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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

A structure theorem for the elementary unimodular vector group


Authors: Selby Jose and Ravi A. Rao
Journal: Trans. Amer. Math. Soc. 358 (2006), 3097-3112
MSC (2000): Primary 13D15, 15A66, 19A15, 19E20, 55Q55
Published electronically: October 31, 2005
MathSciNet review: 2216260
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Abstract | References | Similar Articles | Additional Information

Abstract: Given a pair of vectors $ v,w\in R^{r+1}$ with $ \langle v,w\rangle=v\cdot w^T=1$, A. Suslin constructed a matrix $ S_r(v,w)\in Sl_{2^r}(R)$. We study the subgroup $ SUm_r(R)$ generated by these matrices, and its (elementary) subgroup $ EUm_r(R)$ generated by the matrices $ S_r(e_1\varepsilon,e_1\varepsilon^{T^{-1}})$, for $ \varepsilon\in E_{r+1}(R)$. The basic calculus for $ EUm_r(R)$ is developed via a key lemma, and a fundamental property of Suslin matrices is derived.


References [Enhancements On Off] (What's this?)

  • 1. Jose, S., Rao R. A., A local global principle for the elementary unimodular vector group, Commutative Algebra and Algebraic Geometry (Bangalore, India, 2003), Contemp. Math., vol. 390, Amer. Math. Soc., Providence, RI, 2005, pp. 119-125.
  • 2. Rao, R. A., Jose, S., A fundamental property of Suslin matrices, in preparation.
  • 3. L. N. Vaseršteĭn and A. A. Suslin, Serre’s problem on projective modules over polynomial rings, and algebraic 𝐾-theory, Izv. Akad. Nauk SSSR Ser. Mat. 40 (1976), no. 5, 993–1054, 1199 (Russian). MR 0447245 (56 #5560)
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Additional Information

Selby Jose
Affiliation: Department of Mathematics, Ismail Yusuf College, Jogeshwari(E), Mumbai 400-060, India
Email: selbyjose@rediffmail.com

Ravi A. Rao
Affiliation: School of Mathematics, Tata Institute of Fundamental Research, Dr. Homi Bhabha Road, Mumbai 400 005, India
Email: ravi@math.tifr.res.in

DOI: http://dx.doi.org/10.1090/S0002-9947-05-03794-3
PII: S 0002-9947(05)03794-3
Received by editor(s): January 10, 2004
Received by editor(s) in revised form: July 19, 2004
Published electronically: October 31, 2005
Additional Notes: This article is part of the first author’s doctoral dissertation
Article copyright: © Copyright 2005 American Mathematical Society