Lemma 31.22.1. Let $f : X \to S$ be a morphism of schemes. Let $i : Z \subset X$ be an immersion. Assume

$i$ is an $H_1$-regular (resp. quasi-regular) immersion, and

$Z \to S$ is a flat morphism.

Then for every morphism of schemes $g : S' \to S$ the base change $Z' = S' \times _ S Z \to X' = S' \times _ S X$ is an $H_1$-regular (resp. quasi-regular) immersion.

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