On the non-neutral component of outer forms of the orthogonal group

Uriya A. First

doi:10.1016/j.jpaa.2020.106477

On the non-neutral component of outer forms of the orthogonal group

Uriya A. First

Department of Mathematics

University of Haifa

Research output: Contribution to journal › Article › peer-review

Abstract

Let (A,σ) be a central simple algebra with an orthogonal involution. It is well-known that O(A,σ) contains elements of reduced norm −1 if and only if the Brauer class of A is trivial. We generalize this statement to Azumaya algebras with orthogonal involution over semilocal rings, and show that the “if” part fails if one allows the base ring to be arbitrary.

Original language	American English
Article number	106477
Journal	Journal of Pure and Applied Algebra
Volume	225
Issue number	1
DOIs	https://doi.org/10.1016/j.jpaa.2020.106477
State	Published - Jan 2021

Keywords

Azumaya algebra
Brauer group
Central simple algebra
Involution
Orthogonal group
Reduced norm

All Science Journal Classification (ASJC) codes

Algebra and Number Theory

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10.1016/j.jpaa.2020.106477

Cite this

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abstract = "Let (A,σ) be a central simple algebra with an orthogonal involution. It is well-known that O(A,σ) contains elements of reduced norm −1 if and only if the Brauer class of A is trivial. We generalize this statement to Azumaya algebras with orthogonal involution over semilocal rings, and show that the “if” part fails if one allows the base ring to be arbitrary.",

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AB - Let (A,σ) be a central simple algebra with an orthogonal involution. It is well-known that O(A,σ) contains elements of reduced norm −1 if and only if the Brauer class of A is trivial. We generalize this statement to Azumaya algebras with orthogonal involution over semilocal rings, and show that the “if” part fails if one allows the base ring to be arbitrary.

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KW - Brauer group

KW - Central simple algebra

KW - Involution

KW - Orthogonal group

KW - Reduced norm

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On the non-neutral component of outer forms of the orthogonal group

Abstract

Keywords

All Science Journal Classification (ASJC) codes

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