Remark 49.2.11. Let $f : Y \to X$ be a locally quasi-finite morphism of locally Noetherian schemes. It is clear from Lemma 49.2.3 that there is a unique coherent $\mathcal{O}_ Y$-module $\omega _{Y/X}$ on $Y$ such that for every pair of affine opens $\mathop{\mathrm{Spec}}(B) = V \subset Y$, $\mathop{\mathrm{Spec}}(A) = U \subset X$ with $f(V) \subset U$ there is a canonical isomorphism

$H^0(V, \omega _{Y/X}) = \omega _{B/A}$

and where these isomorphisms are compatible with restriction maps.

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