The Stacks project

Lemma 66.53.7. Let $S$ be a scheme. Let $f : Y \to X$ be a universally injective, integral morphism of algebraic spaces over $S$.

  1. The functor

    \[ f_{small, *} : \mathop{\mathit{Sh}}\nolimits (Y_{\acute{e}tale}) \longrightarrow \mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale}) \]

    is fully faithful and its essential image is those sheaves of sets $\mathcal{F}$ on $X_{\acute{e}tale}$ whose restriction to $|X| \setminus f(|Y|)$ is isomorphic to $*$, and

  2. the functor

    \[ f_{small, *} : \textit{Ab}(Y_{\acute{e}tale}) \longrightarrow \textit{Ab}(X_{\acute{e}tale}) \]

    is fully faithful and its essential image is those abelian sheaves on $Y_{\acute{e}tale}$ whose support is contained in $f(|Y|)$.

In both cases $f_{small}^{-1}$ is a left inverse to the functor $f_{small, *}$.

Proof. Since $f$ is integral it is universally closed (Lemma 66.45.7). In particular, $f(|Y|)$ is a closed subset of $|X|$ and the statements make sense. The rest of the proof is identical to the proof of Lemma 66.13.5 except that we use √Čtale Cohomology, Proposition 59.47.1 instead of √Čtale Cohomology, Proposition 59.46.4. $\square$


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