The Stacks project

59.47 Integral universally injective morphisms

Here is the general version of Proposition 59.46.4.

Proposition 59.47.1. Let $f : X \to Y$ be a morphism of schemes which is integral and universally injective.

  1. The functor

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

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

  2. the functor

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

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

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

Proof. We may factor $f$ as

\[ \xymatrix{ X \ar[r]^ h & Z \ar[r]^ i & Y } \]

where $h$ is integral, universally injective and surjective and $i : Z \to Y$ is a closed immersion. Apply Proposition 59.46.4 to $i$ and apply Theorem 59.45.2 to $h$. $\square$

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