# Commutative Algebra (MATH 205b, Spring 2012)

Day & Time: Tuesday and Friday 9:30am — 10:50am

Location: Goldsmith 0226

Instructor: Joel Bellaiche

Office hours: Tuesdays, 3pm — 4pm

Grade: 40% exercises in class, 40% personal project, 20% final exam

Textbooks: For references, I will essentially use Commutative Algebra (with a view toward algebraic geometry) by David Eisenbud. This is very long, very complete, very detailed book with a lot of explanations and motivations. This book is on reserve at the main library and on sale at the campus bookstore. Other textbooks include Commutative Algebra, and Commutative Ring Theory by Matsumura, which are quite the opposite in style: short, direct, concise; Attiyah-Macdonald's Introduction to commutative algebra; the classic Commutative Algebra of Zariski and Samuel; and of course, Bourbaki.

## Schedule of Lectures

 Class Topic Suggested Reading Exercises Jan 17 Introduction. History. Chapter 1 of Eisenbud; Weil's Number-Theory and Algebraic Geometry Jan 20 Basic Notions: rings, fields, algebras; ideals; prime and maximal ideals; Noetherian rings Exercises 1 Jan 24 Basic Notions Jan 27 The radical of a ring and of an ideal. Reduced rings. Jan 31 Spectrum and Zariski topology, Exercises 2 Feb 3 Spectrum and Zariski topology. The Nullstellensatz Feb 7 The Nullstellensatz and consequences. Beginning of Modules Feb 10 Morphisms of modules. Noetherian modules and noetherian rings. Operations on submodules. Feb 14 Operations on Modules: direct sum, product, Hom Feb 17 Operations on Modules: tensor product. Exercises 3 Feb 17 Operations on Modules: tensor product. Feb 28 Tensor product of algebras. The Cayley-Hamilton theorem Mar 2 Integral elements and integral closure.. Exercises 4 Mar 6 The nullstellensatz for general fields. Mar 9 Rings of fractions Mar 13 No class Mar 16 Localization. Local rings, Nakayama's lemma Exercises 5 Mar 20 Localization, flatness Mar 24 Flatness: criterion with ideals Mar 27 Flatness: criterion with relations. Flat modules over a local rings. Exercises 6