On the non-existence of right almost split maps

TL;DR

This paper characterizes modules that can be codomains of right almost split maps over any ring, answering a long-standing question and providing new conditions for non-precovering modules.

Contribution

It proves that only finitely presented modules with local endomorphism rings can be codomains of right almost split maps, resolving a 40-year-old problem by M. Auslander.

Findings

Modules with local endomorphism rings are precisely the codomains of right almost split maps.

Provides a sufficient condition for modules to be non-precovering.

Demonstrates an application in morphisms determined by objects.

Abstract

We show that, over any ring, a module $C$ is a codomain of a right almost split map if and only if $C$ is a finitely presented module with local endomorphism ring; thus we give an answer to a 40 years old question by M. Auslander. Using the tools developed, we also provide a useful sufficient condition for a class of modules to be non-precovering. Finally, we show a non-trivial application in the general context of morphisms determined by object.