Quasi-Free Resolutions Of Hilbert Modules

The notion of a quasi-free Hilbert module over a function algebra$mathcal{A}$ consisting of holomorphic functions on a bounded domain $Omega$ in complex mspace is introduced. It is shown that quasi-free Hilbert modules correspond tothe completion of the direct sum of a certain number of copies of the algebra$mathcal{A}$. A Hilbert module is said to be weakly regular (respectively, regular) if thereexists a module map from a quasi-free module with dense range (respectively,onto). A Hilbert module $mathcal{M}$ is said to be compactly supported if there exists aconstant $beta$ satisfying $|varphi f| leq beta |varphi | textsl{X} |f |$ for some compact subset X of $Omega$ and$varphi$ in $mathcal{A}$, f in $mathcal{M}$. It is shown that if a Hilbert module is compactly supportedthen it is weakly regular. The paper identifies several other classes of Hilbertmodules which are weakly regular. In addition, this result is extended to yieldtopologically exact resolutions of such modules by quasi-free ones.