Computational Algebraic Geometry

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Paperback | 6 April 2004

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208 Pages

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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 chapters provide an introduction to commutative algebra and connections to geometry. The rest of the book focuses on three active areas of contemporary algebra: Homological Algebra (the snake lemma, long exact sequence inhomology, 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); and Algebraic Geometry (points and curves in projective space, Riemann-Roch, Cech cohomology, regularity).

Industry Reviews

'The book is written in a terse but energetic style - Schenck is clearly in love with the material ... any student who completes this book will be excited about algebraic geometry and well-equipped for further reading.' Bulletin of the American Mathematical Society 'It is a very good introduction to this circle of ideas and it will undoubtedly attract the interest of students to the field.' European Mathematical Society Newsletter 'It is brief, informal and provides many examples - all attributes that many other algebraic geometry books lack! ... its role is to stimulate and outline rather than to give an encyclopedic treatment, and in this it succeeds very well.' Australian Mathematical Society Gazette 'It is a very good introduction to this circle of ideas and it will undoubtedly attract the interest of students to the field.' EMS Newsletter

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Preface	p. xi
Basics of Commutative Algebra	p. 1
Ideals and Varieties	p. 2
Noetherian Rings and the Hilbert Basis Theorem	p. 4
Associated Primes and Primary Decomposition	p. 6
The Nullstellensatz and Zariski Topology	p. 12
Projective Space and Graded Objects	p. 18
Projective Space and Projective Varieties	p. 18
Graded Rings and Modules, Hilbert Function and Series	p. 21
Linear Algebra Flashback, Hilbert Polynomial	p. 26
Free Resolutions and Regular Sequences	p. 34
Free Modules and Projective Modules	p. 35
Free Resolutions	p. 36
Regular Sequences, Mapping Cone	p. 42
Grobner Bases and the Buchberger Algorithm	p. 50
Grobner Bases	p. 51
Monomial Ideals and Applications	p. 55
Syzygies and Grobner Bases for Modules	p. 58
Projection and Elimination	p. 60
Combinatorics, Topology and the Stanley-Reisner Ring	p. 64
Simplicial Complexes and Simplicial Homology	p. 65
The Stanley-Reisner Ring	p. 72
Associated Primes and Primary Decomposition	p. 77
Functors: Localization, Hom, and Tensor	p. 80
Localization	p. 81
The Hom Functor	p. 84
Tensor Product	p. 88
Geometry of Points and the Hilbert Function	p. 92
Hilbert Functions of Points, Regularity	p. 92
The Theorems of Macaulay and Gotzmann	p. 99
Artinian Reduction and Hypersurfaces	p. 100
Snake Lemma, Derived Functors, Tor and Ext	p. 107
Snake Lemma, Long Exact Sequence in Homology	p. 107
Derived Functors, Tor	p. 111
Ext	p. 116
Double Complexes	p. 124
Curves, Sheaves, and Cohomology	p. 126
Sheaves	p. 126
Cohomology and Global Sections	p. 129
Divisors and Maps to P[superscript n]	p. 133
Riemann-Roch and Hilbert Polynomial Redux	p. 139
Projective Dimension, Cohen-Macaulay Modules, Upper Bound Theorem	p. 145
Codimension, Depth, Auslander-Buchsbaum Theorem	p. 145
Cohen-Macaulay Modules and Geometry	p. 149
The Upper Bound Conjecture for Spheres	p. 158
Abstract Algebra Primer	p. 163
Groups	p. 163
Rings and Modules	p. 164
Computational Algebra	p. 168
Complex Analysis Primer	p. 175
Complex Functions, Cauchy-Riemann Equations	p. 175
Green's Theorem	p. 176
Cauchy's Theorem	p. 178
Taylor and Laurent Series, Residues	p. 181
Bibliography	p. 183
Index	p. 189
Table of Contents provided by Ingram. All Rights Reserved.

Computational Algebraic Geometry

At a Glance

Paperback

Industry Reviews

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