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Full Description
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.
Contents
Introduction
Summary of background material
Grothendieck topologies and sites
Fibered categories
Descent and the stack condition
Algebraic spaces
Invariants and quotients
Quasi-coherent sheaves on algebraic spaces
Algebraic stacks: Definitions and basic properties
Quasi-coherent sheaves on algebraic stacks
Basic geometric properties and constructions for stacks
Coarse moduli spaces
Gerbes
Moduli of curves
Glossary of category theory
Bibliography
Index of notation
Index of terminology
