Lemma 47.22.1. Let A \to B be a local homomorphism of Noetherian local rings. Let \omega _ A^\bullet be a normalized dualizing complex. If A \to B is flat and \mathfrak m_ A B = \mathfrak m_ B, then \omega _ A^\bullet \otimes _ A B is a normalized dualizing complex for B.
Proof. It is clear that \omega _ A^\bullet \otimes _ A B is in D^ b_{\textit{Coh}}(B). Let \kappa _ A and \kappa _ B be the residue fields of A and B. By More on Algebra, Lemma 15.99.2 we see that
R\mathop{\mathrm{Hom}}\nolimits _ B(\kappa _ B, \omega _ A^\bullet \otimes _ A B) = R\mathop{\mathrm{Hom}}\nolimits _ A(\kappa _ A, \omega _ A^\bullet ) \otimes _ A B = \kappa _ A[0] \otimes _ A B = \kappa _ B[0]
Thus \omega _ A^\bullet \otimes _ A B has finite injective dimension by More on Algebra, Lemma 15.69.7. Finally, we can use the same arguments to see that
R\mathop{\mathrm{Hom}}\nolimits _ B(\omega _ A^\bullet \otimes _ A B, \omega _ A^\bullet \otimes _ A B) = R\mathop{\mathrm{Hom}}\nolimits _ A(\omega _ A^\bullet , \omega _ A^\bullet ) \otimes _ A B = A \otimes _ A B = B
as desired. \square
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