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Simples in a cotilting heart

Lidia Angeleri Hügel, Ivo Herzog, Rosanna Laking

Abstract

Every cotilting module over a ring R induces a t-structure with a Grothendieck heart in the derived category D(Mod R). We determine the simple objects in this heart and their injective envelopes, combining torsion-theoretic aspects with methods from the model theory of modules and Auslander-Reiten theory.

Simples in a cotilting heart

Abstract

Every cotilting module over a ring R induces a t-structure with a Grothendieck heart in the derived category D(Mod R). We determine the simple objects in this heart and their injective envelopes, combining torsion-theoretic aspects with methods from the model theory of modules and Auslander-Reiten theory.
Paper Structure (22 sections, 32 theorems, 49 equations)

This paper contains 22 sections, 32 theorems, 49 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Notation
  4. Torsion pairs and HRS-tilts
  5. Cotilting modules and cotorsion pairs
  6. Injective envelopes of simples and left almost split morphisms
  7. Localisation in abelian Grothendieck categories
  8. Simple objects in the heart
  9. Injective envelopes in a cotilting heart
  10. Neg-isolated modules
  11. Finitely generated subfunctors of $\mathcal{I}$
  12. The pp-type of a pointed module
  13. Pure-injective modules
  14. Definable subcategories of modules
  15. Neg-isolated pure-injective modules
  16. ...and 7 more sections

Key Result

Proposition 2.2

Let $\tau = (\mathcal{T}, \mathcal{F})$ be a torsion pair in $\mathrm{Mod}\text{-}R$. The two full subcategories of $\mathrm{D}(\mathrm{Mod}\text{-}R)$ form a t-structure.

Theorems & Definitions (82)

  • Definition 2.1
  • Proposition 2.2: happel:reiten:smaloe:1996
  • Lemma 2.3
  • proof
  • Definition 2.4
  • Remark 2.5
  • Definition 2.6
  • Proposition 2.7
  • proof
  • Remark 2.8
  • ...and 72 more