In Bar-Natan's paper: Khovanov’s homology for tangles and cobordisms, he defined a deformation of Khovanov homology. Namely, for any $m\geq 0$, Bar-Natan's homology $BN^{m}(K)$ is obtained by tensoring the original Khovanov complex with a ring $F_{2}[U]/U^{m}$ and then adding an extra term in the Khovanov differential.
Question: Are there relations between the $BN^{m}(K)$ for different $m$? In particular, is it true that by setting $U^{m}=0$ in $BN^{\infty}(K)$, one obtains $BN^{m}(K)$? (Somehow I do not see why this is true from pure algebraic reasoning.)
