Quasi-Tilting Modules and Counter Equivalences

Abstract

Given two rings R and S, we study the category equivalences T T ¡ Y Y, where T T is a torsion class of R-modules and Y Y is a torsion-free class of S-modules. These Ž . equivalences correspond to quasi-tilting triples R, V, S , where V is a bimodule R S which has, ''locally,'' a tilting behavior. Comparing this setting with tilting bimodules and, more generally, with the torsion theory counter equivalences introduced by Colby and Fuller, we prove a local version of the Tilting Theorem for quasi-tilting triples. A whole section is devoted to examples in case of algebras over a field.

Representations of the generalized Kronecker quiver with countably many arrows

“…the article [EE05] and its references). In [CDT97], Example 5.3, the path algebra of Q was introduced as the direct limit of path algebras of finite generalized Kronecker quivers (see [HU91], p. 182). In [D'E00], questions concerning direct products of free and projective representations of infinite generalized Kronecker quivers have been considered.…”

Infinite dimensional tilting modules over finite dimensional algebras

“…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 [22,23,25,26,27,28,29]. Definitions of tilting and cotilting modules of higher were introduced in [3] and [58], where the definition introduced in [3] being the most widely used now.…”

Cotilting versus pure-injective modules

“…Recently they have been introduced [5] in the framework of modules over arbitrary associative rings, acquiring a proper independent role. The cotilting modules generalize the notion of injective cogenerator: They are injectives with respect to short exact sequences of modules cogenerated by them.…”

“…A left R-module R W is said to be cotilting [5] if Cogen R W = ⊥ W . The cotilting modules generalize injective cogenerators: Clearly R W is an injective cogenerator if and only if both the classes Cogen R W and ⊥ W coincide with the whole category of left R-modules.…”