The Stacks project

Situation 79.15.2 (Strong splitting). Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Let $u \in U$ be a point. Assume that

1. $s, t : R \to U$ are separated,

2. $s$, $t$ are locally of finite type,

3. the set $\{ r \in R : s(r) = u, t(r) = u\} $ is finite, and

4. $s$ is quasi-finite at each point of the set in (3).

Note that assumptions (3) and (4) are implied by the assumption that the fibre $s^{-1}(\{ u\} )$ is finite, see Morphisms, Lemma 29.20.7.