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

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