Infinite Dimensional Tilting Modules and Cotorsion Pairs

Classical tilting theory generalizes Morita theory of equivalence of module categories. The key property – existence of category equivalences between large full subcategories of the module categories – forces the representing tilting module to be finitely generated. However, some aspects of the classical theory can be extended to infinitely generated modules over arbitrary rings. In this paper, we will consider such an aspect: the relation of tilting to approximations (preenvelopes and precovers) of modules. As an application, we will present recent connections between tilting theory of infinitely generated modules and the finitistic dimension conjectures. General existence theorems provide a big supply of approximations in the category Mod-R of all modules over an arbitrary ring R. However, the corresponding approximations may not be available in the subcategory of all finitely generated modules. So the usual sharp distinction between finitely and infinitely generated modules becomes unnatural, and even misleading. A convenient tool for the study of module approximations is the notion of a cotorsion pair. Tilting cotorsion pairs are defined as the cotorsion pairs induced by tilting modules. We will present their characterization among all cotorsion pairs, and then apply it to a classification of tilting classes in particular cases (e.g., over Prüfer domains). The point of the classification is that in the particular cases, the tilting classes are of finite type. This means that we can replace the single infinitely generated tilting module by a set of finitely presented modules; the tilting class is then axiomatizable in the language of the first order theory of modules. Most of this paper is a survey of recent developments. We give complete definitions and statements of the results, but most proofs are omitted or replaced by outlines of the main ideas. For full details, we refer to the original papers listed in the references, or to the forthcoming monograph [51]. However, Theorems 3.4, 3.7, 4.14, and 4.15 are new, hence presented with full proofs. In §1, we introduce cotorsion pairs and their relations to approximation theory of infinitely generated modules over arbitrary rings. In §2 and §3, we study infinitely generated tilting and cotilting modules, and characterize the induced tilting and cotilting cotorsion pairs. §4 deals with tilting classes of finite type and cotilting classes of cofinite type, and with their classification over particular rings. Finally, §5 relates tilting approximations to the first and second finitistic dimension conjectures.