Cohomological invariants of algebras with involution
type: Item
instance of: scholarly article
title: Cohomological invariants of algebras with involution
zbMATH Open document ID: 0904.16009
published in: Journal of Algebra
publication date: 1998-11-24T00:00:00Z
review text: Let \(A\) be a finite-dimensional central simple algebra over a field of characteristic different from \(2\). An involution on \(A\) is a ring anti-automorphism of period \(2\). To every involution \(\sigma\) on \(A\), the author attaches the quadratic form \(T_\sigma\colon A\to F\) which maps \(x\in A\) to the reduced trace of \(\sigma(x)x\), where \(F\) is the subfield of the center fixed under \(\sigma\), and the subform \(T_\sigma^+\) which is the restriction of \(T_\sigma\) to the \(F\)-vector space of symmetric elements. The main purpose of the paper is to compute the determinant and Hasse invariant of these quadratic forms, which may be regarded as invariants of the involution \(\sigma\). If the involution is of the second kind, i.e., if the center \(Z\) of \(A\) is a quadratic extension of \(F\), the author also defines a new cohomological invariant \({\mathcal D}(A,\sigma)\) which she calls the determinant class of \(A\). This invariant lies in the quotient of the cohomology group \(H^2(F,Z^1_n)\) by an action of \(\mu_2\), the group of order \(2\), where \(n\) is the degree of \(A\) and \(Z^1_n\) is the \(F\)-algebraic group of \(n\)-th roots of unity of norm \(1\) in \(Z\). The points of \(Z^1_n\) over a separable closure of \(F\) are the \(n\)-th roots of unity, with a twisted Galois action. If \(n\) is even, there is a canonical map \(H^2(F,Z^1_n)\to H^2(F,\mu_2)\) which is compatible with the action of \(\mu_2\). The image of \({\mathcal D}(A,\sigma)\) under this map is called the determinant class mod \(2\) and denoted \(D(A,\sigma)\). This element is explicitly computed in some special cases, and it is related to the Hasse invariant of \(T_\sigma^+\). The determinants of \(T_\sigma\) and \(T_\sigma^+\) are shown to depend only on \(Z\), and the Hasse invariant of \(T_\sigma\) is trivial. If \(\sigma\) is of the first kind, i.e., if \(F\) is the center of \(A\), the author shows that the determinant of \(T_\sigma\) is trivial and that the determinant …
Mathematics Subject Classification ID: 16K20
Mathematics Subject Classification ID: 16W10
Mathematics Subject Classification ID: 11E04
zbMATH DE Number: 1053885
zbMATH Keywords: central simple algebras
zbMATH Keywords: involutions
zbMATH Keywords: trace forms
zbMATH Keywords: quadratic forms
zbMATH Keywords: determinants
zbMATH Keywords: Hasse invariants
zbMATH Keywords: cohomological invariants
zbMATH Keywords: determinant classes
author: Anne Quéguiner-Mathieu
reviewed by: Jean-Pierre Tignol
MaRDI profile type: Publication
full work available at URL: jabr.1996.7015
OpenAlex ID: W2039916405
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DOI: 10.1006/JABR.1996.7015
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description: scientific article; zbMATH DE number 1053885