The Stacks project

Lemma 37.59.19. 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 37.59.4 and Descent, Lemma 35.22.10. 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 37.59.2). 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 37.59.3. Because $\{ \mathbf{A}^ n_{S_ i} \to \mathbf{A}^ n_ S\} $ is a fpqc covering we see from Descent, Lemma 35.22.32 that $j$ is a Koszul-regular immersion as desired. $\square$


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