2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/70454We 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).Added clarifying remarks, and examples of free resolutions. Aside from the examples, there is no substantial change in mathematical contentOperator AlgebrasFunctional Analysis46L07, 47A99text