00 7 Additive Higher Chow Groups of Schemes

We show how to make the additive Chow groups of Bloch-Esnault, Rülling and Park into a graded module for Bloch's higher Chow groups, in the case of a smooth projective variety over a field. This yields a a projective bundle formula as well as a blow-up formula for the additive Chow groups of a smooth projective variety. In case the base-field admits resolution of singularieties, these properties allow us to apply the technique of Guillén and Navarro Aznar to define the additive Chow groups " with log poles at infinity " for an arbitrary finite-type k-scheme X. This theory has all the usual properties of a Borel-Moore theory on finite type k-schemes: it is covariantly functorial for projective morphisms, con-travariantly functorial for morphisms of smooth schemes, and has a projective bundle formula, homotopy property, Mayer-Vietoris and localization sequences. Finally, we show that the regulator map defined by Park from the additive Chow groups of 1-cycles to the modules of absolute Kähler differentials of an algebraically closed field of characteristic zero is surjective, giving an evidence of conjectured isomorphism between these two groups.