Algebraic Spaces and Stacks

Martin Olsson, University of California, Berkeley, CA.

• IntroductionSummary of background materialGrothendieck topologies and sitesFibered categoriesDescent and the stack conditionAlgebraic spacesInvariants and quotientsQuasi-coherent sheaves on algebraic spacesAlgebraic stacks: Definitions and basic propertiesQuasi-coherent sheaves on algebraic stacksBasic geometric properties and constructions for stacksCoarse moduli spacesGerbesModuli of curvesGlossary of category theoryBibliographyIndex of notationIndex of terminology

“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