Finite domination and <scp>Novikov</scp> homology over strongly -graded rings

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