Projectivity and flatness over the graded ring of semi-coinvariants

Abstract

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⊗C, we prove similar results for the graded ring of conormalizing elements of A.