On the vector bundles whose endomorphisms yield Azumaya algebras of cyclic type
type: Item
instance of: scholarly article
title: On the vector bundles whose endomorphisms yield Azumaya algebras of cyclic type
zbMATH Open document ID: 0664.14008
DOI: 10.1016/0021-8693(88)90109-3
author: Hajime Kaji
published in: Journal of Algebra
publication date: 1988-01-01T00:00:00Z
review text: L and M being two n-torsion line bundles over a non-singular quasi- projective variety X over an algebraically closed field of any characteristic, an associative Azumaya algebra A(L,M) of rank \(n^ 2\) is constructed, the elements of which correspond to projective space bundles of rank n over X. It is isomorphic to \({\mathcal E}nd(V)\) for some vector bundle \(V(L,M)\) uniquely determined by \(A(L,M)\) up to tensoring with line bundles, whereas the cup-product \(L\cup M\) is the image in the étale cohomology set \(H^ 2(X,\mu_ n)\) of some line bundle Z under the cycle map. Specializing to the case where X is a product of two elliptic curves and \(n=2\), the \(V(L,M)\)'s are classified and constructed from the Z's, examples are given and there is an application to the existence of rational points on X where the norm of a quaternion Azumaya algebra over X vanishes.
Mathematics Subject Classification ID: 14F05
Mathematics Subject Classification ID: 16H05
Mathematics Subject Classification ID: 14G05
zbMATH DE Number: 4085796
zbMATH Keywords: torsion line bundles
zbMATH Keywords: Azumaya algebra
zbMATH Keywords: existence of rational points
reviewed by: J. H. de Boer
MaRDI profile type: Publication
full work available at URL: 0021-8693(88)90109-3
OpenAlex ID: W2026914269
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description: scientific article; zbMATH DE number 4085796