Lemma 48.17.10. In Situation 48.16.1 let $f : X \to Y$ be a morphism of $\textit{FTS}_ S$. If $f$ is flat, then $f^!\mathcal{O}_ Y$ is a $Y$-perfect object of $D(\mathcal{O}_ X)$ and $\mathcal{O}_ X \to R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(f^!\mathcal{O}_ Y, f^!\mathcal{O}_ Y)$ is an isomorphism.

**Proof.**
Both assertions are local on $X$. Thus we may assume $X$ and $Y$ are affine. Then Remark 48.17.5 turns the lemma into an algebra lemma, namely Dualizing Complexes, Lemma 47.25.2. (Use Derived Categories of Schemes, Lemma 36.35.3 to match the languages.)
$\square$

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