Some remarks on two-periodic modules over local rings

Abstract

In this paper, 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 [Formula: see text] with [Formula: see text] being two-periodic, the natural map [Formula: see text] is an isomorphism. As a consequence, we prove that Auslander’s depth formula holds for such a Tor-independent pair. Tor-independence plays a crucial role for the depth formula to hold. Under certain assumptions on the modules, we show that a pair of modules, over a one-dimensional local ring, is Tor-independent if and only if their tensor product is torsion-free. Celikbas et al. recently showed the Huneke–Wiegand conjecture holds for two-periodic modules over one-dimensional domains. We generalize their result to the case of two-periodic modules with rank over one-dimensional local rings.