Lemma 66.10.1. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $x \in |X|$ be a point. Let $d \in \{ 0, 1, 2, \ldots , \infty \} $. The following are equivalent

for some scheme $U$ and étale morphism $a : U \to X$ and point $u \in U$ with $a(u) = x$ we have $\dim (\mathcal{O}_{U, u}) = d$,

for any scheme $U$, any étale morphism $a : U \to X$, and any point $u \in U$ with $a(u) = x$ we have $\dim (\mathcal{O}_{U, u}) = d$.

If $X$ is a scheme, this is equivalent to $\dim (\mathcal{O}_{X, x}) = d$.

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