Algebraic quotient modules and subgroup depth

In Kadison J Pure Appl Alg 218:367–380, (2014) it was shown that subgroup depth may be computed from the permutation module of the left or right cosets: this holds more generally for a Hopf subalgebra, from which we note in this paper that finite depth of a Hopf subalgebra \(R \subseteq H\) is equivalent to the \(H\) -module coalgebra \(Q = H/R^+H\) representing an algebraic element in the Green ring of \(H\) or \(R\) . This approach shows that subgroup depth and the subgroup depth of the corefree quotient lie in the same closed interval of length one. We also establish a previous claim that the problem of determining if \(R\) has finite depth in \(H\) is equivalent to determining if \(H\) has finite depth in its smash product \(Q^* \# H\) . A necessary condition is obtained for finite depth from stabilization of a descending chain of annihilator ideals of tensor powers of \(Q\) . As an application of these topics to a centerless finite group \(G\) , we prove that the minimum depth of its group \(\mathbb {C}\,\) -algebra in the Drinfeld double \(D(G)\) is an odd integer, which determines the least tensor power of the adjoint representation \(Q\) that is a faithful \(\mathbb {C}\,G\) -module.