The Stacks project

Situation 79.15.3 (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 $\{ g \in G : g\text{ maps to }u\} $ is finite where $G \to U$ is the stabilizer, and

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