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

TL;DR

This paper characterizes finitely dominated chain complexes over Laurent polynomial rings using homological methods, introducing new constructions like multi-complex totalisation and high-dimensional mapping tori.

Contribution

It provides a novel homological characterization of finitely dominated complexes over Laurent polynomial rings, utilizing advanced multi-complex and cubical diagram techniques.

Findings

Homological criteria for finite domination over Laurent rings

Development of a non-standard multi-complex totalisation method

Introduction of high-dimensional mapping torus construction

Abstract

We present a homological characterisation of those chain complexes of modules over a Laurent polynomial ring in several indeterminates which are finitely dominated over the ground ring (that is, are a retract up to homotopy of a bounded complex of finitely generated free modules). The main tools, which we develop in the paper, are a non-standard totalisation construction for multi-complexes based on truncated products, and a high-dimensional mapping torus construction employing a theory of cubical diagrams that commute up to specified coherent homotopies.