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Softcover ISBN:  9781470474805 
Product Code:  COLL/62.S 
List Price:  $99.00 
MAA Member Price:  $89.10 
AMS Member Price:  $79.20 
eBook ISBN:  9781470428655 
Product Code:  COLL/62.E 
List Price:  $89.00 
MAA Member Price:  $80.10 
AMS Member Price:  $71.20 
Softcover ISBN:  9781470474805 
eBook ISBN:  9781470428655 
Product Code:  COLL/62.S.B 
List Price:  $188.00 $143.50 
MAA Member Price:  $169.20 $129.15 
AMS Member Price:  $150.40 $114.80 

Book DetailsColloquium PublicationsVolume: 62; 2016; 298 ppMSC: Primary 14
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 firstyear 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 KeelMori 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 selfcontained 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
ReadershipGraduate students and research mathematicians interested in algebraic spaces and stacks.

Table of Contents

Chapters

Introduction

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. Quasicoherent sheaves on algebraic spaces

Chapter 8. Algebraic stacks: Definitions and basic properties

Chapter 9. Quasicoherent 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


Additional Material

Reviews

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 MathematikerVereinigung 
Graduate students will benefit from the selfcontained and accessible presentation of the entire theory, as well as the numerous exercises. Researchers will like the comprehensive, uptodate 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 
[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 longcontinued gap therein.
Werner Kleinert, Zentralblatt MATH


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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 firstyear 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 KeelMori 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 selfcontained 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.

Chapters

Introduction

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. Quasicoherent sheaves on algebraic spaces

Chapter 8. Algebraic stacks: Definitions and basic properties

Chapter 9. Quasicoherent 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

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 MathematikerVereinigung 
Graduate students will benefit from the selfcontained and accessible presentation of the entire theory, as well as the numerous exercises. Researchers will like the comprehensive, uptodate 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 
[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 longcontinued gap therein.
Werner Kleinert, Zentralblatt MATH