This paper generalizes the notion of local coefficients and fundamental groups of spaces to simplicial coalgebras. We define a Hopf algebra π1(C) from a simplicial coalgebra C as a generalization of a fundamental group, and show that a module overπ1(C) corresponds to a local coefficient of C. As a consequence, the Hoschild cohomology of a Hopf algebraH with a coefficient M coincides with the co...

In this note I give a description of Lyubeznik’s local cohomology invariants for a certain natural class of local rings, namely the ones which have the same local cohomology vanishing as one expects from an isolated singularity. This strengthens our results of [BB04] while at the same time somewhat simplifying the proofs. Through examples I further point out the bad behavior of these invariants...

For a variety where a connected linear algebraic group acts with only finitely many orbits, each of which admits an attractive slice, we show that the stratification by orbits is perfect for equivariant intersection cohomology with respect to any equivariant local system. This applies to provide a relationship between the vanishing of the odd dimensional intersection cohomology sheaves and of t...

Given a family of local systems on the punctured Riemann sphere, its parabolic cohomology is a local system on the base space. We derive exact formulas for the monodromy of this local system. Using the comparison theorem between singular and étale cohomology, we can also (under some conditions) bound its field of moduli. As an application, we prove that the simple groups PSL2(p ) admit regular ...

We study cohomology with coefficients in a rank one local system on the complement of an arrangement of hyperplanes A. The cohomology plays an important role for the theory of generalized hypergeometric functions. We combine several known results to construct explicit bases of logarithmic forms for the only non-vanishing cohomology group, under some nonresonance conditions on the local system, ...

We use stratified Morse theory to construct a complex to compute the cohomology of the complement of a hyperplane arrangement with coefficients in a complex rank one local system. The linearization of this complex is shown to be the Aomoto complex of the arrangement. Using this result, we establish the relationship between the cohomology support loci of the complement and the resonance varietie...

This paper is a survey of our work based on the stratified Morse theory of Goresky and MacPherson. First we discuss the Morse theory of Euclidean space stratified by an arrangement. This is used to show that the complement of a complex hyperplane arrangement admits a minimal cell decomposition. Next we review the construction of a cochain complex whose cohomology computes the local system cohom...

In this paper we prove that over local or global fields of characteristic 0, the Corestriction Principle holds for kernel and image of all maps which are connecting maps in group cohomology and the groups of R-equivalences. Some related questions over arbitrary fields of characteristic 0 are also discussed. AMS Mathematics Subject Classification (1991): Primary 11E72, Secondary 18G50, 20G10 Pla...