Cycle module ($Z^q=\ker d^q$)

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

An element killed by the next differential of a cochain complex. Its class modulo coboundaries is a cohomology class.