Lemma 76.48.9. The property $\mathcal{P}(f) =$“$f$ is a local complete intersection morphism” is fpqc local on the base.

Proof. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $\{ Y_ i \to Y\}$ be an fpqc covering (Topologies on Spaces, Definition 73.9.1). Let $f_ i : X_ i \to Y_ i$ be the base change of $f$ by $Y_ i \to Y$. If $f$ is a local complete intersection morphism, then each $f_ i$ is a local complete intersection morphism by Lemma 76.48.4.

Conversely, assume each $f_ i$ is a local complete intersection morphism. We may replace the covering by a refinement (again because flat base change preserves the property of being a local complete intersection morphism). Hence we may assume $Y_ i$ is a scheme for each $i$, see Topologies on Spaces, Lemma 73.9.5. Choose a scheme $V$ and a surjective étale morphism $V \to Y$. Choose a scheme $U$ and a surjective étale morphism $U \to V \times _ Y X$. We have to show that $U \to V$ is a local complete intersection morphism of schemes. By Topologies on Spaces, Lemma 73.9.4 we have that $\{ Y_ i \times _ Y V \to V\}$ is an fpqc covering of schemes. By the case of schemes (More on Morphisms, Lemma 37.62.19) it suffices to prove the base change

$U \times _ Y Y_ i = U \times _ V (V \times _ Y Y_ i) \longrightarrow V$

of $U \to V$ by $V \times _ Y Y_ i \to V$ is a local complete intersection morphism. We can write this as the composition

$U \times _ Y Y_ i \longrightarrow (V \times _ Y X) \times _ Y Y_ i = V \times _ Y X_ i \longrightarrow V \times _ Y Y_ i$

The first arrow is an étale morphism of schemes (as a base change of $U \to V \times _ Y X$) and the second arrow is a local complete intersection morphism of schemes as a flat base change of $f_ i$. The result follows as being a local complete intersection morphism is syntomic local on the source and since étale morphisms are syntomic (More on Morphisms, Lemma 37.62.20 and Morphisms, Lemma 29.36.10). $\square$

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