The Stacks project

Lemma 100.21.2. Let $\mathcal{X}$ be an algebraic stack. Consider a cartesian diagram

\[ \xymatrix{ U \ar[d] & F \ar[l]^ p \ar[d] \\ \mathcal{X} & \mathop{\mathrm{Spec}}(k) \ar[l] } \]

where $U$ is an algebraic space, $k$ is a field, and $U \to \mathcal{X}$ is locally of finite type. Let $z \in |F|$ be such that $\dim _ z(F) = 0$. Then, after replacing $U$ by an open subspace containing $p(z)$, the morphism

\[ U \longrightarrow \mathcal{X} \]

is locally quasi-finite.

Proof. Since $f : U \to \mathcal{X}$ is locally of finite type there exists a maximal open $W(f) \subset U$ such that the restriction $f|_{W(f)} : W(f) \to \mathcal{X}$ is locally quasi-finite, see Properties of Stacks, Remark 99.9.20 (2). Hence all we need to do is prove that $p(z)$ is a point of $W(f)$. Moreover, the remark referenced above also shows the formation of $W(f)$ commutes with arbitrary base change by a morphism which is representable by algebraic spaces. Hence it suffices to show that the morphism $F \to \mathop{\mathrm{Spec}}(k)$ is locally quasi-finite at $z$. This follows immediately from Morphisms of Spaces, Lemma 66.34.6. $\square$

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