# Commutative Algebra Notes

Welcome to my commutative algebra notes. All of these notes come from courses taught by Craig Huneke at the University of Kansas. All sets of notes were compiled by grad students and I am sure many errors exist. If you find one, please feel free to let me know.

Last updated: April 2013

# Commutative Algebra I (PDF)

Based on a class taken in the Spring of 2002 and of 2007. Topics include the following:

Chapter 1

- Rings, Ideals, and Maps
- Notation and Examples
- Homomorphisms and Isomorphisms
- Ideals and Quotient Rings
- Prime Ideals
- Unique Factorization Domain

Chapter 2

- Modules
- Notation and Examples
- Submodules and Maps
- Tensor Products
- Operations on Modules
- Exercises

Chapter 3

- Localization
- Notation and Examples
- Ideals and Localization
- UFD’s and Localization
- Exercises

Chapter 4

- Chain Conditions
- Noetherian Rings
- Noetherian Modules
- Artinian Rings
- Exercises

Chapter 5

- Primary Decomposition
- Definitions and Examples
- Primary Decomposition
- Exercises

Chapter 6

- Integral Closure
- Definitions and Notation
- Going-Up
- Normalization and Nullstellensatz
- Going-Down
- Examples
- Exercises

Chapter 7

- Krull’s Theorems and Dedekind Domains
- Krull’s Theorems
- Dedekind Domains
- Exercises

Chapter 8

- Completions and Artin-Rees Lemma
- Inverse Limits and Completions
- Artin-Rees Lemma
- Properties of Completions
- Exercises

# Commutative Algebra II (PDF)

Based on a class taken in the Spring of 2011. Topics include the following:

Chapter 1

- Regular Local Rings
- Definitions and Equivalences
- Minimal Resolutions and Projective Dimension
- KoszulComplex
- Corollaries of a RLR
- Exercises

Chapter 2

- Depth, Cohen-Macaulay Modules, and Projective Dimension

Chapter 3

- Gorenstein Rings
- Criteria for Irreducibility
- Injective Modules over Noetherian rings
- Divisible Modules
- Essential Extensions
- Structure of E_R(k)
- Structure of E_R(R/P)
- Minimal Injective Resolutions

# Commutative Algebra III (PDF)

Based on a class taken in the Fall of 2011. Topics include the following:

- Hilbert Functions and Multiplicities
- The Hilbert-Samuel Polynomial
- Multiplicities
- Superficial Elements
- Integral Closure of Ideals
- Associated Graded Ring and Rees Algebra
- Equations defining Rees Algebras
- Grothendieck Groups
- Basic Lemmas and Remarks
- Class Groups