Stratifying Systems via Nested Family of Torsion Pairs

TL;DR

This paper explores the relationship between nested families of torsion pairs and stratifying systems, showing how each induces the other and presenting a novel infinite stratifying system example.

Contribution

It introduces the concept of nested torsion pairs and establishes their strong connection to stratifying systems, including the construction of an infinite example.

Findings

Every stratifying system induces a nested family of torsion pairs

Stratifying systems can be obtained from modules with specific decompositions

An infinite stratifying system not indexed by natural numbers was constructed

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 $\mathbb{N}$.