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An algebraic stack is essentially a geometric stack on the étale site.
Depending on details, this is a Deligne-Mumford stack or a more general Artin stack in the traditional setup of algebraic spaces.
Let be the fppf-site and the (2,1)-topos of stacks over it.
An algebraic stack is
an object ;
such that
the diagonal is representable by algebraic spaces;
there exists a scheme and a morphism which is a surjective and smooth morphism.
This appears in this form as (deJong, def. 47.12.1).
A smooth algebraic groupoid is an internal groupoid in algebraic spaces such that source and target maps are smooth morphisms.
This appears as (deJong, def. 47.16.2).
Notice that every internal groupoid in algebraic spaces represents a (2,1)-presheaf on the fppf-site. We shall not distinguish between the groupoid and the stackification of this presheaf, called the quotient stack of the groupoid.
Every algebraic stack is equivalent to a smooth algebraic groupoid and every smooth algebraic groupoid is an algebraic stack.
This appears as (deJong, lemma 47.16.2, theorem 47.17.3).
Orbifolds are an example of an Artin stack. For orbifolds the stabilizer groups are finite groups, while for Artin stacks in general they are algebraic groups.
A noncommutative generalization for Q-categories instead of Grothendieck topologies, hence applicable in noncommutative geometry of Deligne–Mumford and Artin stacks can be found in (KontsevichRosenberg).
algebraic stack
The original articles:
Pierre Deligne, David Mumford, The irreducibility of the space of curves of given genus, Publications Mathématiques de l’IHÉS (Paris) 36 (1969) 75-109 [doi:10.1007/BF02684599, numdam:PMIHES_1969__36__75_0]
(cf. Deligne-Mumford stack)
Michael Artin, Versal deformations and algebraic stacks, Invent Math 27 (1974) 165–189 [doi:10.1007/BF01390174, eudml:142310, pdf]
(cf. Artin stack)
Early review:
Monographs and review:
Gérard Laumon, Laurent Moret-Bailly, Champs algébriques, Ergebn. der Mathematik und ihrer Grenzgebiete 39, Springer (2000) [doi:10.1007/978-3-540-24899-6]
Angelo Vistoli, Grothendieck topologies, fibered categories and descent theory [math.AG/0412512, MR2223406] in: Fantechi et al. (eds.), Fundamental algebraic geometry. Grothendieck’s FGA explained, Mathematical Surveys and Monographs 123, Amer. Math. Soc. (2005) 1-104 [ISBN:978-0-8218-4245-4, MR2007f:14001]
(focusing on the incarnation of stacks, under the Grothendieck construction, as Grothendieck fibrations)
Frank Neumann, Algebraic Stacks and Moduli of Vector Bundles, impa (2011) [pdf, pdf]
Michael Groechenig, Algebraic Stacks, Lecture notes (2014) [web, pdf, pdf]
Martin Olsson, Algebraic Spaces and Stacks, Colloquium Publications 62 (2016) [doi:10.1090/coll/062, ISBN:978-1-4704-2798-6]
Daniel Halpern-Leistner, Moduli theory, Lecture notes (2020) [pdf, pdf]
See also:
Aise Johan de Jong, The Stacks Project, [tag:026K]
Fredrik Meyer, Notes on algebraic stacks (2013) [pdf, pdf]
Brief overview:
The noncommutative version is discussed in
Last revised on April 16, 2023 at 14:02:32. See the history of this page for a list of all contributions to it.