Jump loci in the equivariant spectral sequence

We study the homology jump loci of a chain complex over an affine \k-algebra. When the chain complex is the first page of the equivariant spectral sequence associated to a regular abelian cover of a finite-type CW-complex, we relate those jump loci to the resonance varieties associated to the cohomology ring of the space. As an application, we show that vanishing resonance implies a certain finiteness property for the completed Alexander invariants of the space. We also show that, generically, a connected, finite-dimensional commutative graded algebra has vanishing resonance.