Finite domination and Novikov rings. Iterative approach

TL;DR

This paper generalizes a criterion for finite domination of chain complexes over Laurent polynomial rings, extending it to multiple indeterminates using Novikov rings, with implications for algebraic topology and K-theory.

Contribution

It introduces a new iterative approach to detect finite domination over Laurent rings with multiple variables, broadening previous single-variable results.

Findings

Generalization of Ranicki's criterion to multiple indeterminates

Development of an iterative method for complex analysis

Enhanced understanding of Novikov rings in algebraic topology

Abstract

Suppose C is a bounded chain complex of finitely generated free modules over the Laurent polynomial ring L = R[x,1/x]. Then C is R-finitely dominated, ie, homotopy equivalent over R to a bounded chain complex of finitely generated projective R-modules, if and only if the two chain complexes C((x)) and C((1/x)) are acyclic, as has been proved by Ranicki. Here C((x)) is the tensor product over L of C with the Novikov ring R((x)) = R[[x]][1/x] (also known as the ring of formal Laurent series in x); similarly, C((1/x)) is the tensor product over L of C with the Novikov ring R((1/x)) = R[[1/x]][x]. In this paper, we prove a generalisation of this criterion which allows us to detect finite domination of bounded below chain complexes of projective modules over Laurent rings in several indeterminates.