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