Group-stack explained

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

l{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 l{V} . It has an action by the affine line 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

\sigma: X x G \to X ,

1. (associativity) a natural isomorphism

\sigma \circ (m x 1 _{X)} \overset{\sim}\to \sigma \circ ( 1 _{X} x \sigma) , where m is the multiplication on G,

1. (identity) a natural isomorphism

1 _{X} \overset{\sim}\to \sigma \circ ( 1 _{X} x 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.

References

• Behrend. K.. Fantechi. B.. 1997-03-01. The intrinsic normal cone. Inventiones Mathematicae. en. 128. 1. 45–88. 10.1007/s002220050136. 0020-9910.

Notes and References

1. Web site: Ag.algebraic geometry - Are Picard stacks group objects in the category of algebraic stacks.