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Stratifying Systems via Nested Family of Torsion Pairs

Edson Ribeiro Alvares, Matheus Vinicius dos Santos

Abstract

In this paper, we introduce the concept of a nested family of torsion pairs and will prove that this concept is strongly related to the existence of stratifying systems. Specifically, every stratifying system induces a nested family of torsion pairs. Moreover, every stratifying system can be obtained as a quotient or submodule of a module that admits a certain direct sum decomposition with respect to the nested family of torsion pairs. Additionally, we present a stratifying system of infinite size that cannot be indexed by $\mathbb{N}$.

Stratifying Systems via Nested Family of Torsion Pairs

Abstract

In this paper, we introduce the concept of a nested family of torsion pairs and will prove that this concept is strongly related to the existence of stratifying systems. Specifically, every stratifying system induces a nested family of torsion pairs. Moreover, every stratifying system can be obtained as a quotient or submodule of a module that admits a certain direct sum decomposition with respect to the nested family of torsion pairs. Additionally, we present a stratifying system of infinite size that cannot be indexed by .

Paper Structure

This paper contains 8 sections, 21 theorems, 19 equations.

Table of Contents

  1. Introduction
  2. Setting and backgrounds
  3. Nested Families of torsion pairs and $\Gamma$-compatibility
  4. Conditions for the existence of a stratifying system
  5. Stratifying system via $\tau$-rigid modules
  6. Expansions in Nested Families
  7. A stratifying system of infinite size not indexed by $\mathbb{N}$
  8. Example

Key Result

Theorem 1

Let $\Gamma = \{(\mathcal{T}_k, \mathcal{F}_k) \}_{k \in G}$ be a nested family of torsion pairs and let $M = \bigoplus\limits_{k \in G}M_k$ be a module.

Theorems & Definitions (34)

  • Theorem : Theorem \ref{['teoresma1']}
  • Theorem : Theorem \ref{['cubo']}
  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Lemma 2.4
  • Definition 3.1: Nested family of torsion pairs
  • Definition 3.2: $\Gamma$-compatibility
  • Proposition 3.3
  • Definition 3.4: Stratum and Substratum
  • ...and 24 more