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). 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