The Stacks project

Lemma 48.2.5. Let $K$ be a dualizing complex on a locally Noetherian scheme $X$. 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 $U \subset X$ be an affine open. Say $U = \mathop{\mathrm{Spec}}(A)$ and let $\omega _ A^\bullet $ be a dualizing complex for $A$ corresponding to $K|_ U$ as in Lemma 48.2.1. By Lemma 48.2.3 the diagram

\[ \xymatrix{ D_{\textit{Coh}}(A) \ar[r] \ar[d]_{R\mathop{\mathrm{Hom}}\nolimits _ A(-, \omega _ A^\bullet )} & D_{\textit{Coh}}(\mathcal{O}_ U) \ar[d]^{R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(-, K|_ U)} \\ D_{\textit{Coh}}(A) \ar[r] & D(\mathcal{O}_ U) } \]

commutes. We conclude that $D$ sends $D_{\textit{Coh}}(\mathcal{O}_ X)$ into $D_{\textit{Coh}}(\mathcal{O}_ X)$. Moreover, the canonical map

\[ L \longrightarrow R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(K, K) \otimes _{\mathcal{O}_ X}^\mathbf {L} L \longrightarrow R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(L, K), K) \]

(using Cohomology, Lemma 20.39.9 for the second arrow) is an isomorphism for all $L$ because this is true on affines by Dualizing Complexes, Lemma 47.15.31 and we have already seen on affines that we recover what happens in algebra. 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. $\square$

[1] An alternative is to first show that $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(K, K) = \mathcal{O}_ X$ by working affine locally and then use Lemma 48.2.4 part (2) to see the map is an isomorphism.

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