The Stacks project

Lemma 48.2.6. Let $X$ be a locally Noetherian scheme. If $K$ and $K'$ are dualizing complexes on $X$, then $K'$ is isomorphic to $K \otimes _{\mathcal{O}_ X}^\mathbf {L} L$ for some invertible object $L$ of $D(\mathcal{O}_ X)$.

Proof. Set

\[ L = R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(K, K') \]

This is an invertible object of $D(\mathcal{O}_ X)$, because affine locally this is true, see Dualizing Complexes, Lemma 47.15.5 and its proof. The evaluation map $L \otimes _{\mathcal{O}_ X}^\mathbf {L} K \to K'$ is an isomorphism for the same reason. $\square$

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