Groupoid scheme

In algebraic geometry, a groupoid scheme is a pair of schemes ${\displaystyle R,U}$ together with five morphisms ${\displaystyle s,t:R\to U,e:U\to R,m:R\times _{U,s,t}R\to R,i:R\to R}$ satisfying ${\displaystyle s\circ e,t\circ e}$ are the identity morphisms, ${\displaystyle s\circ m=s\circ p_{1},t\circ m=t\circ p_{2}}$ and other obvious conditions that generalize the axioms of group action; e.g., associativity.^[1] In practice, it is usually written as ${\displaystyle R\rightrightarrows U}$ (cf. coequalizer.)

Example: Suppose an algebraic group G acts from the right on a scheme U. Then take ${\displaystyle R=U\times G}$, s the projection, t the given action.

The main use of the notion is that it provides an atlas for a stack. More specifically, let ${\displaystyle [R\rightrightarrows U]}$ be the category of ${\displaystyle (R\rightrightarrows U)}$-torsors. Then it is a category fibered in groupoids; in fact, a Deligne–Mumford stack. Conversely, any DM stack is of this form.

Notes

1. Algebraic stacks, Ch 3. § 1.

References

• Behrend, Kai; Conrad, Brian; Edidin, Dan; Fulton, William; Fantechi, Barbara; Göttsche, Lothar; Kresch, Andrew (2006), Algebraic stacks