The Stacks project

Lemma 37.74.5. 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 29.57.2 and 29.57.9. $\square$

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