Periodic String Complexes over String Algebras

TL;DR

This paper introduces combinatorial methods to characterize periodic string complexes over string algebras, providing criteria for infinite global dimension and identifying indecomposable objects in derived categories.

Contribution

It offers a novel combinatorial characterization of periodic string complexes and applies this to determine infinite global dimension and construct indecomposable objects.

Findings

Characterization of periodic string complexes with infinite minimal projective resolution

A sufficient condition for string algebras to have infinite global dimension

Construction of indecomposable objects in derived categories for specific string algebras

Abstract

In this paper we develop combinatorial techniques for the case of string algebras with the aim to give a characterization of string complexes with infinite minimal projective resolution. These complexes will be called \textit{periodic string complexes}. As a consequence of this characterization, we give two important applications. The first one, is a sufficient condition for a string algebra to have infinite global dimension. In the second one, we exhibit a class of indecomposable objects in the derived category for a special case of string algebras. Every construction, concept and consequence in this paper is followed by some illustrative examples.