The textbook for this course may be the best math textbook ever written:

Atiyah, M. F. and MacDonald, I. G. Introduction to Commutative Algebra.

We will also make digressions into a few other topics:

Matsumura, H. Commutative Algebra, 2nd Ed.

Weibel, C. A. An Introduction to Homological Algebra

The requirements of the course are to make regular presentations of exercises in class, and to write a paper or give a presentation about a related project by the end of the semester. Those who prefer not to make presentations in front of the class may submit exercises regularly. Anyone who fulfills these requirements will receive an A in the course.

Review: the definitions of commutative ring, modules, and topological spaces

Project topic (11/28): Dedekind domains (Atiyah—MacDonald, Chapter 9)

Project topic (11/30): extensions of modules

Project topic (12/2): dimension (Atiyah—MacDonald, Chapter 11)

Project topic (12/5): completions (Atiyah—MacDonald, Chapter 10)

Project topic (12/7): flat descent [outline] (Kai Behrend, Brian Conrad, Dan Edidin, Barbara Fantechi, William Fulton, Lothar Göttsche, and Andrew Kresch, §2; Stacks project)

Project topic (12/9): extensions of algebras [outline]

Review: rings, ideals, subrings, quotient rings, polynomial rings

Read: Atiyah-MacDonald, Chapter 1 (pp. 1-10), and Chapter 3 until modules are mentioned (pp. 36-38)

Exercises: §1, #1, 2, 4 (Hint: use 1.9 and exercise 2), 5, 7, 8, 10, 12; Find a universal property for the group ring and prove it.

Presentations: Exercises §1, #1, 2, 4; Exercise §1, #13; Exercise §1, #14

Review: topological spaces

Read: Atiyah-MacDonald, Chapter 1, Exercise 15-22, 26-28 (pp. 12-16)

Presentations: §1, #16, 22, 26

Project topic: Stone-Čech compactification

Project topic: Schemes

Review: modules, kernel, cokernel, direct sum, direct product

Read: Atiyah-MacDonald, Chapter 2 (pp. 17-31)

Project topic: flatness, finite presentation, projectivity, and Lazard's theorem

Project topic: flat descent [outline] (Kai Behrend, Brian Conrad, Dan Edidin, Barbara Fantechi, William Fulton, Lothar Göttsche, and Andrew Kresch, §2; Stacks project)

Exercises: §2, #1-26; I encourage doing some of #13-24, and especially #19 to get comfortable with direct limits

Review: exact sequence, kernel, cokernel

Read: Weibel, Chapters 2 and 3

Project topic: Regular sequences and depth

Project topic: Biextensions

Exercises: Prove that every projective module is a direct summand of a free module and conversely (Weibel, Prop. 2.2.1); Show that the ideal *(2, 1 + x)* in *ℤ[x] / (x^2 + 5)* is projective but not free; Prove that *Ext ^{1}(M,N)* is independent of whether

Read: Atiyah-MacDonald, Chapter 3 (pp. 36-43)

Exercises:

Read: Atiyah-MacDonald, Chapter 6 and Chapter 7

Read: Atiyah-MacDonald, Chapter 8; Matsumura, Section 26

Project topic: smooth morphisms

Read: Atiyah-MacDonald, Chapter 4

Read: Atiyah-MacDonald, Chapter 5

Read: Atiyah-MacDonald, Chapter 9

Read: Atiyah-MacDonald, Chapter 10

Read: Atiyah-MacDonald, Chapter 11