Homological Aspects of the Dual Auslander Transpose

TL;DR

This paper introduces the cotranspose of modules relative to a semidualizing module, explores its properties, and applies it to characterize classes of modules and answer an open question in the context of artin algebras.

Contribution

It defines and studies the $n$-$C$-cotorsionfree modules, establishing their dual properties and applications in module theory and algebra.

Findings

$n$-$C$-cotorsionfree modules characterize the Bass class.

They provide tools for module approximation theory.

Counterexample to an open question by Huang and Huang.

Abstract

As a dual of the Auslander transpose of modules, we introduce and study the cotranspose of modules with respect to a semidualizing module $C$. Then using it we introduce $n$-$C$-cotorsionfree modules, and show that $n$-$C$-cotorsionfree modules possess many dual properties of $n$-torsionfree modules. In particular, we show that $n$-$C$-cotorsionfree modules are useful in characterizing the Bass class and investigating the approximation theory for modules. Moreover, we study $n$-cotorsionfree modules over artin algebras and answer negatively an open question of Huang and Huang posed in 2012.