Double complexes and vanishing of Novikov cohomology

Abstract

We consider a non-standard totalisation functor to produce a cochain complex from a given double complex: instead of sums or products, totalisation is defined via truncated products of modules. We give an elementary proof of the fact that a double complex with exact rows (resp, columns) yields an acyclic cochain complex under totalisation using right (resp, left) truncated products. As an application we consider the algebraic mapping torus T(h) of a self map h of a cochain complex C. We show that if C consists of finitely presented modules then T(h) has trivial negative Novikov cohomology; if in addition h is a quasi-isomorphism, then T(h) has trivial positive Novikov cohomology as well. As a consequence we obtain a new proof that a finitely dominated cochain complex over a Laurent polynomial ring has trivial Novikov cohomology.