Lemma 31.21.11. Let $i : Z \to X$ be a Koszul regular closed immersion. Then there exists a surjective smooth morphism $X' \to X$ such that the base change $i' : Z \times _ X X' \to X'$ of $i$ is a regular immersion.

**Proof.**
We may assume that $X$ is affine and the ideal of $Z$ generated by a Koszul-regular sequence by replacing $X$ by the members of a suitable affine open covering (affine opens as in Lemma 31.20.7). The affine case is More on Algebra, Lemma 15.30.17.
$\square$

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