Computational Algebraic Geometry

This 2003 book investigates interplay between algebra and geometry. Covers: homological algebra, algebraic combinatorics and algebraic topology, and algebraic geometry.

Hal Schenck (Author)

9780521536509, Cambridge University Press

Paperback, published 6 October 2003

208 pages
22.9 x 15.3 x 1.3 cm, 0.278 kg

'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

The 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).

Preface
1. Basics of commutative algebra
2. Projective space and graded objects
3. Free resolutions and regular sequences
4. Gröbner bases
5. Combinatorics and topology
6. Functors: localization, hom, and tensor
7. Geometry of points
8. Homological algebra, derived functors
9. Curves, sheaves and cohomology
10. Projective dimension
A. Abstract algebra primer
B. Complex analysis primer
Bibliography.

Subject Areas: Geometry [PBM], Algebra [PBF]