Exercise 7, § 2

Let $C$ and $C’$ be two complexes, $s$ (resp. $s’$) a splitting of $C$ (resp. $C’$), $u : C' \to C$ a morphism of complexes. One defines an A-endomorphism $\sigma$ of Con (u), graded of degree – 1, by setting

Now prove that for $\sigma$ to be a splitting of Con (u), it is necessary and sufficient that one has $H(u) = 0$.