Syzygies on Path Algebras

Abstract

Let K be a field and KQ be a noetherian path algebra for the quiver Q. Given a left (resp. right) finitely generated ideal I of KQ, we propose a new idea for computing left (resp. right) Groebner bases on KQ. As application, we propose a method for computing the so called left (resp. right) syzygies, that is, given polynomials f 1 ,...,f s ∈KQ \{0} we propose a method for computing the set of all elements (h 1 ,...,h s )∈(KQ) s such that h 1 f 1 + ... + h s f s = 0 (resp. f 1 h 1 + ... + f s h s = 0).