The Free Cover of a Row Contraction
Abstract
We establish the existence and uniqueness of finite free resolutions - and their attendant Betti numbers - for graded commuting d-tuples of Hilbert space operators. Our approach is based on the notion of free cover of a (perhaps noncommutative) row contraction. Free covers provide a flexible replacement for minimal dilations that is better suited for higher-dimensional operator theory. For example, every graded d-contraction that is finitely multi-cyclic has a unique free cover of finite type - whose kernel is a Hilbert module inheriting the same properties. This contrasts sharply with what can be achieved by way of dilation theory (see Remark 2.4).
Cite
@article{arxiv.math/0403231,
title = {The Free Cover of a Row Contraction},
author = {William Arveson},
journal= {arXiv preprint arXiv:math/0403231},
year = {2007}
}
Comments
Added clarifying remarks, and examples of free resolutions. Aside from the examples, there is no substantial change in mathematical content