Greuel / Pfister A Singular Introduction to Commutative Algebra

Erscheinungsjahr 2012
ISBN: 978-3-662-04963-1
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 588 Seiten, eBook

In theory there is no difference between theory and practice. In practice there is. Yogi Berra A SINGULAR Introduction to Commutative Algebra offers a rigorous intro duction to commutative algebra and, at the same time, provides algorithms and computational practice. In this book, we do not separate the theoretical and the computational part. Coincidentally, as new concepts are introduced, it is consequently shown, by means of concrete examples and general proce dures, how these concepts are handled by a computer. We believe that this combination of theory and practice will provide not only a fast way to enter a rather abstract field but also a better understanding of the theory, showing concurrently how the theory can be applied. We exemplify the computational part by using the computer algebra sys tem SINGULAR, a system for polynomial computations, which was developed in order to support mathematical research in commutative algebra, algebraic geometry and singularity theory. As the restriction to a specific system is necessary for such an exposition, the book should be useful also for users of other systems (such as Macaulay2 and CoCoA) with similar goals. Indeed, once the algorithms and the method of their application in one system is known, it is usually not difficult to transfer them to another system.

Zielgruppe

Graduate

Autoren/Hrsg.

Greuel, Gert-Martin

Pfister, Gerhard

Weitere Mitwirkende

Bachmann, O.

Lossen, C.

Schönemann, H.

Inhaltsverzeichnis

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 KoszulComplex 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.