# Math 6150: Fall 2016

## Textbook

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

## Evaluation

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.

## Syllabus

### Before we begin:

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

### Project topics:

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]

### Part 1: Commutative rings

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

### Part 2: Spectra of commutative rings

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

### Part 3: Modules

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

### Part 4: Homological algebra

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 Ext1(M,N) is independent of whether M or N is resolved; Show that Torp(M,N) = 0 for all abelian groups M and N and all p ≥ 2 (Weibel, Prop. 3.1.2); Find an example of a ring A and A-modules M and N such that Torp(M,N) ≠ 0 for all p ≥ 0.

### Part 5: Localization

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

Exercises:

### Part 6: Noetherian rings

Read: Atiyah-MacDonald, Chapter 6 and Chapter 7

### Part 7: Nilpotent extensions and differentials

Read: Atiyah-MacDonald, Chapter 8; Matsumura, Section 26
Project topic: smooth morphisms