Absolute stable rank and Witt cancellation for noncommutative rings

Magurn, Bruce A.; Kallen, Wilberd van der; Vaserstein, L. N.

doi:10.1007/bf01388785

Cited by 42 publications

(43 citation statements)

References 15 publications

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“…Earlier, stability with split coefficient systems of finite degree in the case of R a Dedekind domain, trivial involution, = ±1, and Λ = Λ max (that is, the groups O n,n (A) for = 1 and Sp 2n (A) for = −1) was proved by Charney [15]. In terms of the absolute stable rank of Magurn-van der Kallen-Vaserstein [56] we have asr(R) ≤ 2 for R Dedekind [56,Theorem 3.1], and with trivial involution we have usr(R) ≤ asr(R) [59,Remark 6.4]. Thus for split coefficient systems we get: the map in Theorem 5.15 is an epimorphism for i ≤ n−r−3 2…”

Section: Tomentioning

confidence: 99%

“…, e n+1 , f n+1 for the hyperbolic bases for the copies of H in the target. We follow the proof of [56,Cor. 8.3].…”

Section: It Follows Thatmentioning

confidence: 99%

“…Step 6 in the proof of [56,Theorem 8.1], which gives an explicit sequence of automorphisms of H ⊕n+1 which fix f n+1 and take such a φ(e) to e n+1 . This gives a new isomorphism φ which is the identity on the last copy of H, so restricts to an isomorphism N ∼ = H ⊕n as required.…”

Section: Tomentioning

confidence: 99%

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Homological stability for automorphism groups

Randal-Williams

Wahl

2017

Advances in Mathematics

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Abstract. Given a family of groups admitting a braided monoidal structure (satisfying mild assumptions) we construct a family of spaces on which the groups act and whose connectivity yields, via a classical argument of Quillen, homological stability for the family of groups. We show that stability also holds with both polynomial and abelian twisted coefficients, with no further assumptions. This new construction of a family of spaces from a family of groups recovers known spaces in the classical examples of stable families of groups, such as the symmetric groups, general linear groups and mapping class groups. By making systematic the proofs of classical stability results, we show that they all hold with the same type of coefficient systems, obtaining in particular without any further work new stability theorems with twisted coefficients for the symmetric groups, braid groups, automorphisms of free groups, unitary groups, mapping class groups of non-orientable surfaces and mapping class groups of 3-manifolds. Our construction can also be applied to families of groups not considered before in the context of homological stability.As a byproduct of our work, we construct the braided analogue of the category F I of finite sets and injections relevant to the present context, and define polynomiality for functors in the context of pre-braided monoidal categories.A family of groupsis said to satisfy homological stability if the induced mapsare isomorphisms in a range 0 ≤ i ≤ f (n) increasing with n. In this paper, we prove that homological stability always holds if there is a monoidal category C satisfying a certain hypothesis and a pair of objects A and X in C, such that G n is the group of automorphisms of A ⊕ X ⊕n in C. We show that stability holds not just for constant coefficients, but also for both polynomial and abelian coefficients, without any further assumption on C. The polynomial coefficient systems considered here are functors F : C → Z -Mod satisfying a finite degree condition. They are generalisations of polynomial functors in the sense of functor homology, classically considered in homological stability, and include new examples such as the Burau representation of braid groups. Abelian coefficients are given by functors F : C → ZG ab ∞ -Mod for G ab ∞ the abelianisation of the limit group G ∞ , satisfying the same finite degree condition. These include coefficients such as the sign representation, or determinant-twisted polynomial functors. Such coefficient systems are newer to the subject. One consequence of stability with abelian coefficients is that stability with polynomial coefficients also holds (under the same conditions) for the commutator subgroups G n ≤ G n .Our theorem applies to all the classical examples and gives new stability results with twisted coefficients in particular for symmetric groups, alternating groups, unitary groups, braid groups, mapping class groups of non-orientable surfaces, automorphisms and symmetric automorphisms of free groups, and it proves stability for these groups...

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Section: Tomentioning

confidence: 99%

“…, e n+1 , f n+1 for the hyperbolic bases for the copies of H in the target. We follow the proof of [56,Cor. 8.3].…”

Section: It Follows Thatmentioning

confidence: 99%

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Homological stability for automorphism groups

Randal-Williams

Wahl

2017

Advances in Mathematics

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“…In general, the stable rank of R (denoted by sr(/?)) is less than or equal to the absolute stable rank of R (see [5]). (ii) If R is commutative and the maximal spectrum of R is Noetherian of finite dimension n, then any module-finite /?-algebra A has absolute stable rank at most n+\.…”

Section: Introductionmentioning

confidence: 99%

Commutators in pseudo-orthogonal groups

Arlinghaus

Vaserstein²,

You

1995

J Aust Math Soc A

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We study commutators in pseudo-orthogonal groups O 2n R (including unitary, symplectic, and ordinary orthogonal groups) and in the conformal pseudo-orthogonal groups GO^R. We estimate the number of commutators, c(C>2nR) and c(G02 n R), needed to represent every element in the commutator subgroup. We show that c(O 2n R) < 4 if R satisfies the A-stable condition and either n > 3 or n = 2 and 1 is the sum of two units in R, and that c(GO 2n R) < 3 when the involution is trivial and A = R f . We also show that ciO^R) < 3 and c(GO 2n R) < 2 for the ordinary orthogonal group O 2rl R over a commutative ring R of absolute stable rank 1 where either n > 3 or n = 2 and 1 is the sum of two units in R.

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“…One such prominent condition is the absolute stable rank asr(R), see [28,50,37,47]. In particular, results in terms of the absolute stable rank superceeded the classical results by Anthony Bak [18] and Hyman Bass [23], stated in terms of dimension.…”

Section: Parabolic Factorizations Of Split Classical Groups 639 §1 Pmentioning

confidence: 95%

Untitled

2012

St. Petersburg Math. J.

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Absolute stable rank and Witt cancellation for noncommutative rings

Cited by 42 publications

References 15 publications

Homological stability for automorphism groups

Homological stability for automorphism groups

Commutators in pseudo-orthogonal groups

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