Resolution of singularities for a class of Hilbert modules

TL;DR

This paper provides a new proof of the Rigidity theorem for Hilbert modules, describes the joint kernel for a class of submodules, and introduces an algorithm to construct canonical generators and invariants for homogeneous ideals.

Contribution

It introduces a sheaf theoretic approach to Hilbert modules, offers an explicit construction of canonical generators for homogeneous ideals, and presents computable invariants using monoidal transformations.

Findings

Describes the joint kernel for a large class of submodules.

Provides an algorithm for constructing canonical generators of homogeneous ideals.

Introduces invariants for submodules using monoidal transformations.

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.