## MATH 215A : Commutative Algebra

• Course description: The study of commutative rings, with a view toward algebraic geometry. One goal of the course is to prove the algebraic results used in Math 214AB and in the theory of schemes.
The prime spectrum of a commutative ring. Localization of rings and modules. Noetherian rings, Hilbert basis theorem. Integral closure and normalization. Noether normalization lemma, Hilbert Nullstellensatz. Dedekind domains and discrete valuation rings. Dimension theory. Regular local rings.

• Instructor: Burt Totaro, totaro@math.ucla.edu, MS 6136

• Lecture: MWF 11-11:50, MS 5148.

• Office hour: Friday, 2:00, MS 6136.

• Textbook: Atiyah-Macdonald's "Introduction to commutative algebra" (\$69 at the UCLA Store) is the main book for the class. Compared to Atiyah-Macdonald, I will include more homological algebra (flat modules, projective modules, Ext and Tor) and prove some of the deeper properties of regular local rings. Other useful books are Eisenbud's "Commutative algebra with a view toward algebraic geometry" (Springer, \$44), of which about the first half corresponds to this class, and Reid's "Undergraduate commutative algebra" (Cambridge, \$43), more elementary than this class.

• Prerequisite: Undergraduate algebra is essential (especially fields, rings, and modules). You should know the classification of finitely generated modules over a PID, and the fact that polynomial rings over a field are unique factorization domains -- if not, look these things up in Lang's "Algebra". Galois theory is not required. There is some overlap with Math 210ABC.

• Homework: There will be weekly homework.

• Homework 1 (due October 8, 2018).

• Homework 2 (due October 15, 2018).

• Homework 3 (due October 22, 2018).

• Homework 4 (due October 29, 2018).

• Homework 5 (due November 5, 2018).

• Homework 6 (due November 14, 2018).

• Homework 7 (due November 19, 2018).

• Homework 8 (due November 26, 2018).

• Homework 9 (due December 3, 2018).