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Tag 04ZH

Chapter 58: Morphisms of Algebraic Spaces > Section 58.4: Separation axioms

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 $\mathop{\mathrm{Spec}}(\mathbf{Z})$, 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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