Lemma 49.10.2. Invertibility of the relative dualizing module.

If $A \to B$ is a quasi-finite flat homomorphism of Noetherian rings, then $\omega _{B/A}$ is an invertible $B$-module if and only if $\omega _{B \otimes _ A \kappa (\mathfrak p)/\kappa (\mathfrak p)}$ is an invertible $B \otimes _ A \kappa (\mathfrak p)$-module for all primes $\mathfrak p \subset A$.

If $Y \to X$ is a quasi-finite flat morphism of Noetherian schemes, then $\omega _{Y/X}$ is invertible if and only if $\omega _{Y_ x/x}$ is invertible for all $x \in X$.

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