12 - Infinite dimensional tilting modules over finite dimensional algebras

Summary

Introduction

The theory for tilting and cotilting modules has its roots in the representation theory of finite dimensional algebras (artin algebras) generalizing Morita equivalence and duality. First through reflection functors studied in [16] and a module theoretic interpretation of these in [8], tilting modules of projective dimension at most one got an axiomatic description in [18, 41]. Among others, [6, 17, 59] developed this theory further. These concepts were generalized in [40, 52] to tilting modules of arbitrary finite projective dimension. In the seminal paper [9] tilting and cotilting modules were characterized by special subcategories of the category of finitely presented modules. This paper started among other things the close connections between tilting and cotilting theory and homological conjectures studied further in [23, 42, 44].

Generalizations of tilting modules of projective dimension at most one to arbitrary associative rings have been considered in [4, 27, 32, 51]. As tilting and cotilting modules in this general setting is not necessarily linked by applying a duality, a parallel development of cotilting modules were pursued among others in [25, 24, 26, 28, 29, 30, 31]. Definitions of tilting and cotilting modules of arbitrary projective and injective dimension were introduced in [3] and [61], where the definition introduced in [3] being the most widely used now.