Algebra IV (Commutative Algebra), Spring 2014  

 
Instructor: Andrew Obus
email: obus [at] virginia.edu
office: Kerchof Hall 208
phone: 434-424-4930
website: http://people.virginia.edu/~aso9t/math7754s14.html

 

Lectures

TTh 11:00-12:15, Kerchof Hall 128.  Please ask questions if anything in lecture is unclear. Lectures will run the entire 75 minutes. Please show up on time!



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.

Other references (most of which contain much more material than Atiyah-MacDonald) include:



Content

Commutative algebra is the study of commutative rings, and modules over them.  It is the subject that underlies (among other things) algebraic geometry and algebraic number theory.  While the subject is, on its face, algebraic, we will often find that it is useful to take a geometric perspective, and we will weave this thread throughout the course.  Often, the geometric intuition is useful, even for questions in "pure" algebraic number theory!

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.

Office Hours

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.


Homework

Homework will not be handed in, but you must be prepared to discuss/present solutions at the weekly problem session.  You are encouraged to work together on the homework, although I have found that this is more effective when you have first spent some time thinking about the problems by yourself.  It is a good idea to TeX up full solutions (I still reference my grad school homework solutions every once in a while).  The only way you will learn commutative algebra is by doing the problems (and Atiyah-MacDonald has some excellent ones)!

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).


Exams

There are no exams in this class.


Comments

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