# Reading seminar on Representation theory of Cohen-Macaulay modules

## Course description

#### Course content

The aim of this course is to serve as an introduction to Cohen–Macaulay modules in representation theory. In the first part of the course we will recall some central notions of commutative algebra and representation theory. In the second part of the course we will introduce and study orders and lattices. In particular we will discuss the structure theorems for maximal and hereditary orders. In the third part of the course we will explore several examples: Bäckström orders, ribbon graph orders, tiled orders. If time permits, we will investigate conditions for the category of modules over a commutative noetherian ring and the category of Cohen–Macaulay modules over a local ring of dimension 1 to be of finite type.

#### Learning outcomes

**Knowledge**. The student knows the fundamental concepts introduced in the course, such as complete local rings, completion of commutative rings by ideals, Krull–Schmidt property, orders and lattices. The student can explain the structure theorem for maximal and hereditary orders.**Skills**. The student can give examples of the notions described in 1. The student can also deduce if some categories of representations or Cohen–Macaulay modules have finitely many indecomposable objects.**Competence**. The student will be able to present research-level ideas, concepts and proofs to a specialist audience. The student will also be able to follow research talks and read research papers in representation theory and commutative algebra at a reasonable level.

#### Learning methods and activities

The seminar consists of a 2x45 minute lecture given by a different participant every time on a prescribed topic. The lecture is expected to have a high level of engagement and discussion from the participants. The aim is to make sure that the participants understand as much as possible rather than to cover all the prescribed material in the given time.

## Schedule

We meet on Wednesdays at 10.15-12.00 at Simastuen (656). Here is a schedule with the distribution of talks.

Date | Speaker | Subject |
---|---|---|

Aug 30 | Sondre Kvamme | Introduction |

Sep 20 | Isak Drage | Commutative algebra I |

Sep 27 | Henrik Knudsen | Commutative algebra II |

Oct 04 | Mikkel Elkjær | The Krull–Schmidt property for complete local rings |

Oct 11 | John Aslak Wee Kleven | Auslander–Reiten sequences for orders over complete noetherian local rings |

Oct 18 | Jon Wallem Anundsen | Maximal and hereditary orders |

Oct 25 | Carlo Klapproth | Structure theory for maximal and hereditary orders |

Nov 01 | Henrik Knudsen | Bäckström orders |

Nov 08 | Laertis Vaso | Ribbon graph orders |

Nov 15 | Jacob Fjeld Grevstad | Auslander–Reiten species for Bäckström orders |

Optional topic: Bass orders. You can find details about each topic here.

## Seminar coordinators

For any questions, comments or feedback about the seminar you can contact Laertis Vaso (laertis (dot) vaso (at) ntnu (dot) no ) or Sondre Kvamme (sondre (dot) kvamme (at) ntnu (dot) no ).

## Literature

Our sources for the seminar include the following:

**Michael Atiyah**and**Ian G. Macdonald**,*Introduction to commutative algebra*.**Winfried Bruns**and**Jürgen Herzog**,*Cohen–Macaulay rings*.**Charles W. Curtis**and**Irving Reiner**,*Methods of representation theory*.**David Eisenbud**,*Commutative algebra*.**Wassilij Gnedin**,*Calabi-Yau properties of ribbon graph orders*.**Hiroaki Hijikata**and**Kenji Nishida**,*Classification of Bass orders*.**Hiroaki Hijikata**and**Kenji Nishida**,*Bass orders in nonsemisimple algebras*.**Osamu Iyama**,*Representation theory of orders*.**Osamu Iyama**,*Auslander–Reiten theory revisited*.**Graham J. Leuschke**and**Roger Wiegand**,*Cohen–Macaulay representations*.**Claus Michael Ringel**and**Klaus W. Roggenkamp**,*Diagrammatic methods in the representation theory of orders*.**Klaus W. Roggenkamp**,*Auslander–Reiten quivers for some Artinian torsion theories and integral representations*.**Daniel Simson**,*Cohen–Macaulay modules over classical orders*.

If you do not have access to one of the sources, please contact Laertis or Sondre.

## Examination

To participate in the examination, one has to give at least a lecture and pass an oral exam at the end of the course.

## Cake

We will try to have cake during the seminar. If you want to volunteer to bring cake or if you have any dietary restrictions, please contact Laertis or Sondre.