Remark 49.4.7. Let $f : Y \to X$ be a flat locally quasi-finite morphism of locally Noetherian schemes. Let $\omega _{Y/X}$ be as in Remark 49.2.11. It is clear from the uniqueness, existence, and compatibility with localization of trace elements (Lemmas 49.4.2, 49.4.6, and 49.4.4) that there exists a global section
\[ \tau _{Y/X} \in \Gamma (Y, \omega _{Y/X}) \]
such that for every pair of affine opens $\mathop{\mathrm{Spec}}(B) = V \subset Y$, $\mathop{\mathrm{Spec}}(A) = U \subset X$ with $f(V) \subset U$ that element $\tau _{Y/X}$ maps to $\tau _{B/A}$ under the canonical isomorphism $H^0(V, \omega _{Y/X}) = \omega _{B/A}$.
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