Free resolutions in multivariable operator theory |
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Authors: | Devin C V Greene |
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Institution: | Department of Mathematics, University of Nebraska, Lincoln, NE 68588, USA |
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Abstract: | Let
be the complex polynomial ring in d variables. A contractive
-module is Hilbert space
equipped with an
action such that for any
,||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
modules whose members play the role of free objects. Given a contractive
-module, one can construct a free resolution, i.e. an exact sequence of partial isometries of the following form: where
is a free module for each i0. The notion of a localization of a free resolution will be defined, in which for each λBd there is a vector space complex of linear maps derived from (*):We shall show that the homology of this complex is isomorphic to the homology of the Koszul complex of the d-tuple (1,2,…,d), of where i is the ith coordinate function of a Möbius transform on Bd such that (λ)=0. |
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Keywords: | |
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