Computational Methods In Commutative Algebra And Algebraic Geometry (Google eBook)

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Springer Science & Business Media, May 18, 2004 - Computers - 408 pages
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From the reviews:

"... Many parts of the book can be read by anyone with a basic abstract algebra course... it was one of the author's intentions to equip students who are interested in computational problems with the necessary algebraic background in pure mathematics and to encourage them to do further research in commutative algebra and algebraic geometry. But researchers will also benefit from this exposition. They will find an up-to-date description of the related research ... The reviewer recommends the book to anybody who is interested in commutative algebra and algebraic geometry and its computational aspects."

Math. Reviews 2002

"... a sophisticated notebook, with plenty of suggestions, examples and cross references ... It is a welcome new and deep exploration into commutative algebra and its relations with algebraic geometry. It is full of results, from simple tricks to more elaborate constructions, all having in common a computational and constructive nature..."

Jahresberichte der DMV 1999

  

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Contents

66 Integral Closure of an Ideal
176
67 Integral Closure of a Morphism
184
Ideal Transforms and Rings of Invariants
189
71 Divisorial Properties of Ideal Transforms
190
72 Equations of Blowup Algebras
193
73 Subrings
202
74 Rings of Invariants
209
Computation of Cohomology
219

22 Rings of Endomorphisms
35
23 Noether Normalization
37
24 Fitting Ideals
41
25 Finite and QuasiFinite Morphisms
46
26 Flat Morphisms
49
27 CohenMacaulay Algebras
58
Principles of Primary Decomposition
65
31 Associated Primes and Irreducible Decomposition
67
32 Equidimensional Decomposition of an Ideal
77
33 Equidimensional Decomposition Without Exts
83
34 Mixed Primary Decomposition
85
35 Elements of Factorizers
90
Computing in Artin Algebras
103
41 Structure of Artin Algebras
104
42 ZeroDimensional Ideals
109
43 Idempotents versus Primary Decomposition
113
44 Decomposition via Sampling
115
45 Root Finders
120
Nullstellensatze
127
51 Radicals via Elimination
128
52 Modules of Differentials and Jacobian Ideals
130
53 Generic Socles
134
54 Explicit Nullstellensatze
136
55 Finding Regular Sequences
141
56 Top Radical and Upper Jacobians
146
Integral Closure
149
61 Integrally Closed Rings
151
62 Multiplication Rings
154
63 S2ification of an Affine Ring
159
64 Desingularization in Codimension One
167
65 Discriminants and Multipliers
173
81 Eyeballing
220
82 Local Duality
222
83 Approximation
224
Degrees of Complexity of a Graded Module
227
91 Degrees of Modules
230
92 Index of Nilpotency
244
93 Qualitative Aspects of Noether Normalization
249
94 Homological Degrees of a Module
263
95 Complexity Bounds in Local Rings
273
Primer on Commutative Algebra
281
A2 Krull Dimension
288
A3 Graded Algebras
295
A4 Integral Extensions
298
A5 Finitely Generated Algebras over Fields
305
A6 The Method of Syzygies
309
A7 CohenMacaulay Rings and Modules
321
A8 Local Cohomology
329
A9 Linkage Theory
338
Hilbert Functions
343
B2 The Study of R via grFR
347
B3 The HilbertSamuel Function
352
B4 Hilbert Functions Resolutions and Local Cohomology
356
B5 Lexsegment Ideals and Macaulay Theorem
359
B6 The Theorems of Green and Gotzmann
362
Using Macaulay 2
367
C1 Elementary Uses of Macaulay 2
368
C2 Local Cohomology of Graded Modules
382
C3 Cohomology of a Coherent Sheaf Mathematical Background
387
References
393
Index
405
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About the author (2004)

Author's homepage: http: //www.math.rutgers.edu/~vasconce/

Vasconcelos is a well-known top expert in commutative algebra. Author of Algorithms and Computation in Mathematics 2. ISBN: 3-540-21311-2.

Via the existence of the ACM series he was won as a Springer author in 1995, being a CUP author before.

This new book is a result of the good collaboration with him on ACM 2.

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