On the word length of commutators in \(\text{GL}_n(R)\)
Let \(n\geq 3\) and let \(R\) be a commutative Noetherian ring whose maximal spectrum has dimension \(d\), \(d<\infty\). The authors show: there is an integer \(\ell\) depending only on \(n\) and \(d\) so that for \(a\in\text{E}_n(R)\) and \(b\in\text{SL}_n(R)\) the commutator \([a,b]\) can be written as a product of no more than \(\ell\) elementary transvections. When \(R\) is a localization of an affine algebra over a field, they show that one may also take \(b\) in \(\text{GL}_n(R)\), and they prove an analogue in the Steinberg group with its usual generators. The authors recall that in the stable range \(n\geq d+2\) one may take both \(a\) and \(b\) in \(\text{GL}_n(R)\). The beauty of their main result is thus that \(n\geq 3\) suffices. The proof is based on a localization method of \textit{A. Bak} [K-Theory 4, No. 4, 363--397 (1991; Zbl 0741.19001)]. Recall that \textit{A. A. Suslin} has shown that \([a,b]\) is a product of elementary transvections [Izv. Akad. Nauk SSSR, Ser. Mat. 41, 235--252 (1977; Zbl 0354.13009)], but that the reviewer showed that in general there is no bound on the number of factors that is needed to express an element of \(\text{E}_n(R)\) as a product of these generators [\textit{W. van der Kallen}, \(\text{SL}_3(\mathbb C[X])\) does not have bounded word length. Algebraic \(K\)-theory, Oberwolfach 1980, Lect. Notes Math. 966, 357--361 (1982; Zbl 0935.20501)].
scientific article; zbMATH DE number 1348186