Lemma 67.53.7 (0AEH)—The Stacks project

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Table of contents
Part 4: Algebraic Spaces
Chapter 67: Morphisms of Algebraic Spaces
Section 67.53: Universal homeomorphisms
Lemma 67.53.7 (cite)

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Lemma 67.53.7. Let $S$ be a scheme. Let $f : Y \to X$ be a universally injective, integral morphism of algebraic spaces over $S$.

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
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 67.45.7). 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 67.13.5 except that we use Étale Cohomology, Proposition 59.47.1 instead of Étale Cohomology, Proposition 59.46.4. $\square$

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