A structure theorem for the elementary unimodular vector group

Jose, Selby; Rao, Ravi

doi:10.1090/S0002-9947-05-03794-3

A structure theorem for the elementary unimodular vector group
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by Selby Jose and Ravi A. Rao PDF

Trans. Amer. Math. Soc. 358 (2006), 3097-3112 Request permission

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

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.
Rao, R. A., Jose, S., A fundamental property of Suslin matrices, in preparation.
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
A. A. Suslin, Stably free modules, Mat. Sb. (N.S.) 102(144) (1977), no. 4, 537–550, 632 (Russian). MR 0441949
A. A. Suslin, The structure of the special linear group over rings of polynomials, Izv. Akad. Nauk SSSR Ser. Mat. 41 (1977), no. 2, 235–252, 477 (Russian). MR 0472792