Champs algebriques (Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge A Series of Modern Surveys in Mathematics)

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-?tale" topology. 1. La Categorie Des S-espaces Et Sa Sous-categorie Strictement Pleine Des S-espaces Algebriques -- 2. La 2-categorie Des S-groupoides -- 3. La Sous 2-categorie Strictement Pleine Des S-champs Dans (gr/s) -- 4. La W-categorie Des S-champs Algebriques -- 5. Points D'un S-champ Algebrigue; Topologie De Zariski -- 6. Quelques Resultats De Structure Locale -- 7. Criteres Valuatifs; Morphismes Universellement Fermes, Morphismes Separes, Morphismes Propres -- 8. Caracterisation Des Espaces Algebriques Et Des Champs De Deligne-mumford -- 9. Parenthese Sur Les Topologies Plates -- 10. Les Criteres D'artin Pour Qu'un S-champ Soit Algebrique -- 11. Points Algebriques, Faisceaux Residuels, Gerbes Residuelles, Dimension -- 12. Faisceaux Sur Le Site Lisse-etale D'un S-champ Algebrique -- 13. Modules Quasi-coherents Sur Un S-champ Algebrique -- 14. Constructions Locales -- 15. Modules Coherents Sur Les S-champs Algebriques Localement Noetheriens -- 16. Le Theoreme Principal De Zariski. Applications A La Structure Globale Des Champs De Deligne-mumford -- 17. Le Complexe Cotangent D'un L-morphisme De Champs Algebriques -- 18. Faisceaux Constructibles Sur Un S-champ Algebrique -- 19. Quelquev Prolongements -- App. Complements Sur Les Espaces Algebriques. Gérard Laumon, Luarent Moret-bailly. Includes Bibliographical References (p.[201]-203) And Index. 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.