## Department of Mathematics Syllabus

This syllabus is advisory only. For details on a particular instructor's syllabus (including books), consult the instructor's course page. For a list of what courses are being taught each quarter, refer to the Courses page.

**Approved:**2015-11-01, Greg Kuperberg and Brian Osserman

**Units/Lecture:**

**Suggested Textbook:**(actual textbook varies by instructor; check your instructor)

https://www.amazon.com/Advanced-Algebra-Graduate-Studies-Mathematics/dp/B00N4J2BP6/

**Prerequisites:**

**Suggested Schedule:**

S 7.3-7.5: Semisimple rings, group representations, irreducible representations, regular representation, Schur's lemma, Maschke's theorem, Wedderburn-Artin theorem and Burnside-Molien corollary for group algebras, center of a group algebra, character tables and their properties, dual/contragredient representations** (Ex. 7.31), tensor products of group representations** (Ex. 7.41) (12-15 lectures)

*Optional: S 7.2: Jacobson radical, Hopkins-Levitzki theorem (0-3 lectures)*

S 6.2, 6.3: Categories, functors (3-5 lectures)

S 9.4, 9.6: Free and projective resolutions (vs definition of an injective resolution), homology of a chain complex, Ext functor (4-6 lectures)

Instructor's choice from: S 9.5 Derived functors, S 9.7 Tor, S 9.8 Group cohomology (4-8 lectures).

Total: 23-37 lectures

** These essential points are given as exercises. The instructor can

follow a different treatment.

**Additional Notes:**

Rotman is a thick book with easy sections that can be accelerated and difficult sections that should not be covered completely. Proofs of some of the harder results listed in the syllabus can be taken as optional.

Rotman bases his presentation of homological algebra on derived functors. However, one can skip this machinery in subsequent sections, particularly by assuming existence results. For instance, axioms characterizing Ext and Tor are stated without reference to derived functors.