Injective stability for odd unitary <i>K</i>
                  <sub>1</sub>

Egor, Voronetsky,

doi:10.1515/jgth-2020-0013

Cited by 4 publications

(9 citation statements)

References 22 publications

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“…We discovered odd form algebras in [17]. This objects are more flexible than Petrov's module with odd form parameters.…”

Section: Introductionmentioning

confidence: 93%

“…Isotropic elementary subgroups of reductive groups are defined in [10] by V. Petrov and A. Stavrova, where they also proved normality. In the present paper we show how parabolic subgroups of classical reductive groups may be described purely algebraically, using orthogonal hyperbolic families from [17]. For non-twisted linear groups this objects reduce to families of full orthogonal idempotents.…”

Section: Introductionmentioning

confidence: 93%

“…It turns out that for any quadratic S-module there is a ring R with an involution and an odd form parameter ∆ ≤ Heis(B R ) (where B R (a, b) = a b) such that its unitary group is isomorphic to the unitary group of M , see [16,17]. Such a pair (R, ∆) is the same as a special unital odd form ring according to the definition below if we consider π and ρ are the first and the second projections from ∆ to R, and take φ(a) = (0, a − a).…”

Section: Odd Form Algebrasmentioning

confidence: 99%

“…Fix a commutative ring K. Odd form K-algebras from [16,17] are precisely odd form rings (R, ∆) with a unital action of the odd form ring (K, 0) (with the trivial involution on K) such that ak = ka for all a ∈ R, k ∈ K. Every odd form ring is an odd form Z-algebra with uniquely determined multiplication ∆ × Z → ∆.…”

Section: Odd Form Algebrasmentioning

confidence: 99%

“…Also centrality easily follows from surjective stability of KU 2 . See [17] and [18] for injective stability of KU 1 in the odd unitary case, the proofs from these papers also give surjective stability of KU 2 .…”

Section: Introductionmentioning

confidence: 99%

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Centrality of odd unitary $K_2$-functor

Voronetsky

2020

Preprint

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Let (R, ∆) be an odd form algebra. We show that the unitary Steinberg group StU(R, ∆) is a crossed module over the odd unitary group U(R, ∆) in two major cases: if the odd form algebra has a free orthogonal hyperbolic family satisfying local stable rank condition and if the odd form algebra is sufficiently isotropic and quasi-finite. The proof uses only elementary localization techniques and stability results for KU1 and KU2.

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“…We discovered odd form algebras in [17]. This objects are more flexible than Petrov's module with odd form parameters.…”

Section: Introductionmentioning

confidence: 93%

Section: Introductionmentioning

confidence: 93%

Section: Odd Form Algebrasmentioning

confidence: 99%

Section: Odd Form Algebrasmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Centrality of odd unitary $K_2$-functor

Voronetsky

2020

Preprint

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Commutators of relative and unrelative elementary unitary groups

Vavilov

Zhang

2022

St. Petersburg Math. J.

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In the present paper, which is an outgrowth of the authors’ joint work with Anthony Bak and Roozbeh Hazrat on the unitary commutator calculus [9, 27, 30, 31], generators are found for the mixed commutator subgroups of relative elementary groups and unrelativized versions of commutator formulas are obtained in the setting of Bak’s unitary groups. It is a direct sequel of the papers [71, 76, 78, 79] and [77, 80], where similar results were obtained for G L ( n , R ) GL(n,R) and for Chevalley groups over a commutative ring with 1, respectively. Namely, let ( A , Λ ) (A,\Lambda ) be any form ring and let n ≥ 3 n\ge 3 . Bak’s hyperbolic unitary group G U ( 2 n , A , Λ ) GU(2n,A,\Lambda ) is considered. Further, let ( I , Γ ) (I,\Gamma ) be a form ideal of ( A , Λ ) (A,\Lambda ) . One can associate with the ideal ( I , Γ ) (I,\Gamma ) the corresponding true elementary subgroup F U ( 2 n , I , Γ ) FU(2n,I,\Gamma ) and the relative elementary subgroup E U ( 2 n , I , Γ ) EU(2n,I,\Gamma ) of G U ( 2 n , A , Λ ) GU(2n,A,\Lambda ) . Let ( J , Δ ) (J,\Delta ) be another form ideal of ( A , Λ ) (A,\Lambda ) . In the present paper an unexpected result is proved that the nonobvious type of generators for [ E U ( 2 n , I , Γ ) , E U ( 2 n , J , Δ ) ] \big [EU(2n,I,\Gamma ),EU(2n,J,\Delta )\big ] , as constructed in the authors’ previous papers with Hazrat, are redundant and can be expressed as products of the obvious generators, the elementary conjugates Z i j ( ξ , c ) = T j i ( c ) T i j ( ξ ) T j i ( − c ) Z_{ij}(\xi ,c)=T_{ji}(c)T_{ij}(\xi )T_{ji}(-c) , and the elementary commutators Y i j ( a , b ) = [ T i j ( a ) , T j i ( b ) ] Y_{ij}(a,b)=[T_{ij}(a),T_{ji}(b)] , where a ∈ ( I , Γ ) a\in (I,\Gamma ) , b ∈ ( J , Δ ) b\in (J,\Delta ) , c ∈ ( A , Λ ) c\in (A,\Lambda ) , and ξ ∈ ( I , Γ ) ∘ ( J , Δ ) \xi \in (I,\Gamma )\circ (J,\Delta ) . It follows that [ F U ( 2 n , I , Γ ) , F U ( 2 n , J , Δ ) ] = [ E U ( 2 n , I , Γ ) , E U ( 2 n , J , Δ ) ] \big [FU(2n,I,\Gamma ),FU(2n,J,\Delta )\big ]=\big [EU(2n,I,\Gamma ),EU(2n,J,\Delta )\big ] . In fact, much more precise generation results are established. In particular, even the elementary commutators Y i j ( a , b ) Y_{ij}(a,b) should be taken for one long root position and one short root position. Moreover, the Y i j ( a , b ) Y_{ij}(a,b) are central modulo E U ( 2 n , ( I , Γ ) ∘ ( J , Δ ) ) EU(2n,(I,\Gamma )\circ (J,\Delta )) and behave as symbols. This makes it possible to generalize and unify many previous results, including the multiple elementary commutator formula, and dramatically simplify their proofs.

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Twisted forms of classical groups

Egor

2023

St. Petersburg Math. J.

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Twisted forms of classical reductive group schemes are described in a unified way. Such group schemes are constructed from algebraic objects of finite rank, excluding some exceptions of small rank. These objects, called the augmented odd form algebras, consist of 2 2 -step nilpotent groups with an action of the underlying commutative ring, hence the basic descent theory for them will be developed. Finally, classical isotropic reductive groups are described as odd unitary groups up to an isogeny.

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Injective stability for odd unitary K ₁

Cited by 4 publications

References 22 publications

Centrality of odd unitary $K_2$-functor

Centrality of odd unitary $K_2$-functor

Commutators of relative and unrelative elementary unitary groups

Twisted forms of classical groups

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Injective stability for odd unitary K 1

Cited by 4 publications

References 22 publications

Centrality of odd unitary $K_2$-functor

Centrality of odd unitary $K_2$-functor

Commutators of relative and unrelative elementary unitary groups

Twisted forms of classical groups

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Product

Resources

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Injective stability for odd unitary K ₁