Equivalences and the tilting theory

Abstract

We study a natural generalization of * n -modules (and hence also of * -modules) by introducing the notion of * ∞ -modules. The most important results about * n -modules (and also * -modules) are extended to * ∞ -modules (for example, Theorem 2.7, etc.). An interesting subclass of the class of * ∞ -modules, namely the class of ∞-tilting modules, may be viewed as a more natural generalization of tilting modules of finite projective dimension to infinite projective dimension. We show that the generalization of the Brenner-Butler theorem in the tilting theory holds for ∞-tilting modules (Theorem 3.9).

n-Star modules and n-tilting modules

“…Beginning with Miyashita [10], the defining conditions for a classical tilting module were extended to arbitrary rings by many authors, Wakamatsu [13], Colby and Fuller [5], Colpi and Trlifaj [8], and recently, Angeleri Hügel and Coelho [1], Bazzoni [2] and Wei [14]. Among them, Miyashita [10] considered tilting modules of finitely generated projective dimension n (simply, finitely n-tilting modules), Colpi and Trlifaj [8] investigated (not necessarily finitely generated) tilting modules of projective dimension 1 (simply, 1-tilting modules) and then, Angeleri Hügel and Coelho [1] and Bazzoni [2] necessarily finitely generated) tilting modules of projective dimension n (simply, n-tilting modules).…”

A characterization of the tilting pair

“…All the above mentioned tilting modules are of finite projective dimension. Tilting modules of infinitely projective dimension were studied by Wakamatsu [13] (later called Wakamatsu-tilting modules) and Wei [15] (∞-tilting modules).…”