Equivalence of quotient Hilbert modules

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