For strongly Z^2-graded rings, a chain complex is R_{(0,0)}-finitely dominated iff it is acyclic after base change to eight graded Novikov rings.
K -theory of non-linear projective toric varieties
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Finite domination and Novikov homology over strongly $\mathbb{Z}^2$-graded rings
For strongly Z^2-graded rings, a chain complex is R_{(0,0)}-finitely dominated iff it is acyclic after base change to eight graded Novikov rings.