Lemma 86.9.2. Let $S$ be a scheme. Let $X \to Y$ be a proper, flat morphism of algebraic spaces which is of finite presentation. If $(\omega _{X/Y}^\bullet , \tau )$ is a relative dualizing complex, then $\mathcal{O}_ X \to R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\omega _{X/Y}^\bullet , \omega _{X/Y}^\bullet )$ is an isomorphism and $Rf_*\omega _{X/Y}^\bullet $ has vanishing cohomology sheaves in positive degrees.
Proof. It suffices to prove this after base change to an affine scheme étale over $Y$ in which case it follows from Lemma 86.8.3. $\square$
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