# Higher extensions for gentle algebras

## TL;DR

This paper investigates higher-degree extensions between indecomposable modules over gentle algebras, revealing their vanishing or periodic nature, with geometric interpretations and explicit bases for surface-derived cases.

## Contribution

It provides a comprehensive description of higher extensions in gentle algebras, including geometric insights and explicit bases for surface-related cases.

## Key findings

- Extensions either vanish or are periodic
- Geometric interpretation via surface topology
- Explicit bases for surface-derived gentle algebras

## Abstract

In this paper we determine extensions of higher degree between indecomposable modules over gentle algebras. In particular, our results show how such extensions either eventually vanish or become periodic. We give a geometric interpretation of vanishing and periodicity of higher extensions in terms of the surface underlying the gentle algebra. For gentle algebras arising from triangulations of surfaces, we give an explicit basis of higher extension spaces between indecomposable modules.