Theory of Stein Spaces
1. The classical theorem of Mittag-Leffler was generalized to the case of several complex variables by Cousin in 1895. In its one variable version this says that, if one prescribes the principal parts of a merom orphic function on a domain in the complex
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Editors
S. S. Chern J. L. Doob J. Douglas, jr. A. Grothendieck E. Heinz . F. Hirzebruch E. Hopf S. Mac Lane W. Magnus M. M. Postnikov W. Schmidt D. S. Scott K. Stein J. Tits B. L. van der Waerden Managing Editors B. Eckmann J.K. Moser
H. Grauert R. Remmert
Theory of Stein Spaces Translated by Alan Huckleberry
Springer Science+Business Media, LLC
Hans Grauert Mathematisches Institut der Universităt Gottingen D-3400 Gottingen Federal Republic of Germany
Reinhold Remmert Mathematisches Institut der Westfălischen Wilhelms-Universităt
D-4400 Miinster Federal Republic of Germany
Translator:
Alan Huckleberry Department of Mathematics University of Notre Dame Notre Dame, Indiana 46556 USA
AMS Subject Classifications: 30A46, 32ElO, 32J99, 32-02, 32AlO, 32A20, 32C15
With 5 Figures Library of Congress Cataloging in Publication Data Grauert, Hans, 1930Theory of Stein spaces. (Grundlehren der mathematischen Wissenschaften; 236) Translation of Theorie der Steinschen Răume. Includes index. 1. Stein spaces. 1. Remmert, Reinhold, joint author. II. Title. III. Series: Grundlehren der mathematischen Wissenschaften in EinzeldarstelIungen; 236. QA331.G68313 515'.73 79-1430 AII rights reserved. No part of this book may be translated or reproduced in any form without written permission from Springer-Verlag.
© 1979 by Springer Science+Business Media New York Originally published by Springer Berlin Heidelberg New York in 1979 Softcover reprint ofthe hardcover Ist edition 1979 9 8 7 6 543 2 1 ISBN 978-1-4757-4359-3 ISBN 978-1-4757-4357-9 (eBook) DOI 10.1007/978-1-4757-4357-9
Dedicated to Karl Stein
Contents
Introduction
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . XV
Chapter A. Sheaf Theory
§ 0. Sheaves and Presheaves . . . . . . . . . 1. Sheaves and Sheaf Mappings . . . . . . 2. Sums of Sheaves, Subsheaves, and Restrictions 3. Sections . . . . . . . . . . . . . 4. Presheaves and the Section Functor r . . . 5. Going from Presheaves to Sheaves. The Functor t 6. The Sheaf Conditions 9'1 and 9'2 7. Direct Products . 8. Image Sheaves 9. Gluing Sheaves .
4 4 5
§ 1. Sheaves with Algebraic Structure . 1. Sheaves of Groups, Rings, and at-Modules
5 5
2. 3. 4. 5. 6. 7.
Sheaf Homomorphisms and Subsheaves Quotient Sheaves . . . . Sheaves of Local k-Algebras Algebraic Reduction . . . Presheaves with Algebraic Structure . On the Exactness oft and r . . .
1 1 1
2 2 3 3
6 7 8 8 9 9
§ 2. Coherent Sheaves and Coherent Functors 1. Finite Sheaves 2. Finite Relation Sheaves . . . 3. Coherent Sheaves . . . . . 4. Coherence of Trivial Extensions 5. The Functors ffiP and f\P . . 6. The Functor J't'om and Annihilator Sheaves 7. Sheaves of Quotients . . . . . . . .
10 10 11 11 12 12 13 14
§ 3. Complex Spaces . . . . 1. k-Algebraized Spaces .
14
2. 3. 4. 5. 6. 7. 8.
. . . . . . . . . . Differentiable and Complex Manifolds Complex Spaces and Holomorphic Maps Topological Properties of Complex Spaces Analytic Sets . . . . . . Dimension Theory . . . . Reduction of Complex Spaces Normal Complex Spaces . .
15 15 16
18 18 1