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.)
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.)
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