Resolution of singularities for a class of Hilbert modules

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 $$\{p_i(\frac{\mathrm{\partial }}{\mathrm{\partial }{\bar{w}}_1},\mathrm{.}\mathrm{.}\mathrm{.},\frac{\mathrm{\partial }}{\mathrm{\partial }{\bar{w}}_m})K_{[\mathcal{I}]}(\cdot ,w){\mathrm{\mid }}_{w=0},1\le i\le n\},$$ where $K_{[\mathcal{I}]}$ is the reproducing kernel for the submodule $[\mathcal{I}]$ and $p_1,\mathrm{.}\mathrm{.}\mathrm{.},p_n$ 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.