A Horrocks-Type Theorem for Even Orthogonal $\mathrm{K}_2$

Lavrenov, Andrei; Sinchuk, Sergey

doi:10.25537/dm.2020v25.767-809

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“…The hardest of these to verify is the so-called P 1 -glueing property (also called Horrocks theorem, cf. [12]). Yet another axiom appearing in the statement of Theorem 2.2 is the so-called weak affine Nisnevich excision property for Steinberg groups.…”

Section: Introductionmentioning

confidence: 99%

On the $\mathbb{A}^1$-invariance of $\mathrm{K}_2$ modeled on linear and even orthogonal groups

Lavrenov¹,

Sinchuk²,

Egor³

2021

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Let k be an arbitrary field. In this paper we show that in the linear case (Φ = A , ≥ 4) and even orthogonal case (Φ = D , ≥ 7, char(k) = 2) the unstable functor K2(Φ, −) possesses the A 1invariance property in the geometric case, i. e. K2(Φ, R[t]) = K2(Φ, R) for a regular ring R containing k. As a consequence, the unstable K2 groups can be represented in the unstable A 1 -homotopy category H A 1 k as fundamental groups of the simply-connected Chevalley-Demazure group schemes G(Φ, −). Our invariance result can be considered as the K2-analogue of the geometric case of Bass-Quillen conjecture. We also show for a semilocal regular k-algebra A that K2(Φ, A) embeds as a subgroup into K M 2 (Frac(A)).

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Section: Introductionmentioning

confidence: 99%