Lemma 13.9.15. Let $\mathcal{A}$ be an additive category. Let $A_1^\bullet \to A_2^\bullet \to \ldots \to A_ n^\bullet $ be a sequence of composable morphisms of complexes. There exists a commutative diagram

such that each morphism $B_ i^\bullet \to B_{i + 1}^\bullet $ is a split injection and each $B_ i^\bullet \to A_ i^\bullet $ is a homotopy equivalence. Moreover, if all $A_ i^\bullet $ are in $K^{+}(\mathcal{A})$, $K^{-}(\mathcal{A})$, or $K^ b(\mathcal{A})$, then so are the $B_ i^\bullet $.

## Comments (1)

Comment #297 by arp on