The mathematical influence of Maurice Auslander in Mexico.

Summary

The first visit of Maurice Auslander to Mexico was in the summer of 1975. He lectured on several subjects in the representation theory of algebras. We were impressed mainly by the part of the lectures related to almost split sequences, then recently discovered by M. Auslander and I. Reiten.

At that time we were interested in the Coxeter and reflection functors introduced by Bernstein-Gelfand-Ponomarev [10].

Apparently there were some connections between Coxeter functors and Dtr. Later on [11] these connections were in fact established.

During 1976 – 1977 Roberto Matìnez and the author spent two years at Brandeis University. There, we had the opportunity of knowing, and living in, an exciting atmosphere. We met many people through Maurice who were interested in the representation theory of algebras.

In the following we recall some of the mathematical results in representation theory obtained in Mexico due to the influence of Maurice Auslander.

Almost split sequences and irreducible maps.

Take A an artin algebra, denote by mod △ the full sub category of the category of left △–modules whose objects are finitely generated modules.

We recall the definition of almost split sequence.

Definition: An exact sequence in mod △:

is said to be almost split sequence if:

i)the sequence does not split,

ii) A and C are indecomposables,

iii) if f : X > C is nonsplittable mono, there is some g : X > B such that jg = f.