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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