TITLE: Homotopy theory of schemes examples
QUESTION [22 upvotes]: Is it possible give an example of (or explain) how the Voevodsky et al.'s homotopy theory of schemes computes higher Chow groups?

REPLY [6 votes]: If X is not smooth, then it is possible for the Chow groups and the A^1-represented motivic cohomology theory to disagree.
For instance, if we take X to be two copies of A^1 identified at a point then CH^0(X) has rank 2, but the sheaf represented by X in the A^1 category is contractible, so H^0(X,Z(0)) has rank 1.
To see this last point, we consider X as the colimit of a diagram A^1 <- * -> A^1. Since the maps in this diagrams are monomorphisms of schemes, they are cofibrations. The colimit of the diagram is therefore equivalent to the homotopy colimit, which is invariant under pointwise equivalences of diagrams. Since A^1 is contractible, we are left with the colimit of * <- * -> *, a point.
Morally, A^1 is contractible, so we can shrink down the A^1s without changing the A^1 homotopy type.