Covering classes and $1$-tilting cotorsion pairs over commutative rings

TL;DR

This paper characterizes commutative rings where a specific class related to 1-tilting cotorsion pairs provides covers, linking Gabriel topologies, localizations, and perfect rings.

Contribution

It establishes a precise criterion involving Gabriel topologies and localizations for when the class in a 1-tilting cotorsion pair is covering over commutative rings.

Findings

Covering classes correspond to perfect localizations with projective dimension at most one.

Rings are G-almost perfect if their localizations and quotients are perfect rings.

The characterization links Gabriel topologies, localizations, and covering properties in commutative rings.

Abstract

We are interested in characterising the commutative rings for which a $1$-tilting cotorsion pair $(\mathcal{A}, \mathcal{T})$ provides for covers, that is when the class $\mathcal{A}$ is a covering class. We use Hrbek's bijective correspondence between the $1$-tilting cotorsion pairs over a commutative ring $R$ and the faithful finitely generated Gabriel topologies on $R$. Moreover, we use results of Bazzoni-Positselski, in particular a generalisation of Matlis equivalence and their characterisation of covering classes for $1$-tilting cotorsion pairs arising from flat injective ring epimorphisms. Explicitly, if $\mathcal{G}$ is the Gabriel topology associated to the $1$-tilting cotorsion pair $(\mathcal{A}, \mathcal{T})$, and $R_\mathcal{G}$ is the ring of quotients with respect to $\mathcal{G}$, we show that if $\mathcal{A}$ is covering then $\mathcal{G}$ is a perfect localisation (in Stenstr\"om's sense) and the localisation $R_\mathcal{G}$ has projective dimension at most one. Moreover, we show that $\mathcal{A}$ is covering if and only if both the localisation $R_\mathcal{G}$ and the quotient rings $R/J$ are perfect rings for every $J \in \mathcal{G}$. Rings satisfying the latter two conditions are called $\mathcal{G}$-almost perfect.