Hopf subalgebras and tensor powers of generalized permutation modules. (Q392426)

It was shown by \textit{R. Boltje} et al. [J. Algebra 335, No. 1, 258-281 (2011; Zbl 1250.20001)] that the depth of the group algebra of a subgroup of a finite group \(G\) in the group algebra of \(G\) is always finite. In their paper these authors suggest to find out whether, more generally, the depth of a Hopf subalgebra \(R\) in a finite dimensional Hopf algebra \(H\) is also always finite. The paper under review investigates this question. The main tools are the counit representation \(V\) induced from \(R\) to \(H\), which at the same time is an \(H\)-module coalgebra and a generalized permutation module for the extension \(H\supseteq R\), and the depth of an \(H\)-module \(M\), which is obtained by comparing truncated tensor algebras of \(M\). This yields upper and lower bounds for the depth of \(R\) in \(H\). As an application it is shown that the depth of \(R\) in \(H\) is finite if either \(R\) or \(H\) has finite representation type, or if \(V\) is projective, or if \(V\) is semisimple and the tensor product of two simple \(R\)-modules is semisimple. The author also reproves the result of Boltje et al.\ on the finite depth of extensions of finite dimensional group algebras.