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

Remarks on a conjecture of Huneke and Wiegand and the vanishing of (co)homology