The Stacks project

Lemma 29.23.4. Let $k$ be a field. Let $X$ be a scheme over $k$. The structure morphism $X \to \mathop{\mathrm{Spec}}(k)$ is universally open.

Proof. Let $S \to \mathop{\mathrm{Spec}}(k)$ be a morphism. We have to show that the base change $X_ S \to S$ is open. The question is local on $S$ and $X$, hence we may assume that $S$ and $X$ are affine. In this case the result is Algebra, Lemma 10.40.10. $\square$


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