Combinatorial covers and vanishing of cohomology

Abstract

We use a Mayer-Vietoris-like spectral sequence to establish vanishing results for the cohomology of complements of linear and elliptic hyperplane arrangements, as part of a more general framework involving duality and abelian duality properties of spaces and groups. In the process, we consider cohomology of local systems with a general, Cohen-Macaulay-type condition. As a result, we recover known vanishing theorems for rank-1 local systems as well as group ring coefficients, and obtain new generalizations.

34 pages. To appear in Selecta Mathematica

References in corpus (2)

• The polyhedral product functor: a method of computation for moment-angle complexes, arrangements and related spaces
• Toric complexes and Artin kernels

Cited by in corpus (9)

• Local systems on complements of arrangements of smooth, complex algebraic hypersurfaces
• Around the tangent cone theorem
• Abelian duality and propagation of resonance
• Homology, lower central series, and hyperplane arrangements
• Set of independencies and Tutte polynomial of matroids over a domain
• Action dimensions of some simple complexes of groups
• Complements of hyperplane arrangements as posets of spaces
• Milnor fibrations of arrangements with trivial algebraic monodromy
• From the Mayer-Vietoris spectral sequence to überhomology