The Stacks project

Definition 13.15.3. In Situation 13.15.1.

  1. The right derived functors of $F$ are the partial functors $RF$ associated to cases (1) and (2) of Situation 13.15.1.

  2. The left derived functors of $F$ are the partial functors $LF$ associated to cases (3) and (4) of Situation 13.15.1.

  3. An object $A$ of $\mathcal{A}$ is said to be right acyclic for $F$, or acyclic for $RF$ if $A[0]$ computes $RF$.

  4. An object $A$ of $\mathcal{A}$ is said to be left acyclic for $F$, or acyclic for $LF$ if $A[0]$ computes $LF$.

Comments (3)

Comment #2066 by Hu Fei on

In tag 0157, the last word it may be .

Comment #9856 by on

I want to remark that this definition of “right acyclic for ” might not be equivalent to Lipman's definition of “right--acyclic” [L, 2.2.5 and second paragraph of Sect. 2.7]. Whereas Lipman's definition always implies Definition 13.15.3.3, the converse might not be always true; although the converse holds in with the extra assumption that “has enough right acyclics for (in the sense of Definition 13.15.3.3).” All of this is explained in [GH, Remark 1.17].

[GH] —, A Gospel Harmony of Derived Functors https://eliasguisado.wordpress.com/work/

[L] J. Lipman. “Notes on Derived Functors and Grothendieck Duality”. In: Foundations of Grothendieck Duality for Diagrams of Schemes. Lecture Notes in Mathematics. Springer-Verlag, 2009

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Your email address will not be published.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 0157. Beware of the difference between the letter 'O' and the digit '0'.



View Definition 13.15.3 as pdf