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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