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.[1] 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 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 . It has an action by the affine line 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