The Stacks project

Trivial duality for systems of perfect objects.

Lemma 15.69.15. Let $A$ be a ring. Let $(K_ n)_{n \in \mathbf{N}}$ be a system of perfect objects of $D(A)$. Let $K = \text{hocolim} K_ n$ be the derived colimit (Derived Categories, Definition 13.31.1). Then for any object $E$ of $D(A)$ we have

\[ R\mathop{\mathrm{Hom}}\nolimits _ A(K, E) = R\mathop{\mathrm{lim}}\nolimits E \otimes ^\mathbf {L}_ A K_ n^\vee \]

where $(K_ n^\vee )$ is the inverse system of dual perfect complexes.

Proof. By Lemma 15.69.14 we have $R\mathop{\mathrm{lim}}\nolimits E \otimes ^\mathbf {L}_ A K_ n^\vee = R\mathop{\mathrm{lim}}\nolimits R\mathop{\mathrm{Hom}}\nolimits _ A(K_ n, E)$ which fits into the distinguished triangle

\[ R\mathop{\mathrm{lim}}\nolimits R\mathop{\mathrm{Hom}}\nolimits _ A(K_ n, E) \to \prod R\mathop{\mathrm{Hom}}\nolimits _ A(K_ n, E) \to \prod R\mathop{\mathrm{Hom}}\nolimits _ A(K_ n, E) \]

Because $K$ similarly fits into the distinguished triangle $\bigoplus K_ n \to \bigoplus K_ n \to K$ it suffices to show that $\prod R\mathop{\mathrm{Hom}}\nolimits _ A(K_ n, E) = R\mathop{\mathrm{Hom}}\nolimits _ A(\bigoplus K_ n, E)$. This is a formal consequence of (15.68.0.1) and the fact that derived tensor product commutes with direct sums. $\square$


Comments (0)

There are also:

  • 2 comment(s) on Section 15.69: Perfect 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Your email address will not be published.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 0BKB. Beware of the difference between the letter 'O' and the digit '0'.