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The Stacks project

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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