Tensor products of modules and the rigidity of Tor (Q1337516)

Let \(R\) be a hypersurface domain that is, a ring of the form \(S/(f)\), where \(S\) is a regular local ring and \(f\) is a prime element. Suppose \(M\) and \(N\) are finitely generated \(R\)-modules. We prove two rigidity theorems on the vanishing of Tor. In the first theorem we assume that the regular local ring \(S\) is unramified, that \(M\otimes_ RN\) has finite length, and that \(\dim (M) + \dim (N) \leq \dim (R)\). With these assumptions, if \(\text{Tor}^ R_ j (M,N) = 0\) for some \(j \geq 0\), then \(\text{Tor}^ R_ i (M,N) = 0\) for all \(i \geq j\). The second rigidity theorem states that if \(M \otimes_ RN\) is reflexive, then \(\text{Tor}^ R_ i (M,N) = 0\) for all \(i \geq 1\). We use these theorems to prove the following theorem (valid even if \(S\) is ramified): If \(M \otimes_ RN\) is a maximal Cohen-Macaulay \(R\)-module, then both \(M\) and \(N\) are maximal Cohen-Macaulay modules, and at least one of them is free. Editorial remark: Note that due the the correction [\textit{C. Huneke} and \textit{R. Wiegand}, Math. Ann. 338, No. 2, 291--293 (2007; Zbl 1122.13301)] and a later result of \textit{O. Celikbas} and \textit{R. Takahashi} [Proc. Am. Math. Soc. 147, No. 7, 2733-2739 (2019; Zbl 1411.13022)] one of the conclusions of the depth formula theorem is flawed due to an incorrect convention for the depth of the zero module.