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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