Lemma 66.10.3. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. The following quantities are equal:

The dimension of $X$.

The supremum of the dimensions of the local rings of $X$.

The supremum of $\dim _ x(X)$ for $x \in |X|$.

Lemma 66.10.3. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. The following quantities are equal:

The dimension of $X$.

The supremum of the dimensions of the local rings of $X$.

The supremum of $\dim _ x(X)$ for $x \in |X|$.

**Proof.**
The numbers in (1) and (3) are equal by Definition 66.9.2. Let $U \to X$ be a surjective étale morphism from a scheme $U$. The supremum of $\dim _ x(X)$ for $x \in |X|$ is the same as the supremum of $\dim _ u(U)$ for points $u$ of $U$ by definition. This is the same as the supremum of $\dim (\mathcal{O}_{U, u})$ by Properties, Lemma 28.10.2. This in turn is the same as (2) by definition.
$\square$

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