Math 634 — Spring 2002 Commutative Algebra

Textbooks:

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

Math 634 will be a first course in commutative algebra, at the level of Atiyah-Macdonald. Besides the material of this book, we will also cover Groebner bases and some of their applications. Many examples will also be given.

Tentative list of topics:

• Groebner bases and applications

• We will use these throughout the course to construct interesting examples

• Prime ideals and operations on ideals
   
• Modules, tensor products and operations on modules
   
• Localizations
   
• Primary decompositions
   
• 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.

• Chain conditions (Noetherian rings, Artinian rings)
   
• Primary decomposition for Noetherian rings

• This is the key link between commutative algebra and algebraic geometry

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

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

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