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 ($\Phi=\mathsf{A}_\ell$, $\ell \geq 4$) and even orthogonal case ($\Phi = \mathsf{D}_\ell$, $\ell\geq 7$, $\mathrm{char}(k)\neq 2$) the unstable functor $\mathrm{K}_2(\Phi, -)$ possesses the $\mathbb{A}^1$-invariance property in the geometric case, i. e. $\mathrm{K}_2(\Phi, R[t]) = \mathrm{K}_2(\Phi, R)$ for a regular ring $R$ containing $k$. As a consequence, the unstable $\mathrm{K}_2$ groups can be represented in the unstable $\mathbb{A}^1$-homotopy category $\mathscr{H}_\bullet(k)$ as fundamental groups of the simply-connected Chevalley--Demazure group schemes $\mathrm{G}(\Phi,-)$. Our invariance result can be considered as the $\mathrm{K}_2$-analogue of the geometric case of Bass--Quillen conjecture. We also show for a semilocal regular $k$-algebra $A$ that $\mathrm{K}_2(\Phi, A)$ embeds as a subgroup into $\mathrm{K}^\mathrm{M}_2(\mathrm{Frac}\,A)$.