Group-stack

In algebraic geometry, a group-stack is an algebraic stack whose categories of points have group structures or even groupoid structures in a compatible way.{{Cite web | url=https://mathoverflow.net/q/231313 | title=Ag.algebraic geometry - Are Picard stacks group objects in the category of algebraic stacks}} It generalizes a group scheme, which is a scheme whose sets of points have group structures in a compatible way.

Examples

A group scheme is a group-stack. More generally, a group algebraic-space, an algebraic-space analog of a group scheme, is a group-stack.
Over a field k, a vector bundle stack $\mathcal{V}$ on a Deligne–Mumford stack X is a group-stack such that there is a vector bundle V over k on X and a presentation $V \to \mathcal{V}$ . It has an action by the affine line $\mathbb{A}^1$ corresponding to scalar multiplication.
A Picard stack is an example of a group-stack (or groupoid-stack).

Actions of group-stacks

The definition of a group action of a group-stack is a bit tricky. First, given an algebraic stack X and a group scheme G on a base scheme S, a right action of G on X consists of

a morphism $\sigma: X \times G \to X$ ,
(associativity) a natural isomorphism $\sigma \circ (m \times 1_X) \overset{\sim}\to \sigma \circ (1_X \times \sigma)$ , where m is the multiplication on G,
(identity) a natural isomorphism $1_X \overset{\sim}\to \sigma \circ (1_X \times e)$ , where $e: S \to G$ is the identity section of G,

that satisfy the typical compatibility conditions.

If, more generally, G is a group-stack, one then extends the above using local presentations.

Notes

References

{{Cite journal|last1=Behrend|first1=K.|last2=Fantechi|first2=B.|date=1997-03-01|title=The intrinsic normal cone|journal=Inventiones Mathematicae|language=en|volume=128|issue=1|pages=45–88|doi=10.1007/s002220050136|arxiv=alg-geom/9601010 |bibcode=1997InMat.128...45B |issn=0020-9910}}

Category:Algebraic geometry