Deformation Theory
The basic problem of deformation theory in algebraic geometry involves watching a small deformation of one member of a family of objects, such as varieties, or subschemes in a fixed space, or vector bundles on a fixed scheme. In this new book, Robin Harts
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Deformation Theory
Graduate Texts in Mathematics
257
Editorial Board S. Axler K.A. Ribet
For other titles published in this series, go to http://www.springer.com/series/136
Robin Hartshorne
Deformation Theory
123
Robin Hartshorne Department of Mathematics University of California Berkeley, CA 94720-3840 USA [email protected] Editorial Board S. Axler Mathematics Department San Francisco State University San Francisco, CA 94132 USA [email protected]
K.A. Ribet Mathematics Department University of California at Berkeley Berkeley, CA 94720-3840 USA [email protected]
ISSN 0072-5285 ISBN 978-1-4419-1595-5 e-ISBN 978-1-4419-1596-2 DOI 10.1007/978-1-4419-1596-2 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2009939327 Mathematics Subject Classification (2000): 14B07, 14B12, 14B10, 14B20, 13D10, 14D15, 14H60, 14D20 c Robin Hartshorne 2010 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com).
Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1
First-Order Deformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1. The Hilbert Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2. Structures over the Dual Numbers . . . . . . . . . . . . . . . . . . . . . . . . . 9 3. The T i Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 4. The Infinitesimal Lifting Property . . . . . . . . . . . . . . . . . . . . . . . . . 26 5. Deformations of Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2
Higher-Order Deformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Subschemes and Invertible Sheaves . . . . . . . . . . . . . . . . . . . . . . . . 7. Vector Bundles and Coherent Sheaves . . . . . . . . . . . . . . . . . . . . . . 8. Cohen–Macaulay in Codimension Two . . . . . . . . . . . . . . . . . . . . . 9. Complete Intersections and Gorenstein in Codimension Three . 10. Obstructions to Deformations of Schemes . . . . . . . . . . . . . . . . . . . 11. Obstruction Theory
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