# Squared cycles in monomial relations algebras

Authors
Type
Published Article
Journal
Open Mathematics
Publisher
Versita
Publication Date
Jun 01, 2006
Volume
4
Issue
2
Pages
250–259
Identifiers
DOI: 10.2478/s11533-006-0010-0
Source
De Gruyter
Keywords
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.