# $n$-Cotorsion pairs

## TL;DR

This paper introduces and studies the concept of $n$-cotorsion pairs in abelian categories, generalizing complete cotorsion pairs and exploring their properties and applications in Gorenstein homological algebra.

## Contribution

It defines $n$-cotorsion pairs, analyzes their properties, and relates them to existing cotorsion concepts, with applications to Gorenstein modules and cluster tilting.

## Key findings

- $n$-cotorsion pairs generalize complete cotorsion pairs.
- Established relations between $n$-cotorsion pairs and approximations.
- Applications to Gorenstein modules and cluster categories.

## Abstract

Motivated by some properties satisfied by Gorenstein projective and Gorenstein injective modules over an Iwanaga-Gorenstein ring, we present the concept of left and right $n$-cotorsion pairs in an abelian category $\mathcal{C}$. Two classes $\mathcal{A}$ and $\mathcal{B}$ of objects of $\mathcal{C}$ form a left $n$-cotorsion pair $(\mathcal{A,B})$ in $\mathcal{C}$ if the orthogonality relation $\mathsf{Ext}^i_{\mathcal{C}}(\mathcal{A,B}) = 0$ is satisfied for indexes $1 \leq i \leq n$, and if every object of $\mathcal{C}$ has a resolution by objects in $\mathcal{A}$ whose syzygies have $\mathcal{B}$-resolution dimension at most $n-1$. This concept and its dual generalise the notion of complete cotorsion pairs, and has an appealing relation with left and right approximations, especially with those having the so called unique mapping property. The main purpose of this paper is to describe several properties of $n$-cotorsion pairs and to establish a relation with complete cotorsion pairs. We also give some applications in relative homological algebra, that will cover the study of approximations associated to Gorenstein projective, Gorenstein injective and Gorenstein flat modules and chain complexes, as well as $m$-cluster tilting subcategories.