The Stacks project

Proof. Choose an affine scheme $U$ and a smooth morphism $U \to \mathcal{X}$ such that $x$ is in the image of the open continuous map $|U| \to |\mathcal{X}|$. We may and do replace $\mathcal{X}$ with the open substack corresponding to the image of $|U| \to |\mathcal{X}|$, see Properties of Stacks, Lemma 100.9.12. Thus we may assume $\mathcal{X} = [U/R]$ for a smooth groupoid $(U, R, s, t, c)$ in algebraic spaces where $U$ is a Noetherian affine scheme, see Algebraic Stacks, Section 94.16.

Let $E \subset |U|$ be the inverse image of $\{ x\} \subset |\mathcal{X}|$. Of course $E \not= \emptyset $. Since $|U|$ is a Noetherian topological space, we can choose an element $u \in E$ such that $\overline{\{ u\} } \cap E = \{ u\} $. As usual, we think of $u = \mathop{\mathrm{Spec}}(\kappa (u))$ as the spectrum of its residue field. Let us write

Furthermore, denote $Z = \overline{\{ u\} } \subset U$ with the reduced induced scheme structure. Denote $p : F \to U$ the morphism induced by the second projection (using $s : R \to U$ in the first fibre product description of $F$). Then $E$ is the set theoretic image of $p$. The morphism $R' \to F$ is a monomorphism which factors through the inverse image $p^{-1}(Z)$ of $Z$. This inverse image $p^{-1}(Z) \subset F$ is a closed subscheme and the restriction $p|_{p^{-1}(Z)} : p^{-1}(Z) \to Z$ has image set theoretically contained in $\{ u\} \subset Z$ by our careful choice of $u \in E$ above. Since $u = \mathop{\mathrm{lim}}\nolimits W$ where the limit is over the nonempty affine open subschemes of the irreducible reduced scheme $Z$, we conclude that the morphism $p|_{p^{-1}(Z)} : p^{-1}(Z) \to Z$ factors through the morphism $u \to Z$. Clearly this implies that $R' = p^{-1}(Z)$. In particular the morphism $t' : R' \to u$ is locally of finite presentation as the composition of the closed immersion $p^{-1}(Z) \to F$ of locally Noetherian algebraic spaces with the smooth morphism $\text{pr}_1 : F \to u$; use Morphisms of Spaces, Lemmas 67.23.5, 67.28.12, and 67.28.2. Hence the restriction $(u, R', s', t', c')$ of $(U, R, s, t, c)$ by $u \to U$ is a groupoid in algebraic spaces where $s'$ and $t'$ are flat and locally of finite presentation. Therefore $\mathcal{Z} = [u/R']$ is an algebraic stack by Criteria for Representability, Theorem 97.17.2. As $R'$ is the restriction of $R$ by $u \to U$ we see $\mathcal{Z} \to \mathcal{X}$ is a monomorphism by Groupoids in Spaces, Lemma 78.25.1 and Properties of Stacks, Lemma 100.8.4. Then $\mathcal{Z}$ is (isomorphic to) the residual gerbe by the material in Properties of Stacks, Section 100.11. $\square$