The Stacks project

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

\[ \xymatrix{ X'_ i \ar[d] \ar[r] & W_ i \ar[d] \\ X \ar[r] & \mathbf{A}^ n_ Y } \]

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

\[ \xymatrix{ X' \ar[d] \ar[r] & W \ar[d]^ h \\ X \ar[r] & \mathbf{A}^ n_ Y } \]

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

\[ \xymatrix{ X' \ar[d] \ar[r] & W \ar[d]^ h \ar[r] & \mathbf{A}^{n + m}_ Y \ar[ld] \\ X \ar[r] & \mathbf{A}^ n_ Y } \]

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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