Commutators of elementary subgroups: curiouser and curiouser

DOI10.1007/S00031-021-09662-ZarXiv2004.12870MaRDI QIDQ6339505

Nikolai Vavilov, Zuhong Zhang

Publication date: 27 April 2020

Abstract: Let $R$ be any associative ring with $1$, $nge3$, and let $A,B$ be two-sided ideals of $R$. In our previous joint works with Roozbeh Hazrat [17,15] we have found a generating set for the mixed commutator subgroup $[E(n,R,A),E(n,R,B)]$. Later in [29,34] we noticed that our previous results can be drastically improved and that $[E(n,R,A),E(n,R,B)]$ is generated by 1) the elementary conjugates $z_{ij}(ab,c)=t_{ij}(c)t_{ji}(ab)t_{ij}(-c)$ and $z_{ij}(ba,c)$, 2) the elementary commutators $[t_{ij}(a),t_{ji}(b)]$, where $1leieqjlen$, $ainA$, $binB$, $cinR$. Later in [33,35] we noticed that for the second type of generators, it even suffices to fix one pair of indices $(i,j)$. Here we improve the above result in yet another completely unexpected direction and prove that $[E(n,R,A),E(n,R,B)]$ is generated by the elementary commutators $[t_{ij}(a),t_{hk}(b)]$ alone, where $1leieqjlen$, $1leheqklen$, $ainA$, $binB$. This allows us to revise the technology of relative localisation, and, in particular, to give very short proofs for a number of recent results, such as the generation of partially relativised elementary groups $E(n,A{)}^{E(n,B)}$, %% normality of $E(n,AB+BA)$ inside $[E(n,R,A),E(n,R,B)]$, multiple commutator formulas, commutator width, and the like.

Mathematics Subject Classification ID

Subgroup theorems; subgroup growth (20E07) Other matrix groups over rings (20H25) Commutator calculus (20F12) Other matrix groups over fields (20H20) Linear algebraic groups over adèles and other rings and schemes (20G35)

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