Mondays and Wednesdays 3-4:15 SC 310
This is a graduate-level commutative algebra class. Topics may include, but are not limited to, Hilbert's Basis Theorem and Nullstellensatz, localization, primary decomposition, Artin-Rees Lemma, flat families and Tor, completions of rings, Noether Normalization, systems of parameters, DVRs, dimension theory, Hilbert-Samuel polynomials.
Instructor: Brooke Ullery (bullery@math, office SC 503)
Office hours: Tuesdays 11-1 in my office, SC 503.
Teaching fellow: Yujie Xu (yujiex@math, office hours Wednesdays 7:30-8:30 PM in SC 411)
Text: Commutative algebra with a view toward algebraic geometry, by David Eisenbud.
Recommended prep: Math 122-123, 55, or equivalent. Though no background in algebraic geometry will be necessary, it would be useful to understand some of the basic concepts, such as algebraic sets and coordinate rings. A good reference is Karen Smith's "An Invitation to Algebraic Geometry."
Homework: Problem sets will be assigned every one to two weeks (usually due Wednesdays). I encourage you to work on the problems together, but you must turn in your own solutions and list the names of your collaborators.
Grading: Homework: 90-100%, Participation: 0-10%. Participation can only help your grade, not hurt it. You can get a participation grade by regularly coming to office hours or by regularly asking/answering questions in class.
Problem set 1: pdf file, tex file, due September 25
Problem set 2: pdf file, tex file, due October 9
Problem set 3: pdf file, tex file, due October 23
Problem set 4: pdf file, tex file, due November 6
Problem set 5: pdf file, tex file, due November 20
Problem set 6: pdf file, tex file, due December 4
You can find the lecture notes from class here. I'll post each section after we've covered it, so there may be some notes covering multiple days. Warning: There will inevitably be typos in the notes!
Section 1: Introduction
Section 2: Noetherian rings and the Hilbert Basis Theorem
Section 3: Graded modules and Hilbert functions
Section 4: Localization
Section 5: Hom and tensor
Section 6: The spectrum of a ring
Section 7: The length of a module
Section 8: Associated primes
Section 9: Prime avoidance
Section 10: Primary decomposition
Section 11: Cayley-Hamilton, integrality, and Nakayama's lemma
Section 12: Normal rings and normalization
Section 13: Lying over and going up theorems
Section 14: The Nullstellensatz and a little classical algebraic geometry
Section 15: Filtrations and associated graded modules
Section 16: The blowup algebra and the tangent cone
Section 17: Artin-Rees and the Krull Intersection Theorem
Section 18: Flat families and Tor
Section 19: Completions of rings
Section 20: Hensel's Lemma
Section 21: Introduction to dimension
Section 22: Krull's principal ideal theorem
Section 23: Systems of parameters
Section 24: Going down theorem
Section 25: Regular local rings