Math 8020 -- Commutative Algebra MWF 1:25-2:15 pm, 326 Boyd
Instructor: Assistant Professor Pete L.
Clark, pete (at) math (at) uga (dot) edu
http://math.uga.edu/~pete/MATH8020.html (i.e., right here)
Office Hours: During an hour long discussion section each week in
which all students are strongly encouraged to attend. Otherwise by appointment.
I am quite amenable to booking ``extra'' office hours. The ground rules
are: (i) please give me at least 24 hours notice. (ii) Please send me an email
the night before a morning appointment or the morning of a later appointment to
remind me that we are meeting. (iii) If I do not show for an appointment
(empirically, the chance that I will fail to show seems to be about 5-10%), feel
free to call me on my cell phone. Probably I'm not too far away. (iv) If we do
book an appointment, please do show up or call or email to let me know you're
Course text: none required. Instead, please carefully read the lecture notes (see below). However I strongly recommend that you take a
look at Atiyah and Macdonald's Introduction to Commutative Algebra (Chapters 1-9), which covers similar content but more concisely.
Section: I would like to have an extra discussion section, one hour a week,
for discussion of problems and presentation of projects.
Grade is Computed: From homework, which may include a final project. Stay tuned.
Course Prerequisites: Math 8000 (Graduate Algebra), or equivalent
knowledge. More so than is common for most commutative algebra courses, I wish
to make connections with other areas of mathematics including differential
and general topology and complex analysis. It is hoped that these connections
will make things clearer (and more interesting); if not, they may be ignored (or saved until later). But if you could contrive to learn what a vector bundle on a
topological space is, that would be useful.
Content: Here is a rough outline:
Chapter 1: Introduction to Commutative Rings
Chapter 3: Introduction to Modules
Chapter 4: Ideals
Chapter 5: Examples of Rings (especially rings of functions)
Chapter 6: Swan's Theorem
Chapter 7: Localization
Chapter 8: Noetherian Rings
Chapter 9: Boolean rings
Chapter 11: Affine algebras and the Nullstellensatz
Chapter 13: The spectrum
Chapter 14: Integral extensions
Chapter 15; Factorization
Chapter 21: Dedekind domains
Chapter ??: Picard groups
Here is the entire manuscript of my commutative algebra notes. (pdf)
However, these notes are not quite fit for student consumption, in several ways.
Most importantly they do not contain enough exercises, and none of the exercises
are numbered. Below I give chapter by chapter notes, which are modified versions of the appropriate chapters but should be more polished. The philosophy here is that
the big file above will be continually modified without notice, but once I post an
individual chapter below, it will stay as it it for the duration of the course.
Chapter 1 (15 pages): Commutative Rings (pdf)
Includes 40 exercises.
Chapter 3 (40 pages): Modules
Includes 37 numbered exercises, plus additional (unnumbered) exercises.
Homework Solutions: If you spent a lot of time on any one problem, found it especially interesting or challenging, and/or presented the solution in the problem session, please consider texing up the solution, then send me the tex file. I will (possibly after mild editing, with your permission) post it here. If you do this, please name your tex file as follows: Math8020Ex[chapter number].[exercise number][your name here].
OUR DISTINGUISHED COMPETITION (lecture notes by other people):
The C-Ring Project: a collaborative open source textbook on commutative algebra.
Commutative Algebra, by Robert B. Ash
Algebra Commutative, by Antoine Chambert-Loir.
Notes on Commutative Algebra, by Mel Hochster.
Notes on Topics in Commutative Algebra, by Mel Hochster.
Commutative Rings, by T.Y. Lam. Notes by Anton Geraschenko.
Commutative Algebra, lectures delivered by Jacob Lurie, Harvard Univeristy, Fall 2010. Notes
by Akhil Mathew.
Basic Commutative Algebra, by Keerthi Madapusi
A Primer of Commutative Algebra, by James S. Milne