  We extend Auslander and Buchsbaum's Euler characteristic from the category of
finitely generated modules of finite projective dimension to the category of
modules of finite G-dimension using Avramov and Martsinkovsky's notion of
relative Betti numbers. We prove analogues of some properties of the classical
invariant and provide examples showing that other properties do not translate
to the new context. One unexpected property is in the characterization of the
extremal behavior of this invariant: the vanishing of the Euler characteristic
of a module M of finite G-dimension implies the finiteness of the projective
dimension of M. We include two applications of the Euler characteristic as well
as several explicit calculations.
