##
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).