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Finite domination and Novikov rings. Iterative approach
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.Comment: 15 pages; diagrams typeset with Paul Taylor's "diagrams" macro
package. Version 2: clarified proof of main theorem, fixed minor typos;
Version 3: expanded introduction, now 16 pages; Version 4: corrected mistake
on functoriality of mapping tor
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