Remarks on a conjecture of Huneke and Wiegand and the vanishing of (co)homology (Q6610117)

Let \(R\) be a commutative Noetherian local ring. Assume that all \(R\)-modules are finitely generated. The authors provided some results such that the conjecture of Huneke and Wiegand holds true.\N\NConjecture (Huneke-Wiegand's conjecture): Assume \(R\) is onedimensional and let \(M\) be a nonfree and torsion-free \(R\)-module. Assume \(M\) has rank (e.g., \(R\) is a domain). Then the torsion submodule of \(M\otimes_R M^*\) is nonzero, i.e., \(M\otimes_R M^*\) has (nonzero) torsion, where \(M^* = \mathrm{Hom}_R(M, R).\)\N\NBy using Tate (co)homology, the authors proved that the conjecture is true if \(R\) is a one-dimensional complete intersection domain and \(M\) is two-periodic (Theorem 1.2). This is a different proof from the one of \textit{O. Celikbas} [J. Commut. Algebra 3, No. 2, 169--206 (2011; Zbl 1237.13031)].\N\NNext, Theorem 1.3 is an improvement of Theorem 1.2 by removing the complete intersection domain condition. The authors used some properties of \textit{Tor-rigid} modules in the proof of Theorem 1.3.\N\NWhile the Conjecture holds true for hypersurface rings, it remains an open question for complete intersection rings with a codimension of two or greater. The paper shows some conditions such that the Conjecture is true over each one-dimensional complete intersection local domain.