Introduction To Algebraic Geometry And Commutative Algebra

by Dilip P Patil, Uwe Storch

Overview

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.

Product Details

ISBN-13:	9789814307581
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	04/27/2010
Series:	Iisc Lecture Notes Series , #1
Pages:	220
Product dimensions:	6.00(w) x 8.90(h) x 0.60(d)

Table of Contents

Series Preface v

Preface vii

Chapter 1 Finitely Generated Algebras

1.A Algebras over a Ring 1

1.B Factorization in Rings 2

1.C Noetherian Rings and Modules 4

1.D Graded Rings and Modules 7

1.E Integral Extensions 8

1.F Noether's Normalization Lemma and Its Consequences 12

Chapter 2 The K-Spectrum and the Zariski Topology

2.A The K-Spectrum of a K-Algebra 19

2.B Affine Algebraic Sets 21

2.C Strong Topology 32

Chapter 3 Prime Spectra and Dimension

3.A The Prime Spectrum of a Commutative Ring 41

3.B Dimension 48

Chapter 4 Schemes

4.A Sheaves of Rings 61

4.B Schemes 68

4.C Finiteness Conditions on Schemes 75

4.D Product of Schemes 77

4.E Affine Morphisms 83

Chapter 5 Projective Schemes

5.A Projective Schemes 87

5.B Main Theorem of Elimination 102

5.C Mapping Theorem of Chevalley 107

Chapter 6 Regular, Normal and Smooth Points

6.A Regular Local Rings 111

6.B Normal Domains 118

6.C Normalization of a Scheme 125

6.D The Module of Kähler Differentials 128

6.E Quasi-coherent Sheaves and the Sheaf of Kähler Differentials 139

Chapter 7 Riemann-Roch Theorem

7.A Coherent Modules on Projective Schemes 153

7.B Projective Curves 158

7.C The Projective Line 163

7.D Riemann-Roch Theorem for General Curves 167

7.E Genus of a Projective Curve 175

References 199

List of Symbols 201

Index 203

Biography of Authors 209