Divisibility of ideals and blowing up

Abstract

Let $R$ be a Noetherian integral domain, let $V=\Spec(R)$, and let $I, J$ be nonzero ideals of $R$. Clearly, if $J$ is either a divisor of $I$ or a power of $I$ there is a map $Bl_I(V)\to Bl_J(V)$ of schemes over $V.$ The purpose of this note is to prove, conversely, that if such a map exists, then $J$ must be a fractional ideal divisor of some power of $I$.