## 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 $\mathop{\mathrm{Spec}}(S) \to \mathop{\mathrm{Spec}}(R)$ 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 5.19.6. 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 9369–9375 (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}
```

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