Lemma 107.5.4. If $f: U \to X$ is a smooth morphism of locally Noetherian algebraic spaces, and if $u \in |U|$ with image $x \in |X|$, then

where $\dim _ u (U_ x)$ is defined via Definition 107.5.2.

Lemma 107.5.4. If $f: U \to X$ is a smooth morphism of locally Noetherian algebraic spaces, and if $u \in |U|$ with image $x \in |X|$, then

\[ \dim _ u (U) = \dim _ x(X) + \dim _{u} (U_ x) \]

where $\dim _ u (U_ x)$ is defined via Definition 107.5.2.

**Proof.**
See Morphisms of Spaces, Lemma 67.37.10 noting that the definition of $\dim _ u (U_ x)$ used here coincides with the definition used there, by Remark 107.5.3 (2).
$\square$

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