  In this paper, we prove a decomposition result for the Chow groups of
projectivizations of coherent sheaves of homological dimension $\le 1$. In this
process, we establish the decomposition of Chow groups for the cases of
Cayley's trick and standard flips. Moreover, we apply these results to study
the Chow groups of symmetric powers of curves, nested Hilbert schemes of
surfaces, and the varieties resolving Voisin maps for cubic fourfolds.
