Example 3

Let $f:M\to N$ be a homomorphism of A-modules and $C$ the complex such that $C_1=M$, $C_0=N$, $C_i=0$ for $i\ne0,1$, $d_1=f$, $d_i=0$ for $i\ne1$. Then $C$ is split if and only if $\operatorname{Ker}f$ is a direct factor in $M$ and $\operatorname{Im}f$ a direct factor in $N$.