Algebraic Spaces and Stacks
About this Title
Martin Olsson, University of California, Berkeley, Berkeley, CA
Publication: Colloquium Publications
Publication Year: 2016; Volume 62
ISBNs: 978-1-4704-2798-6 (print); 978-1-4704-2865-5 (online)
MathSciNet review: MR3495343
MSC: Primary 14D23; Secondary 14D22
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.
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 and research mathematicians interested in algebraic spaces and stacks.
Table of Contents
- Chapter 1. Summary of background material
- Chapter 2. Grothendieck topologies and sites
- Chapter 3. Fibered categories
- Chapter 4. Descent and the stack condition
- Chapter 5. Algebraic spaces
- Chapter 6. Invariants and quotients
- Chapter 7. Quasi-coherent sheaves on algebraic spaces
- Chapter 8. Algebraic stacks: Definitions and basic properties
- Chapter 9. Quasi-coherent sheaves on algebraic stacks
- Chapter 10. Basic geometric properties and constructions for stacks
- Chapter 11. Coarse moduli spaces
- Chapter 12. Gerbes
- Chapter 13. Moduli of curves
- Appendix A. Glossary of category theory