Suslin matrices and the Euler characteristic

Jose-Rao proved the Key Lemma for the action of the elementary matrices E-n(R) on the unimodular rows Um(n)(R) of length n. It essentially describes how Sn-1(v epsilon, w epsilon(t-1) ) looks in terms of Sn-1(v, w), n >= 3, when epsilon is an elementary generator E-ij(lambda) of E-n(R), and v, w is an element of Um(n)(R), with (v, w) = 1. In this paper, we prove the generalized Key Lemma for the action of SLn(R) on Um(n)(R), n >= 3, which is inspired by the (unpublished) results of Fossum-Foxby-Iversen.(c) 2023 Elsevier Inc. All rights reserved.