Quasi-tilting modules and counter equivalences

Given two rings R and S, we study the category equivalences T reversible arrow Y, where T is a torsion class of R-modules and Y is a torsion-free class of S-modules. These equivalences correspond to quasi-tilting triples (R, V, S), where V-R(S) is a bimodule which has, ''locally,'' a tilting behavior. Comparing this setting with tilting bimodules and, more generally, with the torsion theory counter equivalences introduced by Colby and Fuller, we prove a local version of the Tilting Theorem for quasi-tilting triples. A whole section is devoted to examples in case of algebras over a field.