In this course we will be covering the basic material in commutative algebra, a subject that has applications to algebraic geometry, number theory, combinatorics, and invariant theory. Below is the (tentative) list of topics I will cover:
Localization, support of a module
Local rings, NAK
Associated primes, primary decomposition
Integrality, Going up/down
Valuation rings, valuative criterion for normality
Discrete valuation rings, Dedekind domains
Hilbert functions, dimension theory
Introduction to Tor/Ext
The Koszul complex
This seems quite ambitious on paper, and we may not get to cover everything, especially some of the things at the end.
Things that we may have to review along the way:
Hom and tensor, exactness properties
Nilpotent elements and the radical
We will follow a combination of the books by Atiyah-Macdonald and Eisenbud, both standard texts in the area. Homeworks will be assigned and collected on a somewhat regular basis (about once every 1-2 weeks). I'll post what we have done and where we intend to go at the end of each class on the blog.