Introduction to Algebraic Geometry and Commutative Algebra

by Patil, Dilip P.; Storch, Uwe

• ISBN13: 9789814304566
• ISBN10: 9814304565
• Format: Hardcover
• Copyright: 2010-03-25
• Publisher: World Scientific Pub Co Inc

Summary

This introductory textbook for a graduate course in pure mathematics provides a gateway into the two difficult fields of algebraic geometry and commutative algebra. Algebraic geometry, supported fundamentally by commutative algebra, is a cornerstone of pure mathematics. Along the lines developed by Grothendieck, this book delves into the rich interplay between algebraic geometry and commutative algebra. A selection is made from the wealth of material in the discipline, along with concise yet clear definitions and synopses.

Table of Contents

Series Preface	p. v
Preface	p. vii
Finitely Generated Algebras
Algebras over a Ring	p. 1
Factorization in Rings	p. 2
Noetherian Rings and Modules	p. 4
Graded Rings and Modules	p. 7
Integral Extensions	p. 8
Noether's Normalization Lemma and Its Consequences	p. 12
The K-Spectrum and the Zariski Topology
The K-Spectrum of a K-Algebra	p. 19
Affine Algebraic Sets	p. 21
Strong Topology	p. 32
Prime Spectra and Dimension
The Prime Spectrum of a Commutative Ring	p. 41
Dimension	p. 48
Schemes
Sheaves of Rings	p. 61
Schemes	p. 68
Finiteness Conditions on Schemes	p. 75
Product of Schemes	p. 77
Affine Morphisms	p. 83
Projective Schemes
Projective Schemes	p. 87
Main Theorem of Elimination	p. 102
Mapping Theorem of Chevalley	p. 107
Regular, Normal and Smooth Points
Regular Local Rings	p. 111
Normal Domains	p. 118
Normalization of a Scheme	p. 125
The Module of Kähler Differentials	p. 128
Quasi-coherent Sheaves and the Sheaf of Kähler Differentials	p. 139
Riemann-Roch Theorem
Coherent Modules on Projective Schemes	p. 153
Projective Curves	p. 158
The Projective Line	p. 163
Riemann-Roch Theorem for General Curves	p. 167
Genus of a Projective Curve	p. 175
References	p. 199
List of Symbols	p. 201
Index	p. 203
Biography of Authors	p. 209
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