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. Then, 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?
Best Answer
I believe Bloch's original insight was something like the following:
First, if $X$ is a regular scheme, you can filter $K_0$ by ``codimension of support''; that is, view $K_0(X)$ as the Grothendieck group of the category of all finitely generated modules and let $F^iK_0(X)$ be the part generated by modules with codimension of support greater than or equal to $i$.
Next, suppose you want to mimic this construction for $K_m$ instead of $K_0$. The first step is to notice that if you patch two copies of $\Delta^m_X$ together along their "boundary" (i.e. the union of the images of the various copies of $\Delta^{m-1}_X$) and call the result $S^m_X$, then Karoubi-Villamayor theory tells you that $K_m(X)$ is a direct summand of $K_0(S^m_X)$. (The complementary direct summand is $K_0(X)$.)
So it suffices to find a "filtration by codimension of support" on $K_0(S^m_X)$.
The usual constructions don't work because $S^m_X$ is not regular (so that in particular, not all modules correspond to $K$-theory classes.)
But: a cycle in $z^i(X,m)$ has a positive part $z_+$ and a negative part $z_-$ which, (if it is homologically a cycle) must agree on the boundary. Therefore you can imagine taking $\Delta^m_X$-modules $M_+$ and $M_-$ supported on these positive and negative parts and patching them along the boundary to get a module on $S^m_X$. If this module has finite projective dimension (which it ``ought'' to because of all the proper-meeting conditions, and as long as it has no bad imbedded components), then it gives a class in $K_0(S^m_X)$, hence a class in $K_m(X)$, and we can take the $i-th$ part of the filtration to be generated by the classes that arise in this way.
The Bloch-Lichtenbaum work largely bypasses this intuition, but this was (I think) the original intuition for why it ought to work.