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Tag 0B7W

Chapter 56: Decent Algebraic Spaces > Section 56.7: Points and specializations

Lemma 56.7.3. Let $S$ be a scheme. Let $f : Y \to X$ be a flat morphism of algebraic spaces over $S$. Let $x, x' \in |X|$ and assume $x' \leadsto x$, i.e., $x$ is a specialization of $x'$. Assume the pair $(X, x')$ satisfies the equivalent conditions of Lemma 56.4.5 (for example if $X$ is decent, $X$ is quasi-separated, or $X$ is representable). Then for every $y \in |Y|$ with $f(y) = x$, there exists a point $y' \in |Y|$, $y' \leadsto y$ with $f(y') = x'$.

Proof. (The parenthetical statement holds by the definition of decent spaces and the implications between the different separation conditions mentioned in Section 56.6.) Choose a scheme $V$ and a surjective étale morphism $V \to Y$. Choose $v \in V$ mapping to $y$. Then we see that it suffices to prove the lemma for $V \to X$. Thus we may assume $Y$ is a scheme. Choose a scheme $U$ and a surjective étale morphism $U \to X$. Choose $u \in U$ mapping to $x$. By Lemma 56.7.2 we may choose $u' \leadsto u$ mapping to $x'$. By Properties of Spaces, Lemma 54.4.3 we may choose $z \in U \times_X Y$ mapping to $y$ and $u$. Thus we reduce to the case of the flat morphism of schemes $U \times_X Y \to U$ which is Morphisms, Lemma 28.24.8. $\square$

    The code snippet corresponding to this tag is a part of the file decent-spaces.tex and is located in lines 1345–1354 (see updates for more information).

    \begin{lemma}
    \label{lemma-generalizations-lift-flat}
    Let $S$ be a scheme. Let $f : Y \to X$ be a  flat morphism of algebraic spaces
    over $S$. Let $x, x' \in |X|$ and assume $x' \leadsto x$, i.e., $x$ is a
    specialization of $x'$. Assume the pair $(X, x')$ satisfies the equivalent
    conditions of Lemma \ref{lemma-UR-finite-above-x} (for example if
    $X$ is decent, $X$ is quasi-separated, or $X$ is representable).
    Then for every $y \in |Y|$ with $f(y) = x$, there exists a point $y' \in |Y|$,
    $y' \leadsto y$ with $f(y') = x'$.
    \end{lemma}
    
    \begin{proof}
    (The parenthetical statement holds by the definition of decent spaces
    and the implications between the different separation conditions
    mentioned in Section \ref{section-reasonable-decent}.)
    Choose a scheme $V$ and a surjective \'etale morphism $V \to Y$.
    Choose $v \in V$ mapping to $y$. Then we see that it suffices to
    prove the lemma for $V \to X$. Thus we may assume $Y$ is a scheme.
    Choose a scheme $U$ and a surjective \'etale morphism $U \to X$.
    Choose $u \in U$ mapping to $x$. By Lemma \ref{lemma-specialization}
    we may choose $u' \leadsto u$ mapping to $x'$. By
    Properties of Spaces, Lemma \ref{spaces-properties-lemma-points-cartesian}
    we may choose $z \in U \times_X Y$ mapping to $y$ and $u$.
    Thus we reduce to the case of the flat morphism of
    schemes $U \times_X Y \to U$ which is
    Morphisms, Lemma \ref{morphisms-lemma-generalizations-lift-flat}.
    \end{proof}

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