The derived category of surface algebras: the case of the torus with one boundary component

TL;DR

This paper refines derived invariants for surface algebras on a torus with one boundary, providing computable tools to distinguish non-derived equivalent gentle algebras and addressing open conjectures.

Contribution

It introduces an easily computable derived invariant for surface algebras on a torus with one boundary, advancing understanding of gentle algebra classifications.

Findings

Identifies gentle algebras with identical AG-invariants that are not derived equivalent.

Provides a partial positive answer to a conjecture on gentle 2-cycle algebras.

Offers a new method for distinguishing derived equivalence classes of surface algebras.

Abstract

In this paper we refine the main result of a previous paper of the author with Grimeland on derived invariants of surface algebras. We restrict to the case where the surface is a torus with one boundary component and give an easily computable derived invariant for such surface algebras. This result permits to give answers to open questions on gentle algebras: it provides examples of gentle algebras with the same AG-invariant (in the sense of Avella-Alaminos and Geiss) that are not derived equivalent and gives a partial positive answer to a conjecture due to Bobi\'nski and Malicki on gentle $2$-cycles algebras.