Tilting modules and Gorenstein rings

We apply tilting theory to study modules of finite projective dimension. We introduce the notion of finite and cofinite type for tilting and cotilting classes of modules, respectively, showing that, for each dimension, there is a bijection between these classes and resolving classes of modules. We then focus on Iwanaga-Gorenstein rings. Using tilting theory, we prove the first finitistic dimension conjecture for these rings. Moreover, we characterize them among noetherian rings by the property that Gorenstein injective modules form a tilting class. Finally, we give an explicit construction of families of (co)tilting modules of (co)finite type for one-dimensional commutative Gorenstein rings. © de Gruyter 2006.