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The Stacks project

Lemma 37.62.7. A composition of local complete intersection morphisms is a local complete intersection morphism.

Proof. Let g : Y \to S and f : X \to Y be local complete intersection morphisms. Let x \in X and set y = f(x). Choose an open neighbourhood V \subset Y of y and a factorization g|_ V = \pi \circ i for some Koszul-regular immersion i : V \to P and smooth morphism \pi : P \to S. Next choose an open neighbourhood U of x \in X and a factorization f|_ U = \pi ' \circ i' for some Koszul-regular immersion i' : U \to P' and smooth morphism \pi ' : P' \to Y. In fact, we may assume that P' = \mathbf{A}^ n_ V, see discussion preceding and following Definition 37.62.2. Picture:

\xymatrix{ X \ar[d] & U \ar[l] \ar[r]_-{i'} & P' = \mathbf{A}^ n_ V \ar[d] \\ Y \ar[d] & & V \ar[ll] \ar[r]_ i & P \ar[d] \\ S & & & S \ar[lll] }

Set P'' = \mathbf{A}^ n_ P. Then U \to P' \to P'' is a Koszul-regular immersion as a composition of Koszul-regular immersions, namely i' and the flat base change of i via P'' \to P, see Divisors, Lemma 31.21.3 and Divisors, Lemma 31.21.7. Also P'' \to P \to S is smooth as a composition of smooth morphisms, see Morphisms, Lemma 29.34.4. Hence we conclude that X \to S is Koszul at x as desired. \square


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