In a finite group, Boltje et al have shown that a measure of induction-restriction properties called depth takes on interesting values on subgroups in zero and positive characteristic. The finiteness of depth of a group algebra extension is a consequence of a permutation module representing an algebraic element of the representation ring of a finite group. We extend some of these results to corings and entwining structures related to a module category of a finite tensor category. As a consequence, we note that S. Danz's results on shrinking depth for certain twisted group algebra extensions in Commun. Alg. (2011) are part of an inequality for depth of crossed products and Hopf subalgebras.
