MATH 602. Homological Algebra (Spring 2007)

Meeting times: MWF, 9:00am-9:50am (MTH 1311)

Instructor: Professor Jonathan Rosenberg. His office is room 2114 of the Math Building, phone extension 55166, or you can contact him by email. His office hours are M and F 1-2, or by appointment.

Text: Charles Weibel, An Introduction to Homological Algebra (Cambridge), Paperback (ISBN-13: 9780521559874 | ISBN-10: 0521559871). The text has a reasonable list price, $43. Errata for the text may be found here or here. Weibel also has an article about the history of homological algebra .

Prerequisite: MATH 600 (graduate algebra). MATH 734 (algebraic topology) or MATH 606 (algebraic geometry) helps in terms of motivation, but won't be assumed.

Catalog description:

Projective and injective modules, homological dimensions, derived functors, spectral sequence of a composite functor. Applications.

This course will introduce the basic methods of homological algebra, and discuss such topics as chain complexes and spectral sequences. This material is essential for algebraic geometry and algebraic topology, and plays a major role in many other subjects as well. We will do applications to subjects such as cohomology of groups and Hochschild cohomology of algebras. If we have time, we may get to some more "modern" topics such as derived and triangulated categories. But at a minimum, I hope you will come out of the course knowing how to compute with spectral sequences.

In terms of Weibel's book, I hope at a minimum to cover chapters 1 (chain complexes), 2 (derived functors), 3 (Ext and Tor), 5 (spectral sequences), 6 (group homology and cohomology), and 9 (Hochschild and cyclic homology), plus the appendix on category theory (as this is quite important and is often not covered in other courses). If time permits we may cover parts of the other chapters also.

1. H. Cartan and S. Eilenberg, Homological Algebra, Princeton Univ. Press. The first book on the subject, and still the definitive reference.
2. S. Mac Lane, Categories for the Working Mathematician, Springer. Still probably the best basic book on category theory.
3. J. McCleary, A User's Guide to Spectral Sequences, 2nd ed., Cambridge. The best guide to computing with spectral sequences.
4. R. Godement, Topologie Algébrique et Théorie des Faisceaux, Hermann. 40 years old, but still the standard reference on sheaves.

Homework will be assigned, collected, and graded, usually once a week. I will probably give a take-home final exam. Grades will have the following meaning:

• A. Did most of the problems correctly, seemed to have a good idea of what's going on.
• B. Did a fair number of problems, but had some trouble with them.
• C. Didn't do much of the homework, or got essentially none of it right.
• D, F. I hope these won't arise. Basically mean "hardly ever showed up".

It is your responsibility to turn the homework in on time.

(Some details to be filled in later if necessary)