The Brauer group and the center of generic matrices
type: Item
instance of: scholarly article
title: The Brauer group and the center of generic matrices
zbMATH Open document ID: 0586.13005
DOI: 10.1016/0021-8693(85)90073-0
author: David Saltman
published in: Journal of Algebra
publication date: 1985-01-01T00:00:00Z
review text: Let k be a field and let R(k,n,r) be the ring generated by r \(n\times n\)- generic matrices. It is a domain with a central localization UD(k,n,r) which is a division algebra of degree n over its center Z(k,n,r). For all n, Z(k,n,r) is unirational and is known to be rational for \(n=2, 3\) and 4. In view of a construction, due to Artin and Mumford, of a field extension K/k, which is unirational but not rational, it is a natural question to ask if there exists a projective k-variety V such that k(V), the function field of V, is Z(k,n,r) and such that the Brauer group Br(V) is unequal to Br(k). In fact, the author shows that for any k,n,r the Brauer group of a V such that \(k(V)=Z(k,n,r)\) must be Br(k). As a step in the argument, he also shows that if W is the Brauer-Severi variety of UD(k,n,r) defined over \(K=Z(k,n,r)\), then K(W)/k is rational. In the last section, the author constructs a smooth subring T of Z(k,n,r) such that Br(T) viewed as in Br(Z(k,n,r)), is generated by the class of UD(k,n,r).
Mathematics Subject Classification ID: 14F22
Mathematics Subject Classification ID: 14M20
Mathematics Subject Classification ID: 16H05
Mathematics Subject Classification ID: 16Kxx
Mathematics Subject Classification ID: 16Rxx
Mathematics Subject Classification ID: 12G05
zbMATH DE Number: 3939468
zbMATH Keywords: Brauer group
zbMATH Keywords: unirational fields
zbMATH Keywords: generic matrices
zbMATH Keywords: Brauer-Severi variety
reviewed by: M.-A. Knus
MaRDI profile type: Publication
full work available at URL: 0021-8693(85)90073-0
OpenAlex ID: W2033044203
cites work: Generic splitting fields of central simple algebras
cites work: Some Elementary Examples of Unirational Varieties Which are Not Rational
cites work: Q5573504
cites work: The Brauer Group of a Commutative Ring
cites work: Separable algebras over commutative rings
cites work: The polynomial identities of matrices
cites work: Invariants and the ring of generic matrices
cites work: A cohomological interpretation of Brauer groups of rings
cites work: PI-algebras. An introduction
cites work: Generic splitting fields
cites work: Rational functions invariant under a finite Abelian group
cites work: Q5611879
cites work: Q3871006
cites work: Q4055755
cites work: On the Galois cohomology of the projective linear group and its applications to the construction of generic splitting fields of algebras
cites work: Q3910639
cites work: Retract rational fields and cyclic Galois extensions
cites work: Q5795154
cites work: Q5732721
cites work: Q3856855
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description: scientific article; zbMATH DE number 3939468