A Course in Homological Algebra

E.L. Lady


In the fall of 1974, I returned to the University of Kansas after spending a year at the University of Illinois. During my time at Illinois, I had sat in on a course on Topos Theory (the most avant-garde form of category theory) given by John Gray, and had also attended the commutative ring theory seminars led by Robert Fossum, Philip Griffith, and Graham Evans. I had also spent a lot of time in the library, as usual reading on a large variety of topics, but most especially trying to understand the most recent commutative ring theory, especially as it related to algebraic geometry.

Back at Kansas, the ring theorists were concerning themselves with the Gilmer-style theory of non-noetherian commutative rings, and were intimidated by any homological approach at all, even the bare mention of Ext.

Paul Conrad who was the head of the algebra department (as it were) at Kansas suggested that I might like to teach a two-semester graduate topics course. I suggested that Homological Algebra might be an appropriate course.

My objective was to educate the faculty as well as whatever students enrolled. And in fact, all three rings theorists -- Brewer, Rutter, and Philip Montgomery, attended regularly.

I wanted to teach essentially everything I knew about homological algebra and category theory, with a large dose of the kind of commutative ring theory that was popular at Illinois and which was then much more fashionable than the Gilmer stuff. I came into class every day with detailed notes, and basically read the notes aloud and copied them onto the board as fast as was physically possible.

The students coped fairly bravely with my furious pace, and at the end of the semester, the graduate students in the Mathematics Department voted to give me the award for the best graduate course taught that year. This award had never before been given for an advanced course, and considering that it was the first graduate course I'd ever taught, I thought I'd done really well.

Despite my award, though, a year later the University decided, over the vociferous objections of the Mathematics Department, that I should be denied tenure on the grounds of inadequate teaching.

By the end of the course, I suppose I had about four or five hundred loose sheets of paper containing the complete notes. I didn't save these when I left Kansas, but I did make an outline including statements of all the non-routine theorems, and proofs for the more difficult ones. That outline is basically what I'm making available here.

Because of the rather schematic quality of these notes, they are probably not very suitable for beginners, even though the original course was an introductory course. (The dates indicate when the notes were put on this web page, or the date of the latest revision.)

A lot of the files listed below are in PDF (Adobe Acrobat) format. Alternate versions are in DVI format (produced by TeX; see see here for a DVI viewer provided by John P. Costella) and postscript format (viewable with ghostscript.) Some systems may have some problem with certain of the documents in dvi format, because they use a few German letters from a font that may not be available on some systems. (Three alternate sites for DVI viewers, via FTP, are CTAN, Duke, and Dante, in Germany.)

Additive and Abelian Categories

Derived Functors

Tor, Flatness, and Purity

April, 1996.
(Click here for dvi version.)
(Click here for Postscript.)
  • Tor
  • Flat modules and algebras.
  • Semi-hereditary rings and Prufer domains. (Flat = torsion-free.)
  • Von Neumann regular rings. (Every module is flat.)
  • Faithful functors.
  • Faithfully flat modules and ring extensions.
  • Pure submodules and subrings.

Faithfully Flat Descent

The word descent had been all the rage among the ring theorists at Illinois, and I had worked hard as hell to figure out what the hell it was about. When I presented it in class though, the ring theorists at Kansas clearly thought it was far too arcane for them to even consider trying to master.

Syzygies, Projective Dimension and Global Dimension

May, 1996
(Click here for dvi version.)
(Click here for Postscript.)
  • Reflexive and torsionless modules.
  • Schanuel's Lemma. Projective dimension.
  • Regular M-Sequences. Depth.
  • Global dimension.
  • Grade and a theorem of Auslander & Buchsbaum.
  • The Koszul Complex.

Gorenstein Rings and Modules

June, 1997
(Click here for dvi version.)
(Click here for Postscript.)
I spent an enormous amount of time in the library working my way through Bass's fundamental paper on Gorenstein rings. It's the kind of paper that appeals to me because it brings some many diverse ideas together.
After I finished teaching this course, the new edition of Kaplansky's book on commutative rings came out with a fairly simple presentation of many of the results here.

Two Papers by Hochster

Generically perfect modules and grade-sensitive modules.

The Tor Inequality

Indecomposable Injective Modules

Here I present briefly a construction given by Robert Fossum in a paper in Math. Scand. 36 (1975), pp. 291-312.
I think that Sharpe and Vamos probably give a better treatment of this material in their book on injective modules.

Spectral Sequences

Auslander's Proof of Roiter's Theorem

June, 1997
(Click here for dvi version.)
(Click here for Postscript.)
This was not actually part of the course I taught at Kansas, but was presented in a seminar here at the University of Hawaii.

Books on Homological Algebra



[ Top of Page | A Graduate Algebra Course | HOME ]