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 \(A \otimes C\), we prove similar results for the graded ring of conormalizing elements of \(A\).