Part I. We start with some elementary theory of homological algebra such as projective, injective, and global dimensions, complexes, the functors Hom and tensor and their derived functors, Ext and Tor. Then we will study Hochschild and cyclic (co)homologies and Poincaré duality for Hochschild (co)homologies, the Auslander condition, and Cohen-Macaulay algebras.

We will work out homological properties and homological invariants for a number of examples: group algebras, the universal enveloping algebra of Lie algebras, noncommutative Iwasawa algebras of compact p-adic analytic groups, quantum groups, etc. The textbook for the first 2/3 quarter is An introduction to homological algebra by C.A. Weibel. I will provide handouts for the last 1/3 quarter. There will be homework every week.

Prerequisite. Math 504/5/6 or equivalent.

Part II. We will go over some recent topics related to homological algebra and some applications.

The first topic is derived categories and derived equivalences, tilting complexes , and dualizing complexes. A good reference for the derived category is Weibel's book. As an application, we will define Tate cohomology by using the stable derived cateogry.

The second topic is differential graded (DG) algebras and DG categories and their derived categories. In the last 1/3 quarter, we will talk about some applications to noncommutative algebraic geometry. In particular, we will study the Proj category and its derived category and prove Serre duality for noncommutative projective schemes. We will mention the twisted homogeneous coordinate rigns of projective varieties and Artin-Schelter regular algebras. Some A-infinity algebras will be used to study Artin-Schelter regular lgebras. There will be no textbook for this quarter. There will be some homework.

Prerequisite. Part I.