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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 alternating sums of these intersections makes $z^i(X,*)$ a chain complex. Bloch's higher Chow groups are defined as the 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 an 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)

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  • $\begingroup$ I think for Chow groups one usually forces Poincare duality by putting $CH^n X := CH_{d-n}X$ when $X$ is of pure dimension $d$ (and similarly for higher Chow groups). The resulting theory behaves much like singular cohomology. Since you expect Poincare duality to hold for varieties, this is a reasonable thing to do. $\endgroup$
    – Pierre
    Commented Aug 7, 2011 at 21:41
  • $\begingroup$ PS the functoriality properties of Chow groups are also compatible with this. In fact only proper maps $X \to Y$ 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. $\endgroup$
    – Pierre
    Commented Aug 7, 2011 at 21:43
  • $\begingroup$ As singular cohomology, we can apply the $Hom(-,\mathbf Z)$ funtor to $z^i(X,*)$ to get a cochain complex. What is the cohomology of the cochain complex? $\endgroup$
    – Liu Hang
    Commented Aug 8, 2011 at 1:50

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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.

  1. Consider the complicated version of equidimensional cycle groups (that uses cdh-sheafification) over $X$; see http://www.math.uiuc.edu/K-theory/0075/

  2. 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).

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    $\begingroup$ Dear Mikhail, There is also the paper of Suslin and Voevodsky in which they show how to compute the actual singular cohomology of a variety, with finite coefficients, using these cycle-type constructions. Best wishes, Matthew $\endgroup$
    – Emerton
    Commented Sep 20, 2011 at 4:52

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