1 0 Ju l 2 00 3 A stability theorem for cohomology of Pure Braid Groups of the series A , B

Consider the ring R := Q[τ, τ−1] of Laurent polynomials in the variable τ . The Artin’s Pure Braid Groups (or Generalized Pure Braid Groups) act over R, where the action of every standard generator is the multiplication by τ . In this paper we consider the cohomology of such groups with corefficients in the module R (it is well known that such cohomology is strictly related to the untwisted integral cohomology of the Milnor fibration naturally associated to the reflection arrangement). We give a sort of stability theorem for the cohomologies of the infinite series A, B and D, finding that these cohomologies stabilize, with respect to the natural inclusion, at some number of copies of the trivial R-module Q. We also give a formula which compute this number of copies.