algebraic stack


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


An algebraic stack is

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

  • such that

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

    2. there exists a scheme USh(C fppf)Sh (2,1)(C fppf)U \in Sh(C_{fppf}) \hookrightarrow Sh_{(2,1)}(C_{fppf}) and a morphism U𝒳U \to \mathcal{X} 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.


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).



A standard textbook reference is

  • G. Laumon, L. Moret-Bailly, Champs algébriques , Ergebn. der Mathematik und ihrer Grenzgebiete 39 , Springer-Verlag, Berlin, 2000

An account is given in chapter 47 of

Course notes are in

A brief overview is in

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

The noncommutative version is discussed in

Last revised on March 29, 2015 at 22:15:09. See the history of this page for a list of all contributions to it.