A Criterion for global dimension two for strongly simply connected schurian algebras

TL;DR

This paper establishes a quiver-based criterion to identify when strongly simply connected schurian algebras have global dimension at most two, by introducing and characterizing a new class of critical algebras.

Contribution

It introduces critical algebras of global dimension three and provides a characterization, enabling a criterion for global dimension two in the studied class of algebras.

Findings

Critical algebras are characterized by quivers with relations.

A strongly simply connected schurian algebra has global dimension at most two if it lacks a critical algebra as a full subcategory.

The main theorem links the absence of critical subcategories to the global dimension bound.

Abstract

The aim of this paper is to provide a criterion to determine, by quivers with relations, when an algebra has global dimension at most two. In order to do that, we introduce a new class of algebras of global dimension three, and we call them critical algebras. Furthermore we give a characterization of critical algebras by quivers with relations. Our main theorem states that if a strongly simply connected schurian algebra does not contain a critical algebra as a full subcategory, then it has global dimension at most two.