Cornell Math - MATH 634, Fall 2005

MATH 634: Commutative Algebra (Fall 2005)

Instructor: Michael Stillman

Meeting Time & Room

Prerequisites: Math 631 or equivalent.

Textbooks:

(1) Eisenbud: Commutative algebra (recommended)
(2) Greuel and Pfister: A Singular introduction to commutative algebra (recommended).

This course is an introduction to commutative algebra. Commutative algebra is a key ingredient of both algebraic number theory and algebraic geometry, and is a lively area of research on its own.

Tentative list of topics:

  • Groebner bases
  • Ideals: localization, prime ideals, and operations on ideals
  • Modules: graded rings and modules, syzygies, free resolution, tensor products, and operations on modules
  • Noether normalization and Hilbert's nullstellensatz, and several other key concepts/theorems in commutative algebra: integral dependence, going up/down and integral closure.
  • Primary decomposition in Noetherian rings
  • Discrete valutation rings
  • Hilbert functions and dimension theory
  • Tor and Ext, and flatness
  • Some other topics in homological commutative algebra: Cohen-Macaulay rings, depth and regular sequences

Throughout the entire course we will include many examples, often using my computer algebra system Macaulay2 (written with Dan Grayson). Since it is crucial to do mathematics in order to learn it, there will also be regular homework assignments.