The Stacks project

Lemma 48.18.4. In Situation 48.16.1 let $f : X \to Y$ be a morphism of $\textit{FTS}_ S$. Assume $f$ is flat. Set $\omega _{X/Y}^\bullet = f^!\mathcal{O}_ Y$ in $D^ b_{\textit{Coh}}(X)$. Let $y \in Y$ and $h : X_ y \to X$ the projection. Then $Lh^*\omega _{X/Y}^\bullet $ is a dualizing complex on $X_ y$.

Proof. The complex $\omega _{X/Y}^\bullet $ is in $D^ b_{\textit{Coh}}$ by Lemma 48.17.9. Being a dualizing complex is a local property. Hence by Lemma 48.18.3 it suffices to show that $(X_ y \to y)^!\mathcal{O}_ y$ is a dualizing complex on $X_ y$. This follows from Lemma 48.17.7. $\square$

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