Lemma 101.32.4. Let \mathcal{X} be an algebraic stack. Assume \mathcal{X} is quasi-DM with separated diagonal (equivalently \mathcal{I}_\mathcal {X} \to \mathcal{X} is locally quasi-finite and separated). Let x \in |\mathcal{X}|. Assume the automorphism group of \mathcal{X} at x is finite (Remark 101.19.3). Then there exists a morphism of algebraic stacks
g : \mathcal{U} \longrightarrow \mathcal{X}
with the following properties
there exists a point u \in |\mathcal{U}| mapping to x and g induces an isomorphism between automorphism groups at u and x (Remark 101.19.5),
\mathcal{U} \to \mathcal{X} is representable by algebraic spaces and étale,
\mathcal{U} = [U/R] where (U, R, s, t, c) is a groupoid scheme with U, R affine, and s, t finite, flat, and locally of finite presentation.
Proof.
Observe that G_ x is a group scheme by Lemma 101.19.1. The first part of the proof is exactly the same as the first part of the proof of Lemma 101.32.3. Thus we may assume \mathcal{X} = [U/R] where (U, R, s, t, c) and u \in U mapping to x satisfy all the assumptions of More on Groupoids in Spaces, Lemma 79.15.13. Our assumption on G_ x implies that G_ u is finite over u. Hence all the assumptions of More on Groupoids in Spaces, Lemma 79.15.12 are satisfied. Hence we can find an elementary étale neighbourhood (U', u') \to (U, u) such that the restriction R' of R to U' is split over u. Note that R' = U' \times _\mathcal {X} U' (small detail omitted; hint: transitivity of fibre products). Replacing (U, R, s, t, c) by (U', R', s', t', c') and shrinking \mathcal{X} as above, we may assume that (U, R, s, t, c) has a splitting over u. Let P \subset R be a splitting of R over u. Apply Lemma 101.32.2 to see that
\mathcal{U} = [U/P] \longrightarrow [U/R] = \mathcal{X}
is representable by algebraic spaces and étale. By construction G_ u is contained in P, hence this morphism defines an isomorphism on automorphism groups at u as desired.
\square
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