{{More citations needed|date=September 2025}} 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.<ref>{{Cite web | url=https://mathoverflow.net/q/231313 | title=Ag.algebraic geometry - Are Picard stacks group objects in the category of algebraic stacks}}</ref> 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''' <math>\mathcal{V}</math> 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 <math>V \to \mathcal{V}</math>. It has an action by the affine line <math>\mathbb{A}^1</math> 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 <math>\sigma: X \times G \to X</math>, # (associativity) a natural isomorphism <math>\sigma \circ (m \times 1_X) \overset{\sim}\to \sigma \circ (1_X \times \sigma)</math>, where ''m'' is the multiplication on ''G'', # (identity) a natural isomorphism <math>1_X \overset{\sim}\to \sigma \circ (1_X \times e)</math>, where <math>e: S \to G</math> 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 == {{reflist}}
== References == *{{Cite journal|last1=Behrend|first1=K.|author-link=Kai Behrend|last2=Fantechi|first2=B.|author-link2=Barbara Fantechi|date=1997-03-01|title=The intrinsic normal cone|journal=Inventiones Mathematicae|language=en|volume=128|issue=1|pages=45–88|doi=10.1007/s002220050136|arxiv=alg-geom/9601010 |bibcode=1997InMat.128...45B |issn=0020-9910}}
Category:Stacks (mathematics) {{algebraic-geometry-stub}}