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

Abstract

Abstract. 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 Rmodules. Our main result characterises 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, and their variants obtained by swapping x and y, and replacing either indeterminate by its inverse.

Finite domination and Novikov rings. Laurent polynomial rings in several variables

“…For two variables (n = 2) a version of this programme has been carried out by the authors in the paper [HQ14]. The present extension to more than two variables is non-trivial as it demands a theory of high-dimensional mapping tori, in turn resting on a theory of homotopy commutative cubical diagrams.…”

Finite domination and Novikov homology over strongly ℤ-graded rings

“…We start with recalling basic concepts from the theory of strongly graded rings. The "if" implication of the main theorem is verified in Part II, building on [HS17] and [HQ15]; the main point is to relate the given chain complex with a complex of diagrams which are the analogues of well-known line bundles on the scheme P 1 × P 1 . Part III focuses on the "only if" implication.…”

“…In the present paper we take the step to strongly Z 2 -graded rings, combining ideas from both of the aforementioned publications [HQ15] and [HS17]. Roughly speaking, a bounded chain complex C of finitely generated free modules over a strongly Z 2 -graded ring is finitely dominated over the degree-0 subring if and only if certain eight complexes induced from C are acyclic.…”

“…Structure of the paper. The paper can be seen as an amalgamation of the two publications [HQ15] and [HS17], combining the graded viewpoint of the latter with the elaborate homological algebra of the former. Replacing central indeterminates by inherently non-commutative structures is a non-trivial task, and it is rather surprising that with the right set-up the overall pattern of proof remains virtually unchanged.…”

“…a vertex is somewhat more demanding in terms of book-keeping. Full details are worked out in the proof of Lemma 4.5.1 in [HQ15], writing R * [x, x −1 ]((y)) in place of R[x, x −1 ]((y)) (and similar for the other rings); the proof carries over mutatis mutandis since the claim can be checked in each degree separately.…”