Abstract. We develop a theory of unbounded derived categories of quasicoherent sheaves on algebraic stacks. In particular, we show that these categories are compactly generated by perfect complexes for stacks that either have finite stabilizers or are local quotient stacks. We also extend Toën and Antieau-Gepner's results on derived Azumaya algebras and compact generation of sheaves on linear categories from derived schemes to derived DeligneMumford stacks. These are all consequences of our main theorem: compact generation of a presheaf of triangulated categories on an algebraic stack is local for the quasi-finite flat topology.

We prove that every algebraic stack, locally of finite type over an algebraically closed field with affine stabilizers, isétale-locally a quotient stack in a neighborhood of a point with a linearly reductive stabilizer group. The proof uses an equivariant version of Artin's algebraization theorem proved in the appendix. We provide numerous applications of the main theorems.

We show that every scheme/algebraic space/stack that is quasi-compact with quasi-finite diagonal can be approximated by a noetherian scheme/algebraic space/stack. More generally, we show that any stack which is etale-locally a global quotient stack can be approximated. Examples of applications are generalizations of Chevalley's, Serre's and Zariski's theorems and Chow's lemma to the non-noetherian setting. We also show that every quasi-compact algebraic stack with quasi-finite diagonal has a finite generically flat cover by a scheme.Comment: 39 pages; complete overhaul of paper; generalized results and simplified proofs (no groupoid-calculations); added more applications and appendices with standard results on constructible properties and limits for stacks; generalized Thm C (no finite presentation hypothesis); some minor changes in 2,1-2.8, 8.2, 8.8 and 8.9; final versio

In this paper, we study quotients of groupoids and coarse moduli spaces of stacks in a general setting. Geometric quotients are not always categorical, but we present a natural topological condition under which a geometric quotient is categorical. We also show the existence of geometric quotients of finite flat groupoids and give explicit local descriptions. Exploiting similar methods, we give an easy proof of the existence of quotients of flat groupoids with finite stabilizers. As the proofs do not use noetherian methods and are valid for general algebraic spaces and algebraic stacks, we obtain a slightly improved version of Keel and Mori's theorem.Comment: 36 pages; the proof of the existence of quasi-finite flat presentations has been excluded (now in 1005.2171); quasi-separatedness assumptions dropped; some parts have been reorganized; title changed from "Existence of quotients by finite groups and coarse moduli spaces

Abstract. -Using the flatification by blow-up result of Raynaud and Gruson, we obtain new results for submersive and subtrusive morphisms. We show that universally subtrusive morphisms, and in particular universally open morphisms, are morphisms of effective descent for the fibered category of étale morphisms. Our results extend and supplement previous treatments on submersive morphisms by Grothendieck, Picavet and Voevodsky. Applications include the universality of geometric quotients and the elimination of noetherian hypotheses in many instances. Résumé (Submersion et descente effective de morphismes étales)On applique le théorème de « platification » de Raynaud et Gruson aux morphismes subtrusifs et obtient le théorème de structure suivant: Tout morphisme universellement subtrusif de présentation finie a un raffinement se factorisant en un recouvrement ouvert suivi d'un morphisme propre. La première application de ce théorème de structure est un théorème de descente effective. On montre que tout morphisme universellement subtrusif est un morphisme de descente effective pour la catégorie fibrée des morphismes étales. Ce résultat réduit l'écart entres schémas et espaces algébriques. Par exemple, on peut montrer que des quotients géométriques sont universels dans la caté-gorie des espaces algébriques. La deuxième application concerne les limites projectives de schémas. On démontre que tout morphisme universellement subtrusif de présenta-tion finie est la limite de morphismes universellement submersifs entre schémas noethériens. Il en découle que la classe de morphismes subtrusifs, introduite par Picavet, Texte reçu le 26 octobre