A stability-like theorem for cohomologies of pure braid groups of the series A, B and D

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 coefficients 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 computes this number of copies.