A characterisation of enveloping 1-tilting 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 G associated to the 1-tilting class T over a commutative ring as illustrated by Hrbek. We prove that a 1-tilting class T is enveloping if and only if G is a perfect Gabriel topology (that is, it arises from a perfect localisation) and R/J is a perfect ring for each J∈G, or equivalently G is a perfect Gabriel topology and the discrete factor rings of the topological ring R=End(RG/R) are perfect rings where RG denotes the ring of quotients with respect to G. Moreover, if the above equivalent conditions hold it follows that p.dimRG≤1 and T arises from a flat ring epimorphism.