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

which produces the isomorphisms (Lemma 48.20.9)

and

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

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