Equivariant K-theory and Higher Chow Groups of Schemes

Abstract

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 $K^G_i(X) \to K_i(X)$ induces an isomorphism $K^G_i(X)/{I_G K^G_i(X)} \cong K_i(X)$ 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.