Lecture plan and exercises
This is a tentative schedule for the course.
Exercises will be posted here every week. You will be expected to have worked on them before the next exercise session, which will be held in class during the last hour of the Friday lecture.
Week | Tuesday | Friday | Exercises | Remarks |
---|---|---|---|---|
2 | Introduction | Quivers and path algebras: definitions | Friday January 11 lecture in 656 | |
3 | Categories: definitions, functors. Representation of quivers: definition. | Categories: natural transformations, equivalences of categories. Representation of quivers: morphisms between representations. | ||
4 | Representation of quivers: Equivalence between Rep_k(Q) and Mod kQ. | Representation of quivers: subrepresentations, quotient, indecomposable representations | Exercise session 1 | |
5 | Representation of quivers: Classifying indecomposables, matrix problems, finite and tame representation type | Representation of quivers: wild representation type | Exercise session 2 | Tuesday January 29 lecture in R60, Friday Feb. 1st lecture in 656 |
6 | Length of a module: composition series, short exact sequences | Length of a module: short exact sequences | Exercise session 3 | |
7 | Length of a module: Jordan-Hölder theorem, links with artinian and noetherian modules | Radical: definition, Nakayama lemma | Exercise session 4 with hints | Not every PDF reader seems to work to see the hints. Adobe Acrobate Reader certainly does. |
8 | Radical: nilpotent ideals, artinian rings | Quivers and path algebras: admissible ideals, representation of bound quivers, radical of a path algebra. Radical: Radical of a module | ||
9 | Radical: Radical of a module, small submodules, radical of a representation, top | Projective and injective modules: definitions | Exercise session 5 with hints | |
10 | Projective and injective modules: essential epimorphism, projective resolutions, projective cover | Projective and injective modules: projective cover | Exercise session 6 with hints | |
11 | Projective and injective modules: projective cover, computing minimal projective resolutions | Projective and injective modules: Socle | Exercise session 7 | |
12 | Krull-Remak-Schmidt theorem: Fitting lemma, local rings | Krull-Remak-Schmidt theorem. Projective and injective modules: Duality functor | ||
13 | Projective and injective modules: injective envelope, minimal injective resolution, computing minimal injective resolutions | Categories: Ext functors | Exercise session 8 | |
14 | Quivers: Extensions between simples | Basic algebras: definition | Exercise session 9 | Tuesday April 2 lecture in R60 |
15 | Basic algebras: basic algebra associated to a finite dimensional algebra, Projectivisation, Morita equivalence | Basic algebras: The quiver of a basic algebra | Exercise session 10 | |
16 | No Lecture | No Lecture | ||
17 | No Lecture | Exercise session and questions | Exercise session 11 |
Lecture notes for April 5-9-12: here