  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 $m$ space is introduced. It is shown that quasi-free Hilbert modules
correspond to the 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 there exists 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 a constant $\beta$ satisfying $\|\phi
f \| \leq \beta \|\phi \|_X \|f\|$ for some compact subset $X$ of $\Omega$ and
$\phi$ in $\mathcal{A}$, $f$ in $\mathcal{M}$. It is shown that if a Hilbert
module is compactly supported then it is weakly regular. The paper identifies
several other classes of Hilbert modules which are weakly regular. In addition,
this result is extended to yield topologically exact resolutions of such
modules by quasi-free ones.
