Mutation of $n$-cotorsion pairs in extriangulated categories

TL;DR

This paper introduces the concept of $n$-cotorsion pairs in extriangulated categories, extending existing theories, and demonstrates that mutations preserve these pairs, with geometric applications in cluster categories.

Contribution

It defines $n$-cotorsion pairs in extriangulated categories and proves mutation invariance, connecting algebraic and geometric perspectives in cluster theory.

Findings

Mutation of $n$-cotorsion pairs remains an $n$-cotorsion pair

Provides a geometric characterization in $n$-cluster categories

Realizes mutations via rotations of $n$-admissible arcs

Abstract

In this article, we introduce the notion of $n$-cotorsion pairs in extriangulated categories, which extends both the cotorsion pairs established by Nakaoka and Palu and the $n$-cotorsion pairs in triangulated categories developed by Chang and Zhou. We further prove that any mutation of an $n$-cotorsion pair remains an $n$-cotorsion pair. As applications, we provide a geometric characterization of $n$-cotorsion pairs in $n$-cluster categories of type $A_{\infty}$, and we realize mutations of $n$-cotorsion pairs geometrically via rotations of certain configurations of $n$-admissible arcs.