School of Mathematics Seminars and Lectures
Different definitions of unstable orthogonal K_2
by Dr. St. Petersburg State University, Russia Andrei Lavrenov
Friday, June 17, 2022
from
to
(Asia/Kolkata)
at AG-77
at AG-77
Description |
Abstract: Many approaches to higher algebraic $\mathrm K$-theory and Hermitian $\mathrm K$-theory are known. For example, stable Quillen's groups $\mathrm K_n(R)$ (defined e.\,g. via the $+$-construction) and stable Karoubi--Villamayor groups $\mathrm{KV}_n(R)$ (defined via standard simplicial scheme) coincide for $n\geq 1$ if $R$ happens to be a regular ring. These theories use infinite-dimensional algebraic groups such as $\mathrm{GL}_\infty(R)$ in their definition. In this talk we will discuss an {\it unstable} analogue of such result for the functor $\mathrm K_2$. The interest for the unstable Quillen's $\mathrm K_2$-groups, in particular, comes from the fact that they appear in Steinberg's presentation of the groups of points of algebraic groups by means of generators and relations. On the other hand,Karoubi--Villamayor $\mathrm K_2$-groups can be interpreted as fundamental groups in the unstable $\mathbb{A}^1$-homotopy category $\mathscr{H}_\bullet(k)$ of F.~Morel and V.~Voevodsky (using results of A.\,Asok, M.\,Hoyois and M.\,Wendt). Conjecturally, for any split simple group $G=\mathrm G(\Phi,R)$ with $\mathrm{rk}\,\Phi\geq5$ and regular ring $R$ holds an equality \begin{align} \label{conj} \pi_1^{\mathbb A^1}(G)(R)=\pi_2(\mathrm BG^+). \end{align} We remark that the Nisnevich localization $\mathrm{a_{Nis}}\,\pi_1^{\mathbb A^1}(G)(R)$ of $\mathbb A^1$-fundamental groups was recently computed by F.\,Morel and A.\,Sawant, and coincides with the unramified Milnor $\mathrm{\underline{K}_2^M}$ or Milnor-Witt $\mathrm{\underline{K}_2^{MW}}$ sheaf depending on $\Phi$. Conjecture~(\ref{conj}) is parallel to the Serre's problem and Bass--Quillen conjecture, and we adopt Quillen--Suslin and Lindel--Popesque results for this case. In particular, for $\Phi=\mathsf A_l,\,\mathsf D_l$ this conjecture is already proven for a regular ring $R$ containing a field $k$ of characteristic $\neq 2$, $l\geq7$. As a corollary, one can obtain the following results. \begin{itemize} \item The group $\mathrm{Spin}_{2l}(k[t_1,\ldots, t_n])$ admits an explicit presentation by means of generators and relation (generalizing Steinberg's presentation in the case $n=0$). \item $\mathrm H_2(\mathrm{Spin}_{2l}(k[t_1,\ldots,t_n]),\,\mathbb Z\big) = \mathrm K^\mathrm{M}_2(k).$ \item $\mathrm H_2 (\mathrm{O}_{2l}(R[t]), \mathbb Z) = \mathrm H_2 (\mathrm{O}_{2l}(R), \mathbb Z).$ \end{itemize} The talk is based on my joint work with Sergey Sinchuk and Egor Voronetsky. |
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