Lemma 13.6.9 (05RK)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 13: Derived Categories
Section 13.6: Quotients of triangulated categories
Lemma 13.6.9 (cite)

Lemma 13.6.9. Let $\mathcal{D}$ be a triangulated category. Let $\mathcal{B}$ be a full triangulated subcategory. The kernel of the quotient functor $Q : \mathcal{D} \to \mathcal{D}/\mathcal{B}$ is the strictly full subcategory of $\mathcal{D}$ whose objects are

$\mathop{\mathrm{Ob}}\nolimits (\mathop{\mathrm{Ker}}(Q)) = \left\{ \begin{matrix} Z \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{D}) \text{ such that there exists a }Z' \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{D}) \\ \text{ such that }Z \oplus Z'\text{ is isomorphic to an object of }\mathcal{B} \end{matrix} \right\}$

In other words it is the smallest strictly full saturated triangulated subcategory of $\mathcal{D}$ containing $\mathcal{B}$ .

Proof. First note that the kernel is automatically a strictly full triangulated subcategory containing summands of any of its objects, see Lemma 13.6.2. The description of its objects follows from the definitions and Lemma 13.5.9 part (4). $\square$

The Stacks project

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