Algebraic Spaces and Stacks

313 Pages·2016·2.231 MB·English

by  Martin Olsson

American Mathematical Society
Colloquium Publications
Volume 62
Algebraic Spaces
and Stacks
Martin Olsson
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Algebraic Spaces
and Stacks
Licensed to Keio University-Hiyoshi Campus. Prepared on Mon Apr 17 21:46:45 EDT 2017for download from IP 131.113.213.130.
License or copyright restrictions may apply to redistribution; see http://www.ams.org/publications/ebooks/terms
Licensed to Keio University-Hiyoshi Campus. Prepared on Mon Apr 17 21:46:45 EDT 2017for download from IP 131.113.213.130.
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http://dx.doi.org/10.1090/coll/062
American Mathematical Society
Colloquium Publications
Volume 62
Algebraic Spaces
and Stacks
Martin Olsson
American Mathematical Society
Providence, Rhode Island
Licensed to Keio University-Hiyoshi Campus. Prepared on Mon Apr 17 21:46:45 EDT 2017for download from IP 131.113.213.130.
License or copyright restrictions may apply to redistribution; see http://www.ams.org/publications/ebooks/terms
EDITORIAL COMMITTEE
Lawrence C. Evans
Yuri Manin
Peter Sarnak (Chair)
2010 Mathematics Subject Classification. Primary 14D23; Secondary 14D20, 14D22.
For additional informationand updates on this book, visit
www.ams.org/bookpages/coll-62
Library of Congress Cataloging-in-Publication Data
Olsson,MartinC.,author.
Algebraicspacesandstacks/MartinOlsson.
Providence,RhodeIsland: AmericanMathematicalSociety,[2016]
pagescm–(AmericanMathematicalSocietyColloquiumPublications;volume62)
Includesbibliographicalreferencesandindex.
ISBN978–1-4704-2798-6(alk.paper)
1.Algebraicspaces. 2.Algebraicstacks. 3.Algebraicgeometry–Foundations–Generalizations
(algebraic spaces, stacks). 4. Algebraic geometry–Families, fibrations–Stacks and moduli prob-
lems. 5.Algebraicgeometry–Families,fibrations–Fineandcoarsemodulispaces.
QA564.0578 2016
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To Jasmine
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Contents
Preface xi
Introduction 1
Chapter 1. Summary of background material 7
1.1. Flatness 7
1.2. Morphisms locally of finite presentation 9
1.3. E´tale and smooth morphisms 13
1.4. Schemes as functors 23
1.5. Hilbert and Quot schemes 29
1.6. Exercises 30
Chapter 2. Grothendieck topologies and sites 35
2.1. Sites 35
2.2. Presheaves and sheaves 38
2.3. Cohomology of sheaves 50
2.4. Simplicial topoi 55
2.5. Exercises 64
Chapter 3. Fibered categories 69
3.1. Definition of fibered category and basic properties 70
3.2. The 2-Yoneda lemma 74
3.3. Splittings of fibered categories 77
3.4. Categories fibered in groupoids 78
3.5. Exercises 84
Chapter 4. Descent and the stack condition 87
4.1. Faithfully flat descent 88
4.2. Generalities on descent 93
4.3. Descent for quasi-coherent sheaves 98
4.4. Examples 103
4.5. Application: Torsors and principal homogenous spaces 108
4.6. Stacks 112
4.7. Exercises 115
Chapter 5. Algebraic spaces 119
5.1. Properties of sheaves and definition of algebraic space 120
5.2. Algebraic spaces as sheaf quotients 124
5.3. Examples of algebraic spaces 127
5.4. Basic properties of algebraic spaces 129
5.5. Algebraic spaces are fppf sheaves 134
vii
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viii CONTENTS
5.6. Exercises 135
Chapter 6. Invariants and quotients 137
6.1. Review of some commutative algebra 137
6.2. Quotients by finite flat groupoids 139
6.3. Topological properties of algebraic spaces 145
6.4. Schematic open subspaces of algebraic spaces 148
6.5. Exercises 149
Chapter 7. Quasi-coherent sheaves on algebraic spaces 151
7.1. The category of quasi-coherent sheaves 151
7.2. Affine morphisms and Stein factorization 155
7.3. Nilpotent thickenings of schemes 162
7.4. Chow’s lemma for algebraic spaces 163
7.5. Finiteness of cohomology 164
7.6. Exercises 167
Chapter 8. Algebraic stacks: Definitions and basic properties 169
8.1. Definition of algebraic stack and fiber products 169
8.2. Properties of algebraic stacks and morphisms between them 175
8.3. Deligne-Mumford stacks 178
8.4. Examples 183
8.5. Exercises 188
Chapter 9. Quasi-coherent sheaves on algebraic stacks 191
9.1. The lisse-´etale site 191
9.2. Comparison with simplicial sheaves and the ´etale topos 197
9.3. Pulling back quasi-coherent sheaves 203
9.4. Exercises 205
Chapter 10. Basic geometric properties and constructions for stacks 209
10.1. Proper morphisms 209
10.2. Relative Spec and Proj 210
10.3. Root stacks 215
10.4. Exercises 218
Chapter 11. Coarse moduli spaces 221
11.1. Basics on coarse moduli spaces 221
11.2. Proof of the main theorem 222
11.3. Applications of the local structure of coarse moduli spaces 230
11.4. Chow’s lemma for Deligne-Mumford stacks and applications 233
11.5. The valuative criterion for properness 235
11.6. Finiteness of cohomology 237
11.7. Exercises 239
Chapter 12. Gerbes 243
12.1. Torsors and H1 243
12.2. Generalities on gerbes 246
12.3. Gerbes and twisted sheaves 250
12.4. Exercises 254
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CONTENTS ix
Chapter 13. Moduli of curves 259
13.1. Moduli of elliptic curves 259
13.2. The stack M . 266
g
13.3. Moduli of stable maps 278
13.4. Exercises 282
Appendix A. Glossary of category theory 285
Bibliography 291
Index of Notation 295
Index of Terminology 297
Licensed to Keio University-Hiyoshi Campus. Prepared on Mon Apr 17 21:46:45 EDT 2017for download from IP 131.113.213.130.
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