The role of gentle algebras in higher homological algebra

TL;DR

This paper explores the significance of gentle algebras in higher homological algebra, providing classifications of certain subcategories and linking algebraic properties to geometric models in derived categories.

Contribution

It classifies weakly d-representation finite gentle algebras and characterizes d-cluster tilting subcategories in derived categories using geometric models.

Findings

Gentle algebras with d-cluster tilting subcategories are radical square zero Nakayama algebras.

Derived categories with d-cluster tilting subcategories closed under [d] are derived equivalent to type A Dynkin algebras.

Provides a geometric characterization of d-cluster tilting subcategories in derived categories.

Abstract

We investigate the role of gentle algebras in higher homological algebra. In the first part of the paper, we show that if the module category of a gentle algebra $\Lambda$ contains a $d$-cluster tilting subcategory for some $d \geq 2$, then $\Lambda$ is a radical square zero Nakayama algebra. This gives a complete classification of weakly $d$-representation finite gentle algebras. In the second part, we use a geometric model of the derived category to prove a similar result in the triangulated setup. More precisely, we show that if $\mathcal{D}^b(\Lambda)$ contains a $d$-cluster tilting subcategory that is closed under $[d]$, then $\Lambda$ is derived equivalent to an algebra of Dynkin type $A$. Furthermore, our approach gives a geometric characterization of all $d$-cluster tilting subcategories of $\mathcal{D}^b(\Lambda)$ that are closed under $[d]$.