The 1-periodic derived category of a gentle algebra : Part 1 -- Indecomposable objects

TL;DR

This paper characterizes the indecomposable objects of the 1-periodic derived category of gentle algebras using geometric models and matrix problems, providing a complete classification in this setting.

Contribution

It introduces a geometric description of indecomposable objects in the 1-periodic derived category of gentle algebras, connecting algebraic and topological methods.

Findings

Indecomposable objects correspond to homotopy classes of curves on surfaces.

Complete classification of indecomposables using geometric and matrix problem techniques.

Extension of existing models to the 1-periodic derived category context.

Abstract

Combining results from Keller and Buchweitz, we describe the 1-periodic derived category of a finite dimensional algebra $A$ of finite global dimension as the stable category of maximal Cohen-Macaulay modules over some Gorenstein algebra $A^\ltimes$. In the case of gentle algebras, using the geometric model introduced by Opper, Plamondon and Schroll, we describe indecomposable objects in this category using homotopy classes of curves on a surface. In particular, we associate a family of indecompoable objects to each primitive closed curve. We then prove using results by Bondarenko and Drozd concerning a certain matrix problem, that this constitutes a complete description of indecomposable objects.