A Singular Introduction to Commutative Algebra
This book can be understood as a model for teaching commutative algebra, taking into account modern developments such as algorithmic and computational aspects. As soon as a new concept is introduced, it is shown how to handle it by computer. The computations are exemplified with the computer algebra...
| Main Authors: | , |
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| Format: | eBook |
| Language: | English |
| Published: |
Berlin, Heidelberg
Springer Berlin Heidelberg
2002, 2002
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| Edition: | 1st ed. 2002 |
| Subjects: | |
| Online Access: | |
| Collection: | Springer Book Archives -2004 - Collection details see MPG.ReNa |
Table of Contents:
- 7.9 Homological Characterization of Regular Rings
- A. Geometric Background
- A.1 Introduction by Pictures
- A.2 Affine Algebraic Varieties
- A.3 Spectrum and Affine Schemes
- A.4 Projective Varieties
- A.5 Projective Schemes and Varieties
- A.6 Morphisms Between Varieties
- A.7 Projective Morphisms and Elimination
- A.8 Local Versus Global Properties
- A.9 Singularities
- B. SINGULAR — A Short Introduction
- B.1 Downloading Instructions
- B.2 Getting Started
- B.3 Procedures and Libraries
- B.4 Data Types
- B.5 Functions
- B.6 Control Structures
- B.7 System Variables
- B.8 Libraries
- References
- Algorithms
- 4.1 The Theory of Primary Decomposition
- 4.2 Zero-dimensional Primary Decomposition
- 4.3 Higher Dimensional Primary Decomposition
- 4.4 The Equidimensional Part of an Ideal
- 4.5 The Radical
- 4.6 Procedures
- 5. Hilbert Function and Dimension
- 5.1 The Hilbert Function and the Hilbert Polynomial
- 5.2 Computation of the Hilbert-Poincaré Series
- 5.3 Properties of the Hilbert Polynomial
- 5.4 Filtrations and the Lemma of Artin-Rees
- 5.5 The Hilbert-Samuel Function
- 5.6 Characterization of the Dimension of Local Rings
- 5.7 Singular Locus
- 6. Complete Local Rings
- 6.1 Formal Power Series Rings
- 6.2 Weierstraß Preparation Theorem
- 6.3 Completions
- 6.4 Standard Bases
- 7. Homological Algebra
- 7.1 Tor and Exactness
- 7.2 Fitting Ideals
- 7.3 Flatness
- 7.4 Local Criteria for Flatness
- 7.5 Flatness and Standard Bases
- 7.6 KoszulComplex and Depth
- 7.7 Cohen-Macaulay Rings
- 7.8 Further Characterization of Cohen-Macaulayness
- 1. Rings, Ideals and Standard Bases
- 1.1 Rings, Polynomials and Ring Maps
- 1.2 Monomial Orderings
- 1.3 Ideals and Quotient Rings
- 1.4 Local Rings and Localization
- 1.5 Rings Associated to Monomial Orderings
- 1.6 Normal Forms and Standard Bases
- 1.7 The Standard Basis Algorithm
- 1.8 Operations on Ideals and Their Computation
- 2. Modules
- 2.1 Modules, Submodules and Homomorphisms
- 2.2 Graded Rings and Modules
- 2.3 Standard Bases for Modules
- 2.4 Exact Sequences and free Resolutions
- 2.5 Computing Resolutions and the Syzygy Theorem
- 2.6 Modules over Principal Ideal Domains
- 2.7 Tensor Product
- 2.8 Operations on Modules and Their Computation
- 3. Noether Normalization and Applications
- 3.1 Finite and Integral Extensions
- 3.2 The Integral Closure
- 3.3 Dimension
- 3.4 Noether Normalization
- 3.5 Applications
- 3.6 An Algorithm to Compute the Normalization
- 3.7 Procedures
- 4. Primary Decomposition and Related Topics