Lemma 37.72.5. Let $f : X \to Y$ be a locally quasi-finite morphism. Then

1. the functions $n_{X/Y}$ of Lemmas 37.27.3 and 37.28.3 agree,

2. if $X$ is quasi-compact, then $n_{X/Y}$ attains a maximum $d < \infty$.

Proof. Agreement of the functions is immediate from the fact that the (geometric) fibres of a locally quasi-finite morphism are discrete, see Morphisms, Lemma 29.20.8. Boundedness follows from Morphisms, Lemmas 29.56.2 and 29.56.9. $\square$

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