Equivariant K-theory and Higher Chow Groups of Schemes

2009/06/17 by Krishna, Amalendu
#14C35 #14C40 #Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.0906.3109

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) → K_i(X) induces an isomorphism K^G_i(X)/I_G K^G_i(X) ≅ 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.

Related

• Notes on equivariant higher Chow groups
• Equivariant Algebraic K-Theory and Derived completions III: Applications
• Completions of Higher Equivariant K-theory
• On a spectral sequence for equivariant K-theory
• Riemann-Roch for equivariant Chow groups