Finite domination and Novikov homology over strongly $\mathbb{Z}^2$-graded rings

TL;DR

This paper characterizes when a bounded chain complex over a strongly -graded ring is finitely dominated over the degree zero part, using tensor products with graded Novikov rings, extending prior results.

Contribution

It extends the characterization of finite domination and Novikov homology from Laurent polynomial rings to strongly -graded rings with formal power series.

Findings

A chain complex is finitely dominated over R_{(0,0)} if and only if it becomes acyclic after tensoring with certain graded Novikov rings.

The result generalizes classical Laurent polynomial ring cases to strongly -graded rings.

Provides a new criterion for finite domination in the context of graded ring structures.

Abstract

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.