p‐essential normality of quasi‐homogeneous Drury–Arveson submodules

The first author and Wang [Math. Ann. 340 (2008) 907‐934] proved that each homogeneous principal submodule of the Drury‐Arveson module H 2 is essentially normal, and hence in dimensions n =2 ,3 each homogeneous submodule of H 2 is essentially normal. For the Bergman modules L 2(Bn) on the unit ball, Douglas and Wang [J. Funct. Anal. 261 (2011) 3155‐3180] recently proved that every principal submodule is essentially normal. In this paper, we develop some new techniques to prove the essential normality of Drury‐ Arveson’s quasi-homogeneous principal submodules, by a combination with the approach of Douglas and Wang. As a consequence, we prove that each quasi-homogeneous submodule of H 2 n is essentially normal for dimensions n =2 ,3, and determine the related K-homology.

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