Equivariant K-theory and higher Chow groups of schemes

For a smooth quasi-projective scheme X over a field k with an action of a reductive group, we establish a spectral sequence connecting the equivariant and the ordinary higher Chow groups of X. For X smooth and projective, we show that this spectral sequence degenerates, leading to an explicit relation between the equivariant and the ordinary higher Chow groups. We obtain several applications to algebraic K-theory. We show that for a reductive group G acting on a smooth projective scheme X, the forgetful map K-i(G)(X) -> K-i (X) induces an isomophism K-i(G)(X)/IGKiG(X) ->(similar or equal to) K-i(X) with rational coefficients. This generalizes a result of Graham to higher K-theory of such schemes. We prove an equivariant Riemann-Roch theorem, leading to a generalization of a result of Edidin and Graham to higher K-theory. Similar techniques are used to prove the equivariant Quillen-Lichtenbaum conjecture.