On the a1-invariance of k2 modeled on linear and even orthogonal groups

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) 6= 2) the unstable functor K2(Φ,−) possesses the Ainvariance 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-homotopy category HA 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 K2 (Frac(A)).