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

Abstract

If A A is a smooth algebra of dimension d ≥ 3 d\geq 3 over a perfect field k k of characteristic different from 2 2 , then we show that the universal Mennicke symbol M S d + 1 ( A ) MS_{d+1}(A) is isomorphic to the K K -cohomology group H d ( A , K d + 1 ) H^d(A,K_{d+1}) . We then prove an analogue of the Bass-Kubota theorem for smooth affine surfaces over the algebraic closure of a finite field.