Let L be a unital Z-graded ring, and let C be a bounded chain complex of finitely generated L-modules. We give a homological characterisation of when C is homotopy equivalent to a bounded complex of finitely generated projective L 0 -modules, generalising known results for twisted Laurent polynomial rings. The crucial hypothesis is that L is a strongly graded ring.have vanishing homology in all degrees.stands for the ring of formal Laurent series in t −1 .The cited paper [Ran95] also contains a discussion of the relevance of finite domination in topology. -The rings R((t)) and R((t −1 )) are Date: September 20, 2018. 2010 Mathematics Subject Classification. Primary 18G35; Secondary 16W50, 55U15. R((t))have vanishing homology in all degrees.

Let $R$ be a strongly $\mathbb {Z}^2$ -graded ring, and let $C$ be a bounded chain complex of finitely generated free $R$ -modules. The complex $C$ is $R_{(0,0)}$ -finitely dominated, or of type $FP$ over $R_{(0,0)}$ , if it is chain homotopy equivalent to a bounded complex of finitely generated projective $R_{(0,0)}$ -modules. We show that this happens if and only if $C$ becomes acyclic after taking tensor product with a certain eight rings of formal power series,  the graded analogues of classical Novikov rings. This extends results of Ranicki,  Quinn and the first author on Laurent polynomial rings in one and two indeterminates.