Unramified cohomology of alternating groups. (Q657406)

Let \(G\) be a finite group, let \(k\) be an algebraically closed field, let \(V\) be a faithful linear representation of \(G\) over \(k\) and let \(K=k(V)^G\). Consider the maps \(s_i\colon H^i(G,\mathbb Z/l)\to H^i(K,\mathbb Z/l)\) induced by the map \(\mathrm{Gal\,}K\to G\). The kernel of \(s_i\) does not depend on the choice of a representation, which allows to define the stable cohomology group \(H^i_{k,s}(G,\mathbb Z/l)=H^i(G,\mathbb Z/l)/\ker s_i\). Then the unramified cohomology \(H^i_{k,\mathrm{un}}(G,\mathbb Z/l)\) is a subgroup of this group, consisting of the elements which map to zero by a composition of \(s_i\) with the residue maps for all divisorial valuations on \(K\). In general, for \(G\) a finite simple group, one conjectures that the groups \(H^i_{k,\mathrm{un}}(G,\mathbb Z/l)\) vanish for all \(i>0\), \(k\) and \(l\). The main result of this article establishes the vanishing in positive degrees of the unramified cohomology groups with \(\mathbb Z/l\)-coefficients for the alternating groups \(A_n\). The computation of the stable cohomology of \(A_n\) is also given.