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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. $\square$

Comments (3)

Comment #11032 by on

The subcategory is always closed under finite direct sums for any subcategory of .

Comment #11034 by on

Sorry for the too many comments (#11032 and #11033 may be deleted). Just to spell it out, the lemma can be generalized to: and are closed under finite direct sums if are.

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