Let $X$ be a regular scheme over a field $k$ and $Δ^m$ be the algebraic $m$-simplex $\mathrm{Spec} k[t_0,...,t_m]/(1-\sum_jt_j)$. The group $z^i(X,m)$ is the free abelian group generated by all closed integral subvarieties on $X×Δ^m$ of codimension $i$ which intersect all faces $X×Δ^j$ properly for all $j < m$. Taking alternative sum of these intersections makes $z^i(X,*)$ a chain complex. Bloch's higher Chow groups is defined as homology groups of these complexes.  

In Jinhyun Park's answer to the question [What do higher Chow groups mean][1], he elaborates that one can see higher Chow groups as algebraic-geometric version of singular **homology** theory.  

Since higher Chow groups are extensively studied. A natural question is

> Can we define the algebro-geometric version of singular **cohomology** theory using above constructions？ Is this a good object to study？What about Chow cohomology(Sorry if this does not make sense, I am new to this stuff)？


  [1]: http://mathoverflow.net/questions/4802/what-do-higher-chow-groups-mean