Lemma 29.50.5. Let $f : X \to Y$ be a morphism of schemes. Assume that $Y$ is locally Noetherian and $f$ is locally of finite type. Then

$\dim (X) \leq \dim (Y) + E$

where $E$ is the supremum of $\text{trdeg}_{\kappa (f(\xi ))}(\kappa (\xi ))$ where $\xi$ runs through the generic points of the irreducible components of $X$.

Proof. Immediate consequence of Lemma 29.50.2 and Properties, Lemma 28.10.2. $\square$

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