## Algebra IV (Commutative Algebra), Spring 2014
Instructor: Andrew Obus
email: obus [at] virginia.edu
office: Kerchof Hall 208 |

Problem session

There will be a weekly one-hour problem session where students will present homework solutions on the board, and where we can discuss the solutions. The problem session will meet on Tuesdays from 2:30 - 3:30 in Kerchof 128.

Textbook

*Introduction to Commutative Algebra*, by M. F. Atiyah and I.
G. MacDonald. I believe the book is out of print, but it is easy
to buy on Amazon. There should also be a copy on reserve in the
library. This is a short, dense book, but the exposition is very
efficient, and it has a wealth of great exercises.

- Commutative Algebra with a View Toward Algebraic Geometry by D. Eisenbud (the Bible of commutative algebra)

- Commutative Algebra and Commutative Ring Theory by H. Matsumura
- Basic Algebra II by N. Jacobson
- Commutative Algebra, Volume I and II by O. Zariski and P. Samuel
- Algèbre Commutative, by N. Bourbaki (also available in English translation--not easy to learn from for the first time, but a good reference)
- A Term of Commutative Algebra, by A. Altman and S. Kleiman. Available at http://web.mit.edu/18.705/www/12Nts-2up.pdf

In particular, I hope to cover essentially all of Atiyah-MacDonald's book (some of which you already know from Algebra I-III). If there is time left over, we may cover some elimination theory, some material on extensions of discrete valuation rings, or some more advanced concepts in dimension theory.

Mondays and Thursdays 2:30 - 3:30. Kerchof Hall 208 (my office). If these times do not work for you, please make an appointment with me.

The homework assignments will be given out in class weekly, and posted on this webpage.

HW #1: (for problem session 1/21): Read Ch. 1 of Atiyah-Macdonald. Do problems 1, 5, 8, 15, 17, 19 in Ch. 1. Read and understand the solution of problem 26 (essentially given in Atiyah-MacDonald).HW #2: (for problem session 1/28): Do problems 21, 22, 28 in Ch. 1. Do problems 4, 5, 8, 10 in Ch. 2. Read and understand the solution of problem 27 in Ch. 1.

HW #3: (for problem session 2/4): Do problems 14, 15, 16, 17, 18, 21 in Ch. 2. Also do the following problem:

Suppose I = N (natural numbers) is viewed as a directed set by i \preceq j iff i | j. Let A_i = Z[1/i], and let \mu_{ij}: A_i \to A_j be given by inclusion. Show that the direct limit over the directed set I of the A_i is Q.

HW #4: (for problem session 2/11): Do problems 1, 5, 12, 16, 17, 18, 19 in Ch. 3. Also read and understand the solution to problem 15 (essentially given in Atiyah-MacDonald).

If you have (anonymous) comments for me about teaching style or anything related to the course, click here for a feedback form.