The Stacks project

Lemma 32.8.8. Notation and assumptions as in Situation 32.8.1. If

  1. $f$ is finite locally free (of degree $d$),

  2. $f_0$ is locally of finite presentation,

then $f_ i$ is finite locally free (of degree $d$) for some $i \geq 0$.

Proof. By Lemmas 32.8.7 and 32.8.3 we find an $i$ such that $f_ i$ is flat and finite. On the other hand, $f_ i$ is locally of finite presentation. Hence $f_ i$ is finite locally free by Morphisms, Lemma 29.48.2. If moreover $f$ is finite locally free of degree $d$, then the image of $Y \to Y_ i$ is contained in the open and closed locus $W_ d \subset Y_ i$ over which $f_ i$ has degree $d$. By Lemma 32.4.10 we see that for some $i' \geq i$ the image of $Y_{i'} \to Y_ i$ is contained in $W_ d$. Then $f_{i'}$ will be finite locally free of degree $d$. $\square$

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