Some remarks on two-periodic modules over local rings

TL;DR

This paper investigates properties of two-periodic modules over local rings, establishing isomorphisms and depth formulas, and generalizing results related to the Huneke-Wiegand conjecture to broader classes of modules.

Contribution

It extends known results about two-periodic modules, proving isomorphisms, depth formulas, and generalizing the Huneke-Wiegand conjecture to modules with rank over one-dimensional local rings.

Findings

The natural map $M ensor_R N o Hom_R(M^*,N)$ is an isomorphism under certain conditions.

The Auslander's depth formula holds for pairs of two-periodic modules.

A pair of modules over a one-dimensional local ring has non-zero torsion iff they are Tor-independent.

Abstract

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.