Graded and Filtered Rings and Modules
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758 C. N&st~.sescu F. Van Oystaeyen
Graded and Filtered Rings and Modules
Springer-Verlag Berlin Heidelberg New York 1979
Authors C. N&st&sescu Faculty of Mathematics University of Bucharest Str. Academiei 14 R - 70109 Bucharest 1 Romania F. Van Oystaeyen Department of Mathematics University of Antwerp UIA Universiteitsplein 1 B - 2610 Wilrijk Belgium
AMS Subject Classifications (1980): Primary: 1 6 A 0 3 Secondary: 16A05, 16A33, 16A45, 16A50, 16A55, 16A63, 1 6 A 6 6
ISBN 3 - 5 4 0 - 0 9 7 0 8 - 2 Springer-Verlag Berlin Heidelberg NewYork ISBN 0 - 3 8 7 - 0 9 7 0 8 - 2 Springer-Verlag NewYork Heidelberg Berlin Library of Congress Catalogingin PublicationData N&st&sescu, Constantin. Graded and filtered rings and modules. (Lecture notes in mathematics;758) Bibliography: p. Includes index. 1. Associativerings. 2. Graded rings. 3. Graded modules.4. Filtered rings. 5. Filtered modules. I. Oystaeyen,F. van, 1947- joint author. II. Title. Ill. Series: Lecture notes in mathematics(Berlin); 758. QA3.L28 no. 758 [QA251.5]510'.8s [512'.4] 79-26585 ISBN 0-387-09708-2 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publishe © by Springer-Verlag Berlin Heidelberg 1979 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210
PREFACE The aim of these lecture notes is to present coherently the latest and less known results concerning the theory of filtered and graded rings and modules.
With an eye
to projective algebraic geometry, the theory of commutative graded rings and Noetherian graded modules over them has been expounded in the large, e.g. using cohomological methods and Serre's theorem, ~ i c h
states that the category of coherent
sheaves over projective n-space is equivalent to a category of graded Noetherian n~dules over a graded polynomial ring, it is possible to prove generalizations of ~lax Noether's and Bezout's theorem within the latter category, cf. [15 ]. In writing these notes we have adopted a completely different point of view, i.e. a purely ring theoretical one, and the only reference to projective geometry will be in II.14, II.15 and II.16 but then over a non-cor~nutative ring.
The material pre-
sented falls naturally into three parts. Chapter I is devoted to the study of the relation between a filtered ring R and the category0f filtered R-modules on one hand, and the graded ring and module category associated to R on the other hand.
In particular we focussed attention on : Krull
dimension in Gabriel's and Rentschler's sense; free, projective and finitely generated objects, homological dimension i.e. projective and flat dimension etc .... ~ s t
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