The spectral sequence of an equivariant chain complex and homology with local coefficients

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 $X$. In the process, we identify the $d^1$ differential in terms of the coalgebra structure of $H_*(X,\Bbbk)$ and the $\Bbbk\pi_1(X)$-module structure on the twisting coefficients. In particular, this recovers in dual form a result of Reznikov on the mod $p$ cohomology of cyclic $p$-covers of aspherical complexes. This approach provides information on the homology of all Galois covers of $X$. It also yields computable upper bounds on the ranks of the cohomology groups of $X$, with coefficients in a prime-power order, rank one local system. When $X$ 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 $H^*(X,\Bbbk)$, thereby generalizing a result of Cohen and Orlik.