The Stacks project

Lemma 81.5.1. Let $S$ be a scheme. Let $f : X \to Y$ be a proper surjective morphism of algebraic spaces over $S$. Any descent datum $(U/X, \varphi )$ relative to $f$ (Descent on Spaces, Definition 74.22.1) with $U$ étale over $X$ is effective (Descent on Spaces, Definition 74.22.10). More precisely, there exists an étale morphism $V \to Y$ of algebraic spaces whose corresponding canonical descent datum is isomorphic to $(U/X, \varphi )$.

Proof. Recall that $U$ gives rise to a representable sheaf $\mathcal{F} = h_ U$ in $\mathop{\mathit{Sh}}\nolimits (X_{spaces, {\acute{e}tale}}) = \mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale})$, see Properties of Spaces, Section 66.18. The descent datum on $U$ relative to $f$ exactly gives a descent datum $(\mathcal{F}, \varphi )$ for étale sheaves with respect to $\{ X \to Y\} $. By Lemma 81.4.5 this descent datum is effective. Let $\mathcal{G}$ be the corresponding sheaf on $Y_{\acute{e}tale}$. By Properties of Spaces, Lemma 66.27.3 we obtain an étale morphism $V \to Y$ of algebraic spaces corresponding to $\mathcal{G}$; we omit the verification of the set theoretic condition1. The given isomorphism $\mathcal{F} \to f_{small}^{-1}\mathcal{G}$ corresponds to an isomorphism $U \to V \times _ Y X$ compatible with the descent datum. $\square$

[1] It follows from the fact that $\mathcal{F}$ satisfies the corresponding condition.

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