  The result of this paper is the determination of the cohomology of Artin
groups of type A_n, B_n and \tilde{A}_{n} with non-trivial local coefficients.
The main result is an explicit computation of the cohomology of the Artin group
of type B_n with coefficients over the module \Q[q^{\pm 1},t^{\pm 1}]. Here the
first (n-1) standard generators of the group act by (-q)-multiplication, while
the last one acts by (-t)-multiplication. The proof uses some technical results
from previous papers plus computations over a suitable spectral sequence. The
remaining cases follow from an application of Shapiro's lemma, by considering
some well-known inclusions: we obtain the rational cohomology of the Artin
group of affine type \tilde{A}_{n} as well as the cohomology of the classical
braid group {Br}_{n} with coefficients in the n-dimensional representation
presented in \cite{tong}. The topological counterpart is the explicit
construction of finite CW-complexes endowed with a free action of the Artin
groups, which are known to be K(\pi,1) spaces in some cases (including finite
type groups). Particularly simple formulas for the Euler-characteristic of
these orbit spaces are derived.
