Remark 48.12.5. Let $Y$ be a quasi-compact and quasi-separated scheme. Let $f : X \to Y$ be a proper, flat morphism of finite presentation. Let $a$ be the adjoint of Lemma 48.3.1 for $f$. In this situation, $\omega _{X/Y}^\bullet = a(\mathcal{O}_ Y)$ is sometimes called the *relative dualizing complex*. By Lemma 48.12.3 there is a functorial isomorphism $a(K) = Lf^*K \otimes _{\mathcal{O}_ X}^\mathbf {L} \omega _{X/Y}^\bullet $ for $K \in D_\mathit{QCoh}(\mathcal{O}_ Y)$. Moreover, the trace map

of Section 48.7 induces the trace map for all $K$ in $D_\mathit{QCoh}(\mathcal{O}_ Y)$. More precisely the diagram

where the equality on the lower right is Derived Categories of Schemes, Lemma 36.21.1. If $g : Y' \to Y$ is a morphism of quasi-compact and quasi-separated schemes and $X' = Y' \times _ Y X$, then by Lemma 48.12.4 we have $\omega _{X'/Y'}^\bullet = L(g')^*\omega _{X/Y}^\bullet $ where $g' : X' \to X$ is the projection and by Lemma 48.7.1 the trace map

for $f' : X' \to Y'$ is the base change of $\text{Tr}_{f, \mathcal{O}_ Y}$ via the base change isomorphism.

## Comments (0)