  In this paper we study a long-standing conjecture of Huneke and Wiegand which
is concerned with the torsion submodule of certain tensor products of modules
over one-dimensional local domains. We utilize Hochster's theta invariant and
show that the conjecture is true for two periodic modules. We also make use of
a result of Orlov and formulate a new condition which, if true over
hypersurface rings, forces the conjecture of Huneke and Wiegand to be true over
complete intersection rings of arbitrary codimension. Along the way we
investigate the interaction between the vanishing of Tate (co)homology and
torsion in tensor products of modules, and obtain new results that are of
independent interest.
