Introduction to Algebraic Geometry and Commutative Algebra
by Patil, Dilip P.; Storch, UweFormat: Hardcover
Pub. Date: 2010-03-25
Publisher(s): World Scientific Pub Co Inc
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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 |
| Table of Contents provided by Ingram. All Rights Reserved. |
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