In this note we settle some technical questions concerning finite rank quasi-free Hilbert modules and develop some useful machinery. In particular, we provide a method for determining when two such modules are unitarily equivalent. Along the way we obtain representations for module maps and study how to determine the underlying holomorphic structure on such modules.

The research of both authors was supported in part by a DST-NSF, S&T Cooperation Programme grant.

The research was begun in July 2003 during a visit by the first author to IHES, funded by development leave from Texas A&M University, and a visit by the second author to Paris VI, funded by a grant from IFCPAR. We thank both institutions for their hospitality.