Projectivity and flatness over the graded ring of normalizing elements

Let $k$ be a field, $H$ a cocommutative bialgebra, $A$ a commutative left $H$-module algebra, $\operatorname{Hom}(H,A)$ the $k$-algebra of the $k$-linear maps from $H$ to $A$ under the convolution product, $Z(H,A)$ the submonoid of $\operatorname{Hom}(H,A)$ whose elements satisfy the cocycle condition and $G$ any subgroup of the monoid $Z(H,A)$. We give necessary and sufficient conditions for the projectivity and flatness over the graded ring of normalizing elements of $A$. When $A$ is not necessarily commutative we obtain similar results over the graded ring of weakly semi-invariants of $A$ replacing $Z(H,A)$ by the set $\chi(H,Z(A)^H)$ of all algebra maps from $H$ to $Z(A)^H$, where $Z(A)$ is the center of $A$.