The Stacks project

Lemma 13.40.6. Let $\mathcal{D}$ be a triangulated category. Let $\mathcal{B}$ be a full triangulated subcategory of $\mathcal{D}$. For an object $X$ of $\mathcal{D}$ consider the property $P(X)$: there exists a distinguished triangle $A \to X \to B \to A[1]$ in $\mathcal{D}$ with $B$ in $\mathcal{B}$ and $A$ in ${}^\perp \mathcal{B}$.

  1. If $X_1 \to X_2 \to X_3 \to X_1[1]$ is a distinguished triangle and $P$ holds for two out of three, then it holds for the third.

  2. If $P$ holds for $X_1$ and $X_2$, then it holds for $X_1 \oplus X_2$.

Proof. Dual to Lemma 13.40.5. $\square$

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