  We discuss the relation between questions regarding the essential normality
of finitely generated essentially spherical isometries and some results and
conjectures of Arveson and Guo-Wang on the closure of homogeneous ideals in the
m-shift space. We establish a general results for the case of two tuples and
ideals with one dimensional zero variety. Further, we show how to reduce the
analogous question for quasi-homogeneous ideals, to those results for
homogeneous ones. Finally, we show that the essential reductivity of positive
regular Hilbert modules is directly related to a generalization of the Arveson
problem.
