Degree three unramified cohomology of adjoint semisimple groups
\par Let \(G\) be an adjoint semisimple algebraic group over a field \(F\), and let \(F(G)\) denote the function field of \(G\). Let \(Z\) be the kernel of a simply connected cover \(\tilde{G} \to G\). Assume that \(|Z|\) is not divisible by the characteristic of \(F\). Let \(\mu_n\) denote the \(n\)th roots of unity in \(F_{\mathrm{sep}}\). For a field extension \(K/F\), let \(H^d(K)\) denote the Galois cohomology \(\operatorname{colim} H^d\left(K,\mu_n^{\otimes(d-1)}\right)\), where the colimit is taken over all \(n\) prime to the characteristic of \(F\). The unramified cohomology \(H^d(K)_{\mathrm{nr}}\) is the subgroup of all elements that map to zero under every residue homomorphsim \(H^d(K) \to H^{d-1}(\kappa(\nu))\) for a discrete valuation \(\nu\) of \(K\) trivial on \(F\), with \(\kappa(\nu)\) denoting the residue field. The goal of the paper is to determine the degree 3 reduced unramified cohomology \(\bar{H}^3(F(G))_{\mathrm{nr}} := H^3(F(G))_{\mathrm{nr}}/H^3(F)\). The author shows that there are precisely two cases, one where \(G\) is a projective unitary group (type \(^2A_n\) with \(n \equiv 3 \mod 4\)) and one where \(G\) is a projective orthogonal group (type \(^2D_n\) with \(n \geq 3\) odd), where \(\bar{H}^3(F(G))_{\mathrm{nr}}\) is non-zero. In both cases the cohomology group is cyclic of order 2. If the group \(G\) was rational, the cohomology would be zero, so these groups with non-trivial cohomology give new examples of non-rational algebraic groups. The proof makes use of the group of homomorphic cohomological invariants on \(G\) (with \(G\) considered as a functor from field extensions of \(F\) to groups). It is first shown that \(\bar{H}^3(F(G))_{\mathrm{nr}}\) lies within the subgroup of degree 3 cohomological invariants in \(\bar{H}^3(F(G))\). Then the degree 3 cohomological invariants are identified with a quotient of the Galois cohomology group \(H^2(F,Z^{\circ})\), where \(Z^{\circ}\) denotes the group dual to \(Z\). This latter connection makes use of classifying spaces of \(G\), \(\tilde{G}\), and \(Z\). A type-by-type analysis is then done to determine \(\bar{H}^3(F(G))_{\mathrm{nr}}\).
scientific article; zbMATH DE number 6928354
scientific article