Representation Theory 2 (Summer semester 2021)

Master course:
This is a 9 credit points course, available for master students in mathematics and also for bachelor students.
Registering to the course in Ilias / Campus is possible.
Responsible for this course: Steffen Koenig

Prerequisites: Linear Algebra 1 and 2. Algebra. Basic representation theory (parts of Grundlagen der Darstellungstheorie or of Algebra 2), basic homological algebra (parts of Representation Theory 1). Not being afraid of abstract structures. Enjoying mathematics.

Format: As in the previous semester, teaching in summer semester has to be online.
Exercises, followed by an office hour: Every Monday at 9:45 on Webex. Starting on the 26th of April.

Contents:
The course will start with an application of the material on first extension groups in part 1 of the course: Auslander's defect formula is about nonexactness of Hom-functors. It implies the Auslander-Reiten formulae, which in turn imply existence of almost split sequences. These are now the standard tool for computing modules and describing categories of finite dimensional modules of finite dimensional algebras.

The main part of the course then will concentrate on derived module categories and tilting theory. After defining derived module categories we will look at examples. Then we will define tilting modules and show that they yield equivalences of derived categories and see examples of such equivalences. The derived categories version of Morita's theorem will be discussed, but not fully proven. Also discussed, with or without proofs, will be examples and applications of derived categories and derived equivalences and invariants under derived equivalences.
If there is time, we also will discuss homotopy categories, stable module categories and triangulated categories in general.

Numbering of chapters and results continues that used in part one of the course.

Chapter 10. Auslander's defect formula and the Auslander-Reiten formulae.
Preview.
Lecture notes with questions. Key concepts: Co- and contravariant defect. Transpose. Auslander's defect formula. Auslander-Reiten formulae.
Exercises for the problem class on April 26, to be continued on May 3.

Chapter 11. Almost split sequences.
Preview.
Lecture notes with questions, first part. Key concepts: Almost split morphisms, minimal morphisms, irreducible morphisms. Almost split sequences. Characterisations and uniqueness of almost split sequences.
Exercises for the problem class on May 3.
The first part of the lecture notes is expected to be read by April 30.
Lecture notes with questions, second part. Key contents: Proof of existence of almost split sequences. Proof of the first Brauer-Thrall conjecture, Harada-Sai lemma.
Lecture notes with questions, third part. Contents: Auslander Reiten quiver. Knitting examples.
The second and third part of the lecture notes is expected to be read by May 7. Working through examples will need further time.
Exercises for the problem class on May 10.
Exercises on almost split sequences: Are indecomposables determined by their dimension vectors?
Exercises on Auslander Reiten quivers: Comparing Gabriel quivers and Auslander Reiten quivers.
The following exercises are about determining Auslander Reiten quivers, in particular of path algebras.
Exercises on Auslander Reiten quivers: The Coxeter transformation.
Exercises on Auslander Reiten quivers: Path algebras.
Exercises on Auslander Reiten quivers: The Kronecker algebra.

Chapter 12. Derived categories.
Preview.
Lecture notes with questions, first part. Key concepts: Cones. Localisation, multiplicative system, calculus of fractions. Defining derived categories.
Lecture notes with questions, second part. Key contents: Definition of triangulated categories. Elementary properties. Stable module categories of self-injective algebras are triangulated. Homotopy categories of complexes are triangulated.
The first and the second part of the lecture notes are expected to be read by June 4.
Exercises on triangulated categories.
More exercises on triangulated categories. (For the problem class on June 21.)
Lecture notes with questions, third part. Key contents: Quasi-isomorphisms form a multiplicative system. Bounded derived categories as homotopy categories.
The third part of the lecture notes is expected to be read by June 18.

Chapter 13. Modules as objects of derived categories.
Preview.
Lecture notes with questions. Key contents: Homomorphisms and extensions between modules in derived categories. Derived categories of hereditary abelian categories.
Exercises on computing extensions in homotopy and derived categories. (For the problem class on July 5.)

Chapter 14. Derived equivalences.
Preview.
Lecture notes with questions, first part. Key contents: Motivating examples. Happel's construction of derived equivalences.
Lecture notes with questions, second part. Key contents: Tilting modules, APR tilting modules. Tilting complexes and statement of Rickard's derived Morita theorem, with comments about the proof. An example of a derived equivalence defined by a tilting complex.
The lecture material is expected to be read by July 12.
Exercises on derived quivalences. (For the problem class on July 12.)

Chapter 15. t-structures.
Preview.
Lecture notes with questions. Key contents: t-structures. Hearts of t-structures are abelian categories.
The lecture material is expected to be read by July 16.
Exercises on t-structures. (For the problem class on July 19.)

Chapter 16. Repetitive algebras.
Preview.
Lecture notes with questions. Key contents: Repetitive algebras, stable categories and derived categories.

Literature:
The following books can be obtained electronically from the university library:
Ibrahim Assem and Flavio Coelho, Basic Representation Theory of Algebras
Ibrahim Assem, Andrzej Skowronski and Daniel Simson, Elements of the Representation Theory of Associative Algebras, Volume 1
Maurice Auslander, Idun Reiten and Sverre Smalø, Representation Theory of Artin Algebras
Michael Barot, Introduction to the Representation Theory of Algebras
Karin Erdmann and Thorsten Holm, Algebras and Representation Theory
Charles Weibel, An Introduction to Homological Algebra

Printed copies are available in the library of:
Charles Curtis and Irving Reiner, Representation theory of finite groups and associative algebras
Charles Curtis and Irving Reiner, Methods of representation theory, volumes I and II
Dieter Happel, Triangulated categories in the representation theory of finite dimensional algebras

Part 1 of the course.

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