We initiate a study of Hilbert modules over the polynomial algebra
A=C[z_1,...,z_d] that are obtained by completing A with respect to an inner
product having certain natural properties. A standard Hilbert module is a
finite multiplicity version of one of these. Standard Hilbert modules occupy a
position analogous to that of free modules of finite rank in commutative
algebra, and their quotients by submodules give rise to universal solutions of
nonlinear relations.
Essentially all of the basic Hilbert modules that have received attention
over the years are standard - including the Hilbert module of the d-shift, the
Hardy and Bergman modules of the unit ball, modules associated with more
general domains in complex d-space, and those associated with projective
algebraic varieties.
We address the general problem of determining when a quotient H/M of an
essentially normal standard Hilbert module H is essentially normal. This
problem has been resistant. Our main result is that it can be ``linearized" in
that the nonlinear relations defining the submodule M can be reduced,
appropriately, to linear relations through an iteration procedure, and we give
a concrete description of linearized quotients.