This is the homepage for my Commutative Algebra book. It is meant to be the first of two volumes, the second being dedicated to homological methods.
The book is available on the AMS Open Math Notes website.
Please, use Github issues to signal errors, incosistencies, typos, or just parts that are written in a way that is not clear. The following is the table of contents.
Introduction
1 Basics
1.1 Rings and ideals
1.2 Quotients
1.3 Modules
1.4 More constructions with modules
1.5 Euclidean rings
1.6 Localization
1.7 Graded rings and modules
1.8 Exercises
2 Finiteness conditions
2.1 Principal ideal domains
2.2 Artinian and Noetherian modules
2.3 Noetherian rings
2.4 Artinian rings
2.5 Length
2.6 Exercises
3 Factorization
3.1 Unique factorization domains
3.2 Primary decomposition
3.3 Primary decomposition for modules
3.4 Factorization in Dedekind rings
3.5 Modules over Dedekind rings
3.6 Exercises
4 Computational methods
4.1 The resultant
4.2 Discriminants
4.3 Gröbner bases
4.4 More algorithmic operations
4.5 Exercises
5 Integral dependence
5.1 Integral extensions
5.2 Going up and down
5.3 Noether normalization
5.4 Integral extensions of Dedekind rings
5.5 Exercises
6 Lattice methods
6.1 Additive structure of number rings
6.2 Prime extensions in number rings
6.3 Prime extensions in Dedekind rings
6.4 Galois extensions of Dedekind rings
6.5 Discriminant and ramification
6.6 Computing prime factorizations
6.7 Geometry of ideal lattices
6.8 Cyclotomic rings
6.9 Exercises
7 Metric and topological methods
7.1 Absolute values
7.2 Valuations and valuation rings
7.3 Discrete valuation rings
7.4 Direct and inverse limits
7.5 Completion of rings and modules
7.6 Hensel’s lemma
7.7 Witt vectors
7.8 Exercises
8 Geometric dictionary
8.1 Affine varieties
8.2 The Nullstellensatz
8.3 The Ax-Grothendieck theorem
8.4 Morphisms
8.5 Local rings and completions revisited
8.6 Graded rings and projective varieties
8.7 A new idea: the dimension
8.8 The Zariski tangent space
8.9 Curves and Dedekind rings
8.10 Exercises
9 Dimension theory
9.1 Dimension of rings and modules
9.2 Hilbert functions
9.3 The main theorem on dimension
9.4 Height
9.5 Properties of dimension
9.6 Dimension of graded rings
9.7 Exercises
10 Local structure
10.1 Regular rings
10.2 Multiplicity and degree
10.3 Formulas for multiplicity
10.4 Multiplicity and valuations
10.5 Superficial elements
10.6 Cohen’s structure theorem
10.7 Exercises
Appendix A Fields
A.1 Algebraic elements
A.2 Finite fields
A.3 Separability
A.4 Normal extensions
A.5 The Galois correspondence
A.6 Some computations
A.7 The trace and norm
A.8 Abelian extensions
A.9 Exercises