Lemma 40.12.4. Let $S$ be a scheme. Let $(U, R, s, t, c, e, i)$ be a groupoid scheme over $S$. Let $G \to U$ be the stabilizer group scheme. Assume $s$ and $t$ are Cohen-Macaulay and locally of finite presentation. Let $u \in U$ be a finite type point of the scheme $U$, see Morphisms, Definition 29.16.3. Assume that $G \to U$ is locally quasi-finite. With notation as in Situation 40.12.1 there exist an affine scheme $U'$ and a morphism $g : U' \to U$ such that

$g$ is an immersion,

$u \in U'$,

$g$ is locally of finite presentation,

the morphism $h : U' \times _{g, U, t} R \longrightarrow U$ is flat, locally of finite presentation, and locally quasi-finite, and

the morphisms $s', t' : R' \to U'$ are flat, locally of finite presentation, and locally quasi-finite.

**Proof.**
Take $g : U' \to U$ as in Lemma 40.12.3. Since $h^{-1}(u) = F'_ u$ we see that $h$ has relative dimension $\leq 0$ at $(u, e(u))$. Hence, by Remark 40.6.3, we obtain an open subscheme $U'' \subset U'$ such that $u \in U''$ and $U'' \times _{g, U, t} R$ is the maximal open subscheme of $U' \times _{g, U, t} R$ on which $h$ has relative dimension $\leq 0$. After replacing $U'$ by $U''$ we see that $h$ has relative dimension $\leq 0$. This implies that $h$ is locally quasi-finite by Morphisms, Lemma 29.29.5. Since it is still locally of finite presentation and Cohen-Macaulay we see that it is flat, locally of finite presentation and locally quasi-finite, i.e., (4) above holds. This implies that $s'$ is flat, locally of finite presentation and locally quasi-finite as a base change of $h$, see Lemma 40.9.2.
$\square$

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