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 15.92.2 we see that

Thus $\omega _ A^\bullet \otimes _ A B$ has finite injective dimension by More on Algebra, Lemma 15.66.7. Finally, we can use the same arguments to see that

as desired. $\square$

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