Ring-induced localizations of nilpotent groups

Abstract

For a commutative ring $R$, we study the $R$-localization functor on the category of groups, defined as localization with respect to the homomorphism $\mathbb{Z}\to R$. Our main result is that, when $R$ is a binomial ring, the $R$-localization of a nilpotent group is again nilpotent. Taking $R=\mathbb{Z}_p$, the ring of $p$-adic integers, yields a new example of a localization functor that preserves nilpotency. To prove this, we characterize $R$-local groups in terms of $R$-groups in the sense of Myasnikov-Remeslennikov. We call an $R$-group a Hall-Petresco $R$-group if it satisfies a version of the Hall-Petresco identity, and show that these form a quasivariety closed under quotients by the center. The crucial input to our main result is that every $R$-local group carries a unique Hall-Petresco $R$-group structure.