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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