Lemma 13.35.3 (0FX3)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 13: Derived Categories
Section 13.35: Operations on full subcategories
Lemma 13.35.3 (cite)

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Lemma 13.35.3. Let $\mathcal{T}$ be a triangulated category. Given full subcategories $\mathcal{A}$ , $\mathcal{B}$ the full subcategories $add(\mathcal{A}) \star add(\mathcal{B})$ and $smd(add(\mathcal{A}))$ are closed under direct sums.

Proof. Namely, if $A \to X \to B$ and $A' \to X' \to B'$ are distinguished triangles and $A, A' \in add(\mathcal{A})$ and $B, B' \in add(\mathcal{B})$ then $A \oplus A' \to X \oplus X' \to B \oplus B'$ is a distinguished triangle with $A \oplus A' \in add(\mathcal{A})$ and $B \oplus B' \in add(\mathcal{B})$ . The result for $smd(add(\mathcal{A}))$ is trivial. $\square$

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