Equivariant K-theory and Higher Chow Groups of Smooth Varieties

Details Equivariant K-theory and Higher Chow Groups of Smooth Varieties Equivariant K-theory and Higher Chow Groups of Smooth Varieties

2009-06-17

arxiv.org/abs/0906.3109v1

Mathematics

Algebraic Geometry

Scientific paper

For a quasi-projective variety $X$ over a field, with the action of a split torus, we construct a spectral sequence relating the equivariant and the ordinary higher Chow groups. We then completely describe the equivariant higher Chow groups of smooth projective varieties in terms of the ordinary higher Chow groups of certain subvarieties. As applications, we show that for a connected reductive group $G$ acting on a smooth variety $X$, the forgetful map from the rational equivariant higher $K$-theory to the ordinary $K$-theory is surjective and we describe its kernel. We also generalize the eqivariant Riemann-Roch theorem of Edididn and Graham to the higher K-theory of such varieties. We finally discuss the equivariant K-theory of these varieties with finite coefficients and prove the equivariant version of the Quillen-Licthenbaum conjecture as a simple application of the techniques involved in proving the above results.

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