Bak's work on the<i>K</i>-theory of rings

Hazrat, Roozbeh; Vavilov, N. A.

doi:10.1017/is008008012jkt087

Cited by 53 publications

(14 citation statements)

References 124 publications

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“…This forces us to take a different route and use second localisation in the spirit of [4], [23]- [26] instead. The resulting bounds are not nearly as good as the estimates in [36]; in fact, now they are exponential in d.…”

Section: An Outline Of the Proofmentioning

confidence: 99%

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On the length of commutators in Chevalley groups

Stepanov

Vavilov

2011

Isr. J. Math.

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Let G = G(Φ, R) be the simply connected Chevalley group with root system Φ over a ring R. Denote by E(Φ, R) its elementary subgroup. The main result of the article asserts that the set of commutators C = { [a, b]|a ∈ G(Φ, R), b ∈ E(Φ, R)} has bounded width with respect to elementary generators. More precisely, there exists a constant L depending on Φ and dimension of maximal spectrum of R such that any element from C is a product of at most L elementary root unipotent elements. A similar result for Φ = A l , with a better bound, was earlier obtained by Sivatski and Stepanov.The width of the linear elementary group En(R) with repect to elementary generators or the set of all commutators was studied by Carter, Keller, Dennis, Vaserstein, van der Kallen and others. It is related to the computation of the Kazhdan constant. For example, the width of (En(R) is finite if R is semilocal (by Gauus decomposition) of R = Z (Carter, Keller), but is infinite for R = C[x] (van der Kallen). Already for R = F [x] over a finite field F this question is open.Van der Kallen noticed that the group E(Φ, R) ∞ /E(Φ, R ∞ ) is an obstruction for the finitness of width of E(Φ, R), were infinte power means the direct product of countably many copies of a ring or a group. The main resutl of the article is equivalent to the fact that this group is central in K 1 (Φ, R ∞ ).

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Section: An Outline Of the Proofmentioning

confidence: 99%

“…However, there are important technical differences. Unlike [36], we cannot invoke decomposition of unipotents and have to rely on a version of Bak's localisation method [4]- [7], [23]- [26] instead. On the other hand, in the above papers induction on dimension takes place at the stage of completion.…”

Section: Introductionmentioning

confidence: 99%

On the length of commutators in Chevalley groups

Stepanov

Vavilov

2011

Isr. J. Math.

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“…It is not feasible even to sketch the development of these results for the classical groups here, see [17], [42], [10], [20], [34], [8] for details and further references. Since in the present paper we are only concerned with the second of these results, let us mention, that for classical groups the first equality was proved by Andrei Suslin and Vyacheslav Kopeiko [36], [37], [23] and the second one by Leonid Vaserstein, Zenon Borewicz and the third author, and Li Fuan, see [40], [13], [24], [25] (and subsequent papers by Vaserstein, for more general related results).…”

Section: Main Structure Theoremsmentioning

confidence: 99%

Relative subgroups in Chevalley groups

Hazrat

Petrov

Vavilov³

2010

J. K-Theory

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Abstract. We finish the proof of the main structure theorems for a Chevalley group G(Φ, R) of rank ≥ 2 over an arbitrary commutative ring R. Namely, we prove that for any admissible pair (A, B) in the sense of Abe, the corresponding relative elementary group E(Φ, R, A, B) and the full congruence subgroup C(Φ, R, A, B) are normal in G(Φ, R) itself, and not just normalised by the elementary group E(Φ, R) and that [E(Φ, R), C(Φ, R, A, B)] = E(Φ, R, A, B). For the case Φ = F 4 these results are new. The proof is new also for other cases, since we explicitly define C(Φ, R, A, B) by congruences in the adjoint representation of G(Φ, R) and give several equivalent characterisations of that group and use these characterisations in our proof.

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“…For further generalisations see [5,10,13,46,49]. An overview on this subject can be found in [7,14,42]. In fact, the proof of centrality of K 2 is based on the same ideas as the proof of normality of the elementary subgroup.…”

Section: Notationsmentioning

confidence: 99%

Another presentation for symplectic Steinberg groups

Lavrenov

2015

Journal of Pure and Applied Algebra

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We solve a classical problem of centrality of symplectic K 2 , namely we show that for an arbitrary commutative ring R, l ≥ 3, the symplectic Steinberg group StSp(2l, R) as an extension of the elementary symplectic group Ep(2l, R) is a central extension. This allows to conclude that the explicit definition of symplectic K 2 Sp(2l, R) as a kernel of the above extension, i.e. as a group of non-elementary relations among symplectic transvections, coincides with the usual implicit definition via plus-construction.We proceed from van der Kallen's classical paper, where he shows an analogous result for linear K-theory. We find a new set of generators for the symplectic Steinberg group and a defining system of relations among them. In this new presentation it is obvious that the symplectic Steinberg group is a central extension.

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Bak's work on theK-theory of rings

Cited by 53 publications

References 124 publications

On the length of commutators in Chevalley groups

On the length of commutators in Chevalley groups

Relative subgroups in Chevalley groups

Another presentation for symplectic Steinberg groups

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