topos theory

# 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 ${C}_{\mathrm{fppf}}$ be the fppf-site and $ℰ={\mathrm{Sh}}_{\left(2,1\right)}\left({C}_{\mathrm{fppf}}\right)$ the (2,1)-topos of stacks over it.

###### Definition

An algebraic stack is

• an object $𝒳\in {\mathrm{Sh}}_{\left(2,1\right)}\left({C}_{\mathrm{fppf}}\right)$;

• such that

1. the diagonal $𝒳\to 𝒳×𝒳$ is representable by algebraic spaces;

2. there exists a scheme $U\in \mathrm{Sh}\left({C}_{\mathrm{fppf}}\right)↪{\mathrm{Sh}}_{\left(2,1\right)}\left({C}_{\mathrm{fppf}}\right)$ and a morphism $U\to 𝒳$ 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).

## 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).

## References

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

The noncommutative version is discussed in

Revised on January 1, 2011 09:37:58 by David Roberts (59.101.32.29)