Lemma 86.2.4. Let S be a scheme. Let X be a locally Noetherian algebraic space over S. Let K be a dualizing complex on X. Then K is an object of D_{\textit{Coh}}(\mathcal{O}_ X) and D = R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(-, K) induces an anti-equivalence
D : D_{\textit{Coh}}(\mathcal{O}_ X) \longrightarrow D_{\textit{Coh}}(\mathcal{O}_ X)
which comes equipped with a canonical isomorphism \text{id} \to D \circ D. If X is quasi-compact, then D exchanges D^+_{\textit{Coh}}(\mathcal{O}_ X) and D^-_{\textit{Coh}}(\mathcal{O}_ X) and induces an equivalence D^ b_{\textit{Coh}}(\mathcal{O}_ X) \to D^ b_{\textit{Coh}}(\mathcal{O}_ X).
Proof.
Let U \to X be an étale morphism with U affine. Say U = \mathop{\mathrm{Spec}}(A) and let \omega _ A^\bullet be a dualizing complex for A corresponding to K|_ U as in Lemma 86.2.1 and Duality for Schemes, Lemma 48.2.1. By Lemma 86.2.3 the diagram
\xymatrix{ D_{\textit{Coh}}(A) \ar[r] \ar[d]_{R\mathop{\mathrm{Hom}}\nolimits _ A(-, \omega _ A^\bullet )} & D_{\textit{Coh}}(\mathcal{O}_{\acute{e}tale}) \ar[d]^{R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_{\acute{e}tale}}(-, K|_ U)} \\ D_{\textit{Coh}}(A) \ar[r] & D(\mathcal{O}_{\acute{e}tale}) }
commutes where \mathcal{O}_{\acute{e}tale} is the structure sheaf of the small étale site of U. Since formation of R\mathop{\mathcal{H}\! \mathit{om}}\nolimits commutes with restriction, we conclude that D sends D_{\textit{Coh}}(\mathcal{O}_ X) into D_{\textit{Coh}}(\mathcal{O}_ X). Moreover, the canonical map
L \longrightarrow R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(L, K), K)
(Cohomology on Sites, Lemma 21.35.5) is an isomorphism for all L in D_{\textit{Coh}}(\mathcal{O}_ X) because this is true over all U as above by Dualizing Complexes, Lemma 47.15.3. The statement on boundedness properties of the functor D in the quasi-compact case also follows from the corresponding statements of Dualizing Complexes, Lemma 47.15.3.
\square
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