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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