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.
- 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
- a morphism ,
- (associativity) a natural isomorphism , where m is the multiplication on G,
- (identity) a natural isomorphism , where 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.
- Behrend, K.; Fantechi, B. (1997-03-01). "The intrinsic normal cone". Inventiones Mathematicae. 128 (1): 45–88. doi:10.1007/s002220050136. ISSN 0020-9910.