Multipliers of improper similitudes
type: Item
instance of: scholarly article
title: Multipliers of improper similitudes
zbMATH Open document ID: 1160.11321
published in: Documenta Mathematica
publication date: 2004-08-31T00:00:00Z
full work available at URL: https://eudml.org/doc/129829
full work available at URL: 11.html
review text: Let \(A\) be a central simple algebra of degree \(2n\) over a field \(k\) (of char. not \(2\)), with an involution \(\sigma\) of orthogonal type. An element \(g \in A^{\times}\) is a similitude if \(\mu(g) = \sigma(g)g\) is central. The reduced norm of such an element is \(\pm \mu(g)^n\); the similitude is proper if its reduced norm is \(\mu(g)^n\), improper otherwise. Let \(G(A)\) denote the image of \(\mu\) in \(k^{\times}\) . Elements of \(G(A)\) are called multipliers. The relation of \(G(A)\) to the discriminant \(d\) of \((A,\sigma)\) was studied by \textit{A. S. Merkurjev} and \textit{J.-P. Tignol} in [J. Reine Angew. Math. 461, 13--47 (1995; Zbl 0819.16015)]. In this paper the similitudes are related to the next invariant, namely the Clifford algebra \(C = C(A,\sigma)\). Let \(Z\) denote the center of \(C\), a quadratic étale \(k\)-algebra. Let \(H^i(k)\) denote the \(i\)th Galois cohomology of \(k\) with coefficients in the \(2\)-primary group of roots of unity (on which the absolute Galois group acts by raising the natural action to the \(i-1\) power). In particular \([d] \in H^1(k)\) and \([C] \in H^2(k)\). Let \(H^2(k)/A\) denote the quotient of \(H^2(k)\) by the subgroup \(\left<[A]\right>\), and \(H^3(k)/A\) the quotient of \(H^3(k)\) by \(k^\times \cup [A]\). Merkurjev and Tignol showed that \(\lambda \cup d = 0\) in \(H^2(k)/A\) for any multiplier \(\lambda\). The main results of this paper are as follows: First, when \(A\) is split by \(Z\), there is an element \(\gamma \in H^2(k)\) whose restriction to \(Z\) equals \(C\); then \(\lambda \cup \gamma = 0\) in \(H^3 / A\) for any multiplier \(\lambda\). Next, let \(\lambda\) be a proper multiplier, then \(\lambda = \text{Norm}_{Z/k}(z)\) for some \(z \in Z^\times\), and \(\text{cor} (z \cup [C]) = 0\) in \(H^3(k)/A\) (here \(\text{cor}\) is the corestriction from \(Z\) to \(k\)). An inverse to these results is given when \(A\) has degree at most \(6\): If \(A\) is split by \(Z\), then \(\lambda \in k^…
reviewed by: Uzi Vishne
Mathematics Subject Classification ID: 11E72
Mathematics Subject Classification ID: 12G05
Mathematics Subject Classification ID: 16K20
zbMATH DE Number: 2095724
zbMATH Keywords: central simple algebras with involutions
zbMATH Keywords: hermitian forms
zbMATH Keywords: Clifford algebra
zbMATH Keywords: similitudes
author: Raman Preeti
author: Jean-Pierre Tignol
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 2095724