Geometric Invariants for Resolutions of Hilbert Modules

The development of complex function theory beyond the one-variable case required new techniques and approaches, not just an extension of what had worked already. The same is proving true of multi—variate operator theory. Still it is reasonable to start by seeking to understand in the larger context results that have proved important and useful in the study of single operator theory. For that reason the first author showed [6] how to frame the canonical model theory of Sz.-Nagy and Foias for contraction operators in the language of Hilbert module resolutions over the disk algebra. This point of view made clear why a straightforward extension of model theory failed in the multi—variate case. Since the appropriate algebra in this case would be higher dimensional, one would expect module resolutions, if they existed, to be of longer length and hence not expressible as a canonical model. While work on this topic has shed light on what one might expect to be true (cf. [7]), useful results are still scarce.