We study the spectral sequence associated to the filtration by powers of the augmentation ideal on the (twisted) equivariant chain complex of the universal cover of a connected CW-complex
X
X
. In the process, we identify the
d
1
d^1
differential in terms of the coalgebra structure of
H
∗
(
X
,
k
)
H_*(X,\Bbbk )
and the
k
π
1
(
X
)
\Bbbk \pi _1(X)
-module structure on the twisting coefficients. In particular, this recovers in dual form a result of Reznikov on the mod
p
p
cohomology of cyclic
p
p
-covers of aspherical complexes. This approach provides information on the homology of all Galois covers of
X
X
. It also yields computable upper bounds on the ranks of the cohomology groups of
X
X
, with coefficients in a prime-power order, rank one local system. When
X
X
admits a minimal cell decomposition, we relate the linearization of the equivariant cochain complex of the universal abelian cover to the Aomoto complex, arising from the cup-product structure of
H
∗
(
X
,
k
)
H^*(X,\Bbbk )
, thereby generalizing a result of Cohen and Orlik.