Computational Algebraic GeometryThe interplay between algebra and geometry is a beautiful (and fun!) area of mathematical investigation. 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 2003 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). |
Contents
Basics of Commutative Algebra | 1 |
11 Ideals and Varieties | 2 |
12 Noetherian Kings and the Hilbert Basis Theorem | 4 |
13 Associated Primes and Primary Decomposition | 6 |
14 The Nullstellensatz and Zariski Topology | 12 |
Projective Space and Graded Objects | 18 |
22 Graded Rings and Modules Hilbert Function and Series | 21 |
23 Linear Algebra Flashback Hilbert Polynomial | 26 |
Geometry of Points and the Hilbert Function | 92 |
72 The Theorems of Macaulay and Gotzmann | 99 |
73 Artinian Reduction and Hypersurfaces | 100 |
Snake Lemma Derived Functors Tor and Ext | 107 |
82 Derived Functors Tor | 111 |
83 Ext | 116 |
84 Double Complexes | 124 |
Curves Sheaves and Cohomology | 126 |
Free Resolutions and Regular Sequences | 34 |
31 Free Modules and Projective Modules | 35 |
32 Free Resolutions | 36 |
33 Regular Sequences Mapping Cone | 42 |
Grobner Bases and the Buchberger Algorithm | 50 |
41 Grobner Bases | 51 |
42 Monomial Ideals and Applications | 55 |
43 Syzygies and Grobner Bases for Modules | 58 |
44 Projection and Elimination | 60 |
Combinatorics Topology and the StanleyReisner Ring | 64 |
51 Simplicial Complexes and Simplicial Homology | 65 |
52 The StanleyReisner Ring | 72 |
53 Associated Primes and Primary Decomposition | 77 |
Functors Localization Hom and Tensor | 80 |
61 Localization | 81 |
62 The Horn Functor | 84 |
63 Tensor Product | 88 |
92 Cohomology and Global Sections | 129 |
93 Divisors and Maps to P | 133 |
94 RiemannRoch and Hilbert Polynomial Redux | 139 |
Projective Dimension CohenMacaulay Modules Upper Bound Theorem | 145 |
102 CohenMacaulay Modules and Geometry | 149 |
103 The Upper Bound Conjecture for Spheres | 158 |
Abstract Algebra Primer | 163 |
A2 Rings and Modules | 164 |
A3 Computational Algebra | 168 |
Complex Analysis Primer | 175 |
B2 Greens Theorem | 176 |
B3 Cauchys Theorem | 178 |
B4 Taylor and Laurent Series Residues | 181 |
| 183 | |
| 189 | |
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Common terms and phrases
affine variety algebraically closed Ass(M associated primes b₁ betti diagram chain complex Chapter codimension coefficients Cohen-Macaulay cohomology cokernel commutative algebra compute the Hilbert coordinate curve defined Definition degree derived functors dimension division algorithm divisor Eisenbud element example Exercise f-vector f₁ finitely free module free resolution geometric global sections graded module Gröbner basis Hilbert function Hilbert polynomial Hilbert series homogeneous ideal homology homomorphism hyperplane I₁ intersection irreducible kernel linear form long exact sequence M₁ Macaulay matrix maximal ideal minimal nice Noetherian nonzero nonzerodivisor Nullstellensatz objects open set P₁ polynomial ring primary decomposition prime ideal projective space projective variety proof Prove quotient R-module regular sequence Riemann-Roch set of points sheaf sheaves short exact sequence simplicial complex simplicial polytope Stanley-Reisner ring submodule Suppose syzygies Theorem topology vector space write Zariski zero