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 if $R$ admits a $λ$-ring structure (for instance, if $R$ is binomial) then the $R$-localization of a nilpotent group is again nilpotent. In particular, taking $R=\mathbb{Z}_p$, the ring of $p$-adic integers, yields a new example of a localization functor that preserves nilpotency. Along the way, we characterize $R$-local groups in terms of the $R$-groups of Myasnikov-Remeslennikov, and show that nilpotent $R$-local groups satisfy a version of the Hall-Petresco identity.