Lemma 67.37.10 (0AFI)—The Stacks project

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Table of contents
Part 4: Algebraic Spaces
Chapter 67: Morphisms of Algebraic Spaces
Section 67.37: Smooth morphisms
Lemma 67.37.10 (cite)

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Lemma 67.37.10. Let $X$ and $Y$ be locally Noetherian algebraic spaces over a scheme $S$ , and let $f : X \to Y$ be a smooth morphism. For every point $x \in |X|$ with image $y \in |Y|$ ,

$\dim _ x(X) = \dim _ y(Y) + \dim _ x(X_ y)$

where $\dim _ x(X_ y)$ is the relative dimension of $f$ at $x$ as in Definition 67.33.1.

Proof. By definition of the dimension of an algebraic space at a point (Properties of Spaces, Definition 66.9.1), this reduces to the corresponding statement for schemes (Morphisms, Lemma 29.34.21). $\square$

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