Lemma 47.16.1. Let $(A, \mathfrak m, \kappa ) \to (B, \mathfrak m', \kappa ')$ be a finite local map of Noetherian local rings. Let $\omega _ A^\bullet$ be a normalized dualizing complex. Then $\omega _ B^\bullet = R\mathop{\mathrm{Hom}}\nolimits (B, \omega _ A^\bullet )$ is a normalized dualizing complex for $B$.

Proof. By Lemma 47.15.8 the complex $\omega _ B^\bullet$ is dualizing for $B$. We have

$R\mathop{\mathrm{Hom}}\nolimits _ B(\kappa ', \omega _ B^\bullet ) = R\mathop{\mathrm{Hom}}\nolimits _ B(\kappa ', R\mathop{\mathrm{Hom}}\nolimits (B, \omega _ A^\bullet )) = R\mathop{\mathrm{Hom}}\nolimits _ A(\kappa ', \omega _ A^\bullet )$

by Lemma 47.13.1. Since $\kappa '$ is isomorphic to a finite direct sum of copies of $\kappa$ as an $A$-module and since $\omega _ A^\bullet$ is normalized, we see that this complex only has cohomology placed in degree $0$. Thus $\omega _ B^\bullet$ is a normalized dualizing complex as well. $\square$

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