## Tag `04DO`

Chapter 53: Étale Cohomology > Section 53.43: Property (B)

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