On the motivic description of truncated fundamental group rings

A topological theorem due to Beilinson (and which appears in a paper by Deligne–Goncharov) states that for a path connected pointed space with a reasonable topology its fundamental group ring with field coefficients and truncated by a power of the augmentation ideal, is naturally isomorphic to a relative cohomology group that is functorial in terms of that space. We generalize this to integral coefficients, give a more directly proof and also express the maps that define the Hopf algebra structure on these truncations in these terms.