The alternating syzygy behavior of monomial algebras (Q1355501)

Let \(\Gamma\) be a finite quiver. The path algebra \(k\Gamma\) over the field \(k\) is the vector space with basis consisting of all finite directed paths in \(\Gamma\), and multiplication defined by concatenation. Denote by \(J\) the two-sided ideal generated by the arrows of \(\Gamma\). The author studies monomial algebras, i.e. algebras of the form \(\Lambda=k\Gamma/I\), where \(I\) is an ideal generated by a finite number of paths and such that \(J^N\subseteq I\subseteq J^2\), for some positive integer \(N\geq 2\). The main result of this paper is the construction of the minimal projective resolution for a monomial algebra, with the aid of noncommutative reduced Gröbner bases. The syzygies for these resolutions exhibit an alternating behaviour induced by certain properties of an associated sequence of paths.