nLab
algebraic stack

Idea

An algebraic stack is essentially a geometric stack on CommRing${}^{\mathrm{op}}$: it denotes either a Deligne-Mumford stack or a more general Artin stack? in the traditional setup of algebraic spaces: they are (Grothendieck) $1$-stacks of groupoids on the étale site? satisfying additional representability conditions.

Definition

An algebraic stack $X$ is required to have an atlas $A\to X$ from an algebraic space (or in a stronger version a scheme) $A$; the morphisms of the cover are required to be

• surjective representable and étale (for a Deligne-Mumford stack);

• surjective representable and smooth (for an Artin stack?).

Here we used the following definition. A map $f:X\to Y$ of 1-stacks on the étale site is a representable morphism of stacks if for any scheme $S$ the fiber product $S{\times }_{Y}X$ is a stack associated to a scheme.

Let $P$ be a property stable under composition and pullback. A representable map $f:X\to Y$ of stacks is said to have a property $P$ if for any scheme $S$ and a morphism of 1-stacks $S\to Y$ the pullback $S{\times }_{Y}X\to X$ has property $P$.

Urs Schreiber: don’t we also need to demand that the diagonal morphism $\Delta :X\to X{\times }_{S}X$ is a) representable morphism of stacks, b) separated, c) quasi compact ?

See for instance def (4.1) in Laumon, Moret/Bailly Champs algébriques.

Generalizations

differentiable and topological stacks

Nowdays people talk about differentiable stacks and topological stacks, meaning analogues of Deligne–Mumford (DM) or Artin stacks defined in the setup of the category of manifolds or some convenient category of topological spaces. Orbifolds are an example of an Artin stack. For orbifolds stabilizer groups are finite, 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

• M. Kontsevich, A. Rosenberg, Noncommutative stacks, preprint MPIM2004-37 dvi ps

References

An introduction to algebraic stacks is being written online in a collaborative project here:

• The Stacks Project

For the topological variant, see topological stack and references therein; and for differentiable stacks see

• Kai Behrend, Ping Xu, Differentiable Stacks and Gerbes