The Stacks project

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. $\square$