Morphisms determined by objects: The case of modules over artin algebras

TL;DR

This paper provides complete proofs and detailed analysis of how morphisms between finitely generated modules over artin algebras are determined by specific modules, refining previous formulas and exploring the role of projective summands.

Contribution

It offers a complete, direct proof of Auslander’s results on right determiners of morphisms and analyzes the influence of projective summands in this context.

Findings

Complete proofs of main results on right determiners

Analysis of indecomposable projective summands

Characterization of morphisms determined by modules without projective summands

Abstract

We deal with finitely generated modules over an artin algebra. In his Philadelphia Notes, M.Auslander showed that any homomorphism is right determined by a module C, but a formula for C which he wrote down has to be modified. The paper includes now complete and direct proofs of the main results concerning right determiners of morphisms. We discuss the role of indecomposable projective direct summands of a minimal right determiner and provide a detailed analysis of those morphisms which are right determined by a module without any non-zero projective direct summand.