An ideal criterion for torsion freeness

Abstract

Auslander and Bridger have shown that, under conditions somewhat weaker than finite projective dimension, the “torsion freeness” properties of a module M (e.g. being reflexive, being the kth syzygy of another module) are determined by certain arithmetic conditions on the Ext i ( M , R ) {\text {Ext}^i}(M,R) . In this paper it is shown that a single ideal, the intersection of the annihilators of these modules, gives this same information. This ideal is then related to the Fitting invariants and invariant factors of M, and a computation is made of certain syzygies of a quotient of M (by a regular M-sequence).