The filtration of a toric variety by equivariant skeletal leads to a spectral sequence converging to the variety's Borel-Moore homology. Its E^1term is the direct sum of exterior algebras associated to the cones contained in the fan defining the toric variety. For rational coefficients this spectral sequence degenerates on the E^2 level. I conjecture that this holds true also over the integers and that there is no composition problem, i. eo, that one can compute the integral Borel-Moore homology of a toric variety from an explicitly known complex. This conjecture is supported by numerous examples. If true, it would be the first method to effectively compute the integral homology of singular toric varieties. The project is related to various topics at the borderline between topology and geometry, in particular to the homology and cohomology of algebraic varieties and their Chow groups. For instance, my conjecture would imply that the canonical map from Chow groups to Borel- Moore homology is injective for all toric varieties. Equivariant cohomology and equivariant Chow groups give additional information and will therefore be considered as well. Here Koszul duality comes in as an important algebraic tool to translate between ordinary and equivariant objects.