The Stacks project

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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