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

Abstract

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)).

A presentation of relative unitary Steinberg groups

“…Such relative Steinberg groups and their generalizations for unstable linear groups and Chevalley groups are used in, e.g., proving centrality of K 2 [7,11], a suitable local-global principle for Steinberg groups [6,7,12], early stability of K 2 [12], and A 1 -invariance of K 2 [8,9]. In [11, theorem 9] S. Sinchuk proved that all relations between the generators z α (a, p) of St(Φ; R, I), where Φ is a root system of type ADE and R is commutative, come from various St(Ψ; R, I) for root subsystems Ψ ⊆ Φ of type A 3 , i.e.…”