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 xi(1), xi(2), . . . , xi(d) is an element of H, parallel toz(1)xi(1) + z(2)xi + ... + z(d)xi(d)parallel to(2) less than or equal to parallel toxi(1)parallel to(2) + parallel toxi(2)parallel to(2) + ... + parallel toxi(d)parallel to(2). Such objects have been shown to be useful for modeling d-tuples of mutually commuting operators acting on a Hilbert space. There is a subclass of the category of contractive A(d) modules whose members play the role of free objects. Given a contractive A(d)-Module, one can construct a free resolution, i.e. an exact sequence of partial isometries of the following form: ... -->(Phi2) F-1 -->(Phi1) F-0 -->(Phi0) H --> 0, where F, is a free module for each i greater than or equal to 0. The notion of a localization of a free resolution will be defined, in which for each lambda is an element of B-d there is a vector space complex of linear maps derived from (*): ... -->(Phi3(lambda)) C-2 -->(Phi2(lambda)) C-1 -->(Phi1(lambda)) --> C-0. We shall show that the homology of this complex is isomorphic to the homology of the Koszul complex of the d-tuple (p(1), p(2), ..., p(d)), of where p(1) is the ith coordinate function of a Mobius transform on B-d such that p(lambda) = 0. (C) 2002 Elsevier Science (USA). All rights reserved.