Applications of Balanced Pairs

TL;DR

This paper explores the properties of balanced pairs in abelian categories, introduces relative cotorsion pairs, and investigates their implications for homotopy and singularity categories, providing new insights into their structure and relationships.

Contribution

It introduces the concept of cotorsion pairs relative to balanced pairs and characterizes their properties, linking resolution dimensions to homotopy and singularity categories.

Findings

Finite resolution dimensions imply inclusion of bounded homotopy categories.

Right and left singularity categories coincide under finite resolution and coresolution dimensions.

Provides equivalent characterizations of hereditary and perfect relative cotorsion pairs.

Abstract

Let $(\mathscr{X}$, $\mathscr{Y})$ be a balanced pair in an abelian category. We first introduce the notion of cotorsion pairs relative to $(\mathscr{X}$, $\mathscr{Y})$, and then give some equivalent characterizations when a relative cotorsion pair is hereditary or perfect. We prove that if the $\mathscr{X}$-resolution dimension of $\mathscr{Y}$ (resp. $\mathscr{Y}$-coresolution dimension of $\mathscr{X}$) is finite, then the bounded homotopy category of $\mathscr{Y}$ (resp. $\mathscr{X}$) is contained in that of $\mathscr{X}$ (resp. $\mathscr{Y}$). As a consequence, we get that the right $\mathscr{X}$-singularity category coincides with the left $\mathscr{Y}$-singularity category if the $\mathscr{X}$-resolution dimension of $\mathscr{Y}$ and the $\mathscr{Y}$-coresolution dimension of $\mathscr{X}$ are finite.