# The Stacks Project

## Tag 01HL

Lemma 25.4.2. Let $f : Z \to X$ be a morphism of locally ringed spaces. In order for $f$ to be a closed immersion it suffices that there exists an open covering $X = \bigcup U_i$ such that each $f : f^{-1}U_i \to U_i$ is a closed immersion.

Proof. Omitted. $\square$

The code snippet corresponding to this tag is a part of the file schemes.tex and is located in lines 356–362 (see updates for more information).

\begin{lemma}
\label{lemma-closed-local-target}
Let $f : Z \to X$ be a morphism of locally ringed spaces.
In order for $f$ to be a closed immersion it suffices
that there exists an open covering $X = \bigcup U_i$ such
that each $f : f^{-1}U_i \to U_i$ is a closed immersion.
\end{lemma}

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

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