The Stacks project

112.3 Books and online notes

• Laumon, Moret-Bailly: Champs Algébriques [LM-B]

  This book is currently the most exhaustive reference on stacks containing many foundational results. It assumes the reader is familiar with algebraic spaces and frequently references Knutson's book [Kn]. There is an error in chapter 12 concerning the functoriality of the lisse-étale site of an algebraic stack. One doesn't need to worry about this as the error has been patched by Martin Olsson (see [olsson_sheaves]) and the results in the remaining chapters (after perhaps slight modification) are correct.

• The Stacks Project Authors: Stacks Project [stacks-project].

  You are reading it!

• Anton Geraschenko: Lecture notes for Martin Olsson's class on stacks [olsson_stacks]

  This course systematically develops the theory of algebraic spaces before introducing algebraic stacks (first defined in Lecture 27!). In addition to basic properties, the course covers the equivalence between being Deligne-Mumford and having unramified diagonal, the lisse-étale site on an Artin stack, the theory of quasi-coherent sheaves, the Keel-Mori theorem, cohomological descent, and gerbes (and their relation to the Brauer group). There are also some exercises.

• Behrend, Conrad, Edidin, Fantechi, Fulton, Göttsche, and Kresch: Algebraic stacks, online notes for a book being currently written [stacks_book]

  The aim of this book is to give a friendly introduction to stacks without assuming a sophisticated background with a focus on examples and applications. Unlike [LM-B], it is not assumed that the reader has digested the theory of algebraic spaces. Instead, Deligne-Mumford stacks are introduced with algebraic spaces being a special case with part of the goal being to develop enough theory to prove the assertions in [DM]. The general theory of Artin stacks is to be developed in the second part. Only a fraction of the book is now available on Kresch's website.

• Olsson, Martin: Algebraic spaces and stacks, [olsson_book]

  Highly recommended introduction to algebraic spaces and algebraic stacks starting at the level of somebody who has mastered Hartshorne's book on algebraic geometry.