Higher Ideal Approximation Theory

TL;DR

This paper extends ideal approximation theory into higher homological algebra by developing concepts like ideal cotorsion pairs and phantom ideals within $n$-exact categories, revealing their suitability for advanced approximation frameworks.

Contribution

It introduces higher ideal approximation concepts such as ideal cotorsion pairs and phantom ideals into $n$-exact categories, expanding the scope of approximation theory.

Findings

Development of higher ideal cotorsion pairs

Introduction of higher phantom ideals

Establishment of higher Salce's and Wakamatsu's Lemmas

Abstract

Let ${\mathscr{C}}$ be an $n$-cluster tilting subcategory of an exact category $({\mathscr{A}}, {\mathscr{E}})$, where $n \geq 1$ is an integer. It is proved by Jasso that if $n> 1$, then ${\mathscr{C}}$ although is no longer exact, but has a nice structure known as $n$-exact structure. In this new structure conflations are called admissible $n$-exact sequences and are ${\mathscr{E}}$-acyclic complexes with $n+2$ terms in ${\mathscr{C}}$. Since their introduction by Iyama, cluster tilting subcategories has gained a lot of traction, due largely to their links and applications to many research areas, many of them unexpected. On the other hand, ideal approximation theory, that is a gentle generalization of the classical approximation theory and deals with morphisms and ideals instead of objects and subcategories, is an active area that has been the subject of several researches. Our aim in this paper is to introduce the so-called `ideal approximation theory' into `higher homological algebra'. To this end, we introduce some important notions in approximation theory into the theory of $n$-exact categories and prove some results. In particular, the higher version of the notions such as ideal cotorsion pairs, phantom ideals, Salce's Lemma and Wakamatsu's Lemma for ideals will be introduced and studied. Our results motivate the definitions and show that $n$-exact categories are the appropriate context for the study of `higher ideal approximation theory'.