The Stacks project

Lemma 81.4.3. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume $f$ is representable (by schemes) and $f$ 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. Each of the properties of morphisms of algebraic spaces mentioned in the statement of the lemma is preserved by arbitrary base change, see the lists in Spaces, Section 65.4. Thus we can apply Lemma 81.4.2 to see that we can work étale locally on $Y$. In this way we reduce to the case where $Y$ is a scheme; some details omitted. In this case $X$ is also a scheme and the result follows from Étale Cohomology, Lemma 59.104.2, 59.104.3, or 59.104.5. $\square$

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