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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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