An Artin-rees Theorem and Applications to Zero Cycles

For the smooth normalization f : X → X of a singular variety X over a field k of characteristic zero, we show that for any conducting subscheme Y for the normalization, and for any i ∈ Z, the natural map Ki(X, X, nY ) → Ki(X, X, Y ) is zero for all sufficiently large n. As an application, we prove a formula for the Chow group of zero cycles on a quasi-projective variety X over k with Cohen-Macaulay isolated singularities, in terms of an inverse limit of relative Chow groups of a desingularization X̃ relative to the multiples of the exceptional divisor. We use this formula to verify a conjecture of Srinivas about the Chow group of zero cycles on the affine cone over a smooth projective variety which is arithmetically Cohen-Macaulay.