The Auslander bijections: How morphisms are determined by modules

TL;DR

This paper surveys Auslander's theory of how morphisms in module categories over artin algebras are determined by modules, highlighting its broader framework beyond classical Auslander-Reiten theory and providing numerous examples.

Contribution

It clarifies Auslander's original concepts, expanding the understanding of morphism determination by modules and illustrating their applications through various examples.

Findings

Provides a comprehensive overview of Auslander's morphism determination theory

Shows how the theory extends classical Auslander-Reiten results

Includes numerous illustrative examples of the theory in practice

Abstract

Let A be an artin algebra. In his seminal Philadelphia Notes published in 1978, M. Auslander introduced the concept of morphisms being determined by modules. Auslander was very passionate about these ivestigations (they also form part of the final chapter of the Auslander-Reiten-Smaloe book and could and should be seen as its culmination), but the feedback until now seems to be somewhat meager. The theory presented by Auslander has to be considered as an exciting frame for working with the category of A-modules, incorporating all what is known about irreducible maps (the usual Auslander-Reiten theory), but the frame is much wider and allows for example to take into account families of modules - an important feature of module categories. What Auslander has achieved is a clear description of the poset structure of the category of A-modules as well as a blueprint for interrelating individual modules and families of modules. Auslander has subsumed his considerations under the heading of "morphisms being determined by modules". Unfortunately, the wording in itself seems to be somewhat misleading, and the basic definition may look quite technical and unattractive, at least at first sight. This could be the reason that for over 30 years, Auslander's powerful results did not gain the attention they deserve. The aim of this survey is to outline the general setting for Auslander's ideas and to show the wealth of these ideas by exhibiting many examples.