Commutative Coherent Rings
This book provides the first extensive and systematic treatment of the theory of commutative coherent rings. It blends, and provides a link, between the two sometimes disjoint approaches available in the literature, the ring theoretic approach, and the ho
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1371 Sarah Glaz
Commutative Coherent Rings
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo
Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann
1371 Sarah Glaz
Commutative Coherent Rings
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo
Author
Sarah Glaz Department of Mathematics, Wesleyan University Middletown, CT 06457, USA
Mathematics Subject Classification (1980): 13-02, 13B99, 13C 11, 13C 13, 13C 15, 13099, 13E99, 18G99, 20M 14 ISBN 3-540-51115-6 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-51115-6 Springer-Verlag New York Berlin Heidelberg
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9, 1965. in its version of June 24, 1985, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law.
© Springer-Verlag Berlin Heidelberg 1989 Printed in Germany Printing and binding: Druckhaus Beltz, Hemsbach/Bergstr. 2146/3140-543210
To my mother Amalia Dauer and
In memory of my father Philip Dauer
Table of Contents INTRODUCTION • • . CHAPTER 1 l.
2. 3. 4.
.
.
vii
PRELIMINARIES Projective and Injective Modules Flatness . . . • . • . . Homological Dimensions Rings of Small Global and Weak Dimensions-Classical Results . . . .
CHAPTER 2 l.
2. 3.
4. 5.
6.
CHAPTER 3 l.
2. 3.
4.
5.
CHAPTER 4 l.
2. 3.
4.
CH.\PTER 5 l.
2. 3.
4.
CHAPTER 6 1. 2. 3. 4. CHAPTER 7 1. 2. 3. 4.
.
1 7 17
23
INTRODUCTION TO COHERENCE Finitely Presented Modules • . . . . . . . Elementary Properties of Coherent Modules . Definitions and Examples of Coherent Rings Ideals, Quotients and Localizations • . . . Homological Dimensions over Coherent Rings Two Homological Characterizations of Coherent Rings
31 41 44 50 55 63
FUNDAMENTAL CONCEPTS Change of Rings and Homological Dimensions Zariski Topology, Projectivity and Rank . . Associated Primes . . . . . . . . . . . . . Fitting Invariants and Euler Characteristic Koszul Complexes . . . . . . . . . . . . .
69 73 85 96 101
RING EXTENSIONS Special Ring Extensions . . • . . . . . . . Min(R) and the Total Ring of Quotients of R The Maximal Flat Epimorphic Extension of R Trivial Ring Extensions and the A Dimension
108 112 130 139
RING CONSTRUCTIONS AND OVERRINGS Cartesian Squares . • . . . . . D + M Constructions . . . . . . Overrings and Integral Closure Coherent Pairs
149 159 170 182
PARTICULAR COHERENT RINGS Uniform Coherent Rings . Regular and Super Regular Coherent Rings Rings of Global and Weak Dimension Two Rings of Small Ng Dimension . . . • . . .
190 200 212 221
POLYNOMIAL RINGS Non-Noetherian Grade Reduction Theorems Stably Coherent Rings Uppers of Zero and the Relation to the Integral Closure . . .
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