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

cohomologytheory 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)？

propermaps $X \to Y$ induces homomorphisms $CH_*X \to CH_* Y$, so that Chow groups are really the analogs of Borel-Moore homology rather than singular homology (ie using compact support). And Poincare duality always holds between Borel-Moore homology and singular cohomology (no compactness assumption!) so really the definition given above is the most reasonable one. $\endgroup$