Lemma 13.6.10. Let \mathcal{D} be a triangulated category. The operations described above have the following properties
S(\mathcal{B}(S)) is the “saturation” of S, i.e., it is the smallest saturated multiplicative system in \mathcal{D} containing S, and
\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
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