nLab algebraic stack

Idea

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.

Definition

Let $C_{fppf}$ be the fppf-site and $\mathcal{E} = Sh_{(2,1)}(C_{fppf})$ the (2,1)-topos of stacks over it.

Definition

An algebraic stack is

• an object $\mathcal{X}\in Sh_{(2,1)}(C_{fppf})$;

• such that

  1. the diagonal $\mathcal{X} \to \mathcal{X} \times \mathcal{X}$ is representable by algebraic spaces;

  2. there exists a scheme $U \in Sh(C_{fppf}) \hookrightarrow Sh_{(2,1)}(C_{fppf})$ and a morphism $U \to \mathcal{X}$ which is a surjective and smooth morphism.

This appears in this form as (deJong, def. 47.12.1).

Definition

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.

Theorem

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).

Properties

• Tannaka duality for geometric stacks.

Examples

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.

Generalizations

Noncommutative spaces

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).

Examples

• projective stack

Related concepts

• stack, moduli stack

• lisse-étale site

• geometric stack

  • algebraic stack

  • topological stack

  • differentiable stack

  • complex analytic stack

• geometric ∞-stack

References

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:

• Angelo Vistoli, Appendix of: Intersection theory on algebraic stacks and on their moduli spaces, Inventiones mathematicae 97 (1989) 613–670 [doi:10.1007/BF01388892]

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)

• Bertrand Toen, Master course on algebraic stacks (2005)

• 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:

• Anatoly Preygel, Algebraic stacks, Seminar notes: Quantization of Hitchin’s integrable system and Hecke eigensheaves, 2009, pdf.

The noncommutative version is discussed in

• Maxim Kontsevich, Alexander Rosenberg, Noncommutative stacks , preprint MPIM2004-37 (dvi ps)