Lemma 37.59.11. Let $f : X \to Y$ be a morphism of schemes. If $f$ is locally of finite type and $X$ and $Y$ are regular, then $f$ is a local complete intersection morphism.

**Proof.**
We may assume there is a factorization $X \to \mathbf{A}^ n_ Y \to Y$ where the first arrow is an immersion. As $Y$ is regular also $\mathbf{A}^ n_ Y$ is regular by Algebra, Lemma 10.163.10. Hence $X \to \mathbf{A}^ n_ Y$ is a regular immersion by Divisors, Lemma 31.21.12.
$\square$

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