Powers of lexsegment ideals with linear resolution (Q384323)

Let \(S=k[x_{1},\dots,x_{n}]\) be the polynomial ring in \(n\) variables over a field \(K\) and, for an integer \(d\geq 2\), let \(\mathcal{M}_{d}\) be the set of all monomials of \(S\) of degree \(d\). A lexsegment ideal of \(S\) is a monomial ideal generated by a lexsegment set \(L(u,v)=\{\omega\in \mathcal{M}_{d}: u\geq_{\mathrm{lex}}\omega\geq_{\mathrm{lex}}v\}\) where \(u\geq_{\mathrm{lex}}v\) are two given monomials of \(\mathcal{M}_{d}\).NEWLINENEWLINE The authors start considering lexsegment ideals that, with respect a suitable order, have linear quotients (equivalently a linear resolution). They analyze when the same property is valid for all powers of these lexsegment ideals. In the main theorem they give a nice characterization for the class of powers of completely lexsegment ideals. After they refine the so-called \(l\)-exchange property, introduced in [Osaka J. Math. 42, No. 4, 807--829 (2005; Zbl 1092.05012)] by \textit{J. Herzog} et al., with the new \(\sigma\)-exchange property, in order to show that the class of all powers of not completely lexsegment ideals with linear resolution have linear quotients too with respect to the decreasing revlexicographic order.