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$R_{\mathcal{G}}\le 1$ and $\mathcal{T}$ arises from a flat ring epimorphism.

Updated version includes improvement of main theorem due to recent results, generalisation of techniques which were moved to Section 4 from Section 5, updates and additions to references and a change of abstract