We  explain  the construction  of an explicit regulator map  at the level of complexes:
\[
\operatorname{Reg} \colon {CH_\Delta^{p}(X, n)}   \longrightarrow H^{2p-n}_{\mathscr{D}}(X;\mathbb{Z}(p)),
\]
from the higher Chow groups of a smooth complex algebraic variety \(  X \),  in their simplicial formulation with \(  \mathbb{Z} \) coefficients, into integral Deligne-Beilinson cohomology.

We start  by   using a suitably  defined \emph{equidimensional cycles} subcomplex  \(  \mathcal{Z}^p_{\Delta, \text{eq}}(X,*) \)
of Bloch's higher Chow complex \( \mathcal{Z}^p_\Delta(X,*)\) to compute the higher Chow groups.  This relies on Suslin's \emph{generic equidimensionality} results.

Next, we use algebraic correspondences to introduce transform operations on a fairly general class of currents. Then we combine these transforms with basic properties of equidimensional cycles to construct a map of complexes
\[
  \operatorname{Reg} \colon \mathcal{Z}^p_{\Delta, \text{eq}}(X,*)  \to \mathbb{Z}(p)_{\mathscr{D}}(X),
  \]
where \(  \mathbb{Z}(p)_{\mathscr{D}}(X) \) is a complex of currents yielding Deligne-Beilinson cohomology.