The Stacks project

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

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. $\square$