Abstract :
Let Ad be the complex polynomial ring in d variables. A contractive Ad-module is Hilbert space H equipped with an Ad action such that for any ξ1,ξ2,…,ξd∈H,||z1ξ1+z2ξ+⋯+zdξd||2⩽||ξ1||2+||ξ2||2+⋯+||ξd||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 Ad modules whose members play the role of free objects. Given a contractive Ad-module, one can construct a free resolution, i.e. an exact sequence of partial isometries of the following form: (∗)⋯→Φ2F1→Φ1F0→Φ0H→0,where Fi is a free module for each i⩾0. The notion of a localization of a free resolution will be defined, in which for each λ∈Bd there is a vector space complex of linear maps derived from (∗):⋯→Φ3(λ)C2→Φ2(λ)C1→Φ1(λ)→C0.We shall show that the homology of this complex is isomorphic to the homology of the Koszul complex of the d-tuple (ϕ1,ϕ2,…,ϕd), of where ϕi is the ith coordinate function of a Möbius transform on Bd such that ϕ(λ)=0.
