In the 1980s Bloch and Srinivas made a number of conjectures concerning the relationship between the zero cycles on a singular variety and those on a chosen desingularisation, which take into account the K-theory of the infinitesimal thickenings of the exceptional locus of the desingularisation. Special cases of these conjectures, in low dimension or for simple types of singularities in characteristic zero, were later settled affirmatively by Krishna and Srinivas. Their work was the initial inspiration for my ``pro cdh descent'' theorem, stating that the failure of K-theory to satisfy cdh descent can be remedied by taking into account infinitesimal thickenings; I will explain this theorem and how it leads to an affirmative resolution of the aforementioned conjectures.