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 ($\Phi ={\mathsf{A}}_{\ell }$, $\ell \ge 4$) and even orthogonal case ($\Phi ={\mathsf{D}}_{\ell }$, $\ell \ge 7$, $\mathrm{char}(k)\ne 2$) 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 ${\mathcal{H}}_{\bullet }(k)$ 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 ${\mathrm{K}}_2^{\mathrm{M}}(\mathrm{Frac}A)$.

References in corpus (2)

• Nice triples and Grothendieck--Serre's conjecture concerning principal G-bundles over reductive group schemes
• Commutators of relative and unrelative elementary subgroups in Chevalley groups

Cited by in corpus (2)

• On the ${\mathbb{A}}^1$-invariance of ${\mathrm{K}}_2$ of Chevalley groups of simply-laced type
• A presentation of relative unitary Steinberg groups