Lemma 66.22.4. 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

the dimension of the local ring of $X$ at $x$ (Definition 66.10.2) is $d$,

$\dim (\mathcal{O}_{X, \overline{x}}) = d$ for some geometric point $\overline{x}$ lying over $x$, and

$\dim (\mathcal{O}_{X, \overline{x}}) = d$ for any geometric point $\overline{x}$ lying over $x$.

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