Essential normality — a unified approach in terms of local decompositions

Abstract

In this paper, we define the asymptotic stable division property for submodules of La2false(Bnfalse). 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 La2false(Bnfalse), ∀p>n, if its associated primary ideals are powers of prime ideals whose zero loci satisfy standard regularity conditions near the sphere.

An index theorem for quotients of Bergman spaces on egg domains

“…. , p k } of homogeneous polynomials in the sense that there exists C > 0 such that every q ∈ I can be written as q = k j=1 r j p j with r j ∈ A and r j p j L 2 (Ω) ≤ C q L 2 (Ω) [43,47]. (6) I is the vanishing ideal of a homogeneous variety smooth away from the origin [27,22,25,48].…”