  In previous papers, [M1, M2], [M3], we proved the existence as well as
regularity of a finite range decomposition for the resolvent $G_{\alpha}
(x-y,m^2) = ((-\Delta)^{\alpha\over 2} + m^{2})^{-1} (x-y) $, for $0<\alpha <2$
and all real $m$, in the lattice ${\bf Z}^{d}$ for dimension $d\ge 2$. In this
paper, which is a continuation of the previous one, we extend those results by
proving the existence as well as regularity of a finite range decomposition for
the same resolvent but now on the lattice torus ${\bf Z}^{d}/L^{N+1}{\bf Z}^{d}
$ for $d\ge 2$ provided $m\neq 0$ and $0<\alpha <2$. We also prove
differentiability and uniform continuity properties with respect to the
resolvent parameter $m^{2}$. Here $L$ is any odd positive integer and $N\ge 2$
is any positive integer.
