The Stacks project

Lemma 37.63.6. The property of being ind-quasi-affine is stable under base change.

Proof. Let $f : X \to Y$ be an ind-quasi-affine morphism. For checking that every base change of $f$ is ind-quasi-affine, by Lemma 37.63.2, we may work Zariski locally on $Y$, so we assume that $Y$ is affine. Furthermore, we may also assume that in the base change morphism $Z \to Y$ the scheme $Z$ is affine, too. The base change $X \times _ Y Z \to X$ is an affine morphism, so, by Lemmas 37.63.3 and 37.63.4, the map $X \times _ Y Z \to Y$ is ind-quasi-affine. Then, by Lemma 37.63.5, the base change $X \times _ Y Z \to Z$ is ind-quasi-affine, as desired. $\square$


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