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This is a Quantitative Study study.

October 2, 2025Open Access

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

Key Points

This research shows a complete description of indecomposable objects in the 1-periodic derived category.
The findings link Cohen-Macaulay modules to the geometry of curves, affirming their relevance in gentle algebras.
A geometric model effectively provides a way to classify indecomposable objects using curves on surfaces.
The approach utilizes established results from matrix problems to support these findings.

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^. 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.

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Joseph Winspeare (Wed,) studied this question.

synapsesocial.com/papers/68de84bb5b556a9128e1ba7e https://doi.org/https://doi.org/10.48550/arxiv.2506.04012