Tilting modules and tilting torsion theories

We generalize basic results about classical tilting modules and partial tilting modules to the infinite dimensional case, over an arbitrary ring R. The methods employed combine classical techniques of representation theory of finite dimensional algebras with new techniques of the theory of *-modules. Using a generalization of the Bongartz lemma, we characterize tilting torsion theories in Mod-R, i.e., torsion theories induced by (infinitely generated) tilting modules. We investigate lattices [Gen(P), P⊥] of torsion classes induced by partial tilting modules P. Applying our results to tilting torsion classes, we prove a version of the Brenner-Butler theorem, and a generalization of the Assem-Smalø theorem to the case when R is artinian.