Suslin matrices and the Euler characteristic

Abstract

Jose-Rao proved the Key Lemma for the action of the elementary matrices En(R) on the unimodular rows Umn(R) of length n. It essentially describes how Sn−1(vε,wεt−1) looks in terms of Sn−1(v,w), n≥3, when ε is an elementary generator Eij(λ) of En(R), and v,w∈Umn(R), with 〈v,w〉=1. In this paper, we prove the generalized Key Lemma for the action of SLn(R) on Umn(R), n≥3, which is inspired by the (unpublished) results of Fossum–Foxby–Iversen.