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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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