The Stacks project

Lemma 81.4.4. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $\pi : X' \to X$ be a morphism of algebraic spaces. Assume

  1. $f \circ \pi $ is representable (by schemes),

  2. $f \circ \pi $ has one of the following properties: surjective and integral, surjective and proper, or surjective and flat and locally of finite presentation.

Then

\[ \mathop{\mathit{Sh}}\nolimits (Y_{\acute{e}tale}) \longrightarrow \text{descent data for étale sheaves wrt }\{ X \to Y\} \]

is an equivalence of categories.

Proof. Formal consequence of Lemma 81.4.3 and Stacks, Lemma 8.3.7. $\square$


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