Applications of Exact Structures in Abelian Categories

TL;DR

This paper explores the correspondence between various exact structures and subfunctors in abelian categories with small Ext groups, establishing new connections and applications in cotorsion theory and module categories.

Contribution

It establishes a one-to-one correspondence between balanced pairs, subfunctors of Ext, and Quillen exact structures, extending the Wakamatsu lemma and constructing cotorsion pairs.

Findings

Established a bijection between balanced pairs, subfunctors, and exact structures.

Extended the Wakamatsu lemma to the exact context.

Constructed (pre)enveloping, (pre)covering classes and hereditary cotorsion pairs.

Abstract

In an abelian category $\mathscr{A}$ with small ${\rm Ext}$ groups, we show that there exists a one-to-one correspondence between any two of the following: balanced pairs, subfunctors $\mathcal{F}$ of ${\rm Ext}^{1}_{\mathscr{A}}(-,-)$ such that $\mathscr{A}$ has enough $\mathcal{F}$-projectives and enough $\mathcal{F}$-injectives and Quillen exact structures $\mathcal{E}$ with enough $\mathcal{E}$-projectives and enough $\mathcal{E}$-injectives. In this case, we get a strengthened version of the translation of the Wakamatsu lemma to the exact context, and also prove that subcategories which are $\mathcal{E}$-resolving and epimorphic precovering with kernels in their right $\mathcal{E}$-orthogonal class and subcategories which are $\mathcal{E}$-coresolving and monomorphic preenveloping with cokernels in their left $\mathcal{E}$-orthogonal class are determined by each other. Then we apply these results to construct some (pre)enveloping and (pre)covering classes and complete hereditary $\mathcal{E}$-cotorsion pairs in the module category.