Rings with q-torsionfree canonical modules

TL;DR

This paper characterizes Noetherian local rings based on properties of their canonical modules, extending Foxby's 1974 results to broader classes of rings and exploring the concept of q-torsionfreeness.

Contribution

It generalizes Foxby's characterization of q-torsionfree canonical modules to arbitrary Noetherian local rings with mild assumptions.

Findings

Characterization of rings with q-torsionfree canonical modules

Extension of Foxby's results beyond Cohen-Macaulay rings

Examples illustrating the theoretical concepts

Abstract

Let A be a Noetherian local ring with canonical module K. We characterize A when K is a torsionless, reflexive, or q-torsionfree module. If A is a Cohen-Macaulay ring, H.-B. Foxby proved in 1974 that the A-module K is q-torsionfree if and only if the ring A is q-Gorenstein. With mild assumptions, we provide a generalization of Foxby's result to arbitrary Noetherian local rings admitting the canonical module. In particular, since the reflexivity of the canonical module is closely related to the ring being Gorenstein in low codimension, we also explore quasi-normal rings, introduced by W. V. Vasconcelos. We provide several examples as well.