  A short proof of the "Rigidity theorem" using the sheaf theoretic model for
Hilbert modules over polynomial rings is given. The joint kernel for a large
class of submodules is described. The completion $[\mathcal I]$ of a
homogeneous (polynomial) ideal $\mathcal I$ in a Hilbert module is a submodule
for which the joint kernel is shown to be of the form $$
\{p_i(\tfrac{\partial}{\partial \bar{w}_1}, ..., \tfrac{\partial}{\partial
\bar{w}_m}) K_{[\mathcal I]}(\cdot,w)|_{w=0}, 1 \leq i \leq n\}, $$ where
$K_{[\mathcal I]}$ is the reproducing kernel for the submodule $[\mathcal I]$
and $p_1, ..., p_n$ is some minimal "canonical set of generators" for the ideal
$\mathcal I$. The proof includes an algorithm for constructing this canonical
set of generators, which is determined uniquely modulo linear relations, for
homogeneous ideals. A set of easily computable invariants for these submodules,
using the monoidal transformation, are provided. Several examples are given to
illustrate the explicit computation of these invariants.
