Lemma 37.22.3 (0AFG)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 37: More on Morphisms
Section 37.22: Cohen-Macaulay morphisms
Lemma 37.22.3 (cite)

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Lemma 37.22.3. Let $f : X \to Y$ be a morphism of locally Noetherian schemes which is locally of finite type and Cohen-Macaulay. 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 Morphisms, Lemma 29.29.4. Then $f$ is flat, locally of finite type and of relative dimension $d$ . Hence the result follows from Morphisms, Lemma 29.29.6. $\square$

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