Lemma 48.2.1. Let $X$ be a locally Noetherian scheme. Let $K$ be an object of $D(\mathcal{O}_ X)$. The following are equivalent

For every affine open $U = \mathop{\mathrm{Spec}}(A) \subset X$ there exists a dualizing complex $\omega _ A^\bullet $ for $A$ such that $K|_ U$ is isomorphic to the image of $\omega _ A^\bullet $ by the functor $\widetilde{} : D(A) \to D(\mathcal{O}_ U)$.

There is an affine open covering $X = \bigcup U_ i$, $U_ i = \mathop{\mathrm{Spec}}(A_ i)$ such that for each $i$ there exists a dualizing complex $\omega _ i^\bullet $ for $A_ i$ such that $K|_{U_ i}$ is isomorphic to the image of $\omega _ i^\bullet $ by the functor $\widetilde{} : D(A_ i) \to D(\mathcal{O}_{U_ i})$.

## Comments (2)

Comment #5530 by Dion Leijnse on

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