Some remarks on two-periodic modules over local rings

In this note, some properties of finitely generated two-periodic modules over commutative Noetherian local rings have been studied. We show that under certain assumptions on a pair of modules $\left(M,N \right)$ with $M$ two-periodic, the natural map $M \otimes_R N \to Hom_R(M^*,N)$ is an isomorphism. As a consequence, we have that the Auslander's depth formula holds for such a pair. Celikbas et al. recently showed the Huneke-Wiegand conjecture holds over one-dimensional domain for two-periodic modules. We generalize their result to the case of two-periodic module with rank over any one-dimensional local ring. More generally, under certain assumptions on the modules, we show that a pair of modules over an one-dimensional local ring has non-zero torsion if and only if they are Tor-independent.