A Bloch-Wigner exact sequence over local rings (Q511870)

In [J. Pure Appl. Algebra 219, 5078--5096 (2015; Zbl 1329.19002)], the author et al. studied a Bloch Wigner exact sequence for local rings with infinite residue field. The present paper considers the case of a local ring \(R\) with more than nine elements in the residue field. This requires revisiting proofs of homology stability for general linear groups, pushing the computations a little further. If \(R\) has more than four elements in its residue field then a presentation of \(K_2(R)\) is given in terms of Steinberg symbols. Previously this presentation was known when there are more than five elements in the residue field. Recall that one also has presentation of \(K_2(R)\) in terms of Dennis-Stein symbols if there are more than three elements in the residue field. See [\textit{W. van der Kallen}, Ann. Sci. Éc. Norm. Supér. (4) 10, 473--515 (1977; Zbl 0393.18012)].