Essential normality — a unified approach in terms of local decompositions

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.