Properties of lexsegment ideals (Q964012)

Let \(S=\Bbbk[x_1,\ldots, x_n]\) be a polynomial ring over a field \(\Bbbk\), \(I\) a monomial ideal and \(G(I)\) the minimal monomial generating set of \(I\). The ideal \(I\) has linear quotients if for some indexing of the elements \( u_1,\ldots, u_m\) of \(G(I)\) the ideals \((u_1,\ldots, u_{j-1}):u_j\) are generated by a subset of \(\{ x_1,\ldots, x_n\}\), for all \(2\leq j\leq m\). It is well known that ideals with linear quotients have a linear minimal free resolution, while the converse might not hold. Lexsegment ideals, i.e.~ideals generated by all monomials that fit lexicographically between two given monomials, have been extensively studied. The main result of this paper is that lexsegment ideals with linear resolution have linear quotients. The main tools are the results of \textit{A.~Aramova, E.~De Negri} and \textit{J.~Herzog} [Ill.~J.~Math.~42, 509--523 (1998; Zbl 0904.13008)] which characterize lexsegment ideals with linear resolution. Lexsegment ideals can be separated into two cases: completely lexsegment ideals (i.e.~ideals generated by a lexsegment \(T\) such that Shad\((T)=\{v x_i:\;v\in T, 1\leq i \leq n\}\) is again lexsegment and so on for all iterared shadows) and not completely lexsegment ideals. The authors of the paper under review give an appropriate indexing of the elements of \(G(I)\) in each case. For the completely lexsegment ideals they show that another nice property is satisfied (the decomposition function is regular) and they apply the description of the minimal resolution of \textit{J.~Herzog} and \textit{Y.~Takayama} [Hom.~Hom.~Appl.~4, 277--294 (2002; Zbl 1028.13008)]. Finally they characterize Cohen-Macaulay lexsegment ideals.