Comparing four definitions of cotilting modules

TL;DR

This paper compares four main definitions of cotilting modules, showing their equivalence under certain algebraic conditions, thus clarifying the dual theory of tilting modules.

Contribution

It provides a unification of different cotilting module definitions in specific algebraic settings, enhancing theoretical understanding.

Findings

Three definitions coincide over right Artinian Noetherian algebras

All four definitions coincide over Artin algebras

Clarifies the dual theory of tilting modules

Abstract

In contrast to the theory of tilting modules, the dual theory lacks a unified definition. Nevertheless, several notions of cotilting modules have been proposed. In this paper, we compare four of the main definitions of cotilting modules that have appeared in the literature. We show that, in the setting of finitely generated right modules, three of these definitions coincide over right Artinian Noetherian algebras, and all four coincide over Artin algebras.