Math 559
Spring 2007

Course Description

Commutative algebra is the engine behind algebraic geometry and algebraic number theory. In addition, problems from other fields such as combinatorics or optimization can sometimes be phrased as commutative algebra problems. This course will be an introduction to the basics of commutative algebra, including localization, primary decomposition, integrality, flatness, and dimension. We will roughly follows Eisenbud's Commutative Algebra with a View Toward Algebraic Geometry. Computational aspects and examples relevant to algebraic geometry will be emphasized, but the only prerequisite is 551/552 or equivalent.

Instructor

Name Office E-mail Phone Office Hours
Diane Maclagan 240 Hill Center maclagan at math.rutgers.edu 732-445-3481 M 1:00-2:00pm and by appointment

Course Time and Location

MW 10:20-11:40 in Hill 525.

Text

Eisenbud Commutative Algebra with a View Toward Algebraic Geometry, Springer, GTM 150, 1995 See also the selection at Abebooks.com (a site that searches thousands of independent and used bookstores - a great resource for math books).

Announcements

Announcements and any handouts will be available here.
Here is a handout on Macaulay 2. This was originally written for a guest lecture in Zeilberger's class, so a few exercises won't make sense.

Have a look at Mel Hochster's Thirteen open questions in commutative algebra. While we won't get to the definition of every word in every problem, they are a good guide. Time spent on the rest of Hochster's expository manuscripts page would not be wasted, either.

Irena Peeva also has a problem list in combinatorial commutative algebra that is worth looking at. Hopefully I will get a link to that soon, or ask me to look at my copy.

Assessment

There will be homework assignments every two weeks. Homework assignments and due dates will be posted one the class schedule, which will also have the reading for the following week. You are encouraged to work on homework together, but you should write up the solutions yourself.
In addition everyone will do a final project extending a topic we studied in class. These will be presented in the last week of class, in the form of talk and accompanying 5-10 page handout. Possible project topics include:
  • Algorithms for primary decomposition
  • Free resolutions of monomial ideals
  • Local cohomology
  • Green's and Macaulay's theorems.
  • Commutative algebra of simplicial complexes
  • Tight closure
  • Complexity of effective commutative algebra.
The above list is very biased towards my interests. You are strongly encouraged to talk to me well early in the semester to choose a topic relevant to your own research interests. Relevant dates are:
Date Activity
2/28 Chose topic
3/26 Annotated bibliography (1/2-1 page)
4/18 Abstract of talk (1/2 - 1 page)
5/7 Talks and presentations