Equivalences and the tilting theory

We study a natural generalization of *n-modules (and hence also of *-modules) by introducing the notion of *∞-modules. The most important results about *n-modules (and also *-modules) are extended to *∞-modules (for example, Theorem 2.7, etc.). An interesting subclass of the class of *∞-modules, namely the class of ∞-tilting modules, may be viewed as a more natural generalization of tilting modules of finite projective dimension to infinite projective dimension. We show that the generalization of the Brenner-Butler theorem in the tilting theory holds for ∞-tilting modules (Theorem 3.9).