Periodic modules over Gorenstein local rings

TL;DR

This paper characterizes when modules over Gorenstein local rings have eventually periodic minimal free resolutions, linking this property to torsion elements in a specific algebraic module J(R).

Contribution

It establishes a structure theorem for J(R) in complete Gorenstein local rings and connects module periodicity with torsion in J(R).

Findings

Periodic modules correspond to torsion elements in J(R).

Structure theorem for J(R) when R is complete Gorenstein.

Main result links periodicity of resolutions to algebraic torsion properties.

Abstract

It is proved that the minimal free resolution of a module M over a Gorenstein local ring R is eventually periodic if, and only if, the class of M is torsion in a certain Z[t,t^{-1}]-module associated to R. This module, denoted J(R), is the free Z[t,t^{-1}]-module on the isomorphism classes of finitely generated R-modules modulo relations reminiscent of those defining the Grothendieck group of R. The main result is a structure theorem for J(R) when R is a complete Gorenstein local ring; the link between periodicity and torsion stated above is a corollary.