On the A1-homotopy groups of Chevalley groups

by Sergey Sinchuk (SPbU)

Thursday 30 Sep 2021, 11:30 → 13:30 Europe/Moscow

818-1526-4739 (Zoom)

The talk is based on the joint work with Egor Voronetsky and Andrei Lavrenov. Let $R$ be a commutative ring and $\Phi$ an irreducible root system of rank $\ge 2$. Steinberg groups $St(\Phi, R)$ are certain groups discovered by R. Steinberg in an attempt to present the simply-connected Chevalley groups $G(\Phi, R)$ by means of generators and relations. The aim of our work is to show that in many cases the kernel $K_2(\Phi, R)$ of the canonical homomorphism $St(\Phi, R) \to G(\Phi, R)$ possesses the $\mathbb{A}^1$-invariance property and therefore can be represented in the unstable $\mathbb{A}^1$-homotopy category as the fundamental group of the scheme $G(\Phi,-)$. Our invariance result can be considered as a certain analogue of (the geometric case of) Bass-Quillen conjecture. With our technique we are also able to establish an analogue of Gersten conjecture for the groups $K_2(\Phi, R)$.

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