Rost's cohomological invariants in positive characteristic
The author studies the Rost invariant \(r_F\colon H^1(F,G)\to H^3(F,\mathbb{Q}/\mathbb{Z}(2))=:H^3F\) in Galois cohomology, where \(G\) denotes a semi-simple simply connected algebraic group over a field \(F\). He provides two constructions of this invariant, one based on the \(\mathcal K\)-cohomology of \(G\) and due to \textit{H. Esnault, B. Kahn, M. Levine} and \textit{E. Viehweg} [J. Am. Math. Soc. 11, No. 1, 73-118 (1998; Zbl 1025.11009)], the other one based on the étale cohomology of \(G\) and due to \textit{J.-P. Serre} [Sém. Bourbaki, Exp. No. 783, Astérisque 227, 229-257 (1995; Zbl 0837.12003)] (the latter one provides such an invariant only for coefficients in \(\mathbb{Q}_\ell/\mathbb{Z}_\ell(2)\) for primes \(\ell\neq\text{char}(F)\)). The author remarks that the invariants defined in these two ways might not be identical, but that they generate in any case the same subgroup of \(H^3F\) which suffices for the applications the author has in mind. Let \(O\) be a complete discrete rank-\(1\) valuation ring, \(K\) its field of fractions with residue field \(F\), and let \(\mathfrak G\) be a semi-simple simply connected group scheme over \(O\) with special fibre \(G_F\). One of the main results of the paper is the commutativity (modulo an automorphism of \(\mathbb{Q}/\mathbb{Z}\)) of the diagram \[ \begin{tikzcd} H^1(K,{\mathfrak G}_K)\ar[r,"r_K"] & H^3K\\ H^1(F,G)\ar[r,"r_F" ']\ar[u] & H^3F\ar[u]\end{tikzcd} \] where both vertical morphisms are injective. This allows to deduce results in positive characteristic (for fields which are not necessarily perfect) from known facts in characteristic \(0\). For example, if \(r_K\) has trivial kernel, then so has \(r_F\). The author mentions that the commutativity would follow readily if Gersten's conjecture in \(K\)-theory were true for all smooth schemes of finite type over \(\text{Spec}(O)\), but this is known to be true only in a few cases. The author continues by generalizing various known results using his methods. For example, let \(D\) be a central simple algebra over \(F\) of index \(n\), let \(G=\text{SL}(D)\). \textit{A. A. Suslin} [J. Sov. Math. 30, 2556-2611 (1985); translation from Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat. 25, 115-207 (1984; Zbl 0558.12013)] showed if \(n\) is square free and invertible in \(F\), than the Rost invariant \(r_F\colon H^1(F,\text{SL}(D))=F^\times/\text{Nrd}(D^\times)\to H^3(F,\mathbb{Z}/n\mathbb{Z}(2))\) is injective. The author proves that this also holds after dispensing with the assumption on \(n\) being invertible. As a consequence, this leads to a characterization of fields of separable cohomological \(p\)-dimension \(\leq 2\) by means of the surjectivity of reduced norms also in positive characteristic. For groups of type \(G_2\), the author provides a new proof in characteristic \(2\) of a result due to Serre [loc. cit.] which states that \(H^1(F,G_2)\) is in bijection with the set of decomposable elements in \(H^3(F,\mathbb{Z}/2\mathbb{Z}(2))\) via the Rost invariant. He also proves the triviality of the kernel of the Rost invariant for groups of type \(F_4\) in all characteristics (this has been known previously only in characteristic \(\neq 2,3\)). All these results imply the validity of Serre's Conjecture II for groups of type \(^1A_n\), \(G_2\) and \(F_4\) in all characteristics.
scientific article; zbMATH DE number 1563218