  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).
