MORPHISMS DETERMINED BY OBJECTS: THE CASE OF MODULES OVER ARTIN ALGEBRAS

摘要

Let A be an artin algebra. In his Philadelphia Notes, M. Auslander showed that any homomorphism between A-mod-ules is right determined by a A-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 SmalΦ. 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 SmalΦ. What we encounter is an intimate relationship to the vanishing of Ext~2.