The Stacks project

Lemma 29.13.5. The base change of a quasi-affine morphism is quasi-affine.

Proof. Let $f : X \to S$ be a quasi-affine morphism. By Lemma 29.13.3 above we can find a quasi-coherent sheaf of $\mathcal{O}_ S$-algebras $\mathcal{A}$ and a quasi-compact open immersion $X \to \underline{\mathop{\mathrm{Spec}}}_ S(\mathcal{A})$ over $S$. Let $g : S' \to S$ be any morphism. Denote $f' : X_{S'} = S' \times _ S X \to S'$ the base change of $f$. Since the base change of a quasi-compact open immersion is a quasi-compact open immersion we see that $X_{S'} \to \underline{\mathop{\mathrm{Spec}}}_{S'}(g^*\mathcal{A})$ is a quasi-compact open immersion (we have used Schemes, Lemmas 26.19.3 and 26.18.2 and Constructions, Lemma 27.4.6). By Lemma 29.13.3 again we conclude that $X_{S'} \to S'$ is quasi-affine. $\square$

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