Covering classes and 1-tilting cotorsion pairs over commutative rings

We are interested in characterising the commutative rings for which a 1-tilting cotorsion pair provides for covers, that is when the class A is a covering class. We use Hrbekšfs 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 is the Gabriel topology associated to the 1-tilting cotorsion pair, and R is the ring of quotients with respect to, we show that if A is covering, then G is a perfect localisation (in Stenstromšfs sense [B. Stenstrom, Rings of Quotients, Springer, New York, 1975]) and the localisation R has projective dimension at most one as an R-module. Moreover, we show that is covering if and only if both the localisation RG and the quotient rings R/J are perfect rings for every J ∈. Rings satisfying the latter two conditions are called G-almost perfect.