Ideals with stable Betti numbers
Let \(\kappa\) be a field of characteristic \(0\) and \(S=\kappa [x_1, \dots,x_n]\) the polynomial ring over \(\kappa\) with each \(\deg x_i=1\) and with the reverse lexicographic term order induced by \(x_1> \cdots >x_n\). For a graded ideal \(I\) of \(S\), let \(\text{Gin} (I)\) denote the corresponding generic initial ideal and \(I_{\langle j\rangle}\) the ideal generated by all homogeneous polynomials of degree \(j\) belonging to \(I\). In this paper the authors prove that the graded Betti numbers of \(I\) and \(\text{Gin}(I)\) are equal if and only if \(I_{\langle j\rangle}\) has a linear resolution for all \(j\).
scientific article; zbMATH DE number 1496431