An Artin-Rees theorem in $K$-theory and applications to zero cycles

For the smooth normalization $f : {\overline X} \to X$ 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 $i \in \mathbb{Z}$, the natural map $K_i(X, {\overline X}, nY) \to K_i(X, {\overline X}, Y)$ 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 the relative Chow groups of a desingularization $\widetilde X$ 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.