Remark 48.20.12. Let $S$ be a Noetherian scheme and let $\omega _ S^\bullet$ be a dualizing complex. Let $f : X \to Y$ be a finite morphism between schemes of finite type over $S$. Let $\omega _ X^\bullet$ and $\omega _ Y^\bullet$ be dualizing complexes normalized relative to $\omega _ S^\bullet$. Then we have

$f_*\omega _ X^\bullet = R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (f_*\mathcal{O}_ X, \omega _ Y^\bullet )$

in $D_\mathit{QCoh}^+(f_*\mathcal{O}_ X)$ by Lemmas 48.11.4 and 48.20.8 and the trace map of Example 48.20.11 is the map

$\text{Tr}_ f : Rf_*\omega _ X^\bullet = f_*\omega _ X^\bullet = R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (f_*\mathcal{O}_ X, \omega _ Y^\bullet ) \longrightarrow \omega _ Y^\bullet$

which often goes under the name “evaluation at $1$”.

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