A characterisation of enveloping 1-tilting 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 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 ( R G / R ) are perfect rings where R G denotes the ring of quotients with respect to G . Moreover, if the above equivalent conditions hold it follows that p . dim R G ≤ 1 and T arises from a flat ring epimorphism .