Lemma 49.10.4. Let $f : Y \to X$ be a morphism of locally Noetherian schemes. If $f$ satisfies the equivalent conditions of Lemma 49.10.1 then $\omega _{Y/X}$ is an invertible $\mathcal{O}_ Y$-module.

**Proof.**
We may assume $A \to B$ is a relative global complete intersection of the form $B = A[x_1, \ldots , x_ n]/(f_1, \ldots , f_ n)$ and we have to show $\omega _{B/A}$ is invertible. This follows in combining Lemmas 49.10.2 and 49.10.3.
$\square$

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