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Tag 02Z4

Chapter 53: Algebraic Spaces > Section 53.14: Examples of algebraic spaces

Lemma 53.14.5. Notation and assumptions as in Lemma 53.14.3. Assume $G$ is finite. Then

  1. if $U \to S$ is quasi-separated, then $U/G$ is quasi-separated over $S$, and
  2. if $U \to S$ is separated, then $U/G$ is separated over $S$.

Proof. In the proof of Lemma 53.13.1 we saw that it suffices to prove the corresponding properties for the morphism $j : R \to U \times_S U$. If $U \to S$ is quasi-separated, then for every affine open $V \subset U$ which maps into an affine of $S$ the opens $g(V) \cap V$ are quasi-compact. It follows that $j$ is quasi-compact. If $U \to S$ is separated, the diagonal $\Delta_{U/S}$ is a closed immersion. Hence $j : R \to U \times_S U$ is a finite coproduct of closed immersions with disjoint images. Hence $j$ is a closed immersion. $\square$

    The code snippet corresponding to this tag is a part of the file spaces.tex and is located in lines 2300–2309 (see updates for more information).

    \begin{lemma}
    \label{lemma-quotient-finite-separated}
    Notation and assumptions as in Lemma \ref{lemma-quotient}.
    Assume $G$ is finite. Then
    \begin{enumerate}
    \item if $U \to S$ is quasi-separated, then $U/G$ is quasi-separated
    over $S$, and
    \item if $U \to S$ is separated, then $U/G$ is separated over $S$.
    \end{enumerate}
    \end{lemma}
    
    \begin{proof}
    In the proof of Lemma \ref{lemma-properties-diagonal}
    we saw that it suffices to prove the
    corresponding properties for the morphism $j : R \to U \times_S U$.
    If $U \to S$ is quasi-separated, then for every affine open $V \subset U$
    which maps into an affine of $S$
    the opens $g(V) \cap V$ are quasi-compact. It follows that $j$ is
    quasi-compact.
    If $U \to S$ is separated, the diagonal $\Delta_{U/S}$ is a closed
    immersion. Hence $j : R \to U \times_S U$ is a finite coproduct
    of closed immersions with disjoint images. Hence $j$ is a closed immersion.
    \end{proof}

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