  We establish the essential normality of a large new class of homogeneous
submodules of the finite rank d-shift Hilbert module. The main idea is a notion
of essential decomposability that determines when an arbitrary submodule can be
decomposed into the sum of essentially normal submodules. We prove that every
essentially decomposable submodule is essentially normal, and using ideas from
convex geometry, we introduce methods for establishing that a submodule is
essentially decomposable. It turns out that many homogeneous submodules of the
finite rank d-shift Hilbert module have this property. We prove that many of
the submodules considered by other authors are essentially decomposable, and in
addition establish the essential decomposability of a large new class of
homogeneous submodules. Our results support Arveson's conjecture that every
homogeneous submodule of the finite rank d-shift Hilbert module is essentially
normal.
