---
_id: '1634902'
abstract:
- lang: eng
text: 'Let G and E stand for one of the following pairs of groups: Either G is the
general quadratic group U(2n, R, Lambda), n >= 3, and E its elementary subgroup
EU(2n, R, Lambda). for an almost commutative form ring (R, Lambda). or G is the
Chevalley group G(Phi, R) of type Phi, and E its elementary subgroup E(Phi, R),
where Phi is a reduced irreducible root system of rank >= 2 and R is commutative.
Using Bak''s localization-completion method in [A. Bak, Nonabelian K-theory: The
nilpotent class of K, and general stability, K-Theory 4 (4) (1991) 363-397], it
was shown in [R. Hazrat, Dimension theory and nonstable K-1 of quadratic modules,
K-Theory 514 (2002) 1-35 and R. Hazrat, N. Vavilov, K-1 of Chevalley groups are
nilpotent, J. of Pure and Appl. Algebra 179 (2003) 99-116] that G/E is nilpotent
by abelian, when R has finite Bass-Serre dimension. In this note, we combine localization-completion
with a version of Stein''s relativization [M.R. Stein, Relativizing functors on
rings and algebraic K-theory, J. Algebra 19 (1) (1971) 140-152], which is applicable
to our situation [A. Bak, N. Vavilov, Structure of hyperbolic unitary groups I,
Elementary subgroups. Algebra Colloq. 7 (2) (2000) 159-196], and carry over the
results in the latter of the two references cited above to the relative case.
In other words, we prove that not only absolute K, functors, but also the relative
K, functors, are nilpotent by abelian. (C) 2008 Elsevier B.V. All rights reserved.'
article_type: original
author:
- first_name: Anthony
full_name: Bak, Anthony
id: '10484'
last_name: Bak
- first_name: Roozbeh
full_name: Hazrat, Roozbeh
last_name: Hazrat
- first_name: N.
full_name: Vavilov, N.
last_name: Vavilov
citation:
ama: 'Bak A, Hazrat R, Vavilov N. Localization-completion strikes again: Relative
K-1 is nilpotent by abelian. *Journal of Pure and Applied Algebra*. 2009;213(6):1075-1085.'
angewandte-chemie: 'A. Bak, R. Hazrat, and N. Vavilov, “Localization-completion
strikes again: Relative K-1 is nilpotent by abelian”, *Journal of Pure and
Applied Algebra*, **2009**, *213*, 1075-1085.'
apa: 'Bak, A., Hazrat, R., & Vavilov, N. (2009). Localization-completion strikes
again: Relative K-1 is nilpotent by abelian. *Journal of Pure and Applied Algebra*,
*213*(6), 1075-1085. doi:10.1016/j.jpaa.2008.11.014'
apa_indent: 'Bak,
A., Hazrat, R., & Vavilov, N. (2009). Localization-completion strikes again:
Relative K-1 is nilpotent by abelian. *Journal of Pure and Applied Algebra*,
*213*(6), 1075-1085. doi:10.1016/j.jpaa.2008.11.014

'
bio1: 'Bak A, Hazrat R, Vavilov N (2009)

Localization-completion strikes again:
Relative K-1 is nilpotent by abelian.

Journal of Pure and Applied Algebra
213(6): 1075-1085.'
chicago: 'Bak,
Anthony, Hazrat, Roozbeh, and Vavilov, N. 2009. “Localization-completion strikes
again: Relative K-1 is nilpotent by abelian”. *Journal of Pure and Applied
Algebra* 213 (6): 1075-1085.

'
default: 'Bak A, Hazrat R, Vavilov N (2009)

*Journal of Pure and Applied
Algebra* 213(6): 1075-1085.'
dgps: 'Bak,
A., Hazrat, R. & Vavilov, N. (2009). Localization-completion strikes again:
Relative K-1 is nilpotent by abelian. *Journal of Pure and Applied Algebra*,
*213*(6), 1075-1085. doi:10.1016/j.jpaa.2008.11.014.

'
frontiers: 'Bak, A., Hazrat, R., and Vavilov, N. (2009). Localization-completion
strikes again: Relative K-1 is nilpotent by abelian. *Journal of Pure and Applied
Algebra* 213, 1075-1085.'
harvard1: 'Bak, A., Hazrat, R., & Vavilov, N., 2009. Localization-completion
strikes again: Relative K-1 is nilpotent by abelian. *Journal of Pure and Applied
Algebra*, 213(6), p 1075-1085.'
ieee: ' A. Bak, R. Hazrat, and N. Vavilov, “Localization-completion strikes again:
Relative K-1 is nilpotent by abelian”, *Journal of Pure and Applied Algebra*, vol.
213, 2009, pp. 1075-1085.'
lncs: ' Bak, A., Hazrat, R., Vavilov, N.: Localization-completion strikes again:
Relative K-1 is nilpotent by abelian. Journal of Pure and Applied Algebra. 213,
1075-1085 (2009).'
mla: 'Bak, Anthony, Hazrat, Roozbeh, and Vavilov, N. “Localization-completion strikes
again: Relative K-1 is nilpotent by abelian”. *Journal of Pure and Applied
Algebra* 213.6 (2009): 1075-1085.'
wels: 'Bak, A.; Hazrat, R.; Vavilov, N. (2009): Localization-completion strikes
again: Relative K-1 is nilpotent by abelian *Journal of Pure and Applied Algebra*,213:(6):
1075-1085.'
date_created: 2010-04-29T13:07:28Z
date_updated: 2019-05-06T13:46:57Z
department:
- _id: '10020'
doi: 10.1016/j.jpaa.2008.11.014
external_id:
isi:
- '000264415800015'
intvolume: ' 213'
isi: 1
issue: '6'
language:
- iso: eng
page: 1075-1085
publication: Journal of Pure and Applied Algebra
publication_identifier:
issn:
- 0022-4049
publication_status: published
quality_controlled: '1'
status: public
title: 'Localization-completion strikes again: Relative K-1 is nilpotent by abelian'
type: journal_article
user_id: '67994'
volume: 213
year: '2009'
...