Sergey Sinchuk | On the A1-homotopy groups of Chevalley groups скачать в хорошем качестве

Sergey Sinchuk | On the A1-homotopy groups of Chevalley groups

Seminar on A1-topology, motives and K-theory, September 30, 2021 Sergey Sinchuk (SPbU) On the A1-homotopy groups of Chevalley groups The talk is based on the joint work with Egor Voronetsky and Andrei Lavrenov. Let R be a commutative ring and Φ an irreducible root system of rank ≥2. Steinberg groups St(Φ,R) are certain groups discovered by R. Steinberg in an attempt to present the simply-connected Chevalley groups G(Φ,R) by means of generators and relations. The aim of our work is to show that in many cases the kernel K2(Φ,R) of the canonical homomorphism St(Φ,R)→G(Φ,R) possesses the A1-invariance property and therefore can be represented in the unstable A1-homotopy category as the fundamental group of the scheme G(Φ,−). 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 K2(Φ,R).