The Stacks project

Lemma 13.25.1. Let $\mathcal{A}$ be an abelian category with enough injectives Let $F : \mathcal{A} \to \mathcal{B}$ be an additive functor into an abelian category. Let $(i, j)$ be a resolution functor, see Definition 13.23.2. The right derived functor $RF$ of $F$ fits into the following $2$-commutative diagram

\[ \xymatrix{ D^{+}(\mathcal{A}) \ar[rd]_{RF} \ar[rr]^{j'} & & K^{+}(\mathcal{I}) \ar[ld]^ F \\ & D^{+}(\mathcal{B}) } \]

where $j'$ is the functor from Lemma 13.23.6.

Proof. By Lemma 13.20.1 we have $RF(K^\bullet ) = F(j(K^\bullet ))$. $\square$

Comments (1)

Comment #723 by Kestutis Cesnavicius on

Right before this lemma a closing parenthesis is missing.

There are also:

  • 1 comment(s) on Section 13.25: Right derived functors via resolution functors

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Your email address will not be published.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



View Lemma 13.25.1 as pdf