The Stacks project

Proof. This follows from Remark 13.18.5. We also give a direct argument here.

Let $\tilde\alpha : K \to \tilde L^\bullet $, $\pi $, $s$ be as in Lemma 13.9.6. If we can show that $\beta _1 \circ \pi $ is homotopic to $\beta _2 \circ \pi $, then we deduce that $\beta _1 \sim \beta _2$ because $\pi \circ s$ is the identity. Hence we may assume $\alpha ^ n : K^ n \to L^ n$ is the inclusion of a direct summand for all $n$. Thus we get a short exact sequence of complexes

which is termwise split and such that $M^\bullet $ is acyclic. We choose splittings $L^ n = K^ n \oplus M^ n$, so we have $\beta _ i^ n : K^ n \oplus M^ n \to I^ n$ and $\gamma ^ n : K^ n \to I^ n$. In this case the condition on $\beta _ i$ is that there are morphisms $h_ i^ n : K^ n \to I^{n - 1}$ such that

Thus we see that

Consider the map $h^ n : K^ n \oplus M^ n \to I^{n - 1}$ which equals $h_1^ n - h_2^ n$ on the first summand and zero on the second. Then we see that

is a morphism of complexes $L^\bullet \to I^\bullet $ which is identically zero on the subcomplex $K^\bullet $. Hence it factors as $L^\bullet \to M^\bullet \to I^\bullet $. Thus the result of the lemma follows from Lemma 13.18.4. $\square$