Group-stack

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 $\mathcal$ 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 $V \to \mathcal$. It has an action by the affine line $\mathbb^1$ 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
# a morphism $\sigma: X \times G \to X$,
# (associativity) a natural isomorphism $\sigma \circ (m \times 1_X) \overset\to \sigma \circ (1_X \times \sigma)$, where ''m'' is the multiplication on ''G'',
# (identity) a natural isomorphism $1_X \overset\to \sigma \circ (1_X \times e)$, 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.

Notes

References

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Algebraic geometry

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