Vector bundles on the projective line and finite domination of chain complexes

Finitely dominated chain complexes over a Laurent polynomial ring in one indeterminate are characterised by vanishing of their Novikov homology. We present an algebro-geometric approach to this result, based on extension of chain complexes to sheaves on the projective line. We also discuss the K-theoretical obstruction to extension.Comment: v1: 11 page

A cohomological interpretation of Brion's formula

A subset K of R^n gives rise to a formal Laurent series with monomials corresponding to lattice points in K. Under suitable hypotheses, this series represents a rational function R(K). Michel Brion has discovered a surprising formula relating the rational function R(P) of a lattice polytope P to the sum of rational functions corresponding to the supporting cones subtended at the vertices of P. The result is re-phrased and generalised in the language of cohomology of line bundles on complete toric varieties. Brion's formula is the special case of an ample line bundle on a projective toric variety. - The paper also contains some general remarks on the cohomology of torus-equivariant line bundles on complete toric varieties, valid over noetherian ground rings.Comment: 15 pages; uses Paul Taylor's "diagrams" and "QED" macro packages; v2: "noetherian" hypothesis removed, minor typos correcte

Total Cofibres of Diagrams of Spectra

If Y is a diagram of spectra indexed by an arbitrary poset C together with a specified sub-poset D, we define the total cofibre \Gamma (Y) of Y as the strict cofibre of the map from hocolim_D (Y) to hocolim_C (Y). We construct a comparison map from the homotopy limit of Y to a looping of a fibrant replacement of Gamma (Y), and characterise those poset pairs (C,D) for which this comparison map is a stable equivalence. The characterisation is given in terms of stable cohomotopy of spaces related to C and D. For example, if C is a finite polytopal complex with underlying space an m-ball with boundary sphere D, then holim_C (Y) and \Gamma(Y) agree up to m-fold looping and up to stable equivalence. As an application of the general result we give a spectral sequence for the homotopy groups of \Gamma(Y) with E_2-term involving higher derived inverse limits of \pi_* (Y), generalising earlier constructions for space-valued diagrams indexed by the face lattice of a polytope.Comment: 11 pages; LaTeX source code uses Paul Taylor's "diagrams" and "QED" macro packages; some diagrams may not display correctly with DVI viewers; paper also available at http://nyjm.albany.edu:8000/j/2005/11-16.htm

Double complexes and vanishing of Novikov cohomology

We consider a non-standard totalisation functor to produce a cochain complex from a given double complex: instead of sums or products, totalisation is defined via truncated products of modules. We give an elementary proof of the fact that a double complex with exact rows (resp, columns) yields an acyclic cochain complex under totalisation using right (resp, left) truncated products. As an application we consider the algebraic mapping torus T(h) of a self map h of a cochain complex C. We show that if C consists of finitely presented modules then T(h) has trivial negative Novikov cohomology; if in addition h is a quasi-isomorphism, then T(h) has trivial positive Novikov cohomology as well. As a consequence we obtain a new proof that a finitely dominated cochain complex over a Laurent polynomial ring has trivial Novikov cohomology.Comment: 6 pages; diagrams typeset with Paul taylors "diagrams" macro package; v2: 7 pages, expanded introduction, minor changes in exposition; v3: minor changes to abstract, typos correcte

Finite domination and Novikov homology over strongly Z-graded rings

Let L be a strongly Z-graded ring, and let C be a bounded chain complex of finitely generated L-modules. We give a homological characterisation of when C is homotopy equivalent, over L_0, to a bounded complex of finitely generated projective L_0-modules, generalising known results for twisted Laurent polynomial rings.Comment: 22 pages; v2: changed example in introduction, and corrected minor misprint