  This paper considers the following conjecture: If $R$ is an unmixed,
equidimensional local ring that is a homomorphic image of a Cohen-Macaulay
local ring, then for any ideal $J$ generated by a system of parameters, the
Chern coefficient $e_1(J)< 0$ is equivalent to $R$ being non Cohen-Macaulay.
The conjecture is established if $R$ is a homomorphic image of a Gorenstein
ring, and for all universally catenary integral domains containing fields.
Criteria for the detection of Cohen-Macaulayness in equi-generated graded
modules are derived.
