MATH 620: HOMOLOGICAL ALGEBRA

DATE: Fall Term, 2003

INSTRUCTOR: John Beachy

COURSE DESCRIPTION:
Categories and functors, projective and injective modules, complexes and homology, Ext, Tor, and dimensions. Applications to cohomology of groups and ring theory.

PREREQUISITE: MATH 521

TEXT: An Introduction to Homological Algebra, by Rotman [Academic Press, 1979]

SYLLABUS:

Chapter 1: Introduction

Some history and motivating problems; categories and functors; tensor products

Chapter 2. Hom and tensor

Direct sums and products; exact sequences; adjoint functors; direct and inverse limits

Chapter 3. Projectives, injectives, and flats

Modules: free, projective, injective, and flat

Extensions of modules

Notes on n-fold extensions of modules

Chapter 6: Homology

Homology functors; derived functors

Chapter 7: Ext

The functor Ext and its relation to extensions of modules

Chapter 8: Tor

The functor Tor and its relation to torsion in modules

Unfortunately, it seems to be impossible to also cover several important chapters:
Chapter 4: Specific Rings;
Chapter 9: Son of Specific Rings; and
Chapter 11: Spectral Sequences.

Here are some references on the WEB:

History of Homological Algebra, by Chuck Weibel, 40 pages, in .dvi format

The Mathematical Atlas: Category theory, homological algebra, by Dave Rusin

A Course in Homological Algebra, by Lee Lady, University of Hawaii