  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.
