Non-commutative localisation and finite domination over strongly Z-graded rings

Let R be a strongly Z-graded ring with degree-0 subring S, and let C be a chain complex of modules over the subring P of elements of non-negative degree. We show that there are non-commutative localisations of P which detect whether the complex C is S-finitely dominated or S-contractible, respectively, and that these localisations are universal among P-rings making S-finitely dominated and S-contractible complexes contractible. This generalises known results for polynomial rings to a much wider class of rings. We show by example that in general C need not be P-homotopy finite even if C is S-finitely dominated; this differs from the case of polynomial rings.