We prove that any element in the Chow ring of the Hilbert scheme Hilb
n of
n points on a smooth surface
![$S$](img21.png)
is a universal class, i.e. the push-forward of a polynomial in the Chern classes of the universal subschemes on Hilb
n x
SK
for some
![$k \in \BN$](img23.png)
N, with coefficients pulled back from the Chow of
![$S^k$](img24.png)
.