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 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.
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