The Stacks project

Lemma 13.9.10. Let $\mathcal{A}$ be an additive category. Let $0 \to A^\bullet \to B^\bullet \to C^\bullet \to 0$ be termwise split exact sequences as in Definition 13.9.9. Let $(\pi ')^ n$, $(s')^ n$ be a second collection of splittings. Denote $\delta ' : C^\bullet \longrightarrow A^\bullet [1]$ the morphism associated to this second set of splittings. Then

\[ (1, 1, 1) : (A^\bullet , B^\bullet , C^\bullet , \alpha , \beta , \delta ) \longrightarrow (A^\bullet , B^\bullet , C^\bullet , \alpha , \beta , \delta ') \]

is an isomorphism of triangles in $K(\mathcal{A})$.

Proof. The statement simply means that $\delta $ and $\delta '$ are homotopic maps of complexes. This is Homology, Lemma 12.14.12. $\square$

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