The theory of algebraic stacks emerged in the late sixties and early seventies in the works of P. Deligne, D. Mumford, and M. Artin. The language of algebraic stacks has been used repeatedly since then, mostly in connection with moduli problems: the increasing demand for an accurate description of moduli "spaces" came from various areas of mathematics and mathematical physics. Unfortunately the basic results on algebraic stacks were scattered in the literature and sometimes stated without proofs. The aim of this book is to fill this reference gap by providing mathematicians with the first systematic account of the general theory of (quasiseparated) algebraic stacks over an arbitrary base scheme. It covers the basic definitions and constructions, techniques for extending scheme-theoretic notions to stacks, Artin's representability theorems, but also new topics such as the "lisse-étale" topology.

Until now, the basic results on algebraic stacks were scattered in the literature and sometimes stated without proofs. This book - written in French -- fills this reference gap by providing mathematicians with the first systematic account of the general theory of (quasiseparated) algebraic stacks over an arbitrary base scheme. It covers the basic definitions and constructions, techniques for extending scheme-theoretic notions to stacks, Artins representability theorems, but also such new topics as the "lisse-tale" topology.

Contenu
Introduction.- La catégorie des S -espaces et sa sous-catégorie strictement pleine des S -espaces algébriques.- La 2-catégorie des S -groupoides.- La sous-2-catégorie strictement pleine des S -champs dans (Gr/S).- La 2-catégorie des S -champs algébriques.- Points d'un S -champ algébrique; topologie de Zariski.- Quelques résultats de structure locale.-Critères valuatifs; morphismes universellement fermés, morphismes séparés, morphismes propres.- Caractérisation des espaces algébriques et des champs de Deligne-Mumford.- Parenthèse sur les topologies plates.- Les critères d'Artin pour qu'un S -champ soit algébrique.- Points algébriques, faisceaux résiduels, gerbes résiduelles, dimension.- Faisceaux sur le site lisse-étale d'un S -champ algébrique.- Modules quasi-cohérents sur un S -champ algébrique.- Constructions locales.- Modules cohérents sur les S -champs algébriques localement noethériens.....