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. 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 [math]\displaystyle{ \mathcal{V} }[/math] 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 [math]\displaystyle{ V \to \mathcal{V} }[/math]. It has an action by the affine line [math]\displaystyle{ \mathbb{A}^1 }[/math] 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

1. a morphism [math]\displaystyle{ \sigma: X \times G \to X }[/math],
2. (associativity) a natural isomorphism [math]\displaystyle{ \sigma \circ (m \times 1_X) \overset{\sim}\to \sigma \circ (1_X \times \sigma) }[/math], where m is the multiplication on G,
3. (identity) a natural isomorphism [math]\displaystyle{ 1_X \overset{\sim}\to \sigma \circ (1_X \times e) }[/math], where [math]\displaystyle{ e: S \to G }[/math] 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.

References

• Behrend, K.; Fantechi, B. (1997-03-01). "The intrinsic normal cone" (in en). Inventiones Mathematicae 128 (1): 45–88. doi:10.1007/s002220050136. ISSN 0020-9910.