Lemma 10.40.10. Let $k$ be a field. Let $R$ and $S$ be $k$-algebras. The map $\mathop{\mathrm{Spec}}(S \otimes _ k R) \to \mathop{\mathrm{Spec}}(R)$ is open.

Proof. Let $f \in S \otimes _ k R$. It suffices to prove that the image of the standard open $D(f)$ is open. Let $S' \subset S$ be a finite type $k$-subalgebra such that $f \in S' \otimes _ k R$. The map $R \to S' \otimes _ k R$ is flat and of finite presentation, hence the image $U$ of $\mathop{\mathrm{Spec}}((S' \otimes _ k R)_ f) \to \mathop{\mathrm{Spec}}(R)$ is open by Proposition 10.40.8. By Lemma 10.40.9 this is also the image of $D(f)$ and we win. $\square$

Comment #4885 by Manuel Hoff on

Small typo: In the first sentence of the proof, $f \in S \otimes_k R$.

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