  We define the notion of a free resolution of a d-tuple $(T_1, T_2, . . .
T_d)$ of mutually commuting operators acting on a Hilbert space H, and that
this invariant gives rise to a class of vector space complexes parametrized by
points in the unit ball $B_d = {z \in C^d: |z| <1}$. We show that for $\lambda
\in B_d$ the homology of the corresponding complex is equivalent to the
homology of the Koszul complex of a Mobius transform of $(T_1, T_2, . . .
T_d)$. The notion of a Mobius transform in multivariable operator is defined,
and some of its properties are investigated.
