The interplay between computation and many areas of algebra is a natural
phenomenon in view of the algorithmic character of the latter. The existence of
inexpensive but powerful computational resources has enhanced these links by the opening
up of many new areas of investigation in algebra. At the same time it made available
the theoretical tools of this area of mathematics to help deal with problems of
interest in physics, engineering and other disciplines. We aim here to discuss how certain
devices that permit the rapid processing of polynomials and matrices make it
possible to examine parts of two areas of algebra - commutative algebra and algebraic
geometry - where those data structures play critical roles.
Among the main tasks in computational algebra are the constructions of
decompositions and closures of objects in the ring of polynomials. Among the former are
finding primary decompositions and modules of syzygies, and among the latter, the
computation of integral closures and of rings of invariants. As a rule, they are
assisted by any a priori knowledge available. Another frequent task is to certify that a
given object has a certain property.
This book is an attempt to deal with these issues, despite the pace of development
in the field (ring?). The material was drawn mostly from the published literature,
both classical and recent, including conference proceedings on computer algebra that
tended to focus on algebraic structures.
Author(s): W.V. Vasconcelos
Series: Algorithms and Computation in Mathematics 2
Publisher: Springer
Year: 1998
Language: English
Pages: 406
Tags: Математика;Общая алгебра;
Algorithms and Computation in Mathematics • Volume 2 ......Page 2
Vasconcelos W.V. Computational Methods in Commutative Algebra and Algebraic Geometry (Springer,1998) ......Page 4
Copyright ......Page 5
Preface ......Page 7
Table of Contents ......Page 9
Introduction 1 ......Page 12
1.Fundamental Algorithms 7 ......Page 18
1.1 GrobnerBasics 8 ......Page 19
1.2 DivisionAlgorithms 12 ......Page 23
1.3 Computation of Syzygies 17 ......Page 28
1.4 Hilbert Functions 21 ......Page 32
1.5 ComputerAlgebraSystems 25 ......Page 36
2.Toolkit 27 ......Page 38
2.1 EliminationTechniques 28 ......Page 39
2.2 RingsofEndomorphisms 33 ......Page 44
2.3 Noether Normalization 35 ......Page 46
2.4 Fitting Ideals 39 ......Page 50
2.5 Finite and Quasi-Finite Morphisms 43 ......Page 54
2.6 Flat Morphisms 46 ......Page 57
2.7 Cohen-MacaulayAlgebras 55 ......Page 66
3.Principles of Primary Decomposition 61 ......Page 72
3.1 Associated Primes and Irreducible Decomposition 63 ......Page 74
3.2 EquidimensionalDecompositionofanIdeal 73 ......Page 84
3.3 EquidimensionalDecompositionWithoutExts 79 ......Page 90
3.4 MixedPrimaryDecomposition 81 ......Page 92
3.5 ElementsofFactorizers 85 ......Page 96
4.ComputinginArtinAIgebras 97 ......Page 108
4.1 Structure of Artin Algebras 98 ......Page 109
4.2 Zero-DimensionalIdeals 103 ......Page 114
4.3 IdempotentsversusPrimaryDecomposition 107 ......Page 118
4.4 Decomposition via Sampling 109 ......Page 120
4.5 RootFinders 114 ......Page 125
5.Nullstellensatze 121 ......Page 132
5.1 Radicals via Elimination 122 ......Page 133
5.2 ModulesofDifferentialsandJacobianIdeals 124 ......Page 135
5.3 Generic Socles 128 ......Page 139
5.4 ExplicitNullstellensatze 130 ......Page 141
5.5 FindingRegularSequences 135 ......Page 146
5.6 Top Radical and Upper Jacobians 139 ......Page 150
6.IntegralClosure 143 ......Page 154
6.1 IntegrallyClosedRings 145 ......Page 156
6.2 Multiplication Rings 148 ......Page 159
6.3 S2-ificati0n0fanAffineRing 152 ......Page 163
6.4 DesingularizationinCodimensionOne 160 ......Page 171
6.5 DiscriminantsandMultipliers 165 ......Page 176
6.6 Integral Closure of an Ideal 168 ......Page 179
6.7 Integral Closure of a Morphism 175 ......Page 186
7.Ideal Transforms and Rings of Invariants 181 ......Page 192
7.1 Divisorial Properties of Ideal Transforms 182 ......Page 193
7.2 EquationsofBlowupAlgebras 185 ......Page 196
7.3 Subrings 194 ......Page 205
7.4 Rings ofInvariants 201 ......Page 212
8.ComputationofCohomology(byDavidEisenbud) 209 ......Page 220
8.1 Eyeballing 210 ......Page 221
8.2 Local Duality 212 ......Page 223
8.3 Approximation 214 ......Page 225
9.DegreesofComplexityofaGradedModule 217 ......Page 228
9.1 DegreesofModules 220 ......Page 231
9.2 IndexofNilpotency 234 ......Page 245
9.3 Qualitative Aspects of Noether Normalization 239 ......Page 250
9.4 Homological Degrees of a Module 253 ......Page 264
9.5 Complexity Bounds in Local Rings 263 ......Page 274
A.1 Noetherian Rings 271 ......Page 282
A.2 Krull Dimension 278 ......Page 289
A.3 Graded Algebras 284 ......Page 295
A.4 Integral Extensions 287 ......Page 298
A.5 Finitely Generated Algebras over Fields 294 ......Page 305
A.6 The Method of Syzygies 299 ......Page 310
A.7 Cohen-Macaulay Rings and Modules 311 ......Page 322
A.8 Local Cohomology 319 ......Page 330
A.9 Linkage Theory 326 ......Page 337
B.1 G-Graded Rings and G-Filtrations 331 ......Page 342
B.2 The Study of R via grF(R) 335 ......Page 346
B.3 The Hilbert-Samuel Function 340 ......Page 351
B.4 Hilbert Functions, Resolutions and Local Cohomology 344 ......Page 355
B.5 Lexsegment Ideals and Macaulay Theorem 347 ......Page 358
B.6 The Theorems of Green and Gotzmann 350 ......Page 361
C.Using Macaulay 2 (by David Eisenbud, Daniel Grayson and Michael Stillman) 355 ......Page 366
C.1 Elementary Uses of Macaulay 2 356 ......Page 367
C.2 Local Cohomology of Graded Modules 370 ......Page 381
C.3 Cohomology of a Coherent Sheaf 375 ......Page 386
Bibliography 381 ......Page 392
Index 391 ......Page 402
cover......Page 1
