Author | Greuel, Gert-Martin. author |
---|---|
Title | A Singular Introduction to Commutative Algebra [electronic resource] / by Gert-Martin Greuel, Gerhard Pfister |
Imprint | Berlin, Heidelberg : Springer Berlin Heidelberg : Imprint: Springer, 2002 |
Connect to | http://dx.doi.org/10.1007/978-3-662-04963-1 |
Descript | XVII, 588 p. 44 illus. online resource |
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-Poincarรฉ 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 Weierstraร 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 โ 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