Equivalence and Duality for Module Categories (with Tilting and Cotilting for Rings),
Tilting and cotilting theory appeared first in representation theory of finite dimensional algebras as a generalization of the classical Morita theory of equivalence and duality. The generalization still has the strength and beauty of the classical theory but it makes it possible to understand new (tilted, or iterated tilted) algebras that are far from being Morita equivalent to original algebras. While the focus in the finite dimensional algebra case is on finite dimensional modules, many results hold for general modules over general rings. This has led to a development of tilting and cotilting theory for rings, in works of the authors of this monograph and of the Padova School, since the late 80's. The monograph provides a unified treatment of the theory, in particular, of its relations to equivalence and duality. The key facts are presented here, often with simplified proofs. Indeed, the authors approach tilting and cotilting modules through more general notions of a comodule and a costar module.
Chapters 2 and 3 of the monograph deal with representable equivalences. Here, modules are shown to be finitely generated, and the notions of a quasi-progenerator and a tilting module to occur naturally as particular instances of * -modules. The main result is a general tilting theorem providing a pair of representable equivalences between large subcategories of modules. Chapters 4 and 5 deal with representable dualities. The focus is on cotilting modules: Bazzoni's proof of their pure-injectivity is presented as well as the Colpi cotilting theorem. Notions of a weak Morita duality and a generalized Morita duality are studied in detail. Working at this level of generality, the authors need interesting classes of non-artinian examples. These are provided by modules over noetherian serial rings. For the convenience of the reader, there is an appendix with preliminaries on these rings, and another appendix on adjoint functors and equivalence. The monograph fills in a gap in the literature by covering tilting and cotilting theory in the general setting of arbitrary modules over arbitrary associative rings. The theory has recently gained importance by its connection to finitistic dimension conjectures. There is no doubt that the monograph will become a basic reference in this active area of abstract algebra.