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.