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Algebraic Spaces and Stacks [Hardback]

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  • Formāts: Hardback, 298 pages, height x width: 254x178 mm, weight: 700 g
  • Sērija : Colloquium Publications
  • Izdošanas datums: 01-Apr-2016
  • Izdevniecība: American Mathematical Society
  • ISBN-10: 1470427982
  • ISBN-13: 9781470427986
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Algebraic Spaces and StacksPalielināt
  • Formāts: Hardback, 298 pages, height x width: 254x178 mm, weight: 700 g
  • Sērija : Colloquium Publications
  • Izdošanas datums: 01-Apr-2016
  • Izdevniecība: American Mathematical Society
  • ISBN-10: 1470427982
  • ISBN-13: 9781470427986
Citas grāmatas par šo tēmu:
Permanent link: https://www.kriso.lv/db/9781470427986.html
Keywords:
This book is an introduction to the theory of algebraic spaces and stacks intended for graduate students and researchers familiar with algebraic geometry at the level of a first-year graduate course. The first several chapters are devoted to background material including chapters on Grothendieck topologies, descent, and fibered categories. Following this, the theory of algebraic spaces and stacks is developed. The last three chapters discuss more advanced topics including the Keel-Mori theorem on the existence of coarse moduli spaces, gerbes and Brauer groups, and various moduli stacks of curves. Numerous exercises are included in each chapter ranging from routine verifications to more difficult problems, and a glossary of necessary category theory is included as an appendix.

Recenzijas

It is splendid to have a self-contained treatment of stacks, written by a leading practitioner. Finally we have a reference where one can find careful statements and proofs of many of the foundational facts in this important subject. Researchers and students at all levels will be grateful to Olsson for writing this book." - William Fulton, University of Michigan

"This is a carefully planned out book starting with foundations and ending with detailed proofs of key results in the theory of algebraic stacks." - Johan de Jong, Columbia University

"Graduate students will benefit from the self-contained and accessible presentation of the entire theory, as well as the numerous exercises. Researchers will like the comprehensive, up-to-date treatment of foundations and key results, together with detailed proofs. Summing up, this is an excellent monograph that fills a large gap in the literature." - Stefan Schröer, Mathematical Reviews

Preface xi
Introduction 1(6)
Chapter 1 Summary of background material
7(28)
1.1 Flatness
7(2)
1.2 Morphisms locally of finite presentation
9(4)
1.3 Etale and smooth morphisms
13(10)
1.4 Schemes as functors
23(6)
1.5 Hilbert and Quot schemes
29(1)
1.6 Exercises
30(5)
Chapter 2 Grothendieck topologies and sites
35(34)
2.1 Sites
35(3)
2.2 Presheaves and sheaves
38(12)
2.3 Cohomology of sheaves
50(5)
2.4 Simplicial topoi
55(9)
2.5 Exercises
64(5)
Chapter 3 Fibered categories
69(18)
3.1 Definition of fibered category and basic properties
70(4)
3.2 The 2-Yoneda lemma
74(3)
3.3 Splittings of fibered categories
77(1)
3.4 Categories fibered in groupoids
78(6)
3.5 Exercises
84(3)
Chapter 4 Descent and the stack condition
87(32)
4.1 Faithfully flat descent
88(5)
4.2 Generalities on descent
93(5)
4.3 Descent for quasi-coherent sheaves
98(5)
4.4 Examples
103(5)
4.5 Application: Torsors and principal homogenous spaces
108(4)
4.6 Stacks
112(3)
4.7 Exercises
115(4)
Chapter 5 Algebraic spaces
119(18)
5.1 Properties of sheaves and definition of algebraic space
120(4)
5.2 Algebraic spaces as sheaf quotients
124(3)
5.3 Examples of algebraic spaces
127(2)
5.4 Basic properties of algebraic spaces
129(5)
5.5 Algebraic spaces are fppf sheaves
134(1)
5.6 Exercises
135(2)
Chapter 6 Invariants and quotients
137(14)
6.1 Review of some commutative algebra
137(2)
6.2 Quotients by finite flat groupoids
139(6)
6.3 Topological properties of algebraic spaces
145(3)
6.4 Schematic open subspaces of algebraic spaces
148(1)
6.5 Exercises
149(2)
Chapter 7 Quasi-coherent sheaves on algebraic spaces
151(18)
7.1 The category of quasi-coherent sheaves
151(4)
7.2 Affine morphisms and Stein factorization
155(7)
7.3 Nilpotent thickenings of schemes
162(1)
7.4 Chow's lemma for algebraic spaces
163(1)
7.5 Finiteness of cohomology
164(3)
7.6 Exercises
167(2)
Chapter 8 Algebraic stacks: Definitions and basic properties
169(22)
8.1 Definition of algebraic stack and fiber products
169(6)
8.2 Properties of algebraic stacks and morphisms between them
175(3)
8.3 Deligne-Mumford stacks
178(5)
8.4 Examples
183(5)
8.5 Exercises
188(3)
Chapter 9 Quasi-coherent sheaves on algebraic stacks
191(18)
9.1 The lisse-etale site
191(6)
9.2 Comparison with simplicial sheaves and the etale topos
197(6)
9.3 Pulling back quasi-coherent sheaves
203(2)
9.4 Exercises
205(4)
Chapter 10 Basic geometric properties and constructions for stacks
209(12)
10.1 Proper morphisms
209(1)
10.2 Relative Spec and Proj
210(5)
10.3 Root stacks
215(3)
10.4 Exercises
218(3)
Chapter 11 Coarse moduli spaces
221(22)
11.1 Basics on coarse moduli spaces
221(1)
11.2 Proof of the main theorem
222(8)
11.3 Applications of the local structure of coarse moduli spaces
230(3)
11.4 Chow's lemma for Deligne-Mumford stacks and applications
233(2)
11.5 The valuative criterion for properness
235(2)
11.6 Finiteness of cohomology
237(2)
11.7 Exercises
239(4)
Chapter 12 Gerbes
243(16)
12.1 Torsors and H1
243(3)
12.2 Generalities on gerbes
246(4)
12.3 Gerbes and twisted sheaves
250(4)
12.4 Exercises
254(5)
Chapter 13 Moduli of curves
259(26)
13.1 Moduli of elliptic curves
259(7)
13.2 The stack
266(12)
13.3 Moduli of stable maps
278(4)
13.4 Exercises
282(3)
Appendix A Glossary of category theory 285(6)
Bibliography 291(4)
Index of Notation 295(2)
Index of Terminology 297
Martin Olsson, University of California, Berkeley, CA, USA.