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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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