DOCUMENTA MATHEMATICA, Vol. 9 (2004), 137-161

William Arveson

The Free Cover of a Row Contraction

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 2.5).

2000 Mathematics Subject Classification: 46L07, 47A99

Keywords and Phrases: Free Resolutions, Multivariable Operator Theory

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