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 anti-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 20.42.9 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.

Comments (2)

Comment #8358 by Haohao Liu on

A small remark, but the last "equivalence" in the statement should be "anti-equivalence".

There are also:

  • 2 comment(s) on Section 48.2: Dualizing complexes on schemes

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0A89. Beware of the difference between the letter 'O' and the digit '0'.



View Lemma 48.2.5 as pdf