Contents

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.

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

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.

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

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

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

• Michael Groechenig, Algebraic Stacks, Lecture notes (2014) [[web](http://individual.utoronto.ca/groechenig/stacks.html), pdf, pdf]

• Daniel Halpern-Leistner, Moduli theory, Lecture notes (2020) [[pdf](http://pi.math.cornell.edu/~danielhl/moduli_theory_notes.pdf), pdf]

• Aise Johan de Jong, The Stacks Project, Tag026K.

Monographs:

• Martin Olsson, Algebraic Spaces and Stacks, Colloquium Publications 62 (2016) [[doi:10.1090/coll/062](https://doi.org/10.1090/coll/062), ISBN:978-1-4704-2798-6]

• Frank Neumann, Algebraic Stacks and Moduli of Vector Bundles, impa (2011) [[pdf](https://impa.br/wp-content/uploads/2017/04/PM_36.pdf), pdf]

Course notes:

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

• Fredrik Meyer, Notes on algebraic stacks (2013) [[pdf](https://fredrikmeyer.net/uio-math/algstacks.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)