In algebraic geometry, a groupoid scheme is a pair of schemes
R
,
U
together with five morphisms
s
,
t
:
R
→
U
,
e
:
U
→
R
,
m
:
R
×
U
,
s
,
t
R
→
R
,
i
:
R
→
R
satisfying
s
∘
e
,
t
∘
e
are the identity morphisms,
s
∘
m
=
s
∘
p
1
,
t
∘
m
=
t
∘
p
2
and other obvious conditions that generalize the axioms of group action; e.g., associativity. In practice, it is usually written as
R
⇉
U
(cf. coequalizer.)
Example: Suppose an algebraic group G acts from the right on a scheme U. Then take
R
=
U
×
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
⇉
U
]
be the category of
(
R
⇉
U
)
-torsors. Then it is a category fibered in groupoids; in fact, a Deligne–Mumford stack. Conversely, any DM stack is of this form.
