Math 740 — Homological Algebra

Spring 2003

Instructor: Michael Stillman

Time: TR 9:45-11:00

Room: MT 230

This course will be a computational introduction to homological algebra, covering what the "working mathematician" should know regarding homological algebra. Many concrete examples will be given, and we will make the material as down to earth as possible. We will cover some recent exciting work including cellular resolutions, and the BGG correspondence. (This is a very cool relationship between the polynomial ring and the exterior algebra, which has many applications, and as a field, is wide open with many research possibilities).

Possible arrangement of topics include:

1. Categories and functors
2. • Functors
   • Abelian Categories
   • Simplicial, cellular complexes
   • Sheaves
   • Homotopy, Mapping cones and cylinders, free resolutions, cellular resolutions, Cech complexes
2. Chain complexes
   • Snake lemma, functorial properties
3. Homology
4. Derived functors
5. Ext and Tor, homological dimensions
6. Spectral sequences
   • BGG correspondence. This is exciting recent, explicit work of Eisenbud, Floystad and Schreyer.
7. Derived categories