Homological and Homotopical Aspects of Torsion Theories
Torsion pairs (originally called ‘torsion theories’) entered module theory in the 60s through the works of Dickson and Gabriel, and were later developed into a powerful tool, e.g. in the monographs of Stenström and Golan. In the 80s, analogues of torsion pairs, called t-structures, were used in the seminal work of Beilinson, Bernstein and Deligne on triangulated categories. At about the same time, tilting theory emerged providing important examples of torsion pairs both in module categories and, later, in bounded derived categories of modules. Yet another source of torsion pairs, this time in stable module categories, came from (co)tilting theory via covariantly and contravariantly finite subcategories, notably in the works of Auslander and Reiten. Moreover, torsion pairs in stable categories were later shown to be closely related to complete cotorsion pairs in the original Abelian categories, and the latter to closed model structures in the sense of Quillen (e.g. in the works of Hovey). Thus it has become clear that torsion pairs tie together a number of important areas of contemporary algebra, topology and geometry.
The Beligiannis-Reiten memoir not only provides a comprehensive treatment of these ties but also finds remarkable generalisations, clarifications and extensions of the results mentioned above. The point is that the authors work in the general setting of pretriangulated categories, which includes both the Abelian and the triangulated setting as special cases. The core of the memoir consists of proving general correspondence theorems (some of which were mentioned above). Moreover, the authors use torsion pairs to develop universal cohomology theories generalising the Tate-Vogel (co)homology theory. There are also a number of concrete applications presented, notably to Gorenstein and Cohen-Macaulay categories (generalizing the classical Gorenstein and Cohen-Macaulay rings). The memoir covers an important area of contemporary pure mathematics; it is particularly recommended to anyone interested in modern representation theory, homological algebra or algebraic topology.