Infinite dimensional tilting modules and cotorsion pairs

Abstract

Classical tilting theory generalizes Morita theory of equivalence of module cat-egories. The key property – existence of category equivalences between large fullsubcategories of the module categories – forces the representing tilting module tobe finitely generated.However, some aspects of the classical theory can be extended to infinitely gen-erated 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 ofinfinitely generated modules and the finitistic dimension conjectures.General existence theorems provide a big supply of approximations in the cate-gory Mod-R of all modules over an arbitrary ring R. However, the correspondingapproximations may not be available in the subcategory of all finitely generatedmodules. So the usual sharp distinction between finitely and infinitely generatedmodules becomes unnatural, and even misleading.A convenient tool for the study of module approximations is the notion of acotorsion pair. Tilting cotorsion pairs are defined as the cotorsion pairs induced bytilting 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., overPruf¨ er domains). The point of the classification is that in the particular cases, thetilting classes are of finite type. This means that we can replace the single infinitelygenerated tilting module by a set of finitely presented modules; the tilting class isthen axiomatizable in the language of the first order theory of modules.Most of this paper is a survey of recent developments. We give complete defi-nitions and statements of the results, but most proofs are omitted or replaced byoutlines of the main ideas. For full details, we refer to the original papers listed inthe 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 the-ory of infinitely generated modules over arbitrary rings. In §2 and §3, we studyinfinitely generated tilting and cotilting modules, and characterize the induced tilt-ing and cotilting cotorsion pairs. §4 deals with tilting classes of finite type andcotilting classes of cofinite type, and with their classification over particular rings.Finally, §5 relates tilting approximations to the first and second finitistic dimensionconjectures.We start by fixing our notation. For an (associative, unital) ring R, Mod-Rdenotes the category of all (right R-) modules. mod-R denotes the subcategoryof Mod-R formed by all modules possessing a projective resolution consisting offinitely generated modules. (If R is a right coherent ring then mod-R is just thecategory of all finitely presented modules).