On the 𝔸1-Invariance of K2 Modeled on Linear and Even Orthogonal Groups

AbstractLet $k$ 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 ($\Phi = \mathsf {D}_{\ell }$, $\ell \geq 7$, $\textrm {char}(k)\neq 2$), the unstable functor $\textrm {K}_{2}(\Phi , -)$ possesses the $\mathbb {A}^{1}$-invariance property in the geometric case, that is, ${\textrm {K}}_{2}(\Phi , R[t]) = {\textrm {K}}_{2}(\Phi , R)$ for a regular ring $R$ containing $k$. As a consequence, the unstable ${\textrm {K}}_{2}$ 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 $\textrm {G}(\Phi ,-)$. Our invariance result can be considered as the ${\textrm {K}}_{2}$-analogue of the geometric case of Bass–Quillen conjecture. We also show for a semilocal regular $k$-algebra $A$ that ${\textrm {K}}_{2}(\Phi , A)$ embeds as a subgroup into ${\textrm {K}}^{\textrm {M}}_{2}(\textrm {Frac}\,A)$.