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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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