# The Stacks Project

## Tag 04ZH

Lemma 58.4.9. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$.

1. If $Y$ is separated and $f$ is separated, then $X$ is separated.
2. If $Y$ is quasi-separated and $f$ is quasi-separated, then $X$ is quasi-separated.
3. If $Y$ is locally separated and $f$ is locally separated, then $X$ is locally separated.
4. If $Y$ is separated over $S$ and $f$ is separated, then $X$ is separated over $S$.
5. If $Y$ is quasi-separated over $S$ and $f$ is quasi-separated, then $X$ is quasi-separated over $S$.
6. If $Y$ is locally separated over $S$ and $f$ is locally separated, then $X$ is locally separated over $S$.

Proof. Parts (4), (5), and (6) follow immediately from Lemma 58.4.8 and Spaces, Definition 56.13.2. Parts (1), (2), and (3) reduce to parts (4), (5), and (6) by thinking of $X$ and $Y$ as algebraic spaces over $\mathop{\rm Spec}(\mathbf{Z})$, see Properties of Spaces, Definition 57.3.1. $\square$

The code snippet corresponding to this tag is a part of the file spaces-morphisms.tex and is located in lines 386–403 (see updates for more information).

\begin{lemma}
\label{lemma-separated-over-separated}
Let $S$ be a scheme.
Let $f : X \to Y$ be a morphism of algebraic spaces over $S$.
\begin{enumerate}
\item If $Y$ is separated and $f$ is separated, then $X$ is separated.
\item If $Y$ is quasi-separated and $f$ is quasi-separated, then
$X$ is quasi-separated.
\item If $Y$ is locally separated and $f$ is locally separated, then
$X$ is locally separated.
\item If $Y$ is separated over $S$ and $f$ is separated, then
$X$ is separated over $S$.
\item If $Y$ is quasi-separated over $S$ and $f$ is quasi-separated, then
$X$ is quasi-separated over $S$.
\item If $Y$ is locally separated over $S$ and $f$ is locally separated, then
$X$ is locally separated over $S$.
\end{enumerate}
\end{lemma}

\begin{proof}
Parts (4), (5), and (6) follow immediately from
Lemma \ref{lemma-composition-separated}
and
Spaces, Definition \ref{spaces-definition-separated}.
Parts (1), (2), and (3) reduce to parts (4), (5), and (6) by thinking
of $X$ and $Y$ as algebraic spaces over $\Spec(\mathbf{Z})$, see
Properties of Spaces, Definition \ref{spaces-properties-definition-separated}.
\end{proof}

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