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; AttiyahMacdonald's Introduction to commutative algebra; the classic Commutative Algebra of Zariski and Samuel; and of course, Bourbaki.
Class 
Topic 
Suggested Reading 
Exercises 
Jan 17 
Introduction. History. 
Chapter 1 of Eisenbud; Weil's NumberTheory 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 CayleyHamilton 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 
