# Essential Normality - a Unified Approach in Terms of Local Decompositions

## TL;DR

This paper introduces a unified approach to essential normality of submodules in the Bergman module using local decompositions, establishing broad conditions for p-essential normality and extending previous results.

## Contribution

It develops a new technique to prove the asymptotic stable division property, unifying and extending known results on essential normality of submodules.

## Key findings

- Submodules with the asymptotic stable division property are p-essentially normal for all p>n.
- Ideal-defined submodules are p-essentially normal under certain regularity conditions.
- The approach unifies proofs of existing results and introduces new cases of essential normality.

## Abstract

In this paper, we define the asymptotic stable division property for submodules of the Bergman module. We show that under a mild condition, a submodule with the asymptotic stable division property is p-essentially normal for all p>n. A new technique is developed to show that certain submodules have the asymptotic stable division property. This leads to a unified proof of most known results on essential normality of submodules as well as new results. In particular, we show that an ideal defines a p-essentially normal submodule of the Bergman module, for all p>n, if its associated primary ideals are powers of prime ideals whose zero loci satisfy standard regularity conditions near the sphere.