Finite domination and Novikov rings. Laurent polynomial rings in two variables

TL;DR

This paper characterizes R-finitely dominated complexes over Laurent polynomial rings in two variables using Novikov cohomology, providing a criterion involving the acyclicity of eight derived complexes.

Contribution

It offers a new homotopy-theoretic characterization of finitely dominated complexes via Novikov cohomology and explicit acyclicity conditions.

Findings

R-finitely dominated complexes are characterized by acyclicity of eight derived complexes.

The characterization involves tensoring with specific Novikov rings and their variants.

Provides a criterion for finite domination in terms of Novikov cohomology.

Abstract

Let C be a bounded cochain complex of finitely generated free modules over the Laurent polynomial ring L = R[x,1/x,y,1/y]. The complex C is called R-finitely dominated if it is homotopy equivalent over R to a bounded complex of finitely generated projective R-modules. Our main result characterises R-finitely dominated complexes in terms of Novikov cohomology: C is R-finitely dominated if and only if eight complexes derived from C are acyclic; these complexes are obtained by tensoring C over L with R[[x,y]][1/xy] and R[x,1/x][[y]][1/y], and their variants obtained by swapping x and y, and replacing either indeterminate by its inverse.