# The Stacks Project

## Tag 0B7L

Lemma 31.11.5. Let $i : Z \to X$ be a closed immersion of schemes inducing a homeomorphism of underlying topological spaces. Then $X$ is quasi-affine if and only if $Z$ is quasi-affine.

Proof. Recall that a scheme is quasi-affine if and only if the structure sheaf is ample, see Properties, Lemma 27.27.1. Hence if $Z$ is quasi-affine, then $\mathcal{O}_Z$ is ample, hence $\mathcal{O}_X$ is ample by Lemma 31.11.4, hence $X$ is quasi-affine. A proof of the converse, which can also be seen in an elementary way, is gotten by reading the argument just given backwards. $\square$

The code snippet corresponding to this tag is a part of the file limits.tex and is located in lines 2645–2650 (see updates for more information).

\begin{lemma}
\label{lemma-thickening-quasi-affine}
Let $i : Z \to X$ be a closed immersion of schemes
inducing a homeomorphism of underlying topological spaces.
Then $X$ is quasi-affine if and only if $Z$ is quasi-affine.
\end{lemma}

\begin{proof}
Recall that a scheme is quasi-affine
if and only if the structure sheaf is ample, see
Properties, Lemma \ref{properties-lemma-quasi-affine-O-ample}.
Hence if $Z$ is quasi-affine, then $\mathcal{O}_Z$ is ample,
hence $\mathcal{O}_X$ is ample by
Lemma \ref{lemma-ample-on-reduction}, hence
$X$ is quasi-affine. A proof of the converse, which
can also be seen in an elementary way, is gotten by
reading the argument just given backwards.
\end{proof}

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