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 {\em 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).