Squared cycles in monomial relations algebras

Abstract

Let $\mathbb{K}$ be an algebraically closed field. Consider a finite dimensional monomial relations algebra $\Lambda = {{\mathbb{K}\Gamma } \mathord{\left/ {\vphantom {{\mathbb{K}\Gamma } I}} \right. \kern-\nulldelimiterspace} I}$ of finite global dimension, where Γ is a quiver and I an admissible ideal generated by a set of paths from the path algebra $\mathbb{K}\Gamma$. There are many modules over Λ which may be represented graphically by a tree with respect to a top element, of which the indecomposable projectives are the most natural example. These trees possess branches which correspond to right subpaths of cycles in the quiver. A pattern in the syzygies of a specific factor module of the corresponding indecomposable projective module is found, allowing us to conclude that the square of any cycle must lie in the ideal I.