Quotients of standard Hilbert modules

Abstract:

We initiate a study of Hilbert modules over the polynomial algebra $\mathcal A=\mathbb C[z_1,\dots ,z_d]$ that are obtained by completing $\mathcal A$ 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 $\mathbb C^d$, 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.