On the growth of deviations (Q2827355)

Let \(S=k[x_1,\dots,x_n]\) be a polynomial ring over a field \(K\), \(I\subseteq(x_1,\dots,x_n)^2\) an homogeneous ideal, and \(R=S/I\). The authors study deviations of the graded algebra \(R\). They show that the deviations of \(R\) are at most equal to the deviations of \(S/\mathrm{in}_{<}(I)\), for any term order \(<\). Moreover, they prove that. among all the homogeneous ideals the rings \(S/I\) with the same Hilbert series, the ring \(S/L\) has the largest deviations, where \(L\) is the lex-segment ideal of \(S/I\). The authors pay attention also to the asymptotic behaviour of the deviations of \(R\). They prove that, if \(R\) is a Golod ring which is not a complete intersection, deviations grow exponentially.