Classification of modules for complete gentle algebras

TL;DR

This paper classifies finitely generated modules over complete gentle algebras, extending known classifications from finite-dimensional cases to more general rings using string and band modules.

Contribution

It introduces a classification of modules over complete gentle algebras, generalizing finite-dimensional results to a broader class of rings.

Findings

Classification in terms of string and band modules

Construction of resolutions for string and band modules

Extension of existing classification methods to complete gentle algebras

Abstract

We classify finitely generated modules over a class of algebras introduced in the authors' Ph.D thesis, called complete gentle algebras. These rings generalise the finite-dimensional gentle algebras introduced by Assem and Skowro\'{n}ski, in such a way so that the ground field is replaced by any complete local noetherian ring. Our classification is written in terms of string and band modules. For the proof we apply the main result from the authors' thesis, which classifies complexes of projective modules with finitely generated homogeneous components up to homotopy. In doing so we construct the resolution of a string or band module, generalising some calculations due to \c{C}anak\c{c}i, Pauksztello and Schroll.