# The Stacks Project

## Tag 04DP

Lemma 50.43.2. Let $f : X \to Y$ be a morphism of schemes. Suppose

1. $V \to Y$ is an étale morphism of schemes,
2. $\{U_i \to X \times_Y V\}$ is an étale covering, and
3. $v \in V$ is a point.

Assume that for any such data there exists an étale neighbourhood $(V', v') \to (V, v)$, a disjoint union decomposition $X \times_Y V' = \coprod W'_i$, and morphisms $W'_i \to U_i$ over $X \times_Y V$. Then property (B) holds.

Proof. Omitted. $\square$

The code snippet corresponding to this tag is a part of the file etale-cohomology.tex and is located in lines 5800–5812 (see updates for more information).

\begin{lemma}
\label{lemma-simplify-B}
Let $f : X \to Y$ be a morphism of schemes. Suppose
\begin{enumerate}
\item $V \to Y$ is an \'etale morphism of schemes,
\item $\{U_i \to X \times_Y V\}$ is an \'etale covering, and
\item $v \in V$ is a point.
\end{enumerate}
Assume that for any such data there exists an \'etale neighbourhood
$(V', v') \to (V, v)$, a disjoint union decomposition
$X \times_Y V' = \coprod W'_i$, and morphisms $W'_i \to U_i$
over $X \times_Y V$. Then property (B) holds.
\end{lemma}

\begin{proof}
Omitted.
\end{proof}

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