Summary: THE FREE COVER OF A ROW CONTRACTION
Abstract. We establish the existence and uniqueness of finite free res-
olutions - and their attendant Betti numbers - for graded commuting
d-tuples of Hilbert space operators. Our approach is based on the no-
tion 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).
The central result of this paper establishes the existence and uniqueness
of finite free resolutions for commuting d-tuples of operators acting on a
common Hilbert space (Theorem 2.6). Commutativity is essential for that
result, since finite resolutions do not exist for noncommuting d-tuples.
On the other hand, we base the existence of free resolutions on a general
notion of free cover that is effective in a broader noncommutative context.
Since free covers have applications that go beyond the immediate needs of