Markus Schmidmeier
Mathematical Sciences
Florida Atlantic University

## Homework assignments

All problem numbers refer to our textbook by Claus Michael Ringel and Jan Schröer, Representation Theory of Algebras I: Modules.

Textbook
Sections

Problems

due

Part III, A-modules.
Section 15: Path algebras
15.7/ 1, 3 Friday, 1/20
Section 15: Path algebras 15.7/ 2, 4, 5 Friday, 1/27
Projective representations Suppose Q is a quiver with no oriented cycles. Show that the representations P(i) introduced in class are all projective, and specify them for the example quiver shown. Friday, 2/3
Projective covers For the example quiver in the previous problem, specify the projective covers of the simple modules S(i) and compute the kernels. Are the kernels projective modules? Make a conjecture about the kernels of the projective covers of the simple modules over a path algebra of a quiver with no oriented cycles! Friday, 2/17
Section 25: Push-out and pull-back Exercise on p.150 (Uniqueness of push-out), exercise on p.151 (Universal property of the pull-back) Monday, 3/26
Section 25: Induced short exact sequence Exercise 25.8/ 2. Friday, 4/ 6
Section 27: Projective resolutions Proof of Lemma 27.1 on page 173 Friday, 4/ 20