  For a moduli space $M$ of stable sheaves over a $K3$ surface $X$, we propose
a series of conjectural identities in the Chow rings $CH_\star (M \times
X^\ell),\, \ell \geq 1,$ generalizing the classic Beauville-Voisin identity for
a $K3$ surface. We emphasize consequences of the conjecture for the structure
of the tautological subring $R_\star (M) \subset CH_\star (M).$ The conjecture
places all tautological classes in the lowest piece of a natural filtration
emerging on $CH_\star (M)$, which we also discuss. We prove the proposed
identities when $M$ is the Hilbert scheme of points on a $K3$ surface.
