  Let H be a finite-dimensional pivotal and unimodular Hopf algebra over a
field k. It was shown in [BBGa] that the projective tensor ideal in H-mod
admits a unique non-degenerate modified trace, a natural generalisation of the
categorical trace. This paper provides an extension of this result to a much
more general setting. We first extend the notion of the modified trace to the
so-called module trace for a given k-linear module category M over a pivotal
category C equipped with a module endofunctor. We provide a non-trivial class
of examples of such module traces. In particular, we show that any
finite-dimensional pivotal Hopf k-algebra, not necessarily unimodular, admits a
non-degenerate module trace on its projective tensor ideal. We also extend this
result to pivotal Hopf group-coalgebras of finite type, and we give explicit
calculations for the family of Taft Hopf algebras and non-restricted Borel
quantum groups at roots of unity.
