Kodaira-Spencer Maps in Local Algebra
The monograph contributes to Lech's inequality - a 30-year-old problem of commutative algebra, originating in the work of Serre and Nagata, that relates the Hilbert function of the total space of an algebraic or analytic deformation germ to the Hilbert fu
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1597
Lecture Notes in Mathematics Editors: A. Dold, Heidelberg B. Eckmann, Ziirich F. Takens, Groningen
1597
Bernd Herzog
Kodaira-Spencer Maps in Local Algebra
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona Budapest
Author Bernd Herzog Matematiska Institutionen Stockholms Universitet S-I13 85 Stockholm, Sweden E-mail: [email protected]
Mathematics Subject Classification (1991): 13D40, 13D10, 14B12, 16S80
ISBN 3-540-58790-X Springer-Verlag Berlin Heidelberg New York CIP-Data applied for 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 any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1994 Printed in Germany Typesetting: Camera-ready TEX output by the author SPIN: 10130247 46/3140-543210 - Printed on acid-free paper
Kodaira-Spencer maps in local algebra
Contents
Introduction Acknowledgments Notation
ix xv xvi
1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11 1.12 1.13 1.14 1.15 1.16 1.17 1.18
Ring filtrations Graded rings Filtered rings Powers of ideals Symbolic powers Weight filtrations Inclusion, intersection, sum filtration Sum filtrations and separatedness Chevalley's theorem Separatedness of the sum in complete local rings Direct and inverse image filtrations The filtration generated by a family Filtered modules, Artin-Rees filtrations Terminology Artin-Rees ring filtrations Artin-Rees ring filtrations of local rings Completion of filtered modules Completion of a cofinitely filtered local ring Closure and Artin-Rees filtrations
2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10
18 Basic lemmas Graded Nakayama lemma 18 Flatness of graded modules 19 20 Local flatness criterion, graded case Refinement of a topology 21 Filtered Grothendieck lemma 21 Free generators of the associated graded module ................•.... 23 24 Lifting free generators to a flat module Length of a tensor product _ 26 Refinement and completion 28 Tjurina's flatness criterion 28
3 3.1
Tangential flatness under base change Strict ideal generators
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1 1 2 3 3 4 4 5 6 6 6 7 8 10 11 11 13 15 17
30 30 v
3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13 3.14 3.15 3.16 3.17
Strictness criterion Liftable syzygies Strict homomorphisms of filtered modules Example of a strict morphism The exact sequence associated with a generating system Exactness of the associated graded sequence Tangential flatness under surjective base change Tangential flatness under polynomial extension Tangential flatness and increased base filtration Convention Minimality of tangentiall
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