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.


  • 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

  1. a morphism ,
  2. (associativity) a natural isomorphism , where m is the multiplication on G,
  3. (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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