Let A be a finite dimensional symmetric Hopf algebra over a field k. We
show that there are A-modules whose Tate cohomology is not finitely generated
over the Tate cohomology ring of A. However, we also construct A-modules
which have finitely generated Tate cohomology. It turns out that if a module in
a connected component of the stable Auslander-Reiten quiver associated to A
has finitely generated Tate cohomology, then so does every module in that
component. We apply some of these finite generation results on Tate cohomology
to an algebra defined by Radford and to the restricted universal enveloping
algebra of sl2​(k).Comment: 12 pages, comments and suggestions are welcome. arXiv admin note:
substantial text overlap with arXiv:0804.4246 by other author