Lemma 31.21.12. Let $i : Z \to X$ be an immersion. If $Z$ and $X$ are regular schemes, then $i$ is a regular immersion.

Proof. Let $z \in Z$. By Lemma 31.20.8 it suffices to show that the kernel of $\mathcal{O}_{X, z} \to \mathcal{O}_{Z, z}$ is generated by a regular sequence. This follows from Algebra, Lemmas 10.106.4 and 10.106.3. $\square$

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