Let $z^i(X, m)$ be the free abelian group generated by all codimension $i$ subvarieties on $X \times \Delta^m$ which intersect all faces $X \times \Delta^j$ properly for all j < m. Thenone gets, by for each i, these groups assemble to give, with the restriction maps to these faces, a simplicial group whose homotopy groups are the higher Chow groups CH^i(X,m) (m=0 gives the classical ones).
Does anyone have an intuition to share about these higher Chow groups? What do they measure/mean? If I pass from the simplicial group to a chain complex, what does it mean to be in the kernel/image of the differential?
Could one say that the higher Chow groups keep track of in how many ways two cycles can be rationally equivalent (and which of these different ways are then equivalent etc.)?
Finally: I don't see any reason why the definition shouldn't make sense over the integers or worse base schemes. Is this true? Does it maybe still make sense but lose its intended meaning?