A Singular Introduction to Commutative Algebra

A Singular Introduction to Commutative Algebra by Gert-Martin Greuel, Gerhard Pfister.

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...

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Bibliographic Details
Main Authors: Greuel, Gert-Martin (Author), Pfister, Gerhard (Author)
Corporate Author: SpringerLink (Online service)
Format: eBook
Language:English
Published: Berlin, Heidelberg : Springer Berlin Heidelberg : Imprint: Springer, 2002.
Edition:1st ed. 2002.
Series:Springer eBook Collection.
Subjects:
Algebra.
Algebraic geometry.
Algorithms.
Computer mathematics.
Computer science—Mathematics.
Electronic resources (E-books)
Online Access:Click to view e-book
Holy Cross Note:Loaded electronically.
Electronic access restricted to members of the Holy Cross Community.
Table of Contents:
  • 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
  • 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 Koszul Complex and Depth
  • 7.7 Cohen-Macaulay Rings
  • 7.8 Further Characterization of Cohen-Macaulayness
  • 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.

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