How modules determine morphisms: The Auslander bijections as a frame for the representation theory of artin algebras.

Claus Michael Ringel (Bielefeld)

Abstract: Let A be an artin algebra. The lecture will report on the work of M. Auslander in his seminal Philadelphia Notes (published already in 1978). Auslander was very passionate about these investigations (they also form part of the final chapter of the Auslander-Reiten-Smalø book and could and should be seen as its culmination) . but the feedback until now is really 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 Auslander's 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 (but even he himself may not have realized it fully) was 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 looks quite technical and unattractive, at least at first sight. This may be the reason that for over 30 years, Auslander's powerful results did not gain the attention they deserve.


Ringel
Last modified: Wed May 23 17:49:20 CEST 2012