The Stacks project

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

  1. 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 $\widetilde{} : D(A) \to D(\mathcal{O}_ U)$.

  2. 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 $\widetilde{} : D(A_ i) \to D(\mathcal{O}_{U_ i})$.

Proof. Assume (2) and let $U = \mathop{\mathrm{Spec}}(A)$ be an affine open of $X$. Since condition (2) implies that $K$ is in $D_\mathit{QCoh}(\mathcal{O}_ X)$ we find an object $\omega _ A^\bullet $ in $D(A)$ whose associated complex of quasi-coherent sheaves is isomorphic to $K|_ U$, see Derived Categories of Schemes, Lemma 36.3.5. We will show that $\omega _ A^\bullet $ is a dualizing complex for $A$ which will finish the proof.

Since $X = \bigcup U_ i$ is an open covering, we can find a standard open covering $U = D(f_1) \cup \ldots \cup D(f_ m)$ such that each $D(f_ j)$ is a standard open in one of the affine opens $U_ i$, see Schemes, Lemma 26.11.5. Say $D(f_ j) = D(g_ j)$ for $g_ j \in A_{i_ j}$. Then $A_{f_ j} \cong (A_{i_ j})_{g_ j}$ and we have

\[ (\omega _ A^\bullet )_{f_ j} \cong (\omega _ i^\bullet )_{g_ j} \]

in the derived category by Derived Categories of Schemes, Lemma 36.3.5. By Dualizing Complexes, Lemma 47.15.6 we find that the complex $(\omega _ A^\bullet )_{f_ j}$ is a dualizing complex over $A_{f_ j}$ for $j = 1, \ldots , m$. This implies that $\omega _ A^\bullet $ is dualizing by Dualizing Complexes, Lemma 47.15.7. $\square$


Comments (2)

Comment #5530 by Dion Leijnse on

In part (2) of the statement of the Lemma, I think the $K|_U$ should be $K|_{U_i}$.

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