Enveloping Classes over Commutative Rings

Given a $1$-tilting cotorsion pair over a commutative ring, we characterise the rings over which the $1$-tilting class is an enveloping class. To do so, we consider the faithful finitely generated Gabriel topology $\mathcal{G}$ associated to the $1$-tilting class $\mathcal{T}$ over a commutative ring as illustrated by Hrbek. We prove that a $1$-tilting class $\mathcal{T}$ is enveloping if and only if $\mathcal{G}$ is a perfect Gabriel topology (that is, it arises from a perfect localisation) and $R\mathrm{/}J$ is a perfect ring for each $J\in \mathcal{G}$, or equivalently $\mathcal{G}$ is a perfect Gabriel topology and the discrete quotient rings of the topological ring $\mathfrak R=$End$(R_ $\mathcal{G}$/R)$areperfectringswhere$R_$\mathcal{G}$$ denotes the ring of quotients with respect to $\mathcal{G}$. Moreover, if the above equivalent conditions hold it follows that pdim$R_{\mathcal{G}}\le 1$ and $\mathcal{T}$ arises from a flat ring epimorphism.