Cycle module
= Cycle module
{title2=$Z^q=\ker d^q$}
The submodule of cocycles in degree $q$. A projective <boundary module> $B^{q+1}$ makes $0\to Z^q\to P^q\to B^{q+1}\to0$ split when $P^q$ is projective, so $Z^q$ is then also projective.
= Cycle module
{title2=$Z^q=\ker d^q$}
The submodule of cocycles in degree $q$. A projective <boundary module> $B^{q+1}$ makes $0\to Z^q\to P^q\to B^{q+1}\to0$ split when $P^q$ is projective, so $Z^q$ is then also projective.