The text starts with the theory of rings and modules and standard bases with
emphasis on local rings and localization. It is followed by the central
concepts of commutative algebra such as integral closure, dimension theory,
primary decomposition, Hilbert function, completion, flatness and homological
algebra. An appendix is devoted to algebraic geometry to show how geometric
problems can be understood using commutative algebra. The book includes a CD
with a distribution of SINGULAR for various platforms (Unix/Linux,
Windows, Macintosh) and which contains all examples and procedures explained
in the text.
| Preface (ps-file) |
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| 1 Rings, Ideals and Standard Bases
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| 1.1 Rings, Polynomials and Ring Maps
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| 1.2 Monomial Orderings
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| 1.3 Ideals and Quotient Rings
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| 1.4 Local Rings and Localization
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| 1.5 Rings Associated to Monomial Orderings
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| 1.6 Normal Forms and Standard Bases
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| 1.7 The Standard Basis Algorithm
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| 1.8 Operations on Ideals and their Computation
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| 1.8.1 Ideal Membership
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| 1.8.2 Intersection with Subrings
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| 1.8.3 Zariski Closure of the Image
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| 1.8.4 Solvability of Polynomial Equations
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| 1.8.5 Solving Polynomial Equations
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| 1.8.6 Radical Membership
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| 1.8.7 Intersection of Ideals
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| 1.8.8 Quotient of Ideals
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| 1.8.9 Saturation
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| 1.8.10 Kernel of a Ring Map
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| 1.8.11 Algebraic Dependence and Subalgebra Membership
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| 2. Modules
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| 2.1 Modules, Submodules and Homomorphisms
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| 2.2 Graded Rings and Modules
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| 2.3 Standard Bases for Modules
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| 2.4 Exact Sequences and free Resolutions
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| 2.5 Computing Resolutions and the Syzygy Theorem
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| 2.6 Modules over Principal Ideal Domains
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| 2.7 Tensor Product
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| 2.8 Operations on Modules and their Computation
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| 2.8.1 Module Membership Problem
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| 2.8.2 Intersection with Free Submodules
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| 2.8.3 Intersection of Submodules
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| 2.8.4 Quotients of Submodules
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| 2.8.5 Radical and Zerodivisors of Modules
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| 2.8.6 Annihilator and Support
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| 2.8.7 Kernel of a Module Homomorphism
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| 2.8.8 Solving Systems of Linear Equations
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| 3. Noether Normalization and Applications
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| 3.1 Finite and Integral Extensions
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| 3.2 The Integral Closure
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| 3.3 Dimension
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| 3.4 Noether Normalization
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| 3.5 Applications
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| 3.6 An Algorithm to Compute the Normalization
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| 3.7 Procedures
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| 4. Primary Decomposition and Related Topics
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| 4.1 The Theory of Primary Decomposition
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| 4.2 Zero-dimensional Primary Decomposition
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| 4.3 Higher Dimensional Primary Decomposition
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| 4.4 The Equidimensional Part of an Ideal
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| 4.5 The Radical
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| 4.6 Procedures
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| 5. Hilbert Function and Dimension
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| 5.1 The Hilbert Function and the Hilbert Polynomial
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| 5.2 Computation of the Hilbert-Poincare Series
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| 5.3 Properties of the Hilbert Polynomial
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| 5.4 Filtrations and the Lemma of Artin-Rees
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| 5.5 The Hilbert-Samuel Function
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| 5.6 Characterization of the Dimension of Local Rings
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| 5.7 Singular Locus
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| 6. Complete Local Rings
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| 6.1 Formal Power Series Rings
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| 6.2 Weierstrass Preparation Theorem
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| 6.3 Completions
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| 6.4 Standard bases
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| 7. Homological Algebra
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| 7.1 Tor and Exactness
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| 7.2 Fitting Ideals
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| 7.3 Flatness
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| 7.4 Local Criteria for Flatness
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| 7.5 Flatness and Standard Bases
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| 7.6 Koszul Complex and Depth
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| 7.7 Cohen-Macaulay Rings
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| 7.8 Further Characterization of Cohen-Macaulayness
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| A. Geometric Background
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| A.1 Introduction by Pictures (ps-file)
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| A.2 Affine Algebraic Varieties
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| A.3 Spectrum and Affine Schemes
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| A.4 Projective Varieties
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| A.5 Projective Schemes and Varieties
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| A.6 Morphisms between Varieties
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| A.7 Projective Morphisms and Elimination
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| A.8 Local versus Global Properties
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| A.9 Singularities
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| B. SINGULAR - A Short Introduction (ps-file)
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| B.1 Downloading Instructions
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| B.2 Getting Started
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| B.3 Procedures and Libraries
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| B.4 Data Types
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| B.5 Functions
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| B.6 Control Structures
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| B.7 System Variables
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| B.8 Libraries
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| B.9 SINGULAR and Maple
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| B.10 SINGULAR and Mathematica
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| B.11 SINGULAR and MuPAD
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| References (ps-file)
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| Index (ps-file)
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| Algorithms
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| SINGULAR Examples
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