Algebraic Spaces and Stacks: (Colloquium Publications)

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.

Table of 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

About the Author :
Martin Olsson, University of California, Berkeley, CA.

Review :
“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 “The book is very carefully written. In the few cases where an argument is not given in full in the book, a precise reference if provided. In addition, the author always explains the relation of his presentation to the existing literature. Hence, the book can also be used as a guide to the rich literature.” - Annette Huber, Jahresbericht der Deutschen Mathematiker-Vereinigung “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 Schroer, Mathematical Reviews “[T]his is an absolutely unique and excellent textbook on modern, highly advanced and abstract topics in algebraic geometry, which has no equal in the current literature, and which finally fills a long-continued gap therein.” - Werner Kleinert, Zentralblatt MATH