  We present a method for deciding when a regular abelian cover of a finite
CW-complex has finite Betti numbers. To start with, we describe a natural
parameter space for all regular covers of a finite CW-complex X, with group of
deck transformations a fixed abelian group A, which in the case of free abelian
covers of rank r coincides with the Grassmanian of r-planes in H^1(X,\Q).
Inside this parameter space, there is a subset \Omega_A^i(X) consisting of all
the covers with finite Betti numbers up to degree i.
  Building on work of Dwyer and Fried, we show how to compute these sets in
terms of the jump loci for homology with coefficients in rank 1 local systems
on X. For certain spaces, such as smooth, quasi-projective varieties, the
generalized Dwyer-Fried invariants that we introduce here can be computed in
terms of intersections of algebraic subtori in the character group. For many
spaces of interest, the homological finiteness of abelian covers can be tested
through the corresponding free abelian covers. Yet in general, abelian covers
exhibit different homological finiteness properties than their free abelian
counterparts.
