ar X iv : m at h / 01 09 11 1 v 2 [ m at h . FA ] 1 8 Se p 20 01 ON FREE RESOLUTIONS IN MULTIVARIABLE OPERATOR THEORY

Let A d be the complex polynomial ring in d variables. A contractive A d-module is Hilbert space H equipped with an A d action such that for any ξ 1 , ξ 2 ,. Such objects have been shown to be useful for modeling d-tuples of mutually commuting operators acting on a Hilbert space. An example is given by H 2 d , which is the Hilbert space of holomorphic functions on B d = {z ∈ C d : |z| ≤ 1} generated by the reproducing kernel k(z, w) = 1 1−z,w , z, w ∈ B d. For a natural subcatagory of contractive A d-modules, whose members are said to be pure, the module H 2 d plays the role of the free module of rank one. In fact, given any pure contractive A d-module H, there is a " free resolution " , i.e. an exact sequence of the following form: