Rost's cohomological invariants in positive characteristic
type: Item
instance of: scholarly article
title: Rost's cohomological invariants in positive characteristic
zbMATH Open document ID: 0993.20031
DOI: 10.1023/A:1007839108933
author: P. Gille
published in: \(K\)-Theory
publication date: 2002-06-16T00:00:00Z
review text: 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 meth…
Mathematics Subject Classification ID: 20G10
Mathematics Subject Classification ID: 11E72
Mathematics Subject Classification ID: 20G15
Mathematics Subject Classification ID: 16K20
Mathematics Subject Classification ID: 19E08
zbMATH DE Number: 1563218
zbMATH Keywords: Galois cohomology
zbMATH Keywords: semi-simple algebraic groups
zbMATH Keywords: Rost invariants
zbMATH Keywords: Bruhat-Tits theory
zbMATH Keywords: Serre's conjecture II
reviewed by: Detlev Hoffmann
MaRDI profile type: Publication
full work available at URL: a:1007839108933
OpenAlex ID: W2039964837
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description: scientific article; zbMATH DE number 1563218