3 problems

Let $R$ be an admissible hypersurface, meaning that $\widehat{R}$ is a quotient of an unramified or equicharacteristic regular local ring by a nonzero element, with an isolated sin…

Let $R$ be a one-dimensional local complete intersection domain, and let $M,N\in\operatorname{mod} R$. Tensor-product torsion-freeness conjecture. If $M\otimes_R N$ is torsion-free…

Tor-rigidity conjecture. The pair $(M,N)$ is $\operatorname{Tor}$-rigid: for any $i>0$, the condition $\operatorname{Tor}_i^R(M,N)=0$ forces $\operatorname{Tor}_j^R(M,N)=0$ for eve…