The Stacks project

Lemma 13.29.6. Let $\mathcal{A}$ be an abelian category. Let $F : K(\mathcal{A}) \to \mathcal{D}'$ be an exact functor of triangulated categories. Then $RF$ is defined at every complex in $K(\mathcal{A})$ which is quasi-isomorphic to a K-injective complex. In fact, every K-injective complex computes $RF$.

Proof. By Lemma 13.15.4 it suffices to show that $RF$ is defined at a K-injective complex, i.e., it suffices to show a K-injective complex $I^\bullet $ computes $RF$. Any quasi-isomorphism $I^\bullet \to N^\bullet $ is a homotopy equivalence as it has an inverse by Lemma 13.29.2. Thus $I^\bullet \to I^\bullet $ is a final object of $I^\bullet /\text{Qis}(\mathcal{A})$ and we win. $\square$


Comments (0)

There are also:

  • 5 comment(s) on Section 13.29: K-injective 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Your email address will not be published.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 070Y. Beware of the difference between the letter 'O' and the digit '0'.