String and band complexes over SAG algebras

TL;DR

This paper provides a combinatorial framework for understanding indecomposable objects in the derived categories of string almost gentle algebras, extending known conditions for infinite projective resolutions and global dimension.

Contribution

It introduces a combinatorial description of string and band complexes for a new class of algebras and extends criteria for infinite resolutions and global dimension.

Findings

Characterization of indecomposable objects as string and band complexes

Necessary and sufficient conditions for infinite minimal projective resolutions

Sufficient conditions for infinite global dimension

Abstract

We give a combinatorial description of a family of indecomposable objects in the bounded derived categories of a new class of algebras: string almost gentle algebras. These indecomposable objects are, up to isomorphism, the string and band complexes introduced by V. Bekkert and H. Merklen. With this description, we give a necessary and sufficient condition for a given string complex to have infinite minimal projective resolution and we extend this condition for the case of string algebras. Using this characterization we establish a sufficient condition for a string almost gentle algebra (or a string algebra) to have infinite global dimension.