Determinantal Rings

Determinantal rings and varieties have been a central topic of commutative algebra and algebraic geometry. Their study has attracted many prominent researchers and has motivated the creation of theories which may now be considered part of general commutat

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1327 Winfried Bruns Udo Vetter

Determinantal Rings

Springer-Verlag Berlin Heidelberg New York London Paris Tokyo

Authors

Winfried Bruns Udo Vetter Universitat Osnabruck - Abteilung Vechta Fachbereich Naturwissenschaften, Mathematik Postfach 1553, 2848 Vechta 1, Federal Republic of Germany

This book is being published in a parallel edition by the Instituto de Maternatica Pura e Aplicada, Rio de Janeiro as volume 45 of the series "Monograffas de Matematica''.

Mathematics Subject Classification (1980): 13-02, 13C05, 13C 13,13010, 13025, 13H 10, 14M 15, 14L30, 20G05, 20G 15 ISBN 3-540-19468-1 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-19468-1 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 1988 Printed in Germany Printing and binding: Druckhaus Beltz, Hemsbach/Bergstr. 2146/3140-543210

Preface

Determinantal rings and varieties have been a central topic of commutative algebra and algebraic geometry. Their study has attracted many prominent researchers and has motivated the creation of theories which may now be considered part of general commutative ring theory. A coherent treatment of determinantal rings is lacking however. We are algebraists, and therefore the subject will be treated from an algebraic point of view. Our main approach is via the theory of algebras with straightening law. Its axioms constitute a convenient systematic framework, and the standard monomial theory on which it is based yields computationally effective results. This approach suggests (and is simplified by) the simultaneous treatment of the coordinate rings of the Schubert subvarieties of Grassmannians, a program carried out very strictly. Other methods have not been neglected. Principal radical systems are discussed in detail, and one section each is devoted to invariant and representation theory. However, free resolutions are (almost) only covered for the "classical" case of maximal minors. Our personal view of the subject is most visibly expressed by the inclusion of Sections 13-15 in which we discuss linear algebra over determinantal rings. In particular the technical details of Section 15 (and perhaps only these) are somewhat demanding. The bibliography contains several titles which have not been cited in the text. They mainly cover topics not discussed: geometric methods and ideals generated by minors of symmetric matrices and Pfaffians of alternating ones. We have tried hard to keep the text as self-contained as possible.

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