  We show that for a K3 surface X the finitely generated subring R(X) of the
Chow ring introduced by Beauville and Voisin is preserved under derived
equivalences. This is proved by analyzing Chern characters of spherical
bundles. As for a K3 surface X defined over a number field all spherical
bundles on the associated complex K3 surface are defined over $\bar\QQ$, this
is compatible with the Bloch-Beilinson conjecture. Besides the work of
Beauville and Voisin, Lazarfeld's result on Brill-Noether theory for curves in
K3 surfaces and the deformation theory developed with Macri and Stellari in
arXiv:0710.1645 are central for the discussion.
