  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.
