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The Stacks project

Lemma 13.12.3. Let \mathcal{A} be an abelian category. Let

\xymatrix{ 0 \ar[r] & A^\bullet \ar[r] & B^\bullet \ar[r] & C^\bullet \ar[r] & 0 }

be a short exact sequences of complexes. Assume this short exact sequence is termwise split. Let (A^\bullet , B^\bullet , C^\bullet , \alpha , \beta , \delta ) be the distinguished triangle of K(\mathcal{A}) associated to the sequence. The \delta -functor of Lemma 13.12.1 above maps the short exact sequences 0 \to A^\bullet \to B^\bullet \to C^\bullet \to 0 to a triangle isomorphic to the distinguished triangle

(A^\bullet , B^\bullet , C^\bullet , \alpha , \beta , \delta ).

Proof. Follows from Lemma 13.9.14. \square


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