Lemma 76.48.7. Let $S$ be a scheme. A Koszul-regular immersion of algebraic spaces over $S$ is a local complete intersection morphism.

**Proof.**
Let $i : X \to Y$ be a Koszul-regular immersion of algebraic spaces over $S$. By definition there exists a surjective étale morphism $V \to Y$ where $V$ is a scheme such that $X \times _ Y V$ is a scheme and the base change $X \times _ Y V \to V$ is a Koszul-regular immersion of schemes. By More on Morphisms, Lemma 37.62.9 we see that $X \times _ Y V \to V$ is a local complete intersection morphism. From Definition 76.48.1 we conclude that $i$ is a local complete intersection morphism of algebraic spaces.
$\square$

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