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

摘要

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 circle times(L) R[[x, y]][(xy)(-1)] and C circle times(L) R[x,x(-1)][[y]][y(-1)], and their variants obtained by swapping x and y, and replacing either indeterminate by its inverse.