Lemma 47.15.5. Let $A$ be a Noetherian ring. If $\omega _ A^\bullet $ and $(\omega '_ A)^\bullet $ are dualizing complexes, then $(\omega '_ A)^\bullet $ is quasi-isomorphic to $\omega _ A^\bullet \otimes _ A^\mathbf {L} L$ for some invertible object $L$ of $D(A)$.

**Proof.**
By Lemmas 47.15.2 and 47.15.4 the functor $K \mapsto R\mathop{\mathrm{Hom}}\nolimits _ A(R\mathop{\mathrm{Hom}}\nolimits _ A(K, \omega _ A^\bullet ), (\omega _ A')^\bullet )$ maps $A$ to an invertible object $L$. In other words, there is an isomorphism

Since $L$ has finite tor dimension, this means that we can apply More on Algebra, Lemma 15.91.2 to see that

is an isomorphism for $K$ in $D^ b_{\textit{Coh}}(A)$. In particular, setting $K = \omega _ A^\bullet $ finishes the proof. $\square$

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