Finite domination and novikov rings. Iterative approach

摘要

Abstract Suppose C is a bounded chain complex of finitely generated free modules over the Laurent polynomial ring L = R[x,x ~(-1)]. Then C is R-finitely dominated, i.e. homotopy equivalent over R to a bounded chain complex of finitely generated projective R-modules if and only if the two chain complexes C - L R((x)) and C-L R((x~(-1))) are acyclic, as has been proved by Ranicki (Ranicki, Finite domination and Novikov rings, Topology 34(3) (1995), 619-632). Here R((x))=R[[x]][x ~(-1)] and R((x ~(-1))) = R[[x ~(-1)]][x] are rings of the formal Laurent series, also known as Novikov rings. 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.