Lemma 29.34.21 (0AFF)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 29: Morphisms of Schemes
Section 29.34: Smooth morphisms
Lemma 29.34.21 (cite)

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Lemma 29.34.21. Let $f : X \to Y$ be a smooth morphism of locally Noetherian schemes. For every point $x$ in $X$ with image $y$ in $Y$ ,

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

where $X_ y$ denotes the fiber over $y$ .

Proof. After replacing $X$ by an open neighborhood of $x$ , there is a natural number $d$ such that all fibers of $X \to Y$ have dimension $d$ at every point, see Lemma 29.34.12. Then $f$ is flat (Lemma 29.34.9), locally of finite type (Lemma 29.34.8), and of relative dimension $d$ . Hence the result follows from Lemma 29.29.6. $\square$

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