Balanced pairs, virtually Gorenstein rings, and cotorsion torsion triples

TL;DR

This paper explores the relationships between balanced pairs of modules, cotorsion triples, and Gorenstein rings, providing classifications and characterizations that deepen understanding of module theory and its applications in representation theory.

Contribution

It characterizes when balanced pairs generate tilting or cotilting cotorsion pairs and classifies cotorsion torsion triples via $1$-resolving subcategories, linking these concepts to virtually Gorenstein rings.

Findings

Balanced pair of Gorenstein modules iff ring is right virtually Gorenstein.

Complete classification of cotorsion torsion triples using $1$-resolving subcategories.

Bijective correspondence between cotorsion torsion triples of right and torsion cotorsion triples of left modules in left noetherian rings.

Abstract

For any ring $R$, we investigate balanced pairs of classes of modules and their relations to cotorsion triples. We characterize the case when a balanced pair generates a tilting cotorsion pair, and dually, when it cogenerates a cotilting cotorsion pair. If $R$ is right noetherian, we prove that the pair consisting of Gorenstein projective modules and Gorenstein injective modules is balanced if and only if $R$ is right virtually Gorenstein. In [4], cotorsion torsion triples in abelian categories were employed in the representation theory of rectangular grids occurring in persistent homology theory. For module categories, we use infinite dimensional tilting theory to completely classify all cotorsion torsion triples by means of $1$-resolving subcategories of $\rfmod R$, and to give an explicit 1-1 correspondence between the formally dual notions of cotorsion torsion triples of right $R$-modules and torsion cotorsion triples of left $R$-modules. This correspondence is bijective in case the underlying ring $R$ is left noetherian, but not in general.