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In this paper, we define the asymptotic stable division property for submodules ofL  a  2    (   B  n   )    . 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 ofL  a  2    (   B  n   )    ,    ∀  p  >  n   , if its associated primary ideals are powers of prime ideals whose zero loci satisfy standard regularity conditions near the sphere.

Essential normality — a unified approach in terms of local decompositions