The Stacks project

Lemma 13.17.1. Let $\mathcal{A}$ be an abelian category. Let $\mathcal{B} \subset \mathcal{A}$ be a weak Serre subcategory. The category $D_\mathcal {B}(\mathcal{A})$ is a strictly full saturated triangulated subcategory of $D(\mathcal{A})$. Similarly for the bounded versions.

Proof. It is clear that $D_\mathcal {B}(\mathcal{A})$ is an additive subcategory preserved under the translation functors. If $X \oplus Y$ is in $D_\mathcal {B}(\mathcal{A})$, then both $H^ n(X)$ and $H^ n(Y)$ are kernels of maps between maps of objects of $\mathcal{B}$ as $H^ n(X \oplus Y) = H^ n(X) \oplus H^ n(Y)$. Hence both $X$ and $Y$ are in $D_\mathcal {B}(\mathcal{A})$. By Lemma 13.4.16 it therefore suffices to show that given a distinguished triangle $(X, Y, Z, f, g, h)$ such that $X$ and $Y$ are in $D_\mathcal {B}(\mathcal{A})$ then $Z$ is an object of $D_\mathcal {B}(\mathcal{A})$. The long exact cohomology sequence ( and the definition of a weak Serre subcategory (see Homology, Definition 12.10.1) show that $H^ n(Z)$ is an object of $\mathcal{B}$ for all $n$. Thus $Z$ is an object of $D_\mathcal {B}(\mathcal{A})$. $\square$

Comments (2)

Comment #930 by correction_bot on

Typos: In the second sentence of the proof, the occurrence of should be replaced with ; also, there's an english problem in the second half of the sentence.

There are also:

  • 9 comment(s) on Section 13.17: Triangulated subcategories of the derived category

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Your email address will not be published.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 06UQ. Beware of the difference between the letter 'O' and the digit '0'.