Projectivity and flatness over the graded ring of semi-coinvariants

Let $k$ be a field, $C$ a bialgebra with bijective antipode, $A$ a right $C$-comodule algebra, $G$ any subgroup of the monoid of grouplike elements of $C$. We give necessary and sufficient conditions for the projectivity and flatness over the graded ring of semi-coinvariants of $A$. When $A$ and $C$ are commutative and $G$ is any subgroup of the monoid of grouplike elements of the coring $A\otimes C$, we prove similar results for the graded ring of conormalizing elements of $A$.