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. In practice, it is usually written as Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): 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 Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): [R \rightrightarrows U]

be the category of Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): (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.