Speaker
Description
Let $k$ be an arbitrary field. The aim of the talk is to give on overview of a recent preprint, in which it is shown that in the linear case ($\Phi=\mathrm{A}_\ell$, $\ell \geq 4$) and even orthogonal case ($\Phi = \mathrm{D}_\ell$, $\ell\geq 7$, $\mathrm{char}(k)\neq 2$) the unstable functor $\mathrm{K}_2(\Phi, R)$ possesses the $\mathbb{A}^1$-invariance property and therefore can be represented in the unstable $\mathbb{A}^1$-homotopy category $\mathcal{H}^{\mathbb{A}^1}_{k}$ as the fundamental group of the simply-connected Chevalley--Demazure group scheme $\mathrm{G}_\mathrm{sc}(\Phi,-)$. This invariance result can be considered as the $\mathrm{K}_2$-analogue of the geometric case of Bass--Quillen conjecture. With our technique we are also able to show for a semilocal regular $k$-algebra $A$ that $\mathrm{K}_2(\Phi, A)$ embeds as a subgroup into the Milnor group $\mathrm{K}^\mathrm{M}_2(\mathrm{Frac}(A))$.
