The Chow of $S^{[n]}$ and the universal subscheme

We prove that any element in the Chow ring of the Hilbert scheme $\text{Hilb}_n$ of $n$ points on a smooth surface $S$ is a universal class, i.e. the pushforward of a polynomial in the Chern classes of the universal subscheme on $\text{Hilb}_n \times S^k$ for some $k \in \mathbb{N}$, with coefficients pulled back from the Chow of $S^k$.

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