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Zariskian Filtrations K-Monographs in Mathematics Zariskian Filtrations K-Monographs in Mathematics VOLUME2 This book series is devoted to developments in the mathematical sciences which h.ave links to K-theory. Like the journal K-theory, it is open to all mathematical disciplines. K-Monographs in Mathematics provides material for advanced undergraduate and graduate programmes, seminars and workshops, as well as for research activities and libraries. The series' wide scope includes such topics as quantum theory, Kac-Moody theory, operator algebras, noncommutative algebraic and differential geometry, cyclic and related (co)homology theories, algebraic homotopy theory and homotopical algebra, controlled topology, Novikov theory, transformation groups, surgery theory, Her­ mitian and quadratic forms, arithmetic algebraic geometry, and higher number theory. Researchers whose work fits this framework are encouraged to submit book proposals to the Series Editor or the Publisher. Series Editor: A. Bak, Dept. of Mathematics, University of Bielefeld, Postfach 8640, 33501 Bielefeld, Germany Editorial Board: A. Connes, College de France, Paris, France A. Ranicki, University ofEdinburgh, Edinburgh, Scotland, UK The titles published in this series are listed at the end ofthis volume. Zariskian Filtrations by Li Huishi Shaanxi Normal University, Xian, People's Republic of China and Freddy van Oystaeyen University ofAntwerp , UIA Antwerp, Belgium SPRINGER-SCIENCE+BUSINESS MEDIA, B.V. Library of Congress Cataloging-in-Publication Data Li, Huishi. Zariskian filtrations 1 by Li Huishi and Freddy van Oystaeyen. p. cm. -- <K-monographs in mathematics ; v. 21 Includes bibliographical references and index. ISBN 978-90-481-4738-0 ISBN 978-94-015-8759-4 (eBook) DOI 10.1007/978-94-015-8759-4 1. Fi ltered rings. 2. Fi ltered modules. I. Oystaeyen, F. van, 1947- II. Title. III. Series. QA251.4.L5 1996 512' .4--dc20 96-32738 ISBN 978-90-481-4738-0 Printed on acid-free paper AII Rights Reserved ©1996 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1996 Softcover reprint ofthe hardcover 1st edition 1996 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner. CONTENTS Introduction vii Chapter I Filtered Ring$ and Modules §1. Topological Prerequisites 1 §2. Filtrations on Rings and Modules 3 §3. Complete Filtered Modules and Completions 10 3.1. Some properties of filtration topology 10 3.2. Equivalence of filtration topologies 11 3.3. Complete filtered modules 13 3.4. The completion of a filtered module 14 3.5. The existence of completions 23 §4. Filtrations and Associated Gradations 27 4.1. Preliminaries on gradations 27 4.2. Associated graded rings and modules 30 4.3. Dehomogenizations of grading to filtrations Rees rings (modules) 33 4.4. Finite intersection property and faithful filtrations 42 §5. Good Filtrations 46 §6. Projective and Injective Objects in R-filt 51 §7. Krull Dimension and Global Dimension of Filtered rings 60 7 .1. Krull dimension of filtered rings 60 7.2. Global dimension of filtered rings 61 Chapter II Zariskian Filtrations §1. Flatness of Completion 70 1.1. Artin-Rees property of filtrations 70 1.2. Flatness of completion 74 §2. Zariskian Filtrations 83 2.1. Definition and various characterizations 83 2.2. Examples of Zariskian filtration 87 §3. Lifting Structures of Zariski Rings 95 3.1. G(R) is gr-Artinian, gr-regular or Von Neuman gr-regular 95 3.2. G(R) is a gr-maximal order 96 §4. Zariskian Filtrations on Simple Artinian Rings 103 vi §5. Global Dimension of Rees Rings Associated to Zariskian Filtrations 110 5.1. Addendum to graded homological algebra 110 5.2. Global dimension of Rees rings 118 §6. Ko of Rings with Zariskian Filtration 121 Chapter III Auslander Regular Filtered (Graded) Rings §i. Spectral Sequence Determined by a Filt-complex 127 1.1. Spectral sequence determined by a filt-complex 127 1.2. Spectral sequence determined by a complete filt-complex 134 1.3. Double complexes and their spectral sequences 137 1.4. An application to Noetherian regular rings 144 §2. Auslander Regularity of Zariskian Filtered Rings 148 2.1. An introduction to Auslander regularity 148 2.2. Auslander regularity of Zariskian filtered rings 150 2.3. Auslander regularity of polynomial rings 153 2.4. Examples of Auslander regular rings 156 2.5. iR(M) = ia(R)(G(M)) = j[i(M) = iJi(M) 157 §3. Auslander Regularity of Graded Rings 161 3.1. Auslander regularity of Rees rings 161 3.2. Good graded filtrations 164 3.3. gr-Auslander regular rings are Auslander regular 168 3.4. Applications to invertible ideals 172 §4. Dimension Theory and Pure Module Theory over Zariskian Filtered Rings with Auslander Regular Associated Graded Rings 176 4.1. Generalized Roos theorem and an application to Zariski rings 176 4.2. Pure (holonomic) modules over Zariski rings 183 4.3. Codimension calculation of Characteristic varieties 195 Chapter IV Microlocalization of Filtered Rings and Modules, Quantum Sections and Gauge Algebras §1. Algebraic Microlocalization 200 §2. Quantum Sections and Microstructure Sheaves 211 §3. Generalized Gauge Algebras 227 §4. Quantum Sections of Enveloping Algebras 241 References 246 Subject Index 251 0. Introduction In Commutative Algebra certain /-adic filtrations of Noetherian rings, i.e. the so-called Zariski rings, are at the basis of singularity theory. Apart from that it is mainly in the context of Homological Algebra that filtered rings and the associated graded rings are being studied not in the least because of the importance of double complexes and their spectral sequences. Where non-commutative algebra is concerned, applications of the theory of filtrations were mainly restricted to the study of enveloping algebras of Lie algebras and, more extensively even, to the study of rings of differential operators. It is clear that the operation of completion at a filtration has an algebraic genotype but a topological fenotype and it is exactly the symbiosis of Algebra and Topology that works so well in the commutative case, e.g. ideles and adeles in number theory or the theory of local fields, Puisseux series etc, .... In Non­ commutative algebra the bridge between Algebra and Analysis is much more narrow and it seems that many analytic techniques of the non-commutative kind are still to be developed. Nevertheless there is the magnificent example of the analytic theory of rings of differential operators and 1J-modules a la Kashiwara-Shapira. In this book we develop an algebraic theory of filtered rings using as main tools : 1. The category equivalence between the category of filtered R-modules and the category of X -torsionfree graded R-modules where R is the Rees ring of the filtration on R and X E R1 is its canonical homogeneous element of degree 1, 2. The dehomogenization from certain graded objects to filtered objects (cf. Proposition 7, Ch. I, § 4.3.) The Rees ring has also been used extensively in commutative algebra where it is sometimes called the "blow-up ring" or the "form-ring", however the simple facts referred to above seem to have been missed. The category equivalence mentioned reduces "filtered problems" to graded problems" over the Rees ring up to keeping track of some "torsionfree" condition. A good example of the simplification this leads to is given in Chapter IV, § 1, where algebraic microlocalization is obtained in a very elegant way from constructions on the Rees ring level. In Chapter I we provide the general theory of filtered rings and modules together with their associated graded objects. The so-called "good filtrations" introduced in § 5 are one of the main ingredients in the theory developed in further chapters. viii Zariskian Filtrations In Chapter II we define (non-commutative) Zariskian filtrations and provide several char­ acterizations of these. It turns out that several notions appearing in the literature (earlier treated as different) are in fact equivalent to our notion of Zariskian filtration. The Zariskian property allows to lift information from the associated graded ring to the filtered ring; for example in Ch. II § 3.2. we lift the property of being a maximal order. In § 4 we study Zariskian filtrations on simple Artinian rings and this yields a link to non-commutative val­ uation theory (cf. also (Vo 1) or (VG 2))~ At the end of the chapter we include an application to the calculation of K0 for a ring with Zariskian filtration, slightly extending results of Quillen (cited in (M-R)). Chapter III is devoted to the theory of Auslander regular filtered rings, in particular to the lifting of Auslander regularity from the associated graded ring to the Rees ring. We have been attracted to the problems dealt with in Chapter III by the work of J.E. Bjork, cf. (Bj. 1), (Bj 2), but it was the use of the categorical relation between filtered modules and X-torsionfree graded Rees-modules that allowed us to obtain a unified treatment and elegant solutions to those problems. In this chapter the interplay between filtered and graded properties is most clear. After the study of Auslander regularity of Zariskian filtered rings we also establish the invariance of the grade number in § 2.5., i.e. j(M) = ia(RJ(G(M)) = hi(M) = jii.(M) for any M with a good filtration and under the condition that G(R) is Auslander regular (here "' denotes the Rees objects, 1\ denotes completion). Section 3.4. deals with holonomic and pure modules over Zariski rings with Auslander regular associated graded ring. It is known that not every simple module needs be holonomic, however, if R is Auslander regular it is true that every simple R-module is pure; so the class of pure modules maybe more interesting that the class of holonomic modules (but for other reasons one may also state the converse !) One of the main results here is Theorem 13 where a.o.
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