Higher Chow Groups for Gl(n) Quotients

Given a topological space Y and a group G that acts on Y, one can define the (Borel) equivariant cohomology of Y as the cohomology of a certain space depending on Y and G. In the case where the G action is on Y is free, the equivariant cohomology is the cohomology of the quotient.

We will outline two approaches to mimicking this construction where Y, G are schemes and the cohomology theory in question is motivic cohomology. One of these is by use of a bar construction, B(G,Y), which lends itself to a spectral sequence. In the easiest of cases the E_2-page of this spectral sequence can be computed, and so we have a method for computing the motivic cohomology of certain quotient schemes, for instance spaces of matrices with rank conditions, or (ideally) Grassmanians.