relative left derived functor Tor(?) (-,-) in the module category, which unifies several related left derived functors. Then we give some criteria for computing the (?)-resolution dimensions of modules in terms of the properties of Torn(?)(-,-). We also construct a complete and hereditary cotorsion pair with respect to exact structures. Some known results are obtained as corollaries.In Chapter 4, we introduce the notion of n-tilting (resp. n-cotilting) objects in functor categories and give some characterizations of n-tilting objects and n-tilting classes (resp. n-cotilting objects and n-cotilting classes). Our results extend results in [15] and [2].