Lemma 31.21.4. Let $i : Z \to X$ be a regular (resp. Koszul-regular, $H_1$-regular, quasi-regular) immersion. Let $X' \to X$ be a flat morphism. Then the base change $i' : Z \times _ X X' \to X'$ is a regular (resp. Koszul-regular, $H_1$-regular, quasi-regular) immersion.

**Proof.**
Via Lemma 31.20.7 this translates into the algebraic statements in Algebra, Lemmas 10.68.5 and 10.69.3 and More on Algebra, Lemma 15.30.5.
$\square$

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