Mutation of $n$-cotorsion pairs in triangulated categories

TL;DR

This paper introduces the concept of $n$-cotorsion pairs in triangulated categories, proves their mutation preserves their structure, and provides geometric characterizations and realizations in specific cluster categories.

Contribution

It generalizes classical cotorsion pairs to $n$-cotorsion pairs, demonstrating mutation invariance and offering geometric insights in $n$-cluster categories.

Findings

Mutation of $n$-cotorsion pairs preserves their structure.

Provides a geometric characterization of $n$-cotorsion pairs in $n$-cluster categories.

Realizes mutation via rotation of $n$-diagonals configurations.

Abstract

In this article, we define the notion of $n$-cotorsion pairs in triangulated categories, which is a generalization of the classical cotorsion pairs. We prove that any mutation of an $n$-cotorsion pair is again an $n$-cotorsion pair. When $n=1$, this result generalizes the work of Zhou and Zhu for classical cotorsion pairs. As applications, we give a geometric characterization of $n$-cotorsion pairs in $n$-cluster categories of type $A$ and give a geometric realization of mutation of $n$-cotorsion pairs via rotation of certain configurations of $n$-diagonals.