Group-stack

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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.

Contents

Examples

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.

Notes

  1. "Ag.algebraic geometry - Are Picard stacks group objects in the category of algebraic stacks".

References