The Stacks project

Remark 48.28.8. Let $X \to S$ be a morphism of schemes which is flat, proper, and of finite presentation. By Lemma 48.28.5 there exists a relative dualizing complex $(\omega _{X/S}^\bullet , \xi )$ in the sense of Definition 48.28.1. Consider any morphism $g : S' \to S$ where $S'$ is quasi-compact and quasi-separated (for example an affine open of $S$). By Lemma 48.28.6 we see that $(L(g')^*\omega _{X/S}^\bullet , L(g')^*\xi )$ is a relative dualizing complex for the base change $f' : X' \to S'$ in the sense of Definition 48.28.1. Let $\omega _{X'/S'}^\bullet $ be the relative dualizing complex for $X' \to S'$ in the sense of Remark 48.12.5. Combining Lemmas 48.28.7 and 48.28.4 we see that there is a unique isomorphism

\[ \omega _{X'/S'}^\bullet \longrightarrow L(g')^*\omega _{X/S}^\bullet \]

compatible with (48.12.8.1) and $L(g')^*\xi $. These isomorphisms are compatible with morphisms between quasi-compact and quasi-separated schemes over $S$ and the base change isomorphisms of Lemma 48.12.4 (if we ever need this compatibility we will carefully state and prove it here).

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