  Let G be a finite group and let k be a field of characteristic p. Given a
finitely generated indecomposable non-projective kG-module M, we conjecture
that if the Tate cohomology $\HHHH^*(G, M)$ of G with coefficients in M is
finitely generated over the Tate cohomology ring $\HHHH^*(G, k)$, then the
support variety V_G(M) of M is equal to the entire maximal ideal spectrum
V_G(k). We prove various results which support this conjecture. The converse of
this conjecture is established for modules in the connected component of k in
the stable Auslander-Reiten quiver for kG, but it is shown to be false in
general. It is also shown that all finitely generated kG-modules over a group G
have finitely generated Tate cohomology if and only if G has periodic
cohomology.
