Lemma 35.21.3. Let $f : U \to V$ be an étale morphism of schemes. Let $u \in U$ and $v = f(u)$. Then $\dim (\mathcal{O}_{U, u}) = \dim (\mathcal{O}_{V, v})$.

**Proof.**
The algebraic statement we are asked to prove is the following: If $A \to B$ is an étale ring map and $\mathfrak q$ is a prime of $B$ lying over $\mathfrak p \subset A$, then $\dim (A_{\mathfrak p}) = \dim (B_{\mathfrak q})$. This is More on Algebra, Lemma 15.44.2.
$\square$

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