Reduction of Frobenius extriangulated categories

TL;DR

This paper introduces a reduction technique for Frobenius extriangulated categories that preserves certain cluster-tilting subcategories, generalizing existing methods and connecting to Calabi-Yau algebras, with applications to frieze patterns.

Contribution

It generalizes Iyama--Yoshino's reduction to Frobenius extriangulated categories and links the reduction to internally Calabi--Yau algebras, providing new insights and proofs.

Findings

Reduction preserves cluster-tilting subcategories containing a given rigid subcategory.

Establishes a connection between the reduction and internally Calabi--Yau algebras.

Provides a conceptual proof of a result on frieze patterns.

Abstract

We describe a reduction technique for stably 2-Calabi--Yau Frobenius extriangulated categories $\mathcal{F}$ with respect to a functorially finite rigid subcategory $\mathcal{X}$. The reduction of such a category is another category $\mathcal{X}^{\perp_1}\subseteq\mathcal{F}$ of the same kind, whose cluster-tilting subcategories are those cluster-tilting subcategories $\mathcal{T}\subseteq\mathcal{F}$ such that $\mathcal{X}\subseteq\mathcal{T}$. This reduction operation generalises Iyama--Yoshino's reduction for 2-Calabi--Yau triangulated categories, which is recovered by passing to stable categories. Moreover, for a certain class of categories $\mathcal{F}$ and rigid objects $M$, we show that the relationship between $\mathcal{F}$ and $M^{\perp_1}$ may also be expressed in terms of internally Calabi--Yau algebras, in the sense of the third author. As an application, we give a conceptual proof of a result on frieze patterns originally obtained by the first author with Baur, Gratz, Serhiyenko, and Todorov.