Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Balanced pairs, virtually Gorenstein rings, and cotorsion torsion triples

Sergio Estrada, Jiangsheng Hu, Jan Trlifaj

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.

Balanced pairs, virtually Gorenstein rings, and cotorsion torsion triples

Abstract

For any ring , 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 is right noetherian, we prove that the pair consisting of Gorenstein projective modules and Gorenstein injective modules is balanced if and only if 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 -resolving subcategories of , and to give an explicit 1-1 correspondence between the formally dual notions of cotorsion torsion triples of right -modules and torsion cotorsion triples of left -modules. This correspondence is bijective in case the underlying ring is left noetherian, but not in general.
Paper Structure (9 sections, 27 theorems, 5 equations)

This paper contains 9 sections, 27 theorems, 5 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Precovering and preenveloping classes
  4. Cotorsion pairs and cotorsion triples
  5. Tilting and cotilting modules and cotorsion pairs
  6. Torsion pairs
  7. Gorenstein modules
  8. Balanced pairs and virtually Gorenstein rings
  9. Cotorsion torsion triples and their duals

Key Result

Lemma 2.1

Let $R$ be a ring and $\mathcal{C}$ be a class of modules closed under isomorphisms and direct summands.

Theorems & Definitions (61)

  • Lemma 2.1
  • Definition 2.2
  • Remark
  • Lemma 2.3
  • Lemma 2.4
  • Remark
  • Example 2.5
  • Lemma 2.6
  • proof
  • Example 2.7
  • ...and 51 more