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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 sequenceassociated to a regular abelian cover of a finite-type CW-complex, we relatethose jump loci to the resonance varieties associated to the cohomology ring ofthe space. As an application, we show that vanishing resonance implies acertain finiteness property for the completed Alexander invariants of thespace. We also show that vanishing resonance is a Zariski open condition, on anatural parameter space for connected, finite-dimensional commutative gradedalgebras.

Jump loci in the equivariant spectral sequence