The Stacks project

Lemma 49.10.2. Invertibility of the relative dualizing module.

  1. 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 $\mathfrak p \subset A$.

  2. 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 $x \in X$.

Proof. Proof of (1). As $A \to B$ is flat, the module $\omega _{B/A}$ is $A$-flat, see Lemma 49.2.9. Thus $\omega _{B/A}$ is an invertible $B$-module if and only if $\omega _{B/A} \otimes _ A \kappa (\mathfrak p)$ is an invertible $B \otimes _ A \kappa (\mathfrak p)$-module for every prime $\mathfrak p \subset A$, see More on Morphisms, Lemma 37.16.7. Still using that $A \to B$ is flat, we have that formation of $\omega _{B/A}$ commutes with base change, see Lemma 49.2.10. Thus we see that invertibility of the relative dualizing module, in the presence of flatness, is equivalent to invertibility of the relative dualizing module for the maps $\kappa (\mathfrak p) \to B \otimes _ A \kappa (\mathfrak p)$.

Part (2) follows from (1) and the fact that affine locally the dualizing modules are given by their algebraic counterparts, see Remark 49.2.11. $\square$

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