Homology of Newtonian Coalgebras

Given a Newtonian coalgebra we associate to it a chain complex. The homology groups of this Newtonian chain complex are computed for two important Newtonian coalgebras arising in the study of flag vectors of polytopes: R〈a,b〉 and R〈c,d〉. The homology of R〈a,b〉 corresponds to the homology of the boundary of the n-crosspolytope. In contrast, the homology of R〈c,d〉 depends on the characteristic of the underlying ring R. In the case the ring has characteristic 2, the homology is computed via cubical complexes arising from distributive lattices. This paper ends with a characterization of the integer homology of Z〈c,d〉.