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. e of the classical braid group Brn this ideal is the principal ideal generated by the nth cyclotomic polynomial. 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).