Syzygies of string modules for special biserial algebras

TL;DR

This paper introduces discrete models for special biserial algebras and their string modules, inspired by cellular automata, to analyze syzygies and address open problems in homological algebra, supported by a new GAP software package.

Contribution

It presents a novel combinatorial model for syzygies of string modules in special biserial algebras and develops an efficient GAP package for their computation.

Findings

Models reveal patterns among syzygies in special biserial algebras.

The SBStrips package outperforms existing tools in computing syzygies.

Applications shed light on homological conjectures and derived categories.

Abstract

We present discrete models of special biserial (SB) algebras and their string modules, drawing inspiration from cellular automata, and cast new light on patterns among syzygies. We explore applications of our models to open questions in homological algebra regarding certain triangulated subcategories of derived categories, with implications for the finitistic dimension conjectures. More pertinently, our models provide the inner workings for a new, original GAP package called SBStrips, written and implemented by the author. Its source code is freely available online and its documentation is included as an appendix. The package calculates syzygies of string modules (and much more besides) using specialised methods much more efficient than the general methods currently employed by the \QPA\ package.