24 research outputs found
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
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
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
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