The Stacks project

Remark 48.20.13. Let $f : X \to Y$ be a flat proper morphism of finite type schemes over a pair $(S, \omega _ S^\bullet )$ as in Situation 48.20.1. The relative dualizing complex (Remark 48.12.5) is $\omega _{X/Y}^\bullet = a(\mathcal{O}_ Y)$. By Lemma 48.20.8 we have the first canonical isomorphism in

\[ \omega _ X^\bullet = a(\omega _ Y^\bullet ) = Lf^*\omega _ Y^\bullet \otimes _{\mathcal{O}_ X}^\mathbf {L} \omega _{X/Y}^\bullet \]

in $D(\mathcal{O}_ X)$. The second canonical isomorphism follows from the discussion in Remark 48.12.5.

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