Group-stack: Difference between revisions – Wikipedia

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 V {\displaystyle {\mathcal {V}}} 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 → V {\displaystyle V\to {\mathcal {V}}} . It has an action by the affine line A 1 {\displaystyle \mathbb {A} ^{1}} corresponding to scalar multiplication.
• A Picard stack is an example of a group-stack (or groupoid-stack).

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 σ : X × G → X {\displaystyle \sigma :X\times G\to X} ,
2. (associativity) a natural isomorphism σ ∘ ( m × 1 X ) → ∼ σ ∘ ( 1 X × σ ) {\displaystyle \sigma \circ (m\times 1_{X}){\overset {\sim }{\to }}\sigma \circ (1_{X}\times \sigma )} , where m is the multiplication on G,
3. (identity) a natural isomorphism 1 X → ∼ σ ∘ ( 1 X × e ) {\displaystyle 1_{X}{\overset {\sim }{\to }}\sigma \circ (1_{X}\times e)} , where e : S → G {\displaystyle 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.