An Algebraic Introduction to Complex Projective Geometry
Commutative Algebra

An excellent textbook for those who seek an efficient and rapid introduction to the geometric applications of commutative algebra.

Christian Peskine (Author)

9780521108478, Cambridge University Press

Paperback / softback, published 9 April 2009

244 pages
22.9 x 15.2 x 1.4 cm, 0.36 kg

'… a detailed study … a solid background.' L'Enseignement Mathématique

In this introduction to commutative algebra, the author leads the beginning student through the essential ideas, without getting embroiled in technicalities. The route chosen takes the reader quickly to the fundamental concepts for understanding complex projective geometry, the only prerequisites being a basic knowledge of linear and multilinear algebra and some elementary group theory. In the first part, the general theory of Noetherian rings and modules is developed. A certain amount of homological algebra is included, and rings and modules of fractions are emphasised, as preparation for working with sheaves. In the second part, the central objects are polynomial rings in several variables with coefficients in the field of complex numbers. After Noether's normalisation lemma and Hilbert's Nullstellensatz, affine complex schemes and their morphisms are introduced; Zariski's main theorem and Chevalley's semi-continuity theorem are then proved. Finally, a detailed study of Weil and Cartier divisors provides a solid background for modern intersection theory. This is an excellent textbook for those who seek an efficient and rapid introduction to the geometric applications of commutative algebra.

1. Rings, homomorphisms, ideals
2. Modules
3. Noetherian rings and modules
4. Artinian rings and modules
5. Finitely generated modules over Noetherian rings
6. A first contact with homological algebra
7. Fractions
8. Integral extensions of rings
9. Algebraic extensions of rings
10. Noether's normalisation lemma
11. Affine schemes
12. Morphisms of affine schemes
13. Zariski's main theorem
14. Integrally closed Noetherian rings
15. Weil divisors
16. Cartier divisors
Subject index
Symbols index.

Subject Areas: Algebraic geometry [PBMW]