Infinite dimensional tilting theory and its applications

Modules (representations) play a key role in the structure theory of associative rings (algebras). Starting from Morita theorem characterizing equivalence of full module categories, we will proceed to Brenner-Butler theorem (1-dimensional case), and the Miyashita theorem (n-dimensional case). In all these theorems, the representing tilting modules are finitely generated. Then we will introduce the recent notion of an infinitely generated tilting module and show its relation to module approximations. Though infinitely generated, the tilting modules are of `finite type', and this fact makes it possible to classify them over particular rings. We will finish by presenting two recent applications of infinitely generated n-tilting modules: (i) for n=1, to describing the structure of Matlis localizations of commutative rings, and (ii) for an arbitrary n, to proving finitistic dimension conjectures for (non-commutative) Iwanaga-Gorenstein rings.