Groupoid scheme

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

Example: Suppose an algebraic group G acts from the right on a scheme U. Then take $R=U\times G$ , 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 $[R\rightrightarrows U]$ be the category of $(R\rightrightarrows U)$ -torsors. Then it is a category fibered in groupoids; in fact, a Deligne–Mumford stack. Conversely, any DM stack is of this form.

Notes

↑ Algebraic stacks, Ch 3. § 1.

References

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

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