Lemma 67.18.9. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$ which is locally of finite type. Let $x \in |X|$ with image $y \in |Y|$. Let $F = f^{-1}(\{ y\} )$ with induced topology from $|X|$. Let $k$ be a field and let $\mathop{\mathrm{Spec}}(k) \to Y$ be in the equivalence class defining $y$. Set $X_ k = \mathop{\mathrm{Spec}}(k) \times _ Y X$. Let $\tilde x \in |X_ k|$ map to $x \in |X|$. Consider the following conditions

$\dim _ x(F) = 0$,

$x$ is isolated in $F$,

$x$ is closed in $F$ and if $x' \leadsto x$ in $F$, then $x = x'$,

$\dim _{\tilde x}(|X_ k|) = 0$,

$\tilde x$ is isolated in $|X_ k|$,

$\tilde x$ is closed in $|X_ k|$ and if $\tilde x' \leadsto \tilde x$ in $|X_ k|$, then $\tilde x = \tilde x'$,

$\dim _{\tilde x}(X_ k) = 0$,

$f$ is quasi-finite at $x$.

Then we have

If $Y$ is decent, then conditions (2) and (3) are equivalent to each other and to conditions (5), (6), (7), and (8). If $Y$ and $X$ are decent, then all conditions are equivalent.

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