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We consider two natural embeddings between Artin groups: the groupG_{tilde{A}_{n-1}} of type tilde{A}_{n-1} embeds into the group G_{B_n} of typeB_n; G_{B_n} in turn embeds into the classical braid group Br_{n+1}:=G_{A_n} oftype A_n. The cohomologies of these groups are related, by standard results, ina precise way. By using techniques developed in previous papers, we giveprecise formulas (sketching the proofs) for the cohomology of G_{B_n} withcoefficients over the module Q[q^{+-1},t^{+-1}], where the action is(-q)-multiplication for the standard generators associated to the first n-1nodes of the Dynkin diagram, while is (-t)-multiplication for the generatorassociated to the last node.  As a corollary we obtain the rational cohomology for G_{tilde{A}_n} as wellas the cohomology of Br_{n+1} with coefficients in the (n+1)-dimensionalrepresentation obtained by Tong, Yang and Ma.  We stress the topological significance, recalling some constructions ofexplicit finite CW-complexes for orbit spaces of Artin groups. In case ofgroups of infinite type, we indicate the (few) variations to be done withrespect to the finite type case. For affine groups, some of these orbit spacesare known to be K(pi,1) spaces (in particular, for type tilde{A}_n).  We point out that the above cohomology of G_{B_n} gives (as a module over themonodromy operator) the rational cohomology of the fibre (analog to a Milnorfibre) of the natural fibration of K(G_{B_n},1) onto the 2-torus.

Cohomology of Artin groups of type tilde{A}_n, B_n and applications