Cohen–Macaulay modules, (co)torsion pairs and virtually Gorenstein algebras

We use torsion pairs in stable categories and cotorsion pairs in modules categories to study, in general infinitely generated, Cohen–Macaulay modules and (a generalization of) modules of finite projective or injective dimension over an Artin algebra. We concentrate our investigation to the study of virtually Gorenstein algebras which provide a common generalization of Gorenstein algebras and algebras of finite representation or Cohen–Macaulay type. This class of algebras on the one hand has rich homological structure and satisfies several representation/torsion theoretic finiteness conditions, and on the other hand it is closed under various operations, for instance derived equivalences and stable equivalences of Morita type. In addition virtual Gorensteinness provides a useful tool for the study of the Gorenstein Symmetry Conjecture and modified versions of the Telescope Conjecture for module or stable categories.