Finite domination and Novikov rings: Laurent polynomial rings in two variables

Let C be a bounded cochain complex of finitely generated free modules over the Laurent polynomial ring L = R[x, x-1, y, y-1]. The complex C is called R-finitely dominated if it is homotopy equivalent over R to a bounded complex of finitely generated projective R-modules. Our main result characterizes R-finitely dominated complexes in terms of Novikov cohomology: C is R-finitely dominated if and only if eight complexes derived from C are acyclic; these complexes are C - L Rx, y[(xy) -1] and C - L R[x, x-1]y[y-1], and their variants obtained by swapping x and y, and replacing either indeterminate by its inverse.