The Stacks project

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

\[ \text{Tr}_{f, \mathcal{O}_ Y} : Rf_*\omega _{X/Y}^\bullet \to \mathcal{O}_ Y \]

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

\[ \xymatrix{ Rf_*a(K) \ar[rrr]_{\text{Tr}_{f, K}} \ar@{=}[d] & & & K \ar@{=}[d] \\ Rf_*(Lf^*K \otimes _{\mathcal{O}_ X}^\mathbf {L} \omega _{X/Y}^\bullet ) \ar@{=}[r] & K \otimes _{\mathcal{O}_ Y}^\mathbf {L} Rf_*\omega _{X/Y}^\bullet \ar[rr]^-{\text{id}_ K \otimes \text{Tr}_{f, \mathcal{O}_ Y}} & & K } \]

where the equality on the lower right is Derived Categories of Schemes, Lemma 36.22.1. 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

\[ \text{Tr}_{f', \mathcal{O}_{Y'}} : Rf'_*\omega _{X'/Y'}^\bullet \to \mathcal{O}_{Y'} \]

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

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