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Tag 01KO

Chapter 25: Schemes > Section 25.21: Separation axioms

Lemma 25.21.7. Let $f : X \to S$ be a morphism of schemes. The following are equivalent:

  1. The morphism $f$ is quasi-separated.
  2. For every pair of affine opens $U, V \subset X$ which map into a common affine open of $S$ the intersection $U \cap V$ is a finite union of affine opens of $X$.
  3. There exists an affine open covering $S = \bigcup_{i \in I} U_i$ and for each $i$ an affine open covering $f^{-1}U_i = \bigcup_{j \in I_i} V_j$ such that for each $i$ and each pair $j, j' \in I_i$ the intersection $V_j \cap V_{j'}$ is a finite union of affine opens of $X$.

Proof. Let us prove that (3) implies (1). By Lemma 25.17.4 the covering $X \times_S X = \bigcup_i \bigcup_{j, j'} V_j \times_{U_i} V_{j'}$ is an affine open covering of $X \times_S X$. Moreover, $\Delta_{X/S}^{-1}(V_j \times_{U_i} V_{j'}) = V_j \cap V_{j'}$. Hence the implication follows from Lemma 25.19.2.

The implication (1) $\Rightarrow$ (2) follows from the fact that under the hypotheses of (2) the fibre product $U \times_S V$ is an affine open of $X \times_S X$. The implication (2) $\Rightarrow$ (3) is trivial. $\square$

    The code snippet corresponding to this tag is a part of the file schemes.tex and is located in lines 3952–3967 (see updates for more information).

    \begin{lemma}
    \label{lemma-characterize-quasi-separated}
    Let $f : X \to S$ be a morphism of schemes.
    The following are equivalent:
    \begin{enumerate}
    \item The morphism $f$ is quasi-separated.
    \item For every pair of affine opens $U, V \subset X$
    which map into a common affine open of $S$ the intersection
    $U \cap V$ is a finite union of affine opens of $X$.
    \item There exists an affine open covering $S = \bigcup_{i \in I} U_i$
    and for each $i$ an affine open covering $f^{-1}U_i = \bigcup_{j \in I_i} V_j$
    such that for each $i$ and each pair $j, j' \in I_i$ the
    intersection $V_j \cap V_{j'}$ is a finite union of affine
    opens of $X$.
    \end{enumerate}
    \end{lemma}
    
    \begin{proof}
    Let us prove that (3) implies (1).
    By Lemma \ref{lemma-affine-covering-fibre-product}
    the covering $X \times_S X = \bigcup_i \bigcup_{j, j'} V_j \times_{U_i} V_{j'}$
    is an affine open covering of $X \times_S X$.
    Moreover, $\Delta_{X/S}^{-1}(V_j \times_{U_i} V_{j'}) = V_j \cap V_{j'}$.
    Hence the implication follows from Lemma \ref{lemma-quasi-compact-affine}.
    
    \medskip\noindent
    The implication (1) $\Rightarrow$ (2) follows from the fact
    that under the hypotheses of (2) the fibre product
    $U \times_S V$ is an affine open of $X \times_S X$.
    The implication (2) $\Rightarrow$ (3) is trivial.
    \end{proof}

    Comments (2)

    Comment #1051 by Wessel Bindt on September 30, 2014 a 11:54 am UTC

    In the penultimate sentence of the proof, it seems that "hypotheses of (1)" is supposed to be "hypotheses of (2)".

    Comment #1061 by Johan (site) on October 5, 2014 a 3:47 pm UTC

    Thanks! Fixed here.

    There are also 6 comments on Section 25.21: Schemes.

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