Lemma 48.28.7. Let $S$ be a quasi-compact and quasi-separated scheme. Let $f : X \to S$ be a proper, flat morphism of finite presentation. The relative dualizing complex $\omega _{X/S}^\bullet $ of Remark 48.12.5 together with (48.12.8.1) is a relative dualizing complex in the sense of Definition 48.28.1.
Proof. In Lemma 48.12.7 we proved that $\omega _{X/S}^\bullet $ is $S$-perfect. Let $c$ be the right adjoint of Lemma 48.3.1 for the diagonal $\Delta : X \to X \times _ S X$. Then we can apply $\Delta _*$ to (48.12.8.1) to get an isomorphism
\[ \Delta _*\mathcal{O}_ X \to \Delta _*(c(L\text{pr}_1^*\omega _{X/S}^\bullet )) = R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_{X \times _ S X}}( \Delta _*\mathcal{O}_ X, L\text{pr}_1^*\omega _{X/S}^\bullet ) \]
The equality holds by Lemmas 48.9.7 and 48.9.3. This finishes the proof. $\square$
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