Constructing projective resolution and taking cohomology for gentle algebras in the geometric model

TL;DR

This paper unifies the geometric models for module and derived categories of gentle algebras, characterizes cohomology via curves, and provides geometric proofs for key theorems and conjectures.

Contribution

It introduces a unified geometric framework for gentle algebras, linking module and derived categories through surface rotations and curve truncations.

Findings

Unified geometric realization of module and derived categories.

Characterization of cohomology via curve truncation.

Geometric proof of the strong Nakayama conjecture.

Abstract

The geometric models for the module category and derived category of any gentle algebra were introduced to realize the objects in module category and derived category by permissible curves and admissible curves respectively. The present paper firstly unifies these two realizations of objects in module category and derived category via same surface for any gentle algebra, by the rotation of permissible curves corresponding to the objects in the module category. Secondly, the geometric characterization of the cohomology of complexes over gentle algebras is established by the truncation of projective permissible curves. It is worth mentioning that the rotation of permissible curves and the truncation of projective permissible curves are mutually inverse processes to some extent. As applications, an alternative proof of ``no gaps" theorem as to cohomological length for the bounded derived categories of gentle algebras is provided in terms of the geometric characterization of the cohomology of complexes. Moreover, we obtain a geometric proof for the strong Nakayama conjecture for gentle algebras. Finally, we contain two examples to illustrate our results.