On the cohomology of Artin groups in local systems and the associated Milnor fiber

Let W be a finite irreducible Coxeter group and let X-W be the classifying space for G(W), the associated Artin group. If A is a commutative unitary ring, we consider the two local systems L-q and L'(q) over X-W, respectively over the modules A[q, q(-1)] and A[[q, q(-1)]] ,given by sending each standard generator of G(W) into the automorphism given by the multiplication by q. We show that H*(X-W, L'(q)) = H*(+1) (X-W, L-q) and we generalize this relation to a particular class of algebraic complexes. We remark that H*(X-W, L'(q)) is equal to the cohomology with trivial coefficients A of the Milnor fiber of the discriminant bundle of the associated reflection group