The top-cohomology of Artin Groups with coefficients in rank-1 local systems over Z

Let W be a Coxeter group and let GW be the associated Artin group. We consider the local system over k(Gw, 1) with coefficients in R = ℤ[q, q-1] which associates to the standard generators of GW the multiplication by q. For the all list of finite irreducible Coxeter groups we calculate the top-cohomology of this local system. It turns out that the ideal which we compute is a sort of Alexander ideal for a hypersurface. In case of the classical braid group Brn this ideal is the principal ideal generated by the nth cyclotomic polynomial. We use these results to calculate the topological category of k(Gw, 1): we prove that it equals the obvious bound given by obstruction theory (so, in case of braid group Brn, it is exactly n). © 1997 Elsevier Science B.V.