Enveloping Classes over Commutative Rings

Abstract: 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 $R/J$ 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$(R_ \mathcal{G}/R)$ are perfect rings where $R_\mathcal{G}$ denotes the ring of quotients with respect to $\mathcal{G}$. Moreover, if the above equivalent conditions hold it follows that pdim$R_\mathcal{G} \leq 1$ and $\mathcal{T}$ arises from a flat ring epimorphism.