The Stacks Project

Tag 04DO

Lemma 53.43.1. Let $f : X \to Y$ be a morphism of schemes. Assume (B) holds. Then the functor $f_{small, *} : \mathop{\mathit{Sh}}\nolimits(X_{\acute{e}tale}) \to \mathop{\mathit{Sh}}\nolimits(Y_{\acute{e}tale})$ transforms surjections into surjections.

Proof. This follows from Sites, Lemma 7.40.2. $\square$

The code snippet corresponding to this tag is a part of the file etale-cohomology.tex and is located in lines 5784–5793 (see updates for more information).

\begin{lemma}
\label{lemma-property-B-implies}
Let $f : X \to Y$ be a morphism of schemes. Assume (B) holds.
Then the functor
$f_{small, *} : \Sh(X_\etale) \to \Sh(Y_\etale)$
transforms surjections into surjections.
\end{lemma}

\begin{proof}
This follows from
Sites, Lemma \ref{sites-lemma-weaker}.
\end{proof}

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