4 problems

Moore–Piepmeyer–Spiroff–Walker conjecture. If $R$ is a complete intersection of codimension $c\geq 2$, then

Huneke–Jorgensen–Wiegand conjecture. If $M\otimes_R N$ is a $(c+1)$st syzygy and $M$ has rank, must

Tor-vanishing conjecture. If

Let $(R,\mathfrak{m})$ be a one-dimensional complete intersection domain. A finitely generated $R$-module is rigid if its higher Tor with every relevant partner vanishes as in the…