The Stacks project

Lemma 37.62.13. Let

\[ \xymatrix{ X \ar[rr]_ f \ar[rd] & & Y \ar[ld] \\ & S } \]

be a commutative diagram of morphisms of schemes. Assume $S$ is locally Noetherian, $Y \to S$ is locally of finite type, $Y$ is regular, and $X \to S$ is a local complete intersection morphism. Then $f : X \to Y$ is a local complete intersection morphism and $Y \to S$ is Koszul at $f(x)$ for all $x \in X$.

Proof. This is a special case of Lemma 37.62.12 in view of Lemma 37.61.6 (and Morphisms, Lemma 29.15.8). $\square$

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