Projectivity and flatness over the graded ring of normalizing elements

Abstract

Let k be a field, H a cocommutative bialgebra, A a commutative left H-module algebra, 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 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 χ(H,Z(A)H) of all algebra maps from H to Z(A)H, where Z(A) is the center of A.