Given a suitable category of
R-modules, a generalized Euler characteristic is defined for each finite sequence of
modules in the category, and is characterized by simple properties. For many
categories, including the category of all finitely generated R-modules, this generalized
characteristic has the following two properties. First, it assigns the same value to
isomorphic sequences. Second, for any chain complex of R-modules in the category,
the characteristic of the sequence of chain modules equals the characteristic of the
sequence of homology modules. For such categories our results imply that any
function having these two properties is itself a function of the characteristic so that
the generalized Euler characteristic is essentially the only such function. For the
special case of the category of all finitely generated modules over a principal ideal
domain the generalized Euler characteristic can be identified with the integer
valued function which is the classical Euler characteristic. By considering the
special case of the category of all finitely generated torsion modules over the
polynomial ring F[x] over a field F we obtain a generalized Euler characteristic for
the case of a linear endomorphism of a finite sequence of finite dimensional
vector spaces over F. In this case we establish the relations between the
characteristic and the sequence of Lefschez numbers of the endomorphism and its
iterates.
