The Stacks project

Lemma 13.15.7. In Situation 13.15.1. Let $\mathcal{P} \subset \mathop{\mathrm{Ob}}\nolimits (\mathcal{A})$ be a subset with the following properties:

  1. every object of $\mathcal{A}$ is a quotient of an element of $\mathcal{P}$,

  2. for any short exact sequence $0 \to P \to Q \to R \to 0$ of $\mathcal{A}$ with $Q, R \in \mathcal{P}$, then $P \in \mathcal{P}$, and $0 \to F(P) \to F(Q) \to F(R) \to 0$ is exact.

Then every object of $\mathcal{P}$ is acyclic for $LF$.

Proof. Dual to the proof of Lemma 13.15.6. $\square$

Comments (0)

There are also:

  • 7 comment(s) on Section 13.15: Derived functors on derived categories

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Your email address will not be published.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 05T9. Beware of the difference between the letter 'O' and the digit '0'.