Mennicke symbols, K-cohomology and a Bass-Kubota theorem

TL;DR

This paper establishes a connection between Mennicke symbols and K-cohomology for smooth algebras over certain fields, and proves an analogue of the Bass-Kubota theorem for curves on smooth surfaces.

Contribution

It demonstrates that the universal Mennicke symbol of length d+1 is isomorphic to a K-cohomology group and extends the Bass-Kubota theorem to smooth surfaces over algebraically closed fields.

Findings

Universal Mennicke symbol is isomorphic to a K-cohomology group.

Established an analogue of the Bass-Kubota theorem for curves.

Extended classical results to higher-dimensional smooth algebras.

Abstract

If A is a smooth algebra of dimension d (greater or equal to 2) over a perfect field k of characteristic different from 2, then we show that the universal Mennicke symbol of length d+1 is isomorphic to some K-cohomology group. When k is algebraically closed and S is a smooth surface, we further prove an analogue of the classical Bass-Kubota theorem for curves.