Stable categories of spherical modules and torsionfree modules

TL;DR

This paper establishes a new equivalence between categories of spherical and torsionfree modules, extending to Gorenstein rings, and provides applications in local cohomology contexts.

Contribution

It constructs an equivalence between stable categories of n-spherical modules and modules of grade at least n, including a Gorenstein analogue, with applications to local rings.

Findings

Equivalence between stable categories of n-spherical and high-grade modules

Gorenstein analogue of the main equivalence

Stable equivalence for torsionfree and spherical modules in Gorenstein rings

Abstract

Auslander and Bridger introduced the notions of n-spherical modules and n-torsionfree modules. In this paper, we construct an equivalence between the stable category of n-spherical modules and the category of modules of grade at least n, and provide its Gorenstein analogue. As an application, we prove that if R is a Gorenstein local ring of Krull dimension d>0, then there exists a stable equivalence between the category of (d-1)-torsionfree R-modules and the category of d-spherical modules relative to the local cohomology functor.