Lemma 37.62.19. The property \mathcal{P}(f) =“f is a local complete intersection morphism” is fpqc local on the base.
Proof. Let f : X \to S be a morphism of schemes. Let \{ S_ i \to S\} be an fpqc covering of S. Assume that each base change f_ i : X_ i \to S_ i of f is a local complete intersection morphism. Note that this implies in particular that f is locally of finite type, see Lemma 37.62.4 and Descent, Lemma 35.23.10. Let x \in X. Choose an open neighbourhood U of x and an immersion j : U \to \mathbf{A}^ n_ S over S (see discussion preceding Definition 37.62.2). We have to show that j is a Koszul-regular immersion. Since f_ i is a local complete intersection morphism, we see that the base change j_ i : U \times _ S S_ i \to \mathbf{A}^ n_{S_ i} is a Koszul-regular immersion, see Lemma 37.62.3. Because \{ \mathbf{A}^ n_{S_ i} \to \mathbf{A}^ n_ S\} is a fpqc covering we see from Descent, Lemma 35.23.32 that j is a Koszul-regular immersion as desired. \square
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