An Algebraic Introduction to Complex Projective Geometry: Commutative Algebra

by Christian Peskine

Overview

In this introduction to commutative algebra, the author choses a route that leads the reader through the essential ideas, without getting embroiled in technicalities. He takes the reader quickly to the fundamentals of complex projective geometry, requiring only a basic knowledge of linear and multilinear algebra and some elementary group theory. The author divides the book into three parts. In the first, he develops the general theory of noetherian rings and modules. He includes a certain amount of homological algebra, and he emphasizes rings and modules of fractions as preparation for working with sheaves. In the second part, he discusses polynomial rings in several variables with coefficients in the field of complex numbers. After Noether's normalization lemma and Hilbert's Nullstellensatz, the author introduces affine complex schemes and their morphisms; he then proves Zariski's main theorem and Chevalley's semi-continuity theorem. Finally, the author's 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.

Product Details

ISBN-13:	9780521480727
Publisher:	Cambridge University Press
Publication date:	05/02/1996
Series:	Cambridge Studies in Advanced Mathematics , #47
Edition description:	New Edition
Pages:	244
Product dimensions:	6.18(w) x 9.33(h) x 0.75(d)

Table of Contents

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.

Editorial Reviews

An introductory text presenting a cohesive set of methods in commutative algebra for use in geometry written for students with a basic knowledge of linear and multilinear algebra and some elementary group theory. The topics cover rings, homomorphisms, ideals, modules, noetherian and artinian rings, integral and algebraic extensions, affine schemes, and proofs for Zariski's main theorem and Chevalley's semi-continuity theorem. Modern intersection theory is detailed in a study of Weil and Cartier divisors. Annotation c. Book News, Inc., Portland, OR (booknews.com)

Booknews

"Useful, apt advice and numerous expository niceties make this compact volume a pleasure to read. Highly recommended." Choice

"...a good and systematic introduction to complex projective geometry....an excellent textbook." Mathematical Reviews

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