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

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	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

Bibliographical note

Publisher Copyright:
© 2020 Elsevier B.V.

Keywords

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

ASJC Scopus subject areas

Algebra and Number Theory

Access to Document

10.1016/j.jpaa.2020.106477

Cite this

@article{0a1ca4c07a9d4c73b7db06bd4b59b9a0,

title = "On the non-neutral component of outer forms of the orthogonal group",

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.",

keywords = "Azumaya algebra, Brauer group, Central simple algebra, Involution, Orthogonal group, Reduced norm",

author = "First, {Uriya A.}",

note = "Publisher Copyright: {\textcopyright} 2020 Elsevier B.V.",

year = "2021",

month = jan,

doi = "10.1016/j.jpaa.2020.106477",

language = "English",

volume = "225",

journal = "Journal of Pure and Applied Algebra",

issn = "0022-4049",

publisher = "Elsevier",

number = "1",

}

TY - JOUR

T1 - On the non-neutral component of outer forms of the orthogonal group

AU - First, Uriya A.

PY - 2021/1

Y1 - 2021/1

N2 - 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.

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.

KW - Azumaya algebra

KW - Brauer group

KW - Central simple algebra

KW - Involution

KW - Orthogonal group

KW - Reduced norm

UR - http://www.scopus.com/inward/record.url?scp=85086772648&partnerID=8YFLogxK

U2 - 10.1016/j.jpaa.2020.106477

DO - 10.1016/j.jpaa.2020.106477

M3 - Article

AN - SCOPUS:85086772648

SN - 0022-4049

VL - 225

JO - Journal of Pure and Applied Algebra

JF - Journal of Pure and Applied Algebra

IS - 1

M1 - 106477

ER -

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

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

Access to Document

Other files and links

Fingerprint

Cite this