Simple Morphisms in Algebraic Geometry
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935
Richard Sot
Simple Morphisms in Algebraic Geometry
Springer-Verlag Berlin Heidelberg New York 1982
Author
Richard Sot School of Mathematics, The Institute for Advanced Study Princeton, NJ 08540, USA
AMS Subject Classifications (1980): 14-XX ISBN 3-540-11564-1 ISBN 0-387-11564-1
Springer-Verlag Berlin Heidelberg New York Springer-Verlag New York Heidelberg Berlin
Library of Congress Cataloging in Publication Data Sot, Richard, 1948- Simple morphisms in algebraic geometry. (Lecture notes in mathematics; 935) Bibliography: p.lncludes indexes. 1. Geometry, Algebraic. 2. Morphisms (Mathematics) I. Title. II. Series: Lecture notes in mathematics (Springer-Verlag); 935. QA3.L28 no. 935 [QA5641510s [516.3'5182-10303 ISBN 0-387-11564-1 (U.S.)
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© by Springer-Verlag Berlin Heidelberg 1982 Printed in Germany
Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210
CONTENTS Chapter 1
The Zariski topology, the Jacobian criterion and examples of simple algebras over a field k
Chapter 2
The Kahler
Chapter 3
Every k-algebra A which is essentially of finite type over k and simple is a regular local ring
35
Chapter 4
Brief discussion of unramified and homomorphisms
45
Chapter 5
Some corollaries to Theorem 3. 5
54
Chapter 6
Fitting ideals
57
Chapter 7
Proof of the Jacobian criterion and some characterizations of simple k-algebras and A-algebras
73
Characterization of simple A-algebras in terms of homomorphisms; invariance of the property of being a simple algebra under composition and change of base
89
Chapter 8
Chapter 9
Chapter 10
1-differentials
18
Descent of simple homomorphisms and removal of all noetherian assumptions in Chapter 7 and Chapter 8
103
Simple morphisms of preschemes and translation of previous theorems into the language of preschemes
117
APPENDIX
128
BIBLIOGRAPHY
145
INDEX TO TERMINOLOGY
146
INDEX TO SYMBOLS
146
Supported in part by NSF grant MCS 77-18723 A04.
CHAPTER 1 The Zariski topology, the Jacobian criterion and examples of simple algebras over a field
k
Introduction. This text treats in detail the concepts of simple algebra over a field
k,
simple homomorphism of rings,
simple algebraic variety, simple morphism of algebraic varieties and simple morphism of preschemes, which all reduce (in our treatment, by definition) to the concept of a simple algebra over a field
k.
For the first nine chapters it is only assumed that the reader is acquainted with basic algebra, a few elementary notions in general topology, and a few notions in commutative algebra, the appendix supplying a reference for several of the
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