This book stems from lectures on commutative algebra for 4th-year university students at two French universities (Paris and Rennes). At that level, students have already followed a basic course in linear algebra and are essentially fluent with the language of vector spaces over fields. The topics introduced include arithmetic of rings, modules, especially principal ideal rings and the classification of modules over such rings, Galois theory, as well as an introduction to more advanced topics such as homological algebra, tensor products, and algebraic concepts involved in algebraic geometry.
More than 300 exercises will allow the reader to deepen his understanding of the subject.
The book also includes 11 historical vignettes about mathematicians who contributed to commutative algebra.
Author(s): Antoine Chambert-Loir
Series: Universitext
Edition: 1
Publisher: Springer Nature Switzerland
Year: 2021
Language: English
Pages: 466
City: Cham
Tags: Algebra, Rings, Ideals, Modules, Galois Theory, Homological Algebra
Preface
Contents
Chapter 1 Rings
1.1.
Definitions. First Examples
1.2.
Nilpotent Elements; Regular and Invertible Elements; Division Rings
On William Hamilton
1.3.
Algebras, Polynomials
1.4.
Ideals
1.5.
Quotient Rings
1.6.
Fraction Rings of Commutative Rings
1.7.
Relations Between Quotient Rings and Fraction Rings
1.8.
Exercises
Chapter 2 Ideals and Divisibility
2.1.
Maximal Ideals
2.2.
Maximal and Prime Ideals in a Commutative Ring
2.3.
Hilbert's Nullstellensatz
On David Hilbert
2.4.
Principal Ideal Domains, Euclidean Rings
2.5.
Unique Factorization Domains
2.6.
Polynomial Rings are Unique Factorization Domains
On Carl Friedrich Gauss
2.7.
Resultants and Another Theorem of Bézout
2.8.
Exercises
Chapter 3 Modules
3.1.
Definition of a Module
3.2.
Morphisms of Modules
3.3.
Operations on Modules
3.4.
Quotients of Modules
3.5.
Generating Sets, Free Sets; Bases
3.6.
Localization of Modules (Commutative Rings)
3.7.
Vector Spaces
3.8.
Alternating Multilinear Forms. Determinants
On Arthur Cayley
3.9.
Fitting Ideals
3.10.
Exercises
Chapter 4
Field Extensions
4.1.
Integral Elements
4.2.
Integral Extensions
4.3.
Algebraic Extensions
4.4.
Separability
4.5.
Finite Fields
4.6.
Galois's Theory of Algebraic Extensions
On Évariste Galois
4.7. Norms and Traces
4.8.
Transcendence Degree
4.9.
Exercises
Chapter 5
Modules Over Principal Ideal Rings
5.1.
Matrix Operations
5.2.
Applications to Linear Algebra
5.3.
Hermite Normal Form
5.4.
Finitely Generated Modules Over a Principal Ideal Domain
On Ferdinand Frobenius
5.5.
Application: Finitely Generated Abelian Groups
5.6.
Application: Endomorphisms of a Finite-Dimensional Vector Space
5.7.
Exercises
Chapter 6
Noetherian and Artinian Rings. Primary Decomposition
6.1.
Nakayama's Lemma
6.2.
Length
6.3.
The Noetherian Property
6.4.
The Artinian Property
On Emil Artin
6.5.
Support of a Module, Associated Ideals
6.6.
Primary Decomposition
On Emmy Noether
6.7
Exercises
Chapter 7
First Steps in Homological Algebra
7.1. Diagrams, Complexes and Exact Sequences
7.2.
The "Snake Lemma''. Finitely Presented Modules
7.3.
Projective Modules
7.4.
Injective Modules
7.5.
Exactness Conditions for Functors
7.6.
Adjoint Functors
7.7.
Differential Modules. Homology and Cohomology
7.8.
Exercises
Chapter 8
Tensor Products and Determinants
8.1.
Tensor Products of Two Modules
8.2.
Tensor Products of Modules Over a Commutative Ring
8.3.
Tensor Algebras, Symmetric and Exterior Algebras
8.4.
The Exterior Algebra and Determinants
8.5.
Adjunction and Exactness
8.6.
Flat Modules
8.7.
Faithful Flatness
8.8.
Faithfully Flat Descent
On Alexander Grothendieck
8.9.
Galois Descent
8.10.
Exercises
Chapter 9
The Normalization Theorem, Dimension Theory and Dedekind Rings
9.1.
Noether's Normalization Theorem
9.2.
Finiteness of Integral Closure
9.3.
Dimension and Transcendence Degree
9.4.
Krull's Hauptidealsatz and Applications
On Wolfgang Krull
9.5.
Heights and Dimension
9.6.
Dedekind Rings
On Richard Dedekind
9.7.
Exercises
Appendix
A.1.
Algebra
A.2.
Set Theory
A.3.
Categories
Credits
References
Index
