Lemma 107.5.4 (0DRI)—The Stacks project

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Table of contents
Part 7: Algebraic Stacks
Chapter 107: The Geometry of Algebraic Stacks
Section 107.5: Dimension theory of algebraic stacks
Lemma 107.5.4 (cite)

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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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