Lemma 39.25.1. 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. Let $\varphi$ be a descent datum as in Descent, Definition 35.34.1. Recall that the functor from schemes over $Y$ to descent data relative to $\{ X \to Y\}$ is fully faithful, see Descent, Lemma 35.35.11. Thus using Constructions, Lemma 27.2.1 it suffices to prove the lemma in the case that $Y$ is affine. Some details omitted (this argument can be avoided if $Y$ is separated or has affine diagonal, because then every morphism from an affine scheme to $X$ is affine).

Assume $Y$ is affine. If $V$ is also affine, then we have effectivity by Descent, Lemma 35.37.1. Hence by Descent, Lemma 35.35.13 it suffices to prove that every point $v$ of $V$ has a $\varphi$-invariant affine open neighbourhood. Consider the groupoid $(X, X \times _ Y X, \text{pr}_1, \text{pr}_0, \text{pr}_{02})$. By Lemma 39.21.3 the descent datum $\varphi$ determines and is determined by a cartesian morphism of groupoid schemes

$(V, R, s, t, c) \longrightarrow (X, X \times _ Y X, \text{pr}_1, \text{pr}_0, \text{pr}_{02})$

over $\mathop{\mathrm{Spec}}(\mathbf{Z})$. Since $X \to Y$ is finite locally free, we see that $\text{pr}_ i : X \times _ Y X \to X$ and hence $s$ and $t$ are finite locally free. In particular the $R$-orbit $t(s^{-1}(\{ v\} ))$ of our point $v \in V$ is finite. Using the equivalence of categories of Lemma 39.21.3 once more we see that $\varphi$-invariant opens of $V$ are the same thing as $R$-invariant opens of $V$. Our assumption shows there exists an affine open of $V$ containing the orbit $t(s^{-1}(\{ v\} ))$ as all the points in this orbit map to the same point of $Y$. Thus Lemma 39.24.1 provides an $R$-invariant affine open containing $v$. $\square$

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