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Torsion pairs via the Ziegler spectrum

Lidia Angeleri Hügel, Rosanna Laking, Francesco Sentieri

Abstract

We establish a bijection between torsion pairs in the category of finite-dimensional modules over a finite-dimensional algebra A and pairs (Z, I) formed by a closed rigid set Z in the Ziegler spectrum of A and a set I of indecomposable injective A-modules. This can be regarded as an extension of a result from $τ$-tilting theory which parametrises the functorially finite torsion pairs over A. We also obtain a one-one-correspondence between finite-dimensional bricks and certain (possibly infinite-dimensional) indecomposable modules satisfying a rigidity condition. Our results also hold when A is an artinian ring.

Torsion pairs via the Ziegler spectrum

Abstract

We establish a bijection between torsion pairs in the category of finite-dimensional modules over a finite-dimensional algebra A and pairs (Z, I) formed by a closed rigid set Z in the Ziegler spectrum of A and a set I of indecomposable injective A-modules. This can be regarded as an extension of a result from -tilting theory which parametrises the functorially finite torsion pairs over A. We also obtain a one-one-correspondence between finite-dimensional bricks and certain (possibly infinite-dimensional) indecomposable modules satisfying a rigidity condition. Our results also hold when A is an artinian ring.
Paper Structure (16 sections, 44 theorems, 37 equations)

This paper contains 16 sections, 44 theorems, 37 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation.
  4. Torsion pairs
  5. T-structures.
  6. Essentials on purity.
  7. Cosilting complexes and t-structures
  8. Cosilting modules and torsion pairs
  9. Closed sets associated to cosilting complexes
  10. Cosilting pairs in the module category
  11. Critical modules
  12. Brick--critical correspondence
  13. The Ziegler spectrum
  14. The spectrum of a locally coherent Grothendieck category
  15. Purity and the Ziegler spectrum of the derived category
  16. ...and 1 more sections

Key Result

Proposition 2.2

Let $R$ be a ring.

Theorems & Definitions (92)

  • Definition 2.1
  • Proposition 2.2
  • Definition 2.3
  • Example 2.4
  • Definition 2.5
  • Lemma 2.6
  • Proposition 2.7
  • Definition 2.8
  • Theorem 2.10
  • Proposition 2.11
  • ...and 82 more