The Stacks project

Example 48.20.11. Let $S$ be a Noetherian scheme and let $\omega _ S^\bullet $ be a dualizing complex. Let $f : X \to Y$ be a proper morphism of finite type schemes over $S$. Let $\omega _ X^\bullet $ and $\omega _ Y^\bullet $ be dualizing complexes normalized relative to $\omega _ S^\bullet $. In this situation we have $a(\omega _ Y^\bullet ) = \omega _ X^\bullet $ (Lemma 48.20.8) and hence the trace map (Section 48.7) is a canonical arrow

\[ \text{Tr}_ f : Rf_*\omega _ X^\bullet \longrightarrow \omega _ Y^\bullet \]

which produces the isomorphisms (Lemma 48.20.9)

\[ \mathop{\mathrm{Hom}}\nolimits _ X(L, \omega _ X^\bullet ) = \mathop{\mathrm{Hom}}\nolimits _ Y(Rf_*L, \omega _ Y^\bullet ) \]


\[ Rf_*R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(L, \omega _ X^\bullet ) = R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ Y}(Rf_*L, \omega _ Y^\bullet ) \]

for $L$ in $D_\mathit{QCoh}(\mathcal{O}_ X)$.

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