MATH 6340: Commutative Algebra (Spring 2010)

Instructor: W. Frank Moore

Prerequisite: Math 6310 or equivalent.

Textbook:

• Eisenbud: Commutative algebra (recommended)
• Atiyah and MacDonald: 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 valuation rings
• Hilbert functions and dimension theory
• Tor and Ext, and flatness
• Some other topics in homological commutative algebra (time permitting): Cohen-Macaulay rings, depth and regular sequences, Gorenstein rings, duality

Since it is crucial to do mathematics in order to learn it, there will also be regular homework assignments.