The Stacks project

Remark 49.16.3. Let $S$ be a Noetherian scheme endowed with a dualizing complex $\omega _ S^\bullet $. Let $f : Y \to X$ be a quasi-finite Gorenstein morphism of compactifyable schemes over $S$. Assume moreover $Y$ and $X$ Cohen-Macaulay and $f$ ├ętale at the generic points of $Y$. Then we can combine Duality for Schemes, Remark 48.23.4 and Remark 49.16.2 to see that we have a canonical isomorphism

\[ \omega _ Y = f^*\omega _ X \otimes _{\mathcal{O}_ Y} \omega _{Y/X} = f^*\omega _ X \otimes _{\mathcal{O}_ Y} \mathcal{O}_ Y(R) \]

of $\mathcal{O}_ Y$-modules. If further $f$ is finite, then the isomorphism $\mathcal{O}_ Y(R) = \omega _{Y/X}$ comes from the global section $\tau _{Y/X} \in H^0(Y, \omega _{Y/X})$ which corresponds via duality to the map $\text{Trace}_ f : f_*\mathcal{O}_ Y \to \mathcal{O}_ X$, see Lemma 49.15.2.

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