Degree four cohomological invariants for certain central simple algebras
type: Item
instance of: scholarly article
title: Degree four cohomological invariants for certain central simple algebras
zbMATH Open document ID: 1475.16017
published in: Journal of Algebra
publication date: 2019-03-25T00:00:00Z
review text: The author introduces in this paper an invariant of bi-quaternion algebras \(A=(\alpha,\beta)_{2,F} \otimes (\gamma,\delta)_{2,F}\) over fields \(F\) of \(\operatorname{char}(F)\neq 2\) under the condition that \(-1\) is the sum of two squares in \(F\) and \((-1)\cup A\) is trivial as an element of \(H^3(F,\mu_2^{\otimes 3})\). This invariant is simply the symbol \((\alpha,\beta,\gamma,\delta)\) in \(H^4(F,\mu_2^{\otimes 4})\), and the proof relies on the chain lemma that the author provided in an earlier work [J. Algebra 350, No. 1, 170--173 (2012; Zbl 1253.16020)]. This invariant coincides with the familiar divided power operator when \(\sqrt{-1}\in F\). Another invariant the author introduces concerns sums \(C\otimes \omega\), where \(\omega \in Br_2(F)\) and \(C\) is a central simple algebra of degree dividing 4. This invariant is \((C^{\otimes 2})\cup \omega\) as an element of \(H^4(F,\mu_2^{\otimes 4})\).
reviewed by: Adam Chapman
Mathematics Subject Classification ID: 16E99
Mathematics Subject Classification ID: 11R52
Mathematics Subject Classification ID: 16H05
zbMATH DE Number: 7041664
zbMATH Keywords: Brauer group
zbMATH Keywords: biquaternion
zbMATH Keywords: cup-product
zbMATH Keywords: cyclic elements
zbMATH Keywords: divided power operation
author: A. S. Sivatski
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2019.01.037
OpenAlex ID: W2922023574
cites work: Q3516073
cites work: Some applications of Jordan norms to involutorial simple associative algebras
cites work: Q5484519
cites work: Comparison of some field invariants
cites work: Q4698634
cites work: Q3160258
cites work: Biquaternion algebras and quartic extensions
cites work: FIELDS OF DEFINITION FOR DIVISION ALGEBRAS
cites work: $ K$-COHOMOLOGY OF SEVERI-BRAUER VARIETIES AND THE NORM RESIDUE HOMOMORPHISM
cites work: Zero Divisors in Tensor Products of Division Algebras
cites work: The trace form of a central simple algebra of degree 4.
cites work: The chain lemma for biquaternion algebras.
cites work: Corps a Involution Neutralises par une Extension Abelienne Elementaire
cites work: Operations in Milnor \(K\)-theory
Wikidata QID: Q128226085
DOI: 10.1016/J.JALGEBRA.2019.01.037
description: scientific article; zbMATH DE number 7041664