## The free cover of a row contraction

### Summary

Summary: 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.

46L07, 47A99

### Keywords/Phrases

free resolutions, multivariable operator theory