Essential Normality of Homogenous Quotient Modules Over the Polydisc: Distinguished Variety Case

Abstract

In the present paper, we investigate the essential normality of quotient modules over the polydisc. Let I be a homogeneous ideal in $$\mathbb {C}[z_1,\ldots ,z_d]$$ , we show that if the homogeneous quotient module $$[I]^\bot $$ of $$H^2(\mathbb D^d)$$ is essentially normal, then $$\dim _{\mathbb {C}}Z(I)\le 1$$ . It is shown that if Z(I) is distinguished, then $$[I]^\bot $$ is $$(1,\infty )$$ -essentially normal, i.e. the $$[S_{z_i}^*, S_{z_j}]$$ ’s are not necessarily trace class operators but indeed belong to the interpolation ideal $$\mathcal L^{(1,\infty )}$$ , see the monograph “Noncommutative Geometry” of Connes. This result leads to the answer to the polydisc version of Arveson–Douglas problem. Moreover, we study the boundary representation of $$[I]^\bot $$ .