Auslander-Reiten theory and applications in representation theory and commutative algebra, under the code of Elective course 1 for Mathematical Structures (NMAG498) - information about the course in the winter semester 2024/2025.

Basic information

If one has a finite dimensional algebra or a commutative noetherian algebra over a field, one might ask what the category of finitely generated modules looks like. Although this is a very difficult problem in general, in many interesting cases one can describe the category in terms of generating homomorphisms and relations between them, using techniques developed by Maurice Auslander and Idun Reiten. These generating morphisms form a directed graph, which is called the Auslander-Reiten quiver, so one even can depict these categories graphically.

The aim of the course is to explain this piece of theory and illustrate it on examples of finite dimensional algebras and coordinate rings of isolated singularities. The plan is as follows:

  1. Motivation.
  2. Auslander-Reiten theory: the Jacobson radical of an additive category, irreducible morphisms, almost split morphisms and Auslander-Reiten sequences.
  3. Examples from representation theory of finite dimensional algebras.
  4. Background in commutative algebra: isolated singularities, Gorenstein rings, Cohen-Macaulay modules.
  5. Examples from commutative algebra.

Basic information about the course can be also found in the Student Information System.

Schedule (to be found also in the Student Information System): Wednesdays at 12:20 in the seminar room of the Department of Algebras. The first lecture starts on October 9.

Exam

The exam will be oral, please contact us (both Jan Stovicek and Souvik Dey) to agree on a time.

Requirements for the exam:

  • Basics of the Auslander-Reiten theory as covered by Sec. IV.1-IV.4 of [ASS]. The proof of Auslader-Reiten formulas was lectured from [Kr1], but any correct proof counts (see e.g. Theorem IV.2.13 in [ASS]).
  • Substantial enough part of the material covered in Chapters 5, 10 and 11 of [Yo] and Chapters 13 and 15 of [LW]. Please consult Souvik Dey if unsure whether your selection of topics suffices for the exam.

Program of the course

A brief overview of what has been taught will appear here.

DateWhat has been lecturedSources
Oct 9Motivation: How to depict a category of modules. Preadditive categories as rings with several objects. Ideals in (pre)additive categories. The Jacobson radical.[ASS], app. A.3
Oct 16Irreducible morphisms and the relation to the radical of a module category. Minimal almost split morphisms and their properties.[ASS], sec. IV.1
Oct 23Almost split (aka Auslander-Reiten) sequences – definition and characterization. The transpose of a module.[ASS], sec. IV.1 and IV.2
Oct 30A crash course on projective covers. Properties of the transpose of a module. Projectively and injectively stable module categories.[ASS], sec. I.5 and IV.2
Nov 06Transpose as a duality between projectively stable module categories. The Auslander-Reiten translation and its action on stable module categories. Auslander’s defect formula.[ASS], sec. IV.2,
[Kr1],
[ARS], sec. IV.1 and IV.4
Nov 13The Auslander-Reiten formulas and the existence of almost split sequences. The Auslander-Reiten quiver and how it is related to almost split sequences.[ASS], sec. IV.3 and IV.4,
[Kr1]
Nov 20Preliminaries on commutative algebra - depth, Krull dimension, maximal Cohen-Macaulay modules, Cohen-Macaulay rings, Henselian and complete rings. Illustrative results which one can obtain with the Auslander-Reiten theory.[Yo], Ch. I
Nov 27Basic properties of Auslander-Reiten quivers, translation quivers and their full subquivers, Auslander-Reiten quivers of KQ for Q finite acyclic (mostly without proofs)[ASS], sec. IV.4, VII.5 and VIII.2,
[Kr2], sec. 6,
[ARS], Ch. VIII,
[Ke]
Dec 4Normal commutative noetherian rings, characterisation of Cohen-Macaulay isolated singularities of dimension 2. The fundamental module and its decomposition. Almost split sequences for 2-dimensional isolated singularities.

Literature

[ASS] I. Assem, D. Simson, A. Skowroński, Elements of the representation theory of associative algebras, Vol. 1,London Math. Soc. Stud. Texts, vol. 65, Cambridge University Press, Cambridge, 2006.
[Au1] M. Auslander, Isolated singularities and existence of almost split sequences, in: Representation Theory, II, Ottawa, Ont., 1984, in: Lecture Notes in Math., vol. 1178, Springer, Berlin, 1986.
[Au2] M. Auslander, Representation theory of Artin algebras II, Comm. Algebra 1 (1974) 269--310.
[ARS] M. Auslander, I. Reiten, S. Smalø, Representation theory of Artin algebras, Cambridge University Press, 1997.
[BH] W. Bruns, J. Herzog, Cohen-Macaulay rings, Cambridge Stud. Adv. Math. 39, Cambridge University Press, 1993.
[En1] H. Enomoto, Classifications of exact structures and Cohen–Macaulay-finite algebras, Adv. Math. 335 (2018), pp. 838-877.
[En2] H. Enomoto, Relations for Grothendieck groups and representation-finiteness, J. Alg. 539 (2019), pp. 152-176.
[Ke]O. Kerner, Representations of wild quivers, p. 65-107. In: Representation theory of algebras and related topics (Mexico City, 1994), CMS Conf. Proc 19, AMS, Providence, RI, 1996
[Kr1] H. Krause, A short proof for Auslander's defect formula, Special issue on linear algebra methods in representation theory, Linear Algebra Appl. 365 (2003), pp. 267-270.
[Kr2] H. Krause, Representations of quivers via reflection functors, arXiv:0804.1428v2. [PDF]
[LW] G. J. Leuschke, R. Wiegand, Cohen-Macaulay Representations, Mathematical Surveys and Monographs, vol. 181, American Mathematical Society.
[Sta] Stacks Project, https://stacks.math.columbia.edu.
[Yo] Y. Yoshino, Cohen-Macaulay Modules over Cohen–Macaulay Rings, London Mathematical Society, Lecture Note Series, vol. 146, Cambridge University Press, Cambridge, 1990.