MATH 6340: Commutative Algebra (Spring 2011)

Instructor: Michael Stillman

Textbooks:

  1. Atiyah and Macdonald: Introduction to commutative algebra
  2. Eisenbud: Commutative algebra (recommended)

MATH 6340 will be a first course in commutative algebra. We assume that you have taken MATH 6310, or an equivalent course.

We will cover most of the material in Atiyah-Macdonald. Besides the material of this book, we will also cover Groebner bases and Cohen-Macaulay modules and free resolutions. Many examples will also be given.

We will assume that you know about prime ideals, tensor products, Noetherian rings (Hilbert basis theorem), although we will try to provide a deeper understanding of tensor products and their relationship to geometry.

Tentative list of topics:

  • Groebner bases and applications

We will use these throughout the course to construct interesting examples

  • Localizations (and Nakayama's lemma)
  • Primary decompositions, prime avoidance lemmas, ideal quotients

This is a key link between commutative algebra and algebraic geometry

  • Integral dependence

This section includes some key theorems in commutative algebra: the going-up and going-down theorems, Hilbert's Nullstellensatz, and Noether's normalization theorem. If time permits, we will also describe an elegant algorithm for computing the integral closure of a ring.

  • Rings of dimension zero and one: Artinian rings and discrete valuation rings.
  • Completions
  • Dimension theory

This is the culmination of Atiyah-Macdonald. We define dimension several different ways and then show that they are the same. This is a very basic, powerful set of results.

  • Cohen-Macaulay rings, depth, regular sequences
  • Free resolutions of ideals and modules, Ext and Tor, Koszul complex

There will be weekly homeworks and probably also a project/presentation.