Algebraic Spaces
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203 Donald Knutson Columbia University in the City of New York, New York, NY/USA
Algebraic Spaces
Springer-Verlag Berlin· Heidelberg· NewYork 1971
AMS Subject Classifications (1970): 14-02, 14A 15, 14A20, 14F20, 18F 10
ISBN 3-540-05496-0 Springer-Verlag Berlin . Heidelberg . New York ISBN 0-387-05496-0 Springer-Verlag New York . Heidelberg . Berlin 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 with the publisher.
© by Springer-Verlag Berlin' Heidelberg 1971. Library of Congress Catalog Card Nurnber73-164957.Printed in Germany. Offsetdruck: Julius Beltz, Hernsbach
PREFACE The core of this book is the author's thesis, Algebraic Spaces, written under Michael Artin at the Massachusetts Institute of Technology. The object there as here was to work out the foundations
a
la EGA for the theory of algebraic spaces, and hence give the
necessary background for Artin's fundamental papers Algebraization of Formal Moduli I, II. While working on this book, I was supported by M.I.T., Boston College, Columbia University, and the Advanced Science Summer Seminar at Bowdoin College, sponsored by the National Science Foundation. To all these institutions, I extend my gratitude. My special thanks goes to Professor Michael Artin both for many helpful discussions and for his initial suggestion that I undertake this project. Donald Knutson
CONTENTS Introduction Chapter One: The Etale Topology of Schemes 1. 2. 3. 4. 5.
29
Grothendieck Topologies and Descent Theory The Zariski Topology of Schemes The Flat Topology of Schemes The Etale Topology of Schemes Etale Equivalence Relations
29 38 52
59 72
Chapter Two: Algebraic Spaces
91
1. The Category of Algebraic Spaces ••.• 91 2. The Etale Topology of Algebraic Spaces • • • • • 101 3. Descent Theory for Algebraic Spaces • • • • . • • • 106 4. Quasicoherent Sheaves and Cohomology • 113 • • • 120 5. Local Constructions . 6. Points and the Zariski Topology • • 129 7. Proper and Projective Morphisms • 139 8. Integral Algebraic Spaces • • 144 Chapter Three: Quasicoherent Sheaves on Noetherian Locally Separated Algebraic Spaces . . • 1. 2. 3. 4. 5.
The Completeness / Extension Lemma The Serre Criterion Schemehood and Nilpotents Chevalley's Theorem. Devissage • • • • •
Chapter Four: The Finiteness Theorem 1. 2. 3. 4.
Actions of a Finite Group •• Symmetric powers of Projective Spaces Chow's Lemma The Finiteness Theorem
• • 153 153 • • 159 • 165 • • 169
• 173 • • • • • 176 . . . • • • . 177
• 185 • • • 192 • 202
VI Chapter Five: Formal Algebraic Spaces
• 204
1. Affine Formal Schemes . . . . •
2. Formal Algebraic Spaces • • . • •
• 204 • • 215
3. The Theorem of Holomorphic Func
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