The Stacks project

Lemma 47.15.3. Let $A$ be a Noetherian ring. If $\omega _ A^\bullet $ is a dualizing complex, then the functor

\[ D : K \longmapsto R\mathop{\mathrm{Hom}}\nolimits _ A(K, \omega _ A^\bullet ) \]

is an anti-equivalence $D_{\textit{Coh}}(A) \to D_{\textit{Coh}}(A)$ which exchanges $D^+_{\textit{Coh}}(A)$ and $D^-_{\textit{Coh}}(A)$ and induces an anti-equivalence $D^ b_{\textit{Coh}}(A) \to D^ b_{\textit{Coh}}(A)$. Moreover $D \circ D$ is isomorphic to the identity functor.

Proof. Let $K$ be an object of $D_{\textit{Coh}}(A)$. From Lemma 47.15.2 we see $R\mathop{\mathrm{Hom}}\nolimits _ A(K, \omega _ A^\bullet )$ is an object of $D_{\textit{Coh}}(A)$. By More on Algebra, Lemma 15.98.2 and the assumptions on the dualizing complex we obtain a canonical isomorphism

\[ K = R\mathop{\mathrm{Hom}}\nolimits _ A(\omega _ A^\bullet , \omega _ A^\bullet ) \otimes _ A^\mathbf {L} K \longrightarrow R\mathop{\mathrm{Hom}}\nolimits _ A(R\mathop{\mathrm{Hom}}\nolimits _ A(K, \omega _ A^\bullet ), \omega _ A^\bullet ) \]

Thus our functor has a quasi-inverse and the proof is complete. $\square$


Comments (4)

Comment #3623 by Janos Kollar on

A small remark, but if you say "anti-equivalence" in line 3 then probably should do the same in line 4.

Comment #8374 by Haohao Liu on

I am sorry if this question is naïve. Is it trivial that sends to ? The proof ignores this part.

Comment #8376 by Nicolás on

Maybe this follows from a spectral sequence argument? Something like Here has finite injective dimension, hence the is zero for outside of a fixed interval.

There are also:

  • 2 comment(s) on Section 47.15: Dualizing complexes

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Your email address will not be published.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 0A7C. Beware of the difference between the letter 'O' and the digit '0'.