Basic Commutative Algebra

The following sections are included:

• Preface

• Contents

The following sections are included:

• Recollection and Preliminaries

• Prime and Maximal Ideals

• Sums, Products and Colons

• Radicals

• Zariski Topology

• Exercises

The following sections are included:

• Modules

• Homomorphisms

• Direct Products and Direct Sums

• Free Modules

• Exact Sequences

• Algebras

• Fractions

• Graded Rings and Modules

• Homogeneous Prime and Maximal Ideals

• Exercises

The following sections are included:

• Polynomial Rings

• Power Series Rings

• Exercises

The following sections are included:

• Categories and Functors

• Exact Functors

• The Functor Hom

• Tensor Product

• Base Change

• Direct and Inverse Limits

• Injective, Projective and Flat Modules

• Exercises

The following sections are included:

• Tensor Product of Algebras

• Tensor Algebras

• Symmetric Algebras

• Exterior Algebras

• Anticommutative and Alternating Algebras

• Determinants

• Exercises

The following sections are included:

• Modules of Finite Length

• Noetherian Rings and Modules

• Artinian Rings and Modules

• Locally Free Modules

• Exercises

The following sections are included:

• Primary Decomposition

• Support of a Module

• Dimension

• Exercises

The following sections are included:

• Filtrations and Associated Graded Rings and Modules

• Linear Topologies and Completions

• Ideal-adic Completions

• Initial Submodules

• Completion of a Local Ring

• Exercises

The following sections are included:

• Numerical Functions

• Hilbert Function of a Graded Module

• Hilbert–Samuel Function over a Local Ring

• Exercises

The following sections are included:

• Principal Ideal Theorem

• Dimension of a Local Ring

• Exercises

The following sections are included:

• Integral Extensions

• Prime Ideals in an Integral Extension

• Integral Closure in a Finite Field Extension

• Exercises

The following sections are included:

• Unique Factorization Domains

• Discrete Valuation Rings and Normal Domains

• Fractionary Ideals and Invertible Ideals

• Dedekind Domains

• Extensions of a Dedekind Domain

• Exercises

The following sections are included:

• Transcendental Extensions

• Separable Field Extensions

• Lüroth's Theorem

• Exercises

The following sections are included:

• Noether's Normalization Lemma

• Hilbert's Nullstellensatz

• Dimension of an Affine Algebra

• Dimension of a Graded Ring

• Dimension of a Standard Graded Ring

• Exercises

The following sections are included:

• Derivations

• Differentials

• Exercises

The following sections are included:

• Valuations Rings

• Valuations

• Extensions of Valuations

• Real Valuations and Completions

• Hensel's Lemma

• Discrete Valuations

• Exercises

The following sections are included:

• Derived Functors

• Uniqueness of Derived Functors

• Complexes and Homology

• Resolutions of a Module

• Resolutions of a Short Exact Sequence

• Construction of Derived Functors

• The Functors Ext

• The Functors Tor

• Local Cohomology

• Homology and Cohomology of Groups

• Exercises

The following sections are included:

• Injective Dimension

• Projective Dimension

• Global Dimension

• Projective Dimension over a Local Ring

• Exercises

The following sections are included:

• Regular Sequences and Depth

• Depth and Projective Dimension

• Cohen–Macaulay Modules over a Local Ring

• Cohen–Macaulay Rings and Modules

• Exercises

The following sections are included:

• Regular Local Rings

• A Differential Criterion for Regularity

• A Homological Criterion for Regularity

• Regular Rings

• A Regular Local Ring is a UFD

• The Jacobian Criterion for Geometric Regularity

• Exercises

The following sections are included:

• Divisor Class Groups

• The Case of Fractions

• The Case of Polynomial Extensions

• The Case of Galois Descent

• Galois Descent in the Local Case

• Exercises

The following sections are included:

• Bibliography

• Index