The Stacks project

Lemma 49.2.7. Let $A \to B$ and $A \to C$ be quasi-finite maps of Noetherian rings. Then $\omega _{B \times C/A} = \omega _{B/A} \times \omega _{C/A}$ as modules over $B \times C$.

Proof. Choose factorizations $A \to B' \to B$ and $A \to C' \to C$ such that $A \to B'$ and $A \to C'$ are finite and such that $\mathop{\mathrm{Spec}}(B) \to \mathop{\mathrm{Spec}}(B')$ and $\mathop{\mathrm{Spec}}(C) \to \mathop{\mathrm{Spec}}(C')$ are open immersions. Then $A \to B' \times C' \to B \times C$ is a similar factorization. Using this factorization to compute $\omega _{B \times C/A}$ gives the lemma. $\square$

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