A Course in Commutative Algebra

A Course in Commutative Algebra


0 Ring Theory Background
0.1 Prime Avoidance
0.2 Jacobson Radicals, Local Rings, and Other Miscellaneous Results
0.3 Nakayama’s Lemma

1 Primary Decomposition and Associated Primes
1.1 Primary Submodules and Ideals
1.2 Primary Decomposition
1.3 Associated Primes
1.4 Associated Primes and Localization
1.5 The Support of a Module
1.6 Artinian Rings

2 Integral Extensions
2.1 Integral Elements
2.2 Integrality and Localization
2.3 Going Down

3 Valuation Rings
3.1 Extension Theorems
3.2 Properties of Valuation Rings
3.3 Discrete Valuation Rings

4 Completion
4.1 Graded Rings and Modules
4.2 Completion of a Module
4.3 The Krull Intersection Theorem

5 Dimension Theory
5.1 The Calculus of Finite Differences
5.2 Hilbert and Hilbert-Samuel Polynomials
5.3 The Dimension Theorem
5.4 Consequences of the Dimension Theorem
5.5 Strengthening of Noether’s Normalization Lemma
5.6 Properties of Affine k-Algebras

6 Depth
6.1 Systems of Parameters
6.2 Regular Sequences

7 Homological Methods
7.1 Homological Dimension: Projective and Global
7.2 Injective Dimension
7.3 Tor and Dimension
7.4 Application

8 Regular Local Rings
8.1 Basic Definitions and Examples

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