Daniel Bissinger

An Introduction to Auslander-Reiten Theory

 

Lecturer

Daniel Bissinger

 

Class Times

We 10:15 - 11:45, LMS4 - R.312

 

Prerequisites

The lectures Representations of Associative Algebras or Homological algebra

or an idea of the basic concepts of representation theory like exact sequences, projective covers, Hom- and Tensor-functors, natural equivalences.

 

Contents (tentative)

Aim of these lectures is to provide a gentle introduction to the theory of almost split sequences and translations quivers for finite-dimensional k-algebras.

We start by proving the Theorem of Krull-Remak-Schmidt, which allows us to reduce the task of "understanding the module category of a finite-dimensional algebra" to the problem of determining all its indecomposable modules and morphism between them.
The Auslander-Reiten Theory then can be seen as a combinatorial tool that helps us to organize these indecomposable objects in an oriented graph (quiver) Q. The quiver Q consists of the following data:

  •  Vertices correspond to the isomorphism classes of indecomposable modules.
  •  Oriented edges (arrows) correspond to so-called irreducible morphisms.


Moreover Q is a so-called translation quiver; roughly speaking there exists a quiver morphism T on Q such that the direct predecessors of every vertex v correspond to the direct successors of T(v). This fact is encoded in the existence of Auslander-Reiten sequences, which are short exact sequences that "almost split". We devote a big part of this lecture to prove the existence of almost split sequences and clarify the term "almost split".

Lecture Notes

  • Basic concepts
    1. Preliminaries
    2. The radical of the module category
    3. The Theorem of Krull-Remak-Schmidt
  • Auslander-Reiten sequences
    1. Definition and first properties
    2. The Auslander-Reiten translation
    3. The defect of an exact sequence (a weak version)
    4. Pushouts and pullbacks
    5. The existence of Auslander-Reiten sequences
    6. A symmetric definition
    7. Irreducible morphisms
  • The Auslander-Reiten quiver
    1. Translation quivers
    2. Finite representation type
    3. Stable translation quivers
    4. The RIedtmann Structure Theorem

 

References

  • M. Auslander, I. Reiten, S. Smalo: Representation Theory of Artin Algebras. Cambridge studies in advanced mathematics 36. Cambridge University Press, 1995.
  • I. Assem, D. Simson, A. Skowronski: Elements of the Representation Theory of Assciative Algebras. London Mathematical Society Student Texts 65, Cambridge University Press, 2006.
  • A. Skowronski, K. Yamagata: Frobenius Algebras I. EMS Textbooks in Mathematics, European Mathematical Society Publishing House, 2001
  • D. Benson: Representations and Cohomology, I. Cambridge studies in advanced mathematics 30. Cambridge University Press, 1991.