For the smooth normalization f : X ¯ → X f : {\overline X} \to X of a singular variety X X over a field k k of characteristic zero, we show that for any conducting subscheme Y Y for the normalization, and for any i ∈ Z i \in \mathbb {Z} , the natural map K i ( X , X ¯ , n Y ) → K i ( X , X ¯ , Y ) K_i(X, {\overline X}, nY) \to K_i(X, {\overline X}, Y) is zero for all sufficiently large n n . As an application, we prove a formula for the Chow group of zero cycles on a quasi-projective variety X X over k k with Cohen-Macaulay isolated singularities, in terms of an inverse limit of the relative Chow groups of a desingularization X ~ \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.