MATH 7400: Homological Algebra (fall 2008)

Instructor: Mike Stillman

This course will be an introduction to homological algebra and some of its applications. Examples will be stressed throughout.

The main text will be a book that I am writing with Greg Smith: "Computational Homological Algebra". We will also refer to other texts, such as Rotman's book and/or Weibel's book.

Tentative list of topics:

  1. Basics: Chain Complexes, homotopies, snake lemma, connecting homomorphisms, mapping cones and cylinders, triangles (long exact sequences).
  2. Examples: Simplicial complexes, Koszul complexes, Eagon-Northcott type complexes, determinants and resultants.
  3. Resolutions: free resolutions, Cech complexes, projective, and injective resolutions.
  4. Derived categories and functors. The basic ideas of derived categories will be developed, based on resolutions of complexes. This will be done in a concrete and explicit manner.
  5. Torsion products: tensor products, Tor, Kunneth formla, Koszul cohomology, flatness, symmetry of Tor.
  6. Extension modules: Hom, Ext, universal coefficient theorem, Yoneda extensions, group cohomology, global Ext, relationship to solutions of differential equations.

We will also try to cover at least some of the following:

  1. Local cohomology. Definition, basic properties, local duality, sheaf cohomology.
  2. Koszul duality. The Bernstein-Gelfand-Gelfand correspondence. This is an explicit extremely useful construction! Applications to sheaf cohomology, hyperplane arrangements.
  3. Spectral sequences.

Throughout, we will use or introduce basic category theory as needed to understand these topics.