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, 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)？
Best Answer
I think you should read 'motivic' papers (Voevodsky, Friedlander, and others). There seems to be two reasonable answers to your question that are closely related.
Consider the complicated version of equidimensional cycle groups (that uses cdh-sheafification) over $X$; see http://www.math.uiuc.edu/K-theory/0075/
Calculate Hom-groups between the motif of $X$ to $\mathbb Z[i](j)$ (in Voevodsky's triangulated category; you may also consider relative motives by Cisinski and Deglise).