The Stacks project

Proof. 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. Observe that $u = \mathop{\mathrm{Spec}}(\kappa (u)) \to \mathcal{X}$ is quasi-compact, see Properties of Stacks, Lemma 100.14.1. Consider the cartesian diagram

Since $U$ is an affine scheme and $F \to U$ is quasi-compact, we see that $F$ is quasi-compact. Since $U \to \mathcal{X}$ is locally quasi-finite, we see that $F \to u$ is locally quasi-finite. Hence $F \to u$ is quasi-finite and $F$ is an affine scheme whose underlying topological space is finite discrete (Spaces over Fields, Lemma 72.10.8). Observe that we have a monomorphism $u \times _\mathcal {X} u \to F$. In particular the set $\{ r \in R : s(r) = u, t(r) = u\}$ which is the image of $|u \times _\mathcal {X} u| \to |R|$ is finite. we conclude that all the assumptions of More on Groupoids in Spaces, Lemma 79.15.11 hold.

Thus we can find an elementary étale neighbourhood $(U', u') \to (U, u)$ such that the restriction $R'$ of $R$ to $U'$ is strongly 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 strong splitting over $u$. Let $P \subset R$ be a strong splitting of $R$ over $u$. Apply Lemma 101.32.2 to see that

is representable by algebraic spaces and étale. Since $P \subset R$ is open and contains $\{ r \in R : s(r) = u, t(r) = u\}$ by construction we see that $u \times _\mathcal {U} u \to u \times _\mathcal {X} u$ is an isomorphism. The statement on residual gerbes then follows from Properties of Stacks, Lemma 100.11.14 (we observe that the residual gerbes in question exist by Lemma 101.31.2). $\square$