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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