The Stacks project

Lemma 13.6.10. Let $\mathcal{D}$ be a triangulated category. The operations described above have the following properties

  1. $S(\mathcal{B}(S))$ is the “saturation” of $S$, i.e., it is the smallest saturated multiplicative system in $\mathcal{D}$ containing $S$, and

  2. $\mathcal{B}(S(\mathcal{B}))$ is the “saturation” of $\mathcal{B}$, i.e., it is the smallest strictly full saturated triangulated subcategory of $\mathcal{D}$ containing $\mathcal{B}$.

In particular, the constructions define mutually inverse maps between the (partially ordered) set of saturated multiplicative systems in $\mathcal{D}$ compatible with the triangulated structure on $\mathcal{D}$ and the (partially ordered) set of strictly full saturated triangulated subcategories of $\mathcal{D}$.

Proof. First, let's start with a full triangulated subcategory $\mathcal{B}$. Then $\mathcal{B}(S(\mathcal{B})) = \mathop{\mathrm{Ker}}(Q : \mathcal{D} \to \mathcal{D}/\mathcal{B})$ and hence (2) is the content of Lemma 13.6.9.

Next, suppose that $S$ is multiplicative system in $\mathcal{D}$ compatible with the triangulation on $\mathcal{D}$. Then $\mathcal{B}(S) = \mathop{\mathrm{Ker}}(Q : \mathcal{D} \to S^{-1}\mathcal{D})$. Hence (using Lemma 13.4.9 in the localized category)

\begin{align*} S(\mathcal{B}(S)) & = \left\{ \begin{matrix} f \in \text{Arrows}(\mathcal{D}) \text{ such that there exists a distinguished} \\ \text{triangle }(X, Y, Z, f, g, h) \text{ of }\mathcal{D}\text{ with }Q(Z) = 0 \end{matrix} \right\} \\ & = \{ f \in \text{Arrows}(\mathcal{D}) \mid Q(f)\text{ is an isomorphism}\} \\ & = \hat S = S' \end{align*}

in the notation of Categories, Lemma 4.27.21. The final statement of that lemma finishes the proof. $\square$

Comments (0)

There are also:

  • 2 comment(s) on Section 13.6: Quotients of triangulated 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Your email address will not be published.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 05RL. Beware of the difference between the letter 'O' and the digit '0'.