Resolution of singularities for a class of Hilbert modules

Abstract

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.