The Stacks project

Lemma 76.48.2. Let $S$ be a scheme. Let $f : X \to Y$ be a local complete intersection morphism of algebraic spaces over $S$. Let $P$ be an algebraic space smooth over $Y$. Let $U \to X$ be an étale morphism of algebraic spaces and let $i : U \to P$ an immersion of algebraic spaces over $Y$. Picture:

\[ \xymatrix{ X \ar[rd] & U \ar[l] \ar[d] \ar[r]_ i & P \ar[ld] \\ & Y } \]

Then $i$ is a Koszul-regular immersion of algebraic spaces.

Proof. Choose a scheme $V$ and a surjective étale morphism $V \to Y$. Choose a scheme $W$ and a surjective étale morphism $W \to P \times _ Y V$. Set $U' = U \times _ P W$, which is a scheme étale over $U$. We have to show that $U' \to W$ is a Koszul-regular immersion of schemes, see Definition 76.44.2. By Definition 76.48.1 above the morphism of schemes $U' \to V$ is a local complete intersection morphism. Hence the result follows from More on Morphisms, Lemma 37.62.3. $\square$

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