A commutative regulator map into Deligne-Beilinson cohomology

In 1986, Spencer Bloch gave an abstract definition of a (regulator) map from higher Chow groups to Deligne-Beilinson cohomology. This map can be defined on the underlying complexes, and Kerr, Lewis and M\"uller-Stach gave an explicit description of this map in terms of currents. Using a multiplicative version of the Deligne complex, we give a commutative version of this map.