Lemma 37.59.20. The property $\mathcal{P}(f) =$“$f$ is a local complete intersection morphism” is syntomic local on the source.

**Proof.**
We will use the criterion of Descent, Lemma 35.25.4 to prove this. It follows from Lemmas 37.59.8 and 37.59.7 that being a local complete intersection morphism is preserved under precomposing with syntomic morphisms. It is clear from Definition 37.59.2 that being a local complete intersection morphism is Zariski local on the source and target. Hence, according to the aforementioned Descent, Lemma 35.25.4 it suffices to prove the following: Suppose $X' \to X \to Y$ are morphisms of affine schemes with $X' \to X$ syntomic and $X' \to Y$ a local complete intersection morphism. Then $X \to Y$ is a local complete intersection morphism. To see this, note that in any case $X \to Y$ is of finite presentation by Descent, Lemma 35.13.1. Choose a closed immersion $X \to \mathbf{A}^ n_ Y$. By Algebra, Lemma 10.136.18 we can find an affine open covering $X' = \bigcup _{i = 1, \ldots , n} X'_ i$ and syntomic morphisms $W_ i \to \mathbf{A}^ n_ Y$ lifting the morphisms $X'_ i \to X$, i.e., such that there are fibre product diagrams

After replacing $X'$ by $\coprod X'_ i$ and setting $W = \coprod W_ i$ we obtain a fibre product diagram of affine schemes

with $h : W \to \mathbf{A}^ n_ Y$ syntomic and $X' \to Y$ still a local complete intersection morphism. Since $W \to \mathbf{A}^ n_ Y$ is open (see Morphisms, Lemma 29.25.10) and $X' \to X$ is surjective we see that $X$ is contained in the image of $W \to \mathbf{A}^ n_ Y$. Choose a closed immersion $W \to \mathbf{A}^{n + m}_ Y$ over $\mathbf{A}^ n_ Y$. Now the diagram looks like

Because $h$ is syntomic and hence a local complete intersection morphism (see above) the morphism $W \to \mathbf{A}^{n + m}_ Y$ is a Koszul-regular immersion. Because $X' \to Y$ is a local complete intersection morphism the morphism $X' \to \mathbf{A}^{n + m}_ Y$ is a Koszul-regular immersion. We conclude from Divisors, Lemma 31.21.8 that $X' \to W$ is a Koszul-regular immersion. Hence, since being a Koszul-regular immersion is fpqc local on the target (see Descent, Lemma 35.22.32) we conclude that $X \to \mathbf{A}^ n_ Y$ is a Koszul-regular immersion which is what we had to show. $\square$

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