Stable categories of spherical modules and torsionfree modules

Abstract

Auslander and Bridger [Stable module theory, Memoirs of the American Mathematical Society, No. 94, American Mathematical Society, Providence, R.I., 1969] introduced the notions of n n -spherical modules and n n -torsionfree modules. In this paper, we construct an equivalence between the stable category of n n -spherical modules and the category of modules of grade at least n n , and provide its Gorenstein analogue. As an application, we prove that if R R is a Gorenstein local ring of Krull dimension d > 0 d>0 , then there exists a stable equivalence between the category of ( d − 1 ) (d-1) -torsionfree R R -modules and the category of d d -spherical modules relative to the local cohomology functor.