Commutative Algebra in Four Movements
Authors
Description
The book is an introduction to the study of commutative rings aimed at readers who are at the beginning of their journey through the fascinating area of mathematics and is not designed to be read linearly, from end to end, but rather “navigated” according to the interests and needs of each reader.
The prerequisites for this book are few: in essence, a good undergraduate algebra course covers far more than is necessary to read it. In any case, the appendices present these prerequisites in a telegraphic way and the reader can consult them to remember this or that result or even learn it “on the fly” if they are unfamiliar with it.
Name: Commutative Algebra in Four Movements
Author(s): Eduardo Tengan e Herivelto Borges
Pages: 480
Publication: IMPA, 2020
ISBN: 978-65-990528-8-0
Edition: 2
I Nocturne
1 Naming names
1.1 Notations, definitions and conventions
1.2 Domains, reduced and indecomposable rings
1.3 Ideals
1.3.1 Eigen and maximal ideas
1.3.2 Operations with ideals
1.4 Quotient ring
1.5 Chinese remainder theorem
1.6 Modules
1.6.1 Exact sequences
1.6.2 Operations on modules
1.7 Graded rings and modules
1.8 Exercises
2 Rings that appear in nature
2.1 Formal series
2.2 Algebraic integers
2.3 Algebraic varieties
2.3.1 Affine algebraic sets
2.3.2 Morphisms and rings of regular functions
2.3.3 Equivalence of categories
2.3.4 Projective algebraic sets
2.4 P-adic integers
2.5 Exercises
II Scherzo
3 Prime ideals and spectrum
3.1 Prime ideals
3.2 Krull dimension
3.3 Zariski topology
3.4 Exercises
4 Localization
4.1 Construction and universal property
4.2 The localization functor
4.3 How to murder primes
4.4 Connectedness and irreducibility
4.5 Local rings and Nakayama’s lemma
4.6 Minimal bases
4.7 Exercises
5 Tensor product
5.1 Construction and basic properties
5.2 The base change functor
5.3 Tensor product of algebras
5.4 Fibers
5.5 Modules and plane algebras
5.6 Exercises
6 Noetherian rings and modules
6.1 Definitions and basic properties
6.2 Hilbert basis theorem
6.3 Finite presentation algebras and modules
6.4 Exercises
7 Artinian rings and modules
7.1 Basic definitions and properties
7.2 Module lengths
7.3 Structure of Artinian rings
7.4 Exercises
III Passacaglia
8 Finite extensions and integrals
8.1 Basic definitions and properties
8.2 Fibres of finite extensions and integrals
8.3 Normal rings and normalization
8.4 Exercises
9 Noether normalization and Nullstellensatz
9.1 Noether normalization theorem
9.2 Dimension of finitely generated domains over bodies
9.3 Nullstellensatz
9.4 NullstellensatZ
9.5 Exercises
10 Dedekind domains and discrete valuations
10.1 Discrete valuations
10.2 Dedekind domains
10.3 Order
10.4 Exercises
11 Group action and Going-down
11.1 Groups acting on a ring
11.2 Going-down
11.3 Decomposition and inertia groups
11.4 Applications in Galois theory
11.5 Exercises
12 Zero divisors and associated primes
12.1 Support and annuli of a module
12.2 Zero divisors and associated primes
12.3 Serre’s normality criterion
12.4 Primary decomposition
12.5 Exercises
IV Burlesque
13 Complete rings
13.1 α-adic topology and the Artin-Rees theorem
13.2 Complete and Henselian rings
13.3 Completeness of Noetherian rings
13.4 Weierstraß preparation theorem
13.5 Exercises
14 Dimension
14.1 Some binomial identities
14.2 Hilbert-Samuel polynomial
14.3 Krull’s dimension theorem
14.4 Fiber dimension
14.5 Regular local rings
14.6 Exercises
15 Schemes
15.1 Geometry with category
15.1.1 Pre-bundles and bundles
15.1.2 Locally annular spaces
15.2 Schemes
15.2.1 Structural bundle of a ring
15.2.2 Affine schemes
15.2.3 Examples
15.2.4 Projective schemes
15.3 Point functor and fibrated product
15.3.1 Point functor of schemes
15.3.2 Fibrated product of schemes
15.4 Properties of schemes
15.5 Exercises
V Appendices
A Fundamentals
A.1 General topology
A.1.1 Building new topologies
A.1.2 Metric spaces
A.1.3 Properties
A.1.4 Topological groups
A.2 Categories and functors
A.3 Limits
A.4 Exercises
B Single factorization
B.1 Euclidean domains, domains of principal ideas and domains of single factorization
B.2 Example: Gauß integers
B.3 Gauß’s lemma
B.4 Structure of finitely generated modules over domains of principal ideals
B.5 Exercises
C Theory of bodies
C.1 Finite and algebraic extensions of bodies
C.2 Simple extensions and algebraic closure
C.3 Quasi-Galois extensions and fundamental lemma
C.4 Separability
C.5 Galois theory
C.6 Infinite Galois theory
C.7 Trace and norm
C.8 Discriminant
C.9 Transcendental extensions
C.10 Exercises
Bibliography