The Stacks project

I think there is a mistake in the proof. Basically what is done in the proof is the following: first find $p$ homotopic to zero such that the left square commutes when $b$ is replaced by $b+p$ as in the first part of lemma 8.4, then find $q$ homotopic to zero such that the right square commutes when $b$ is replaced by $b+q$ as in the second part of lemma 8.4, then let $b' = b + p + q$. But then since the left square commutes with $b+p$ and with $b+p+q$ we must have $q \circ \alpha_1 = 0$ and $0 = c \circ \beta_1 \circ \alpha_1 = \beta_2 \circ (b+q) \circ \alpha_1 = \beta_2 \circ b \circ \alpha_1$. However, as far as I understand, the last equality does not always hold, because we only required that initial squares commute up to homotopy.

I believe the right solution is to first replace $b$ by homotopic $b'$ so that the left square would commute, then the right square would still commute up to another homotopy $g'$ and then replace $b'$ by $b'' = b' + d \circ k + k \circ d$ where $k$ is $s_2 \circ g' \circ s_1 \circ \beta_1$ (we postcompose with $s_1 \circ \beta_1$ so that the addition will not break the commutativity of the left square). In one step, just set $h^n = -f^n \circ \pi_1^n + s_2^{n-1} \circ g^n \circ s_1^n \circ \beta_1^n$.

OK, this is an actual honest mistake. The lemma is just wrong. Damn! I turned it into a remark explaining why it was wrong. Hopefully this is now correct. Thanks very much for your comment.

Yes, you are right. The problem with my solution is that the projection on the $C_1^\bullet$ part ($s_1 \circ \beta_1$) is not a morphism of complexes, so composition with it does not commute with formation of homotopy.