Computational algebraic geometry / Hal Schenck.

Format:

Book

Author/Creator:

Schenck, Hal.

Series:

London Mathematical Society student texts ; 58.

London Mathematical Society student texts ; 58

Language:

English

Subjects (All):

Geometry, Algebraic--Data processing--Congresses.

Geometry, Algebraic.

Geometry, Algebraic--Data processing.

Congresses and conventions.

Physical Description:

xiv, 193 pages : illustrations ; 24 cm.

Place of Publication:

Cambridge, UK ; New York : Cambridge University Press, 2003.

Summary:

The interplay between algebra and geometry is a beautiful (and fun!) area of mathematical investigation. Recent advances in computing and algorithms make it possible to tackle many classical problems in a down-to-earth and concrete fashion. This opens wonderful new vistas and allows us to pose, study and solve problems that were previously out of reach.

Suitable for graduate students, the objective of this book is to bring advanced algebra to life with lots of examples. The first three chapters provide an introduction to commutative algebra and connections to geometry. The rest of the book then focuses on three active areas of contemporary algebra: * Homological Algebra - the snake lemma, long exact sequence in homology, functors and derived functors (Tor and Ext), and double complexes. * Algebraic Combinatorics and Algebraic Topology - simplicial complexes and simplicial homology, Stanley-Reisner rings, upper bound theorem and polytopes. * Algebraic Geometry - points and curves in projective space, Riemann-Roch theorem, Cech cohomology, regularity.

Contents:

1 Basics of Commutative Algebra 1

1.1 Ideals and Varieties 2

1.2 Noetherian Rings and the Hilbert Basis Theorem 4

1.3 Associated Primes and Primary Decomposition 6

1.4 The Nullstellensatz and Zariski Topology 12

2 Projective Space and Graded Objects 18

2.1 Projective Space and Projective Varieties 18

2.2 Graded Rings and Modules, Hilbert Function and Series 21

2.3 Linear Algebra Flashback, Hilbert Polynomial 26

3 Free Resolutions and Regular Sequences 34

3.1 Free Modules and Projective Modules 35

3.2 Free Resolutions 36

3.3 Regular Sequences, Mapping Cone 42

4 Grobner Bases and the Buchberger Algorithm 50

4.1 Grobner Bases 51

4.2 Monomial Ideals and Applications 55

4.3 Syzygies and Grobner Bases for Modules 58

4.4 Projection and Elimination 60

5 Combinatorics, Topology and the Stanley-Reisner Ring 64

5.1 Simplicial Complexes and Simplicial Homology 65

5.2 The Stanley-Reisner Ring 72

5.3 Associated Primes and Primary Decomposition 77

6 Functors: Localization, Hom, and Tensor 80

6.1 Localization 81

6.2 The Hom Functor 84

6.3 Tensor Product 88

7 Geometry of Points and the Hilbert Function 92

7.1 Hilbert Functions of Points, Regularity 92

7.2 The Theorems of Macaulay and Gotzmann 99

7.3 Artinian Reduction and Hypersurfaces 100

8 Snake Lemma, Derived Functors, Tor and Ext 107

8.1 Snake Lemma, Long Exact Sequence in Homology 107

8.2 Derived Functors, Tor 111

8.3 Ext 116

8.4 Double Complexes 124

9 Curves, Sheaves, and Cohomology 126

9.1 Sheaves 126

9.2 Cohomology and Global Sections 129

9.3 Divisors and Maps to P[superscript n] 133

9.4 Riemann-Roch and Hilbert Polynomial Redux 139

10 Projective Dimension, Cohen-Macaulay Modules, Upper Bound Theorem 145

10.1 Codimension, Depth, Auslander-Buchsbaum Theorem 145

10.2 Cohen-Macaulay Modules and Geometry 149

10.3 The Upper Bound Conjecture for Spheres 158

A Abstract Algebra Primer 163

A.1 Groups 163

A.2 Rings and Modules 164

A.3 Computational Algebra 168

B Complex Analysis Primer 175

B.1 Complex Functions, Cauchy-Riemann Equations 175

B.2 Green's Theorem 176

B.3 Cauchy's Theorem 178

B.4 Taylor and Laurent Series, Residues 181.

Notes:

Includes bibliographical references (pages 183-187) and index.

ISBN:

052182964X

0521536502

OCLC:

52203132

Online:

Publisher description