Approximations and locally free modules

TL;DR

This paper investigates the existence of precovers in module theory, proving their existence under certain conditions and showing their non-existence in specific cases involving locally free modules and tilting theory.

Contribution

It establishes new conditions for the existence of precovers and demonstrates non-precovering results for locally free modules induced by tilting modules.

Findings

Precovers exist for modules with bounded C-resolution dimension.

Locally free modules induced by non-sum-pure-split tilting modules are not precovering.

The class of locally Baer modules is not precovering for certain hereditary artin algebras.

Abstract

For any set of modules S, we prove the existence of precovers (right approximations) for all classes of modules of bounded C-resolution dimension, where C is the class of all S-filtered modules. In contrast, we use infinite dimensional tilting theory to show that the class of all locally free modules induced by a non-sum-pure-split tilting module is not precovering. Consequently, the class of all locally Baer modules is not precovering for any countable hereditary artin algebra of infinite representation type.