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

Abstract

Let $\Lambda$ be an artin algebra. In his Philadelphia Notes, M. Auslander showed that any homomorphism between $\Lambda$-modules is right determined by a $\Lambda$-module $C$, but a formula for $C$ which he wrote down has to be modified. The paper presents corresponding counter-examples, but also provides a quite short proof of Auslander’s assertion that any homomorphism is right determined by a module. Using the same methods, we describe the minimal right determiner of a morphism, as discussed in the book by Auslander, Reiten and Smalø. In addition, we look at the role of indecomposable projective direct summands of a minimal right determiner and provide a detailed analysis of the kernel-determined morphisms: these are those morphisms which are right determined by a module without any non-zero projective direct summand. In this way, we answer a question raised in the book by Auslander, Reiten and Smalø. What we encounter is an intimate relationship to the vanishing of $\operatorname{Ext}^{2}$.