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Tag 00I1

Chapter 10: Commutative Algebra > Section 10.40: Going up and going down

Proposition 10.40.8. Let $R \to S$ be flat and of finite presentation. Then $\mathop{\rm Spec}(S) \to \mathop{\rm Spec}(R)$ is open. More generally this holds for any ring map $R \to S$ of finite presentation which satisfies going down.

Proof. Assume that $R \to S$ has finite presentation and satisfies going down. It suffices to prove that the image of a standard open $D(f)$ is open. Since $S \to S_f$ satisfies going down as well, we see that $R \to S_f$ satisfies going down. Thus after replacing $S$ by $S_f$ we see it suffices to prove the image is open. By Chevalley's theorem (Theorem 10.28.9) the image is a constructible set $E$. And $E$ is stable under generalization because $R \to S$ satisfies going down, see Topology, Lemmas 5.19.2 and 5.19.5. Hence $E$ is open by Lemma 10.40.7. $\square$

    The code snippet corresponding to this tag is a part of the file algebra.tex and is located in lines 9339–9345 (see updates for more information).

    \begin{proposition}
    \label{proposition-fppf-open}
    Let $R \to S$ be flat and of finite presentation.
    Then $\Spec(S) \to \Spec(R)$ is open.
    More generally this holds for any ring map $R \to S$ of
    finite presentation which satisfies going down.
    \end{proposition}
    
    \begin{proof}
    Assume that $R \to S$ has finite presentation and satisfies
    going down.
    It suffices to prove that the image of a standard open $D(f)$ is open.
    Since $S \to S_f$ satisfies going down as well, we see that
    $R \to S_f$ satisfies going down. Thus after replacing
    $S$ by $S_f$ we see it suffices to prove the image is
    open. By Chevalley's theorem
    (Theorem \ref{theorem-chevalley})
    the image is a constructible set $E$. And $E$ is stable
    under generalization because $R \to S$ satisfies going down,
    see Topology, Lemmas \ref{topology-lemma-open-closed-specialization}
    and \ref{topology-lemma-lift-specializations-images}.
    Hence $E$ is open by
    Lemma \ref{lemma-constructible-stable-specialization-closed}.
    \end{proof}

    Comments (2)

    Comment #519 by Fred Rohrer on April 4, 2014 a 5:25 am UTC

    In the statement of the result the map between the spectra goes the wrong way.

    Comment #522 by Johan (site) on April 4, 2014 a 8:18 pm UTC

    Thanks, fixed here.

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