Equivalence of quotient Hilbert modules

Let \cl{M} be a Hilbert module of holomorphic functions over a natural function algebra $\mathcal{A}(\Omega)$, where \Omega \subseteq \bb{C}^m is a bounded domain. Let \cl{M}_0\subseteq \cl{M} be the submodule of functions vanishing to order $k$ on a hypersurface \cl{Z} \subseteq \Omega. We describe a method, which in principle may be used, to construct a set of complete unitary invariants for quotient modules \cl{Q}=\cl{M} \ominus \cl{M}_0. The invariants are given explicitly in the particular case of $k = 2$.Comment: 11 page