Zero cycles on singular varieties and their desingularisations

We use pro cdh-descent of $K$-theory to study the relationship between the zero cycles on a singular variety $X$ and those on its desingularisation $X^{\mathrm{\prime }}$. We prove many cases of a conjecture of S. Bloch and V. Srinivas, and relate the Chow groups of $X$ to the Kerz–Saito Chow group with modulus of $X^{\mathrm{\prime }}$ relative to its exceptional fibre.