Hostname: page-component-76fb5796d-2lccl Total loading time: 0 Render date: 2024-04-26T20:12:29.337Z Has data issue: false hasContentIssue false
  • English
  • Français

Injective stability for unitary K1, revisited

Published online by Cambridge University Press:  06 March 2013

S. Sinchuk
S. Sinchuk*
Affiliation:
Department of Mathematics and Mechanics, Saint-Petersburg State University, Saint-Petersburg, Russiasinchukss@yandex.ru
Get access

Abstract

We prove the injective stability theorem for unitary K1 under the usual stable range condition on the ground ring. This improves the stability theorem of A. Bak, V. Petrov and G. Tang where a stronger Λ-stable range condition was used.

Keywords

Unitary groupsparabolic subgroupsstable range conditionsDennis-Vaserstein type decompositions19B9919C99
Type
Research Article
Information
Journal of K-Theory , Volume 11 , Issue 2 , April 2013 , pp. 233 - 242
Copyright
Copyright © ISOPP 2013 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Bak, A., K-theory of forms, Ann. Math. Stud. 98, 1981Google Scholar
2.Bak, A., Petrov, V. A., Tang, G., Stability for quadratic K1, K-Theory 30 (1) (2003) 1–11Google Scholar
3.Bak, A., Tang, G., Stability for Hermitian K1, J. Pure Appl. Algebra 150 (2000) 107–121Google Scholar
4.Bak, A., Vavilov, N., Structure of hyperbolic unitary groups. I. Elementary subgroups, Algebra Colloq. 7 (2) (2000) 159–196Google Scholar
5.Bass, H., Algebraic K-theory, Benjamin, New York. 1968Google Scholar
6.Hahn, A. J., O'Meara, O. T., The classical groups and K-theory, Springer, Berlin et. al. 1989CrossRefGoogle Scholar
7.Hazrat, R., Dimension Theory and Nonstable K1 of Quadratic Modules, K-Theory 27 (2002) 293–328Google Scholar
8.Hazrat, R., Vavilov, N., Bak's work on the K-theory of rings, J. K-Theory 4 (1) (2009) 1–65Google Scholar
9.Knus, M.-A., Quadratic and Hermitian Forms over Rings, Grundl. Math. Wiss. 294, Springer, Berlin, 1991Google Scholar
10.Petrov, V.A., Overgroups of unitary groups, K-Theory 29 (2003) 147–174CrossRefGoogle Scholar
11.Petrov, V.A., Odd unitary groups, Journal of Math. Sciences 130 (3) (2005) 4752–4766CrossRefGoogle Scholar
12.Plotkin, E., Stein, M. R., Vavilov, N., K1-stability for classical groups revisited. (to appear)Google Scholar
13.Smolensky, A., Sury, B., Vavilov, N., Gauss Decomposition for Chevalley groups, revisited, International Journal of Group Theory 1 (1) (2012) 3–16Google Scholar
14.Stein, M. R., Stability theorems for K1, K2 and related functors modeled on Chevalley groups, Japan J. Math. 4 (1) (1978) 77–108CrossRefGoogle Scholar
15.Tang, G., Hermitian Groups and K-Theory, K-Theory 13 (1998) 209–267Google Scholar
16.Vaserstein, L. N., Stabilization of unitary and orthogonal groups over a ring, Math. USSR Sbornik 10 (1970) 307–326Google Scholar
17.Vaserstein, L. N., You, H., Normal subgroups of classical groups over rings, J. Pure and Applied Algebra 105 (1) (1995) 93–105Google Scholar
18.Vavilov, N. A., Sinchuk, S. S., Dennis-Vaserstein type decompositions, Journal of Math. Sciences 171 (3) (2010) 331–337Google Scholar
19.Vavilov, N. A., Sinchuk, S. S., Parabolic factorizations of split classical groups, Algebra and Analysis 23 (4) (2011) 1–31Google Scholar
20.Zhang, Z., Subnormal structure of non-stable unitary groups over rings, J. Pure Appl. Algebra 214 (5) (2010) 622–628Google Scholar