Projection to cohomology

Question

I have a differential graded algebra $(A,d)$ and $H^*$ be a graded ring.

Let $Z^*=Ker(d)$ and $\alpha:H^*(A)->Z^*$ be a right inverse for the projection to cohomology.

I have the following questions:

1.What is $Z^*$?

(usually with * in the paper the author denotes complexes, like $H^*$, but here...).

2.What is this projection?

Thanks!