Lemma 39.25.2. Let $X \to Y$ be a surjective finite locally free morphism. Let $V$ be a scheme over $X$ such that for all $(y, v_1, \ldots , v_ d)$ where $y \in Y$ and $v_1, \ldots , v_ d \in V_ y$ there exists an affine open $U \subset V$ with $v_1, \ldots , v_ d \in U$. Then any descent datum on $V/X/Y$ is effective.
Proof. As in the proof of Lemma 39.25.1, we may assume that $Y$ is affine, in particular quasi-compact. Thus there exists an integer $d$ such that all geometric fibers of $X \to Y$ have at most $d$ points, see the discussion in Morphisms, Section 29.57. Our assumption on $V$ trivially implies that we can apply Lemma 39.25.1, which completes the proof. $\square$
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