Cohomology and homotopy
In higher category theory
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
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.
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).
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
Revised on March 29, 2015 22:15:09
by Adeel Khan